摘要:

Collision Complex Dynamics in Alkali Halide Exchange Reactions* BY P. BRUMERt AND M. KARPLUS Department of Chemistry, Harvard University, Cambridge, Massachusetts 02 138 Received 19th January, 1 973 Some results of a classical trajectory calculation on NaBr + KCbNaCI + KBr, a reaction which proceeds via a long-lived collision complex, are presented. The dynamics of the complex are briefly considered and complex lifetimes and the equilibration of vibrational energy are described. The origin of the nonstatistical ratio of reactive to nonreactive product is examined. Its dependence on angular momentum is pointed out and related to evidence for the participation of linear K-CI-Na-Br. Evidence from crossed molecular beam studies for the existence of long-lived collision complexes in simple neutral-neutral reactions was first presented at the last Faraday Society meeting on Chemical Dynamics.' Since that time, several other neutral-neutral reactions have been shown to proceed via long-lived collision com- plexes,2 among them the alkali halide exchange reaction CsCl+ KI+(CsClKI)-+CsI + KCl.(0 Of particular interest in this reaction is the observation that, although long-lived complexes are definitely formed and many attributes are consistent with simple statistical models, the ratio of reactive to nonreactive product deviates significantly from the statistical value. One way to increase our understanding of such chemical reactions is by means of classical trajectory techniques. They provide a direct method of ascertaining the validity of statistical models and can be used to obtain information on many properties which are difficult to study experimentally ; e.g., the lifetime distribution of the collision complex, the cross section and criteria for complex formation and the rate of energy equilibration.In this paper we report some results from a detailed trajectory study of the reaction NaBr + KCl+(NaBrKCl)-+NaCl+ KBr which is a member of the same family as reaction (I). In Section 1 we outline certain aspects of the trajectory technique, including the choice of potential surface, method of numerical integration and selection of initial conditions. Some aspects of the reaction dynamics are discussed in Section 2. Section 3 considers the calculated ratio of reactive to nonreactive scattering and provides evidence for the participation of a linear intermediate that accounts for the observed nonstatistical behaviour.Other results will be reported in a future p~blication.~ * Supported in part by a grant from the National Science Foundation. t Work done as National Science Foundation Predoctoral Fellow. Present address ; Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel. Address after July 1973 ; Smithsonian Astrophysical Observatory, Cambridge, Massachusetts. 80P . BRUMER AND M . KARPLUS 81 1. TRAJECTORY TECHNIQUE The dynamics of NaBr + KCl were studied by the well-documented quasiclassical trajectory method.6 We outline below aspects of the method which are specific for this system, as well as modifications of the integration technique required to treat long trajectories.A. POTENTIAL SURFACE A potential surface of the ionic model type was developed for this system. The method used is a modification of the Rittner model * which includes electric-field dependent repulsive interactions; that is, the repulsion between each pair ( i , j ) is altered due to the polarization of i and j by the electric-field of all the ions. It was found necessary to introduce this type of effect to obtain consistent results for alkali halide monomers? dimers and crystals. The resulting potential gives satisfactory values for the known (NaCl), and (KCl), dimer dissociation energies (& 4 %) and an exothermicity of 1.85 kcal/mol for reaction (11), as compared with an experimental estimate of 0.5k2.0 kcal/mol.For details of the form of the surface and parameter values, see ref. (7). The lowest energy configuration on the potential surface is shown in fig. 1. The molecule is a planar rhomboid which is 46 kcal/mol more stable than the reactants, and has bond lengths - 0.2 A larger than the corresponding diatomics. In addition? the lowest energy linear configurations of the system have also been computed and are shown in fig. 2. Of the various possible linear configurations K-Cl-Na-Br is the lowest energy local minimum and lies only 5.4 kcal/mol above the absolute minimum. The linear configurations K-Br-Na-CL and Na-Br-K-C1 are also local minima, while Na-CI-K-Br is at a saddle point. It is of interest to note that for a dimer potential of the Rittner typeY8 without the field-dependent repulsions, all linear configurations are just saddle point^.^ Evidence for stable linear structures in alkali halide dimers is available from gaseous (LiF), spectra l o and the possibility of linear dimers in (NaF), and LiNaF, has also been suggested." 40 V = -284.7900 Ere= = -46.2572 = - 44.3995 FIG.1 .-Absolute minimum of the surface. V is the potential energy with respect to separated free ions, Ereac and Eprod are the dissociation energies to the separated reactants (NaBr+KCI) and products (KBr+ NaCI) respectively. All energies are in kcal/mol and distances are in A, B. NUMERICAL INTEGRATION TECHNIQUE Two factors make accurate numerical integration of Hamilton's equations for this system considerably more difficult than for most systems previously studied.First, since the trajectories are long, often requiring more than 18 000 integration steps, small truncation errors accumulate and are amplified during the calculation. Second, the trajectories are highly seiisjtive to small changes in the initial conditions? i.e., the82 COLLISION COMPLEX DYNAMICS system displays dynamic instability.' Numerical studies on such systems require considerable care, and must be made with an accurate integrator that has certain stability properties. Comprehensive tests on a variety of integration techniques demonstrated that the sixth order hybrid method due to Gear l4 was satisfactory for this study. It yields a stable integrator with a truncation error proportional to h7. AN'+ @ @ @ V = -265.2659 = -26.7331 I I 2.38074 I 2.39768 ' 3.08602 ' Eprod = -24.8754 I I ' 2.62895 2.44654 2.49374 I Eprod = -34.1082 6" @ @ @ V = -259.0425 Ere= = -20.5097 2.54549 ' 2.S8914 2.88609 ' Eprd = -.18.6520 I I t I I 2.44504 ' 2.29327 ' 2.66331 ' Eprod = - 39.0229 FIG. 2.-Lowest energy linear configurations. Notation as in fig. 1. A variable step algorithm was developed, step size halving and doubling being used to control the local truncation error. Accurate results (as confirmed by repeated " back integration " tests) were obtained for trajectories with lifetimes up to 350 000 a.u. of time ( ~ 8 . 5 x s). Trajectories not completing within this time limit were terminated and denoted as " incomplete ". c. INITIAL CONDITIONS Sets of trajectories were run with fixed initial relative translational energy ER, initial orbital angular momentum L, diatomic rotational states J1, J2 and diatomic vibrational states u l , v2 ; the remaining variables specifying the initial conditions were averaged over using Monte Carlo techniques.Several values of L, E R , ul, Ji (i = 1,2) were considered, although the majority of trajectories were carried out with (J, z ~ ) ~ ~ ~ = (46,O) and (J, u),,, = (50,0), the most probable states at 950 K, the temperatureP . BRUMER AND M . KARPLUS 83 corresponding to the experiments on reaction (I).3 By considering sets of trajectories with fixed ER and L, we study the complex dynamics as a function of the total energy and initial orbital angular momentum, two quantities which play a major role in simple models of complex dynamics.Table 1 presents a summary of the trajectory series computed for the surface described above. The values of ER, L, vi are given in columns 2 to 5. The average energy ( E ) of the system with respect to the separated reactants is given in column 6. The initial reactant separation R, (column 7) was chosen to be large enough such that the diatomics can be regarded as freely rotating ; i.e., all diatomic-diatomic orient- ations are essentially equally probable. In addition, R, was chosen to be larger than RB, the position of the angular momentum barrier in the incoming channel. The number of trajectories done in each series is given in column 8. Although in several series only 100-250 trajectories were calculated, t h s number of trajectories was found to provide reliable information on a wide variety of reaction attributes.Column 9 lists the free passage time (FPT), i.e., the time required to pass through the reaction zone if the diatomics are assumed noninteracting. Column 10 indicates the fraction of incomplete trajectories fI in each series. In most cases, fI does not exceed t. An extreme exception is the low energy, low angular momentum series 11, in which a majority of the trajectories are incomplete ; consequently, studies on this series were terminated after 55 trajectories. Individual trajectories required up to 15 win of IBM 360165 machine time. TABLE 1 .-TRAJECTORY SERIES-CHARACTERISTICS number of f l Rs trajectories FPTd series b ER L UN~EI UKC1 < e > 1 2 3 4 5 6 7 8 9 10 11 0.086968 1600 0 0 0.086968 1250 0 0 0.086968 350 0 0 0.036844 1000 0 0 0.036844 600 0 0 0.036844 350 0 0 0.018 422 600 0 0 0.018422 350 0 0 0,018422 600 7 8 0.018422 350 7 8 0.009211 350 0 0 0.091 244 25 399 0.091 272 25 388 0.091 227 25 356 0.041 106 30 245 0.041 117 30 477 0.041 122 30 387 0.022717 33 443 0.022727 33 121 0.041 579 3 3 370 0.041 581 33 126 0.013 497 41 55 27 993 30 268 33 262 55 488 59 585 61 036 90 404 94 195 90 404 94 195 164 507 0.0 0.0 0.0 0.0 0.006 0.036 0.212 0.256 0.021 0.047 0.709 0 All quantities in atomic units.An atomic unit of time is 2.418 89 x lo-’’ s. b All series were done with JNaBr = 46, JKCI = 50. c Slight variations in <E> for the same values of ER, UNaBr, UKCI are due to the potential energy contributions at R,. <E’>, the total energy with respect to the products is given by <E’> = (E) + 0.002 960 a.u.d Free passage time (see text). 2. COMPLEX DYNAMICS The time dependence of the six internuclear distances for a sample (nonreactive) NaBr + KC1 trajectory involving a long-lived complex is shown in fig. 3. This is to be contrasted with the analogous results l6 for direct reactions ; e.g., the H + H2 reaction yields plots much simpler in structure, with a reaction time on the order of the free passage time. In fig. 3 the trajectory required 300 000 a.u. as compared with the free passage time of - 60 500 a.u. The onset of complex formation is at t - 30 000 a.u. with subsequent decomposition at t-220 000 a.u. During its lifetime of 190 000 a.u. (-4.5 x 10-l2 s) the complex oscillates between tightly bound rhomboidal configura- tions (corresponding to minima in the envelope of the figure) and various near linear configurations (corresponding to maxima in the envelope of the figure) ; it exists for - 384 COLLISION COMPLEX DYNAMICS rotational periods.Both the onset of complex formation and breakup are sharply defined. Moreover, the oscillation between the two types of complex structures has a rather regular period of -30 OOO a.u. ; this is not, however, a general phenomenon. There appears to be no indication in fig. 3 and corresponding figures for other trajec- tories as to the conditions under which complex formation and breakup occur. However, a number of approximate criteria for complex formation and dissociation have been found.7b An analysis of the coordinates and momenta in complex forming trajectories suggests that the system traverses a large part of the available phase space.This is the required behaviour for the validity of statistical descriptions based upon a microcanonical ensemble. Further insight into the reaction dynamics may be obtained by considering the probability of complex formation. Since a " collision complex " is not a uniquely defined entity, we introduce a time T, equal to the difference between the last closest 0 50 I00 I50 200 250 2 90 O l ! I ! ! ' I ! ! ! ' I ! . ! ! " ! I ! ! . ! ! ! I " * * ' time/a.u. x lo3 FIG. 3.-The six interionic distances during the course of a long-lived trajectory from series 5 (see table 1). Each of the six interionic distances is marked with a different symbol identified in the figure key.The marks are separated by approximately one hundred data points. Abscissa in units of lo3 a.u. of time (-2.42 x 10-14 s). The ordinate ranges from 0 to 20 a.u. accounting for the flat regions in the figures where distances exceed 20 a.u. a, NaBr ; A, KBr ; 'I, KCl ; x , NaCl ; +, NaK ; .. BrCI. collision of the products and the first closest collision of the reactants, and divide the trajectories into two groups, those with T,< 18 000 a.u. (4.35 x 10-13 s - 3 vibrational periods) and those with 7'' > 18 000 a.u. The former can be unambiguously regarded as non-complex forming trajectories. The results of this partition are shown in fig. 4. The values of ER and L for all trajectories in this study are such that complex formation would be expected in all cases, other than series I, if one uses a criterion based on the requirement of sufficient energy to pass over the centrifugal barrier in the presence of an effective spherically averaged long-range dipole-dipole interaction.' In marked contrast with such a model, fig. 4 shows that a significant number of trajectories do not form complexes. For example, for (6) = 0.0227a.u., an energy twice that available in thermal experiments, over 30 % of the trajectories are short-lived. For (e) = 0.0135a.u. however, only 7.2 % are short-lived, indicating that arguments based on the passage over a two-body incoming centrifugal barrier may be more reliable at lower energies. One source of the short-lived nonreactive trajectories is collisions in which there is, in the incoming channel, a repulsive interaction between the reactants due to configurations of the form Na-Br-Cl-K and Br-Na-K-Cl.The fraction of short-lived trajectories increases with increasing (E) (being over 0.7 for series 3, the highest energy studied) and with increasing L. The fraction of short-livedP. BRUMER AND M. KARPLUS 85 trajectories for initially vibrationally excited trajectories (denoted by A and B in fig. 4) is greater than for the corresponding series with the same initial translational energy ; that is, in contrast to the simple statistical model, vibrational excitation reduces the cross section for complex formation. Also of note is that the fraction of short-lived reactiue scattering increases with increasing ( E ) for a given L, being fully one-half of F' I 500 0.172 7 El SL 0.489 LL .17 0.002 4 0 e i k 0 . 2 ~ LLb,~,m;, I , o . ~ i ; , , ~ LL 4 7 LL 0.172 SL 0.6 1 SL 3 SL 4 300 sL ~ 0 0 - 2 0.006 :'I q+-y SL 0.349 0.031 -u- L 00135 00221 0 0411 0 0912 <c>/a.u. FIG. 4.-Partitioning of trajectory results into short-lived (SL) (Tc< 18 OOO a.u.) and longer lived trajectories (LL). The results for each trajectory series are located at the intersection of its <E> and L values. For each series two bars are shown, the top bar corresponding to the long-lived species and the bottom to the short-lived species, These bars are divided into two parts, the first is the prob- ability of nonreactive scattering, the second of reactive ; incomplete trajectories are assumed half reactive and half nonreactive. Thus, for example, for series 7, <E> = 0.0227a.u.and L = 600h, 39.2 % of the trajectories are short-lived of which 39 % are nonreactive and 0.2 % reactive. The remaining 60.8 % of the trajectories have intermediates living longer than 18 OOO a.u. of which 41.6 % are nonreactive and 19.2 % reactive. The boxes labelled A and B correspond to series 9 and 10 whose initial conditions include vibrationally excited reactants. the total reactive scattering for (E> = 0.0912 a.u. L = 350 R. This indicates that at higher energies a direct reaction takes place which does not contribute significantly at lower energies. Furthermore, short-lived reactive scattering is significantly greater for trajectories which are initially excited vibrationally, as compared to trajectories with the same translational energy but with the diatomics initially in their ground states.The ratio of the " long-lived " reactive to " long-lived " nonreactive trajec- tories is on the order of 3 to +, although for ( E } = 0.041 1 a.u., L = 350 Ji the ratio is close to one.86 1 .oo 0.90 - om- 0.70 - 0.60 - 0.50 - 0.40 - 0.30- - I I 1 1 I I - - - - - - - COLLISION COMPLEX DYNAMICS LIFETIMES At the foundation of statistical theories of chemical reactions is the random life- time assumption; that is, that the gap distribution of a long-lived species decays exponentially. Studies of the lifetime distributions of the intermediates for FIG. 4-4 5.-Logarithmic plot of the fraction of nonreactive trajectories in the interval [T,, Tc+ATcI against Tc for series 7, (L = 6OOh ; (E) = 0.022 717 a.u.; T, > 2.0 x lo4). trajectories which are ultimately non-reactive indicate that, for the majority of the series considered, the fall off is exponential at long times but is contaminated by a short-lived component. A sample result is shown in fig. 5, which is a logarithmic plot series 1 2 3 4 5 6 7 8 9 10 11 TABLE 2.-INTERMEDIATE LIFETIMES (IN lo-' * S) H. %"J 7: 7TRJ a C a 0.58-0.70 c a 1.95-2.3 1 c a 1.44-1.59 1.11 2.29 1.56-1.74 3.51 2.1 1 2.37 7.74 b (1.69) 18.7 b 1.37 1.08 (2.26) a 117.0 a 1.69 0.70 1.42-1.81 1.36-2.18 3.41 1.50-1.98 7% c C 0.62 0.90 2.67 4.84 0.88 2.61 C 11.4 49.8 a-Too few long-lived trajectories ; b-unable to fit reliably with exponential ; c-negative numerator in eqn (1).P .BRUMER A N D M . KARPLUS 87 r z 1' n U U 1988 COLLISION COMPLEX DYNAMICS of the fraction, NAT/NT, of trajectories with lifetimes T, in a given time interval (T,, T, + ATc) against 7''. The short-lived trajectories (T, < 20 000 a.u.) have been eliminated to prevent their biasing the fit. Although an accurate value of the lifetime is difficult to determine from this plot, it is clear that the range of possible fits is rather small. Lifetimes T?& obtained from such plots for the nonreactive mode of dissocia- tion are given in coIumn 2 of table 2. Cases where insufficient information was avail- able for a reliable estimate have been enclosed in parenthesis. In addition, if several fits seemed possible, the range of values has been included in the table.In the reactive trajectories many more series did not display exponential behaviour ; T?~, is given in column 4 of table 2 for the cases where a fit was possible. The reason for the non- exponential behaviour is not clear, although it may result from an insufficient number of reactive trajectories to provide the true distribution. For comparison, we give the statistical lifetimes from RRKM theory assuming a loose transition state. The nonreactive statistical lifetime z!tt is where z0 is a characteristic vibrational period (- 10-13 s), Y is the potential energy of the (rhombic) intermediate relative to the separated reactants, B, is the rotational energy of the (rhombic) intermediate and B is the rotational energy at the centrifugal barrier maximum in the non-reactive channel.B, is approximately equal to L2/21, where Z, is the intermediate principal axis of the rhombic complex. For the reactive statistical lifetime &at, terms in eqn (1) are replaced by the corresponding quantities for the outgoing reactive channel. The statistical lifetimes are given in columns 3 and 5 of table 2. It is seen that TTRJ and TStat are generally of the same order of magnitude. Surprising, however, is the constancy of zFRJ and T& with changes in the energy ; e.g., whereas Tgtt increases by a factor of 7 from series 5 to series 7, only small changes are seen in 7?fJ. Further studies of these results are in progress. VIBRATIONAL ENERGY EQUILIBRATION To obtain information on the time required to equilibrate the energy amongst the vibrational modes, we consider the energy in each alkali halide bond at fixed times during the course of the trajectories.The average vibrational energy (Evib) in each MX bond for series 7 is shown in fig. 6. The standard error about (Evib) is also shown. These provide a measure of the width of the vibrational energy distribution at fixed times. It is clear from fig. 6 that the energy is reasonably constant in each of the bonds, other than KBr, after t-240 000 a.u. ( 5 . 8 0 ~ 10-l2 s). This can be compared with the value of 10-l1 s estimated by Bunker for the equilibration of a coupled oscillator system. In addition, we note that at large times the vibrational energies in all but the KBr bond are quite similar, consistent with equipartition of energy; (&b) of KBr at long times is considerably larger.This is consistent with the conclusion given below that for high angular momenta the participation of linear K-Cl-Na-Br will cause (Ev,b) for KBr to be considerably larger than (k&) in the other bonds. A similar analysis of the L = 350 h systems indicates that the energy tends to equalize amongst all four bonds, which is consistent with the observation that the rhombic configuration predominates for low angular momentum conditions (see Section 3). 3. NON-STATISTICAL PRODUCT RATIO In table 3 we present the computed ratio r = rR/rN of reactive to nonreactive Two values of r are given in columns 2 and 3. product for each trajectory series.P . BRUMER AND M . KARPLUS 89 T,(UB) is an upper bound to r, obtained by making the (unreasonable) assumption that all incomplete trajectories are reactive.A more reasonable value, rc(+), [column 31 results from assuming that the outcome of the incomplete trajectories is equally TABLE 3.-GAMMA RATIOS series rc(UB) rc(t) r(stat) rdstat) I’c(Tc> 60 000 1 0.0 0.0 b b c a& 0.002 0.002 b b C 3 0.294 0.294 1.13 1.10 0.51 4 0.0 0.0 b b 0.0 5 0.207 0.203 1.23 1.12 0.67 6 0.423 0.387 1.31 1.27 0.92 7 0.429 0.241 1.62 1.34 0.72 8 0.657 0.368 1.63 1.88 0.90 9 0.171 0.156 I .23 1.56 0.58 10 0.482 0.432 1.30 1.36 1.25 11 a a 2.3 1 a a 3 a-Insufficient information to provide reliable values due to the large number of incomplete trajectories, see table 1 ; b-((~’> - B’) or ((E) - B), in eqn (2), is negative ; c-insufficient number of trajectories with Tc > 60 OOO.divided between reactive and nonreactive product. In determining T,(UB) and rC(+), all trajectories are included ; that is, no separation has been made into complex forming and non-complex forming trajectories (see fig. 4). The statistical value of r, r(stat), is obtained from where ( E ) and ( e l ) are the available energies in the nonreactive and reactive channels (see table 1) and B, B’ are the energies in centrifugal motion at the critical point (the barrier maximum) for the nonreactive and reactive decay. The proportionality constant is the ratio of vibrational frequencies at the two similar critical configurations and is taken as unity. A loose transition state has been assumed, which accounts for the exponent of four.An alternative statistical value, r,(stat), is given in column 5 of table 3 and is obtained with B and B’ taken as the average centrifugal energy at the actual barrier maxima as calculated directly from the trajectories. This removes any bias introduced by assuming a fixed centrifugal barrier for a given ( E ) and L. Al- though the energy in centrifugal motion calculated by these two methods often differs by as much as a factor of ten, r(stat) and r,(stat) are very similar. This is a conse- quence of the initial energies and angular momenta i.e., ( E ) , (E’)>>B, B’ so that there is only weak dependence on the B, B’ values. A comparison of columns 2-5 of table 3 clearly indicates that the calculated gamma ratio is considerably less than statistical.The values of r,(+) are in good agreement with the nolistatistical experimental values obtained for the related CsCl+ KI ~ystem.~ The ratio Tc(+) is seen to increase with decreasing L and, for L = 350 h, increase with decreasing total energy. For fixed total energy, the dependence onL is strong, a change from L = 350 h to L = 600 rZ results in a decrease in r,(+) by a factor of more than 1.5, in disagreement with the statistical predictions. To eliminate the bias introduced into the gamma ratio by the shorter lived species, we present in table 3, column 6 , the gamma ratios for the trajectories with T,> 60 000 a.u. (1.45 x s- 12 vibrational periods). Although these values are greater than F,(+) [column 3 of table 31, they still are generally smaller than the statistical values90 COLLISION COMPLEX DYNAMICS by a factor of two.The short-lived species are therefore not capable of explaining the deviation from the statistical predictions. We note also that the strong depend- ence of Tc on L still remains after the shorter lived trajectories have been disregarded. Detailed studies of the complex dynamics, such as that in fig. 3, indicate that the intermediate oscillates between the linear and rhombic forms of the dinier. The " percentage " of linear and rhombic configurations is obtained by considering the fraction,J;[R(MX)], of the complex lifetime spent by the MX bond in the interval [R(MX), R(MX) + AR(MX)] and averaging over all trajectories within each series to find (f,[R(MX)]). Additional information is obtained by averaging independently over trajectories which are nonreactive (NR), reactive (R) and incomplete (INC), giving(fFR[R(MX)]), (fP[R(MX)]) and (f:Nc[R(MX)]).In determining these averages only trajectories with T, > 60 000 a.u. have been included, eliminating any bias due to short-lived trajectories. The results for the KBr bond are shown in fig. 7 and 8 for series 7 ( ( E ) = 0.0227 a.u., L = 600 h) and series 8 ((8) = 0.0227 a.u., L = 350 k). Fig. 7a-c are bimodal with peaks at - 6 and N 14 a.u. Reference to fig. 1 and 2 shows that the peak at - 14 a.u. corresponds to that expected from the linear K-C1-Na-Br configuration, whereas the peak at - 6 a.u. can result from the rhombic configuration or any of the remaining three linear configurations.Similar studies of the other alkali halide bonds, however, indicate that linear configurations other than K-Cl-Na- Br do not strongly participate. Thus the peaks in fig. 7 and 8 at R(KBr)-6 a.u. result primarily from a rhombic intermediate A comparison of fig. 7a, b and c shows 0 3 - n 3 0.2 W 5 p: 0 1 - h % * 0 I I I I I I I I Series 7 (Cl - T, > 60,000 L I I i Series 7 T, > 60,000 0 I I I I I I I I 0 2 4 6 8 10 12 14 1 6 > 1 7 L 01 I 1 I 1 I I I I=3n71 0 2 4 6 8 10 12 14 16>17P . BRUMER A N D M . KARPLUS 91 r-l I 0 2 4 6 8 10 12 14 1 6 > 1 7 I Series 8 T, >60000 0. I 0 2 4 6 8 10 13, 14 16 >I7 0.3 I I I I I I 1 L I Series 8 T C > 60,000 P-4- o r ] 0 2 1) 6 8 10 12 14 16 >I7 R(KBr)/a.u. FIG. 8.-Fraction of the time Tc spent in the interval [R(KBr), R(KBr)+AR(KBr)] for series 8 trajectories with Tc > 60 000 a.u.Notation as fig. 7. that nonreactive long-lived trajectories spend considerably more time in the linear configuration than those which are reactive ; the incomplete trajectories spend approximately equal time in the two configurations. For lower angular momentum (fig. 8, L = 350 h), the intermediate tends to exist primarily in the rhombic form, in marked contrast with the higher angular momentum results (fig. 7, L = 600 ti). Studies of the other series indicate qualitatively similar behaviour. In summary, linear K-C1-Na-Br participates more in trajectories which are nonreactive than those which are reactive, and more for higher L values, independent of whether the traject- ories are reactive or nonreactive.These observations provide an interpretation of the observed low gamma ratio and its behaviour with increasing L. The essential point, in accordance with the suggestion of Miller, Safron and Her~chbach,~ is that the complex consists not only of the rhombic form of the dimer which can dissociate either reactively or nonreactively, but also of linear K-Cl-Na-Br, which can only dissociate nonreactively. Other linear dimers, such as K-Br-Na-Cl, which could lead to reaction, do not contribute significantly. In addition, the decreasing gamma ratio with increasing L, which cannot be observed experimentally, results from the increasing participation of the linear intermediate. These results appear to be a consequence of two, somewhat interrelated, aspects of the dynamics.The first is that in forming the complex, near-liilea~-nonreactive92 COLLISION COMPLEX DYNAMICS configurations contribute significantly (for T‘> 60 000 a.u. systems, 20-40 % are formed as near-linear K-Cl-Na-Br) and that, with increasing angular momentum, they tend to be “ trapped ” more effectively in these configurations, that is, the angular momentum stabilizes linear relative to rhombic configurations (see fig. 7 and 8) due to the difference in rotational energy contributions. The second is that reactive linear configurations, which are formed from the rhombic geometry by bond breaking, make a small contribution because they have a higher potential energy. Of the two effects only the first one, arising from formation of the complex in a linear geometry, will be reduced in the longer, more equilibrated trajectories.As to the second, it is expected to be eliminated in the reverse of reactions (I) or (IT), so that an experimental or trajectory investigation of the relative importance of the two effects might be feasible. (a) W . B. Miller, S. A. Safron and D. R. Herschbach, Disc. Faraday SOC., 1967, 44, 108 ; (b) G. A. Fisk, J. D. McDonald and D. R. Herschbach, Disc. Faraday Soc., 1967, 44, 228; (c) D. 0. Ham, J. L. Kinsey and F. S. Klein, Disc. Favaday SOC., 1967, 44, 174. (a) J. M. Parson and Y. T. Lee, J. Chem. Phys., 1972,56,4658 ; (6) S . A. Freund, G. A. Fisk, D. R. Herschbach and W. A. Klemperer, J. Chem. Phys., 1971,54,2510 ; (c) G. H. Kwei, A. B. Lees and J.A. Silver, J. Chem. Phys., 1971, 55,456. W . B. Miller, S. A. Safron and D. R. Herschbach, J. Chem. Phys., 1972, 56, 3581. * K. Morokuma and M. Karplus, J. Chem. Phys., 1971,55, 63. P. Brumer and M. Karplus, to be published. ti (a) M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Phys., 1965,43, 3259; (b) D. L. Bunker in Methods in Computational Physics, vol. 10, ed. B. Alder, S. Fernbach and M. Rotenberg (Academic Press, New York, 1971), p. 287. ’ (a) P. Brumer and M. Karplus, Perturbation Theory and Ionic Models for Alkali Halide Systems 11. Dimers (to be published) ; see also (b) P. Brumer Ph.D. Thesis, (Harvard University, 1972). * (a) E. S . Rittner, J. Chem. Phys., 1951, 19, 1030; (6) C. T. O’Konski and W. I. Higuchi, J. Chem. Phys., 1955,23, 1175 ; (c) J.Berkowitz, J. Chem. Phys., 1958,29,1386 ; ( d ) P. Brumer and M. Karplus, Perturbation Theory and Ionic Models for Alkali Halide Systems, I. Diatomics. J. Chem. Phys., 1973, in press. The result that there are linear species which are not local minima seems to contradict the “ proof” of N. L. Yarym-Agaev, Zhur. Fiz. Khim., 1964, 38, 2579, that alkali halide dimers have both rhombic and linear minima. However, he demonstrates that the linear form must be a saddle point, which is obvious from physical considerations. lo S. Abromowitz, N. Acquista and I. W . Levin, J. Res. Nat. Bur. Stand., 1968, 72A, 487. * A. Snelson, J. Phys. Chem., 1969,73, 1919 ; S. J. w i n , B. N. Cyvin and A. Snelson, J. Phys. Chem., 1970,74,4338. l 2 Dynamic instability has also been observed in numerical studies on the N-body gravitational problem [e.g., (a) R. H. Miller, Astrophys. J., 1964, 140, 2501 and coupled oscillator systems, [e.g., (b) J. Ford and G. H. Lunsford, Phys. Rev. A, 1970, 1, 591. Indeed there is reason to expect that for systems displaying ergodic behaviour, initially adjacent points in phase space diverge exponentially in time. See, for example, (c) C. Froeschle, Asfron. Astrophys., 1970, 9, 15 and references therein. l3 P. Brumer, Stability Concepts in the Numerical Solittion to Classical Atomic and Molecular Scattering Problems (to be published). l4 C. W. Gear, J. SIAM Numer. Anal., 1964, 2B, 69. Is For phase space theory see, for example, (a) R. A. White and J. C. Light, J. Chem. Phys., 1971, 55, 379 and references therein. For a variety of simple models see (6) W. B. Miller, Ph.D. Thesis (Harvard University, 1969) ; and (c) S. A. Safron, Ph. D. Thesis (Harvard University, 1969) and references therein. For a discussion of the interrelationship between various models see (d) N. D. Weinstein, Ph. D. Thesis (Harvard University, 1972). l6 See fig. 8, 9 in Special Results of Trajectory Studies by M. Karplus in the International School ofPhysics “ Enrico Fermi ”, XLZV Course, ed. D. Beck (Academic Press, New York, 1970). l7 Ref. (3), (15b), (1%). Also, S . A. Safron, N. D. Weinstein, D. R. Herschbach and J. C. Tully, Chem. Physics Letters, 1972, 12, 564. l8 (a) N. B. Slater, Theory of Unimolecular Reactions (Methuen and Co. Ltd., London, 1959) ; (b) R. D. Levine, Quantum Theory of Molecular Rate Processes (Clarendon Press, Oxford, 1969), Chap. 3.6. See also (c) D. L. Bunker, J. Chem. Phys., 1964, 40, 1946.

ISSN:0301-7249    

DOI:10.1039/DC9735500080

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Alkali-Methyl Iodide Reactions BY DON L. BUNKER AND ELIZABETH A. GORING-SIMPSON Dept. of Chemistry, University of California, Irvine, CaIifornia 92664, U.S.A. Received 14th December 1972 Using empirical Monte Carlo trajectory procedures, we fitted a detailed 6-atom potential energy surface to molecular beam data for Rb+CHJ -+ RbI+CH,. Measurements at a single energy of product recoil velocity and scattering angle distributions, and of the effect of aligning the CH31, were employed. We then used the potential surface to predict the energy dependence of the reactive cross-section, and to ascertain the consequences of omitting the CH, hydrogens from the calculations. Experimental data with which to compare our results are expected to appear. Molecular beam studies of the alkali+CH,I reactions have a long history of interaction with trajectory calculations. The first comparison of a Monte Carlo calculation with experimental results was made in 1962.Re-assessments of the situation have appeared at intervals. 3-5 The sophistication of the experiments and of their interpretation have increased, and the trajectory procedures have become more general. There has been a steady growth in the reliability of the potential energy hypersurface that can be extracted from the experimental data by this empirical approach. Recently there have been complementary further advances in the experiments and the calculations. The spatial orientation of CH,I has become an experimental variable,6 and cross-sections have been measured as a function of reactant relative translational en erg^.^ In the trajectory calculations, the necessity for treating CH3 as a single structureless particle disappeared when the first study of a CH4 reaction was made.' This makes possible a more detailed comparison than has been done before.We record here the current status of our efforts to provide one. ETHOD The technology of the calculat features of the present study, viz., method of selecting Monte Carlo symmetric top. s been re~iewed.~ We describe the unique pirical potential function we used, and the , conditions when one of the reactants is a The potential energy is u = UHMF+ U3H-k u R * (1) This was pieced together from several sources. UHMF is a standard three-particle A+BC type of function that has been fully described eIsewhere.'O It provides the aikali-I-C part of the interaction.The U,, term represents the methyl hydrogens and their collapse to a planar configuration at the end of the reaction. It was taken from the first CH, calculation.' There are three alkali-H repulsions in U,, each of the form DR exp [- PR(P - rR)l. (2) All the potential parameters are identified in table 1 . Bonding parameters for stable 9394 ALKALI-METHYL IODIDE REACTIONS reactant and product molecules are also present, and can be identified from the listed references. For starting conditions, we begin with a randomly oriented CH31 molecule. (This involves three random rotations from an initially fixed position, if a general molecular beam experiment is being simulated. For experiments with oriented TABLE 1 .-PARAMETERS OF THE EMPIRICAL POTENTIAL parameter purpose ref.final value KEACTION PATH : ( urnF) “0 sharpness of corner 10, 3 1 /9 A” U O location of downhill 10, 3 - 1 A2 a steepness of downhill 10,3 4 A-” U* location 10 -4.5 A’ L” characteristic 0” 10 9 A-2 B* width 10 6 n directivity 3 1 d intensity 3 23.9 kcal YV CH3 collapse rate 8 2.7 A rs softness 8 4.2A D, intensity eqn. (2) 100 kcal P R suddenness eqn(2) 4A-1 YR characteristic distance eqn (2) 2.8 A ENTRANCE VALLEY CONSTRICTION (u,,,) : ANGLE-DEPENDENT FORCES (&,) : UMBRELLA FUNCTION (u3~) : ALKALI-H REPULSIONS (U,) : CHJ we omit two of them. Simulation of this case is for extrapolated perfect alignment. The alignment axis is 2.) The CH31 angular momentum and its z component are sampled from the probability distributions P(L,) = exp (- L:/21zkT), 0 \< L, < co ; (3) P(L) = L ~ X P ( - LZJ21,kT), L, < L < CO.(4) ( 5 ) (6) Eqn (3) can be sampled by rejection, and eqn (4) by the cdf formula R being a random number OGR < 1. The vector with cos a = LJL, /3 random 0Gp,<27r7 and e unit vectors, is subjected to the same rotation as CH31 was. The rotational atomic momentum contributions are then given by the usual pi = mi@ A ri. (For oriented molecule experiments L = L, and most of this is unnecessary.) L = [L: - 21,kT In (1 - R)]*, o = (Lz/Z,)ez + (L sin a sin P/Z,)e, + (L sin a cos p/I,)e,, The C-I bond has initial vibrational momentum, pu = S(-2kTp In R)+, (7) with S a random sign and ,u the reduced mass of T and CH,. There is also an initial displacement, p- In [l +(p:/2pD)+] sin [n(X -91, (8)D.L . BUNKER AND E . A . GORING-SIMPSON 95 obtained by sampling the classical phase distribution between turning points, with and .D Morse parameters for the C--I bond. These contributions are distributed vectorially over the molecule in the way appropriate to its actual orientation. Lastly, the impact parameter and alkali velocity vector orientation are chosen by standard methods. RESULTS Our results take the form of a tentative potential surface for Rb+CH31, based on molecular beam evidence at fixed reactant energy, plus a prediction of the, as yet, unmeasured energy-dependent cross-section for reactive scattering. The reason for this is that although the cross-sections as a function of energy have been measured for the K reaction, the corresponding aligned-CHJ data are insufficiently precise for our purposes. The final parameters for Rb + CH31 are given in table 1.We determined these by requiring that the RbI scattering be peaked into the backward hemisphere, the product recoil velocity be as large as possible, and the cross-section ratio for extra- polated perfect alignment/anti-alignment be about 4.6*7 The peak scattering angle for the surface finally selected was 110", the average recoil energy was a little over collinear = 180' 135O 90" 4 5 O 0" FIG. 1.-Potential energy surface for Kb+ CHJ, with several angles of approach. Contour intervals are 5 kcal, descending from lower right to upper left for 180, 135 and 90" ascending for the others.96 N s ALKALI-METHYL IODIDE REACTIONS I 1 2 3 4 relative energy/eV FIG.2.-Predicted cross-section as a function of relative energy for Rb+ CH31. 2 1 0 .? 0.4 0 8 2 0 .I FIG. 3.-Distributions of scattering angle 6, product recoil energy vibration and rotation) and CH3 internal energy, at various reactant energies. RbI internal energy (ERbI, 6 = 0 corresponds to forward scattering.D. L . BUNKER A N D E . A . GORING-SIMPSON 97 0.8 of exoergicity with a perceptible tail towards lower values, and the alignment/ anti-alignment reactivity ratio was 3 We rejected 32 potential surfaces in the course of reaching it. (A list of their parameters and individual defects can be supplied, but its presentation here would not be illustrative, because the logical sequence was disrupted by changes in the experimenters' interpretation of their data.The ranges of parameter variation were vo = +_ 1, uo = 3 to 1/10, a = 1 to 6, n = 1 to 4, d = 0 to 47.8, /?* = 3 to 8.) To obtain the scattering and alignment/ anti-alignment ratio simultaneously correct involves a delicate combination of valence-type forces (n, d ) and Rb-H repulsions. At the same time the cross-section has to be kept low (a, p*) and the recoil velocity high (uo, mostly). The parameter variations interact with one another to a much larger extent than has been found in 3-par ticle calculations. The potential surface has been plotted in fig. 1. The plots are for collinear Rb approach, and for Rb--I-C angles of 135, 90, 45 and 0". The difference between the eclipsed and staggered configurations of Rb-I-C--H is perceptible on a large-scale map, but not on a figure of the size printed here.The predicted reactive cross-section as a function of reactant relative translational energy is displayed in fig. 2. The point at 0.1 eV corresponds to the data used to establish the potential surface. The curve shown is arbitrary and is provided only as a guide. Fig. 3 shows how the scattering and the energy disposal appear to change as the reactant energy increases. There is a slow shift from backward to sideways scattering -the reaction would probably approach the stripping regime at high enough energy, but 4 eV is not enough for this. Although most of the exoergicity appears as product recoil at low incident energy, the main effect of increasing reactant translation seems €0 be augmentation of the RbI internal energy. Product relative energy increases slowly, and CH3 internal energy hardly at all.At 4 eV reactant approach energy, vibrationally cold CH3 is still being produced. 1. It was not easy to attain this result. COMPARISON WITH 3-P A R T I C L E C A L C U LA TI 0 NS At 0.5 and 4 eV relative energy, we made calculations with the methyl hydrogens " suppressed ". This was done by setting the R b H repulsions and all the C-H force constants to zero, making the hydrogens into free particles that dispersed with- out interacting with the rest of the system. The calculation is for comparison with - - - 05 eV L 0 180 Q 5 0 4 8 ERel/eV ERbIIeV FIG. 4.-Effect of suppressing the methyl hydrogens. The solid histograms are similar to those in fig.3. The broken ones are with the hydrogens suppressed. 55-D98 ALKALI-METHYL IODIDE REACTIONS the 3-particle study recently made by LaBudde, Kuntz, Bernstein, and Levine. They treated K rather than Rb. Using K in our calculation leads to essentially the same comparative results as are exhibited here for Rb. In fig. 4 we show the effect of hydrogen suppression (broken histograms) on the kind of data displayed in fig. 3. There is a moderately small forward shift in the scattering, and at high energies an increased internal energization of RbI. The other data are unaffected, within the noise level of the calculation. The cross- sections are also about the same when the hydrogens are removed, but-and most importantly-alignment discrimination is lost, and the unfavourably aligned CH31 are as reactive as the favourably aligned ones.This last item is from an additional independent calculation at 0.1 eV relative energy. DISCUSSION Our results must be considered provisional. We expect to have to refine our potential surface further, if the anticipated (cross-section, energy) measurements for Rb+CH31 do not support our prediction. To this end, we have kept the resolution of our calculations minimal, to provide a reserve if it is needed. our calculation of the K + CH31 cross-section as a function of energy, on the assumption that the K and Rb potential surfaces are similar. (This assumption is demonstrably incorrect, at least as regards the ability of the same angle-dependent forces to provide reasonable alignment /anti-alignment reactivity ratios for both reactions.) Our cross-sections for K peak where the experimental ones do, but do not decrease nearly as rapidly with increasing energy.It remains to be seen whether the Rb comparison will also have this feature. If it does, and if further manipulation of the potential parameters does not reproduce it, we may have to consider allowing transitions between ionic and covalent surfaces to play a role in the trajectories. We have rejected so many surfaces in the course of the work reported here that our remaining flexibility of choice cannot be large. employs the locally lower of two potential surfaces, ionic and covalent, with the crossing near the closest approach of all the atoms (roughly, vo = 0 in our terms).Our potential also has a sharp transition (large a), but this arose naturally from the experimental data rather than by advance specification. Our crossing-if that is indeed what it corresponds to-is much earlier in the reactant channel than is true in the 3-particle representations. However, before we began trying to encompass the aligned-CH,I data, our transition points also tended to be near uo = 0. It is too early to assess the relationship between these two trajectory calculations. The 3-particle calculation is much less empirical than the one described here. Comparison is mainly with the cross-section energy dependence, and the other data take second priority. Our work does just the opposite. The aligned-CHJ data are of crucial importance for us and have not been used in the 3-particle procedure, probably for the reasons given earlier.The K-Rb mismatch of experimental subject matter continues to be a nuisance. But there is hope for clarification, possibly by the time this paper is published. We have published The 3-particle calculation of LaBudde, Kuntz, Bernstein and Levine We thank the National Science Foundation for supporting this work. N. C. Blais and D. L. Bunker, J. Chem. Phys., 1962,37,2713. D. R. Herschbach, Disc. Faraduy SOC., 1962, 33, 149. D. L. Bunker and N. C. Blais, J. Chem. Phys., 1964,41,2377. L M. Raff and M. Karplus, J. Chem. Phys., 1966,44,1212.D . L . BUNKER AND E . A . GORING-SIMPSON 99 C . Ottinger, J. Chem. Phys., 1969, 51, 1170. R. J. Beuhler and R. B. Bernstein, J . Chern. Phys., 1969, 51, 5305. M. E. Gersh and R. B. Bernstein, J. Chern. Phys., 1972,56,6131. D. L. Bunker and M. D. Pattengill, J. Chem. fhys., 1970,53, 3041. D. L. Bunker, Methods in Cumpututiond Physics (Academic Press, New York, 1971), vol. 10, p. 287. D. L. Bunker and C. A. Parr, J. Chem. Phys., 1970,52, 5700. cf. N. R. Davidson, Statistical Mechanics (McGraw-Hill, New York, 1962), p. 173. R. A. LaBudde, P. J. Kuntz, R. B. Bernstein and R. D. Levine, private communication of a manuscript (WIS-TCI-407). l 3 D. L. Bunker and E. A. Goring, Chem. Phys. Letters, 1972, 15, 521.

ISSN:0301-7249    

DOI:10.1039/DC9735500093

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Energy Disposal and Energy Requirements for Elementary Chemical Reactions * BY R. D. LEVINE-~ Department of Physical Chemistry, The Hebrew University, Jerusalem, Israel AND R. B. BERNSTEIN Theoretical Chemistry Insti t Ute and Chemistry Depart men t, University of Wisconsin, Madison, Wisconsin 53706 U.S.A. Receiwd 29th December, I9 72 The role of energy in promoting chemical change and the reverse problem of the energy distri- bution of the reaction products are of fundamental interest in the study of chemical dynamics. Since the earliest atomic flame experiments it has been known that exoergic elementary reactions often lead to products with high internal excitation. More reccntly, direct evidence from molecular and ion beam experiments indicates that, for reactions with an activation barrier, reagent internal energy is often more effective in promoting reaction than is translational energy.In addition, detailed product state distributions for exoergic reactions determined by chemiluminescence, chemical laser and molecular beam mcthods have frequently indicated substantial population inversion. Both phenomena (obviously related, through microreversibility) can be qualitatively understood from potential surface considerations (via classical trajectory analysis), but several problems remain. We address ourselves to the following questions : (a) optimal means of characterization of the product (or reactant) energy state distribution, (b) compaction of the voluminous body of data (or computer-simulation thereof) comprising such distributions, and (c) development of measures of the specificity of the energy release, of the selectivity of the energy consumption (and their mutual dependence). As a link between the reactive scattering and the non-equilibrium statistical mechanics we use the concept of the entropy (or information) of the product (or reactant) state distribution.The required measure of specificity (or of selectivity) is provided by the concept of the surprisul of a particular outcome. This approach leads to a means of dealing with molecular bcam measurements of product angular and translational energy distributions. The molecular beam scattering technique provides the possibility of measure- ment of the angular distribution for specified product states as well as the influence of reactant state selection upon the detailed reaction cross sections.Even in its present state of development, the molecular beam method has revealed many remarkable features of the dynamics of reactive collisions. From these results, together with those obtained via the chemiluminescence technique (which has yielded the most extensive information on product state distributions) and with the availability of confirmatory chemical laser experiment^,^ there is now emerging a picture of element- ary reactions which can be characterized by two words : spec$cify and selectivity. * Supported by the U.S. Air Force Office of Scientific Research (AFSC), Grant AFOSR-72-2272 (Ohio State University) ; also NASA Grant NGL 50-002-001 and NSF Grant GP 35848X (University of Wisconsin).t Also : Department of Chemistry, Ohio State University, Columbus, Ohio 43210. 100R. D. LEVINE AND R. B. BERNSTEIN 101 Exoergic reactions of atoms with diatomic molecules generally produce quite specific product state distributions (frequently yielding extensive population inversion). When competing reaction products are possible, the branching ratio is often found to be far different from expectation based on simple equilibrium or statistical consider- ations. Not only are these exoergic reactions highly spec$% in their mode of energy release, their endoergic counterparts are highly selective in their energy requirements. For reactions with an activation barrier, internal excitation of the reactants is often more effective than translation in surmounting the potential barrier.6 This is, of course, as expected ’ on the basis of microscopic reversibihty ; specificity of release and selectivity of consumption are different manifestations of the same (time-reversal- invariant) molecular dynamics on a given potential surface.To take advantage of this inherent symmetry, one makes use’ of the concept of the yield function lo or the averaged (state-to-state) reaction probability, Kinsey’s * The information-theoretic approach ’ furnishes a link between scattering and non-equilibrium statistical mechanics, via the entropy of the product (or reactant) state distribution. The entropy deficiency, i.e., the difference between the entropy of an observed product distribution and that of a statistical, “ phase space-governed ”, distribution is an integral measure of the specificity.’ ’ The differential measure of specificity is the surprisal of a particular outcome,12 an expression of its deviation from (statistical) expectation.Analysis of the surprisal can, in favourable cases, yield a simple characterization of the entire nonequilibrium product distribution in terms of translational,’ vibrational and rotational l3 “ temperature-parameters ”. In the present paper we present extensions of this approach which are pertinent to the problem of characterizing molecular beam measurements of product angular and translational energy distributions. Also considered is the influence of sekctivity upon speciJcity and vice-versa, i.e., the relevance of the reactant energy distribution to the energy disposal in the products.qr, r’). YIELD, TRANSLATIONAL EXO- AND ENDOERGICITY, AND THE SURPRISAL The yield lo is the rate of all transitions from a group of reactant states to a group of states of product. If rc(T+T’) is the measured rate at a given total energy E (averaged over the initial group r and summed over the final group r’) then the yield is y(r, r’) = P(r)K(r+rl) (1) where p ( f ) is the density of states (per unit volume) of the reactants. Kinsey’s * averaged transition probability is w(r, r’) = q=+rf)/p(ri) = K(r‘+r)/p(r) = w, r‘)lP(r)P(r‘). (2) In the simple statistical limit, i.e., all energetically allowed final states equi- probable, K(r‘-”r‘’)cfp(r‘’), so the average transition rate is the same for all final states of either the forward or reverse reaction, i.e., w ( r , r‘) is a constant, and there- fore y r , r’) cf P(r)p(r’).(3) This simple result requires modification to take account of the restriction of totalI02 ENERGY DISPOSAL AND ENERGY REQUlREMENTS angular momentum ( J ) conservation. If d(T--+r') is the rate for a given J, then in the statistical limit rcJ(r-T')oCpJ(I'') so that In practice, this additional complication is not very serious. It can be readily dealt with by first computing K(I'-+r') by summation, using the more correct statist- ical approximation s (which imposes the requirement of conservation of total J), and then calculating o(T, I"). In applications to molecular beam scattering, the most direct specification of the initial and final states is in terms of the fractionfT of the total available energy E in trans1ation.l' In ref.(12) the appropriate density of states expressions have been presented to allow calculation of the a-priori " expected " (i.e., microcanonical equi- librium, equiprobable states) translational distribution function P"(f,lE). Two alternative approximate expressions were developed, one based on the vibrating-rotor (VR) level scheme (in which a continuous distribution is used for the rotational states, but vibrational states quantized) and the cruder RRHO approximation (which assumes a continuous distribution of both vibrational and rotational states). The result in the latter approximation (for the probability distribution at a given total E), is l 2 Unfortunately this is a poor approximation forfT-+ 1 due to the incorrect handling of the v = 0 state, which does in fact contribute significantly to the density atf,+l.r 1 I I I 1 1 I I I 1 - \ - \ 0 0.2 0.4 0.6 0.8 1.0 FIG. 1 .-Comparison of the RRHO approximation (dashed) and that for the VR level scheme (solid curve) applied to the calculation of the translational energy distribution when the product diatomic has large vibrational spacings. The VR calculation is for the reaction Cl+HI+I+HCI'(. at E = 34 kcal mol-I. The sharp drops correspond to the cutoff of the different vibrational manifolds of fT HCI. A comparison of the VR results with the RRHO approximation is shown in fig. 1 for the reaction C1+ HI+I + HClt at E = 34 kcal mol-'. (Obviously the discrepancy is worst for reactions producing hydrogenic diatomics, with large vibrational spacings.) The most precise Po is obtained by the full statistical theory (employing conservation of total angular momentum).Fig. 2 shows the results of such calculations (once again compared with the RRHO curve), which are seen to be not very different than the VR-approximation (cf. fig. 1).R . D . LEVINE AND R . B . BERNSTEIN 103 1 I I . - \ - - \ I I I 0 0.2 04 06 0.8 1.0 fT FIG. 2.-Comparison of the RRHO approximation (dashed) and a calculation by a statistical theory (with conservation of total angular momentum) which includes smoothing of the discrete distribution by averaging over a small energy interval. The normalized yield at a given E, in the equiprobable expectation approximation (here neglecting the J-conservation restraint) is given by This function is shown in fig.3, in the limiting case of the RRHO approximation. It is noted that the distribution is not uniform in thefT,fTt plane. The rapid rise from po(fT, fT') = d f T I E ) p ( f Y I E ) * (6) FIG. 3.-Contour diagram of the normalized yield Po(fT, fr) in the RRHO approximation. Shown are equi-yield contours, with the most-probable point1at f~ = f ~ e = 3, P" = 25/12. the (0,O) corner is due to the increase of the translational density of states with fT while the decline toward the (1, 1) corner arises from the decrease in the number of available internal states with increasing fT. Fig. 4 shows a hypothetical contour map of specific reaction rates, i.e., 7c(fT4fTt) at a given E. Such a map serves as a summary (here simulated) of results of an xtensive translational energy study.(In terms of the fractions fI, fIt in internalI 04 ENERGY DISPOSAL AND ENERGY REQUIREMENTS energy, the diagram would be the same providedfT were replaced by 1 -fl andfT# by 1 -fi#.) Here KdfTdfT# corresponds to the rate from reactants with & in the range ET, & f dE, tO products E y , ETn + d&. 1- 1 0.G I ' "I ' ' FIG. 4.-contour diagram of K(fT-ffT') for a reaction with a barrier in the entrance valley (schematic only). fT FIG. 5.-Contour diagram of ~ ( f ~ , f ~ t ) corresponding to the K diagram of fig. 4. It represents the uncoupled approximation, o = exp[ - AT~T- h ~ ' f T * ) : here AT = - 1, AT' = 4. A more useful representation is shown in fig. 5, in which we exhibit not the average reaction rate (nor the reaction cross section) but rather Kinsey's averaged transition probability in the form of a so-called global inclusive representation (" poor-man's plot ").The o display amntuates the deviation from (statistical) expectation and hence d f T , = P(fn fT')/P*(fT, A*), (7)R. D . LEVINE AND R. B . BERNSTEIN 105 isolates those features of specificity (or selectivity) introduced by the reaction dyn- amics. The rather simple dependence of co uponfT,fT# (fig. 5) is discussed below. Full global-inclusive knowledge of any reaction is still in the future (experiment- ally). At present one is restricted to cases in which translational analysis experiments are available either for reactants or for products. Even for such " averaged inclusive" experiments we can still consider the deviation in either thefT or thefTt distribution (as the case may be) with respect to the a-priori statistical expectation. Thus we have, for the case of product distributions cou(fT) = p(fT')/po(fT' (8) where Po(fT#) = p(fT.IE) is the expected distribution in fT# in the absence of any dynamical bias, computed as previously discussed.From the information theory viewpoint l 1 we are interested in the surprisal of a given, observed fractional energyfx with respect to that expected on a statistical basis : In the present case, e.g., considering product translational distributions, (X = T') : W X ) = - log C~(fx>/~"(fX)l = - log Offx). (9) P(fT') = Po( fT') exp [-I(&*)] = q,'P"( fT*) exp [ - & p fT'+ . .I.(11) Here we have expanded I(fx) in a (rapidly converging) Taylor series about the origin, with and qx = exp [1(0)1 Ax = [dMx)/dfxlo* Higher order terms in thefx expansion are neglected, based on our observation that for nearly all cases investigated I(fx) is essentially a linear function offX.12 In this approximation (12) i.e., AT* = dZ(fTe)/dfTt is simply a constant, termed l 2 a translational temperature parameter. It is a differential measure of the specificity, i.e., the deviation of the observed distribution P(fTp) from the " reference ", a-priori expected distribution Po(fTt). The larger the ATt the more specific is the energy release, most of it going into product excitation rather than translational recoil. A negative AT, implies an " abnormally high " translational exoergicity, while AT# = 0 corresponds to the statist- ical expectation.Similarly, we can define the surprisal with respect to the selectivity of the energy requirements of the reactants, (13) jZT = dz(fT) ldfT. (14) In contrast to experiment, present computational techniques (via classical tra- lectories) are perfectly adequate to produce the full set of ~ ( f T + f T t ) contours ana- jogous to fig. 4. From the rates we obtain ci)(fT,fT*) : p(fT') = po(fT') exp [-'h'fT'l, I(fT) = -log!p(fT)/po(fT)l = -log O(fT). ThefT-derivative of Z(fT) is the differential measure of the selectivity : 4 f T , f T # ) = ~(fi- + fTt)/~(fT') (15)106 ENERGY DISPOSAL AND ENERGY REQUIREMENTS (cf. eqn (2)) and hence the T, T’ surprisal, defined I(fT, fT‘) = -log [O(fT, fT’)] = -log Ep(fT, fT‘)lPo(fT, fT’)l* (16) The simplest assumption about the form of Z(fT,f’*) would be the “ uncoupled approximation ”, i.e., so that p(fT, fT’) p(fT)p(fT’), (17) I(fT, fTt) = I(fT) -k I(fT‘)- (18) Fig.4 and 5 were calculated using this uncoupled model, assuming also the simple linear Z(fx) relation (i.e., that AT and AT* are constants over the entirefT,fTr range). The uncoupled approximation implies that velocity selection of reactants would not influence the surprisal of the products. Since this is not expected to be true in general we must learn to deal with the coupling problem (as discussed below). ENTROPY DEFICIENCY AND PRODUCT STATE DISTRIBUTIONS The surprisal of any particular final state provides a local measure of the deviation of the observed probability (i.e., fraction in that state) from the statistical expectation.The temperature parameter AT = dZ(fT)/dfT, (a near-constant, only slowly varying with fT) is the differential measure of the deviation, which can be used for extrapolation or interpolation. However, one still needs an integral measure of the “ overall ” aver- age deviation of P from Po, which is provided by the entropy (the average value of the surprisal) : (For simplicity we shall use a dimensionless entropy, dropping the Boltzmann constant factor and lg/log ratio. 2, The entropy dejciency (relative to the equiprobable statistical expectation) is a more useful measure of the overall specificity ; it is given by l1 = sequi.-s[fT] = dfTP(fT) log co 2 O* (20) J: It can be shown l2 that the entropy deficiency of a detailed product state distri- bution (with vibrotational states fully resolved) is always greater than that obtained from results of velocity analysis, i.e., AScfTO] c AS[fRl,fve].When the contribution from the different vibrational manifolds can be fully resolved, as in the reaction l4 F+D,-+D+DF(v’), then P(fTt) = XVtP(fT#, u’) and considerably more information can be extracted. The areas of the band contours, i.e., P(v‘) = iP(fTp, u’) dfTt, determine the product vibrational state distribution and thus the vibrational temperature and entropy deficiency. From thefTt-dependence of the band contours one can obtain P(fR.]u’) = P(fTt, u’)/P(z;’), the distribution of rotational energy within a given vibrational manifold. The a-priori distribution, for comparison, is best obtained from a full statistical calculation, i.e., Po( fR0lv‘) = (da”(V’)/d fRf)/ao(v‘) (21) where oo(u’) is the statistical-theoretical calculated reaction cross section into theR.D. LEVINE AND R. B . BERNSTEIN 107 vibrational manifold u’. The associated rotational entropy deficiency, summed over all manifolds, is A s [ f R ’ l v ’ ] = 1 p(u’) c p ( f R ’ l v ’ ) log [ ~ ( f ~ ’ l v ’ > / p o ( f R ’ l u ’ ) ] . (22) 0‘ j ‘ (In ref. (1 3) an expression is given for the P/P” ratio in terms of a rotational tempera- ture parameter.) Next one can evaluate the entropy deficiency of the product internal energy distribution l2 : As[ fR’, v’] = AS[v’] +AS[ fRtlV‘] = c c p ( f R ‘ , v’) log [p(.fR’, v ’ > / p o ( f R ’ , v’>].(23) v’ j ‘ The entropy deficiency of the translational distribution, AS[fTt] is always smaller than this ASVRP, v’], as follows from a basic theorem of information theory.15 (Any averaging reduces the entropy deficiency ; since all vibrational manifolds are summed the resulting distribution is nearer to an equilibrium one.) The difference between these two quantities is the equivocation (following the usage of Shannon 15) : As[V’lf~.] = AS[f&, V ’ ] -AS[fy] > 0, (24) a measure of how much more information is gained when the resolution of P(fTt) into vibrational band contours is possible, i.e., when one can extract P(fT*,v’) and P ( ~ R + J ’ ) from the P(f+). Eqn (24) can be expressed in terms of P’s as follows : Defining P(v’lfTt) as the probability (fraction) of products withfTt in the range fTt,fTp +dfTp, in the vibrational state v’ : p(fT’, v‘> = p(v’lfT’)p(fT’), (26) eqn (25) becomes In the RRHO approximation the Po takes a simple form : P ” ( V ’ l f T , ) = yf$,/[yf+,(l - f T ’ ) ] = (1 -fT,)-’.(28) Another identity, closely related to eqn (24), expresses the gain in information when one has not only P(u’) but also P(fT#, 0’) : As[ fT#lV’] = As[ fT’, v‘] -AS[V’] > 0. (29) Returning to eqn (24) we see that the equivocation AS[v’IJ;p] is the additional information gained when vibrational resolution is achieved. The resulting P&, u’) is then equivalent to that of a chemiluminescence experiment (except that since the latter does not contain information on the v’ = 0 state, the resulting P(fRt, v’) actually yields less information than P(fTt, u’).When only limited data are available from a chemiluminescence experiment, one may utilize the temperature parameter concept to extrapolate, e.g., to obtain P(v’ = 0),108 ENERGY DISPOSAL AND ENERGY REQUIREMENTS or to interpolate as desired. One can thus “ synthesize ” the translational spectrum PUT#) which would be observable via molecular beam velocity analysis. Thus P(fT’) = P(fT’, v’) = p(u’)p(fT’lu’) (30) 0’ V‘ where (from ref. (12) and (13)) P(v’) = P”(v’) exp [ - Av. fvf]/Qv. and The Po factors in eqn (31,32) are, in the RRHO approximation P”(v‘) = 3(1- fVy Po(fR‘lu’) = w1 -fV’-.fR’)*/[$(l -fV‘)’l (33) (34) (Slightly more complicated results for the Po factors are obtained in the more exact VR level scheme treatment.12 The best Po’s are those computed by the complete statistical theory.) Eqn (30) (with eqn (31)-(34)) enables us to make an approximate “ synthesis ” of the product translational distribution, P(fT$), given 3Lvt and OR# at the specified E.Of course it will not, in general, be of the simple form of eqn (12), but the general expo- nential character of the o(fTr), i.e., the near-linearity of the translational surprisal, I(fTt), is retained. ENTROPY DEFICIENCY AND ANGULAR DISTRIBUTIONS The most revealing molecular beam scattering measurements yield the detailed differential cross section (angular and recoil energy distribution) at a given E : The entropy deficiency is From ref. (12), Po( fTt, a) = PO(nl fT’)Po(fT’) = (4n)- ‘Po(fT*).AS[aI fTe] = As[ f T ’ , a] -AS[ fTt] > 0. (37) (38) The additional information gained by the angular resolution of the scattering is If previously only the angular distribution were known, the additional information gained by velacity analysis is As[ fT,IQ] = As[ fT‘, 01 -AS[Q] (39) where AS[sZ] = J d2RP(n) log [P(~)/PO(~)], 4 nR . D . LEVINE AND R . B . BERNSTEIN 109 with For the primitive, “ uncoupled approximation ” , 1 6 i.e., assuming P(fT’, 0) to be factorizable as the product P(fT*)P(O), we obtain from eqn (36) the simple result AS[ fT1, a] = AS[O] -I- AS[ f ~ p ] . (42) Thus, using eqn (42), AS[Q/fT’] = A,!?@) and A,!?[fTtlO] = AS[fT.]. general the angular and recoil distributions are coupled and However, in As[fTp, 01‘2 As[0] +As[f~.], (43) AS[fT’, 01 = As[Q] +AS[fT’] +M[fT‘, 01 (44) i.e., where M , the mutual information is here defined by It is the amount of information on the fTg distribution provided by the measurement of the angular distribution (or vice versa).Clearly M>O, with the equality in the uncoupled limit. The concept of mutual information can be applied to other joint distributions, e.g., where RELEVANCE As techniques improve it should become possible not only to measure P(fTf) but also (via detailed experiments) to determine P( fT’,fT), thus gaining information on the relevance of the reactant energy distribution to the energy disposal in the products. In quantitative terms we define relevance by so that, using Shannon’s inequality l 7 R 0 with equality only in the uncoupled limit Note that relevance is again an average value of the appropriate surprisal.It provides a quantitative measure of how far one can alter the energy disposal by modifying the reactant energy distribution. Since by construction P(fT,fT‘) is a symmetric function of the initial and final channel, so is R. The relevance of the reactant to the product channels is the same for both the forward and reverse reactions, at the same total energy E. p(fT‘, f T ) = p(fT’)p(fT)- Using the representation p(fT 3 f T ’ ) = P(f T’ I fTlp(f TI* (49)= s[fT’] -s[fT’lfT1 = As[fT’lfT1 -As[fT’l* (51) Thus the relevance is the additional information gained from “ detailed ” We also note that over “ inclusive ” experiments. AS[~T’, f T ] = J 1 J ’ dfT’ dfTP(fT’’ f ~ ) log [P(~T‘, fT)/po(fT’)yo(.fT)l 0 = AS[fTp] + As[f~] + R = As[fT1 +As[fT’lfT] = As[fTp] +As[fTIfT’I* (52) A diagramatic summary of these relations is presented in fig.6. 4 f T I f T l l A S [ f.fI IT] FIG. &-Schematic relationships (Venn diagram) among measures of specificity and selectivity. Symbols defined in text. CONCLUDING REMARKS Increasing experimental sophistication and computational capability in the field of chemical dynamics have led to a considerable gain in understanding of both specificity and selectivity of reactive iiiolecular collisions. The wealth of new data (and theirR. D. LEVINE A N D R. B . BERNSTEIN 111 considerable detail) makes it imperative that results be compacted and codified in a systematic fashion. The present paper and the preceding ones in the series 11-13 represent attempts in this direction.The concepts of entropy deficiency AS as an overall measure of specificity (and selectivity), and of the surprisal I(fx) as a local measure thereof, provides ample means of characterization of the findings. In many cases the main features of the detailed distributions can be accounted for (and the distributions thereby " synthesized " in terms of a very few parameters, e.g., AT, A", 8,. In particular, we have focussed on translational and angular distributions, accessible only by the molecular beam scattering technique. Introduced in the present paper is the concept of the relevance R, i.e., a quantitative measure of the influence of the reactant energy distribution upon that of the products and vice versa.This is becoming important in the light of newer experiments in chemiluminescence,19 molecular 2o and ion Consideration has been limited thus far to the simplest atom-diatomic exchange reactions, mainly of the " direct " rather than " complex " type, where none of the additional complications (and compensatory simplifications !) of polyatomic systems arise. Work in progress includes extension of the present methods to such systems as well as the detailed analysis of hydrogenic systems (for which the RRHO approxi- mation is inadequate and J-conservation important), the construction of " entropy cycles ", and a quantitative investigation of relevance (using both experimental and computational data). beams. The authors appreciate valuable discussions with Dr. A.Ben-Shaul, Dr. B. R. Johnson and Prof. G. L. Hofacker. They also acknowledge with thanks the kind help of Dr. Johnson who carried out several of the computations presented in the figures. For a recent review of the field, see J. L. Kinsey, chapter in MTP International Review of Science, Vol. 9, Chemical Kinetics, ed. J. C. Polanyi (Butterworths, London, 1972), p. 173. See also D. R. Herschbach, Ado. Chem. Phys., 1966, 10, 319 for earlier work. See review by T. Carrington and J. C. Polanyi, chapter in MTP, Vol. 9 (cf. ref. (l)), p. 135, and references cited therein. A recent example is : J. C. Polanyi and J. J. Sloan, J. Chem. Phys., 1972,57,4988. See review by C. B. Moore, Ann. Reu. Phys. Chem., 1971, 22, 387 and references therein. See, e.g., J. D. McDonald, P.R. LeBreton, Y. T. Lee and D. R. Herschbach, J. Chem. Plzys., 1972, 56, 769. See, e.g., (a) J. C. Light, Disc. Faraday SOC., 1967, 44, 14; (6) E. E. Nikitin, Theor. Expt'l. Chem., 1965, 1, 83, 90; (c) D. G. Truhlar, J. Chem. Phys., 1971, 54, 2635. See, e.g., (a) D. H. Stedman, D. Steffenson and H. Niki, Chem. Phys. Letters, 1970, 7, 173 ; (b) F. P. Tully, Y. T. Lee and R. S. Berry, Chem. Phys. Letters, 1971,9, 80 ; (c) T. J. Odiorne, P. R. Brooks and J. V. Kasper, J. Chem. Phys., 1971,55,1980 ; some of the early evidence has been reviewed by (d) R. B. Bernstein, Israel J. Chem., 1971, 9, 615. 'See, e.g., K. G. Anlauf, D. H. Maylotte, J. C. Polanyi and R. B. Bernstein, J. Chem. Phys., 1969,51,5716. J. L. Kinsey, J. Chem. Phys., 1971, 54, 1206. For background, see J.Ross, J. C. Light and K. E. Shuler, Chap. 8 in Kinetic Processes in Gases and Plasmas, ed. A. R. Hochstim (Academic Press, N.Y., 1969). (a) R. D. Levine, F. A. Wolf and J. A. Maus, Chem. Phys. Letters, 1971,10,2 ; (6) R. D. Levine and R. B. Bernstein, Chem. Phys. Letters, 1971,11, 552 ; (c) R. D. Levine and R. B. Bernstein, J. Chem. Phys., 1972, 56,2281. lo C. A. Coulson and R. D. Levine, J. Chem. Phys., 1967,47,1235. R. B. Bernstein and R. D. Levine, J. Chenr. Phys., 1972, 57, 434. (a) A. Ben-Shaul, R. D. Levine and R. B. Bernstein, J. Chem. Phys., 1972,57,5427 ; (b) Chem. Phys. Letters, 1972, 15, 160. l 3 R. D. Levine, B. R. Johnson and R. B. Bernstein, Chem. Phys. Letters, 1973, 18. l4 T. P. Schafer, P. E. Siska, J. M. Parson, F. P. Tully, Y. C. Wong and Y. T. Lee, J. Chem. Phys., 1970.53, 3385.112 ENERGY DISPOSAL AND ENERGY REQUIREMENTS l 5 C. E. Shannon, Bell System Tech. J., 1948, 27, 379, 623, reprinted in C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (Univ. of Illinois Press, Urbana, 1949). See also A. Katz, Principles of Statistical Mechanics-Information Theory Approach (W. H. Freeman, San Francisco, 1967). If Xj and yj are normalized, Cxi lOg(Xj/yj)> 0 with equality if xj = yi. Since PcfTt,f~) is a normalized yield, P ( j + , f ~ ) = P(fT’lfT)f(fT), where P(f~nlf~) is the normal- ized rate of transitions from a group of initial states (with translation in the rangefT,fT + dfT) to a group of final states (translation in the rangefT, f+ + df+). I 9 See, for example, L. T. Cowley, D. S. Horne and J. C. Polanyi, Chem. Phys. Letters, 1971, 12, 144 and L. J. Kirsch and J. C. Polanyi, J. Chem. Phys., 1972,58,4498. 2o Ref. (4) and A. M. Rulis and R. B. Bernstein, J. Chem. Phys., 1972, 57, 5497. 21 See, for example, R. B. Cohen, J. Chem. Phys., 1972, 57, 676 and references therein. 22 See, for example, (a) W. B. Miller, S. A. Safron and D. R. Herschbach, Disc. Faraduy Soc., 1967, 44,108,292 and J. Chem. Phys., 1972,56,3581; (6) S. M. Freund, G. A. Fisk, D. R. Herschbach and W. Klemperer, J. Chem. Phys., 1971, 54,2510; (c) H. G. Bennewitz, R. Haerten and G. Miiller, Chem. Phys. Letters, 1971, 12, 335. l6 See, for example, T. T. Warnock and R. B. Bernstein, J. Chem. Phys., 1968,49, 1878. i

ISSN:0301-7249    

DOI:10.1039/DC9735500100

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. D. R. Herschbach (Harvard University) said : The four-centre collision complex studied by Brumer and Karplus nicely demonstrates how nonstatistical decay can arise from geometrical isomerism and high centrifugal momentum. Three-atom reactions can show the same phenomenon, and indeed it is very prominent in exchange reactions of alkali atoms with alkali halide molecules, A+B+X- +(BA)*x- -+ A+X-+B. More than 20 such reactions have been studied in crossed-beam experiments.’. Statistical models give good agreement with the product velocity and angular distri- butions but in each case overestimate the ratio of reactive to nonreactive decay of the collision complex, often by a factor of 3 to 5 or more. Since these systems have only one electron outside the closed shells of the A+, B+, and X- ions, the pseudopotential approximation can be expected to provide accurate potential surfaces.Roach and Child and Struve have calculated such surfaces for several cases. These surfaces predict the most stable configuration of the complex is triangular but the preferred direction of approach is collinear, with the incoming alkali atom attacking the “ wrong end ” of the salt molecule. This configuration corresponds to linear (AB)+X- and reflects the stability of the dialkali ion. Only a low potential energy ridge, typically N 2 kcal/mol, separates the collinear and tri- angular configurations. However, as a consequence of the strong long-range attraction, most complexes are formed in collisions with large impact parameters and thus the centrifugal momentum often restrains the roughly collinear (AB)+X- con- figurations from bending into the triangular configurations required for reaction. A trajectory study of the K + NaCl reaction has recently beeh reported by Kwei, Boffardi, and Sun.S This employed an analytic form for the potential surface which conforms approximately to the Roach-Child surface for the triangular configura- tions corresponding to the potential basin but lacks the long-range anisotropy favouring the “wrong-end” approach.Kwei, et al. find for their surface the ratio of non- reactive to reactive decay of the complex is statistical. This suggests the anisotropy indeed produces the observed nonstatistical decay. Kwei is now making a direct test in a new round of trajectory calculations. Dr.S. M. Lin and Dr. R. Grice (Cambridge University) said: We should like to report calculations of potential surfaces for M2X+ ions, where M denotes an alkali atom, X a halogen atom. A generalisation of Rittner’s theory for alkali halide molecules was employed, assuming the electronic structure to be a cluster of three ions, M+, X-, M+. The binding energies of M2X+ with respect to M++MX were W. B. Miller, S. A. Safron, G . A. Fisk, J. D. McDonald and D. R. Herschbach, Disc. Faraday G. H. Kwei, A. B. Lees and J. A. Silver, J. Chem. Phys., 1971,55,456; 1973,58, 1710. A. C. Roach and M. S. ChiId, Mol. Phys., 1968, 14, 1. W. S. Struve, Mol. Phys., 1973, 25, 777. G. H. Kwei, B. P. Boffardi and S . F. Sun, J. Chenz. Phys., 1973, 58, 1722. S. M.Lin, J. Wharton and R. Grice, Mol. Phys., 1973, 26, in press. Soc., 1967, 44, 108, 292 and J. Cheni. Phys., 1973, to be published. ’ E. S. Rittner, J. Chem. Phys., 1951, 19, 1030. 113114 GENERAL DISCUSSION found to be substantial ( N 1.5-2.0 ev) and to be in accord with the mass spectro- metric determinations of Chupka. The most stable geometry was found to be linear for the case of heavy M, light X, e.g., Cs+F-Csf, but to become bent for the case of light My heavy X, e.g., Li+I-Li+. Since the corresponding M2X molecules are more weakly bound ( N 0.5 eV) with respect to M + MX, they have surprisingly low adiabatic ionisation potentials-lower than or comparable to the alkali atom, Iad(M2X)qI(M). Although the M2X molecule geometry is more bent than that of the corresponding M2X+ ion, the vertical ionisation potential remains low, comparable to that of the alkali atom Iv(M2X)-Z(M), over a considerable range of bent geometry for K2X, Na,X. This low vertical ionisation potential Iv(M2X) has important implications for the reaction dynamics 3-6 of alkali dimers M2 with polyhalide molecules AX,. Even in reactive collisions with large impact parameters, it provides a rapid mechanism for a second electron jump 3-6 resulting in the capture of both alkali atoms M of the dimer M2 as alkali halide molecules MX.Thus, the reaction dynamics may be represented schematically as : K2 + AX,, + Kz +AX, -+ KZX- +AX,- -+ K2X++AXn-.1 -+ 2KX+AX,,-2. (1) Dr. P. Brumer (Harvard Uniuersity) said : In response to Lin and Grice, I should like to know if their generalization of the Rittner model included three body repulsive contributions.As we have remarked it was necessary to use electric-field dependent repulsive interactions, which are four body in nature, in order to obtain a reliable model potential for the NaBrKCl system. The equilibrium geometry of the dimer was found to be quite sensitive to terms of this kind. Prof. S. A. Rice (University of Chicago) said : I am interested in the comment in the opening paragraph of the paper by Brumer and Karplus, referred to again later, that the results of their trajectory calculations deviate somewhat from the predictions of the statistical model of reaction rate. As I am sure the authors are aware, there are available a number of studies,'both analytic *-l and numerica1,l 2-1 of the motion of N nonlinear coupled oscillators hescribed by the Hamiltonian H = Ho(p"', 9"') + A V(p"', q"').(As usual the notation is such that the integrable part of the Hamiltonian is Ho and AV is everything else). Trajectory computations 2-1 of the motion in phase space of such systems show two interesting features : W. A. Chupka, J. Chem. Phys., 1959,30,458. J. C. Whitehead, D. R. Hardin and R. Grice, Mol. Phys., 1973, 25, 515. P. B. Foreman, G. M. Kendall and R. Grice, Mol. Phys., 1973, 25, 529. P. B. Foreman, G. M. Kendall and R. Grice, Mol. Phys., 1973, 25, 551. * A. C. Roach and M. S. Child, Mol. Phys., 1968, 14, 1. ti D. R. Hardin, K. B. Woodall and R. Grice, Mol. Phys., 1973, in press. ' P. Brumer and M. Karplus, this Discussion.V. I. Arnol'd, Russ. Math. Survey, 1963, 18, 85. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954,98, 527. K. C. Mo, Physica, 1972, 57, 445. lo J. Moser, Math. Annalen, 1969, 169, 136. l 2 M. Henon and C. Heiles, Astronom. J., 1964, 69, 73. l3 B. Barbanis, Astronom. J., 1968, 71, 415. l 4 G. H. Walker and J. Ford, Phys. Ret.., 1969, 188,416. R. H. Miller, Astrophys. J., 1964, 140, 250. J. Ford, Ado. Chem. Phys., 1973, 24, in press.GENERAL DISCUSSION 115 (i) When A is small the motion of the system is conditionally periodic, but when 3, exceeds a threshold value, A,, the motion becomes essentially stochastic. (ii) Consider two initially adjacent trajectories. In the region of phase space where motion is conditionally periodic (A < A,) the distance between corresponding points on the trajectories grows linearly in time, but in the region of phase space where motion is stochastic (A > A,) this distance grows exponentially in time.These features are confirmed by analytic studies 1-4 of the motion. The transition from conditionally periodic to stochastic behaviour is, I believe, related to the passage from an energy domain wherein randomization of vibrational energy is slow to a region wherein it is fast. With this in mind, can Dr. Brumer say if he has mapped out, in energy space, the regions where trajectories separate linearly and exponentially in time, respectively? Does one find many trajectories that do not separate exponentially, and to what conditions do they correspond? Does he agree that if all trajectories separate exponentially a statistical theory of reaction rate is valid ? With respect to my question to Dr.Brumer, I note that in his introductory remarks Prof. Marcus cited the Brumer-Karplus work as supporting the statistical model of reaction rate, yet they point out deviations. Can he clarify what is meant by that citation ? Would he care to comment on my questions ? Finally, I address myself to Dr. Menzinger. There are at least two examples, in addition to those cited, for which there exists evidence that the molecule excited above threshold lives much longer than predicted by the statistical theory of reaction rate. As noncontroversial cases I cite the photodissociation of chloro- and bromo-acetyl- ene 5 * By noncontroversial I mean that the observation of fluor- escence under collision-free conditions from individually excited vibronic levels on a many nanosecond time scale implies, without uncertainties of interpretation, that the molecule remains intact for that long.Other cases for which it is suspected that energy randomization is slow, cited for example in our paper and some of its references, are more controversial because of the assumptions which must be added to the statistical theory of reactions in order that product recoil energy distributions be predictable. It would be foolish to argue on the basis of the above that energy randomization is never rapid relative to reaction. Some of the interpretations of data which infer that randomization of energy is slow are subject to uncertainty, and special arguments concerning energy transfer in the exit channel can be invoked to “ rescue ” energy randomization in the reaction complex.My position is that the limited available evidence cannot tell us whether rapid randomization of internal energy is character- istic of most reactions (likely) or of only a few. Accordingly, I think it equally foolish to always invoke special considerations and to “ explain away ” non statistical behaviour thereby completely ignoring the possibility that the class of reactions in which energy randomization is slow might be larger than now suspected. and of S02.7 Prof. R. A. Marcus (University of ZZZinois) said : In response to Rice’s first question, the agreement that I referred to in Brumer and Karplus’ paper was that for the complex- ’ V.I. Arnol’d, Russ. Math. Survey, 1963, 18, 85. 35. Moser, Math. Annalen, 1959, 169, 136. ’ K. Evans and S. A. Rice, Chem. Phys. Letters, 1972, 14, 8. A. N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1945,98, 527. K. C. Mo, Physica, 1972, 57, 445. K. Evans, R. Scheps, S. A. Rice and D. Heller, J.C.S. Faraday 11, 1973, 69. M. H. Hui and S. A. Rice, Chem. Phys. Letters, 1972, 17, 474. J. Parson, K. Shobatake, Y. T. Lee and S. A. Rice, this Discussion.116 GENERAL DISCUSSION forming trajectories in table 2 of their paper. In response to the second question the rigorous analytical results of Moser and others apply only to low A’s, namely for the regime of conditional periodicity. At high A’s the purely numerical results point to, in effect, a stochastic behaviour.Since high energy systems sample configurations of large AV, I, too, believe that such results do tend to support the chemist’s usually held intuitive view of energy randomization at high energies and nonrandomization at low energies. An example of a high A model would be RRKM, while an example of a low A system would be that in my paper of this Discussion on bound states and resonances. One would expect that the actual situation is more complicated for most molecules of chemical interest, in that there may be several “ A’s ”. When two frequencies of modes in H,, or better yet in H, are linearly-dependent, or nearly so, one has a resonance or a near-resonance. In such cases, the behaviour with respect to the subset of these two modes might range from largely nonstochastic at low values of some parameter Al to largely stochastic at high Al.At the same time, the behaviour in the other modes could still be conditionally periodic unless some other ;1, A2, is Iarge. With the latter view in mind, one might expect that conditional periodicity (and so no energy-randomization) prevails at low energies. Then, energy randomization begins to occur among a subset of modes for which resonances occur (or nearly occur). Finally, at high enough energies complete randomization might occur, in effect, if the system lives long enough. While RRKM theory might be used for this third region, if the complex lives long enough, and RRKM (with a subset of the modes) for the second region, the region between them would, of course, require additional dynamics.Dr. P. Brumer (Harvard University) said: Although we did study the mode of separation of initially adjacent trajectories in phase space we did so only for a small number of trajectories. The presence of exponentially separating trajectories is the cause of the remark in our paper that the NaBrKCl system displayed dynamic instability making numerical integration unusually difficult.2 We believe that the origin of the non-statistical behaviour we observed is, as we have stated, a result of the participation of both linear and cyclic structures in the complex. The relationship between the presence of two structures and the behaviour of initially adjacent trajectories has not, however, been investigated. As to whether exponential separation of all trajectories guarantees the validity of a statistical theory I can only say that one is intuitively led to believe that this should be true.I am unaware, however, of any proof that this is the case.3 Both exponential and linear separation were observed. Mr. D. S. Y. Hsu and Prof. D. R. Herschbach (Harvard University) said : Trajectory studies and other theoretical treatments of reaction dynamics have focussed primarily on product energy distributions and angular distributions since quantitative experi- mental data are most abundant for these properties. The work of Goring-Simpson and Bunker emphasizes the need for other properties in order to enhance “ resolution ” in determining the potential-energy surface. In particular, the dependence of reaction For an example of exponentially separating NaBr + KC1 trajectories see figure V-10 in P.Brumer, Ph. D. Tltesis (Harvard University, 1972). P Brumer and M. Karplus, this Discussion. In this regard see, for example, footnote 21 of G. H. Walker and J. Ford, Phys. Reu., 1969,188, 416. See, in particular, ref. (12).GENERAL DISCUSSION 117 probability on orientation of the reagent molecule is important in characterizing the entrance valley of the surface. Reactive scattering experiments using oriented molecules thus far have been feasible only for symmetric top molecules (CHJ and CF,I), however. We wish to mention complementary experiments which deal with TABLE 1 .-ANALYSIS OF POLARIZATION DATA react ants 4 a4 (cos* x> ( ~ 0 ~ 4 X> K+HBr Cs+ HBr Cs+HI K+ Br2 Cs+ Br2 CS+ CH3I Cs+ CF3I cs + cc14 CS + SF4 CS+ SFs - 1.6 - 1.5 - 1.7 - 0.6 - 0.5 - 0.6 N - 0.6 - 0.3 -0.1 - 0.2 0.4 0.12 0.027 0.7 0.14 0.052 0.9 0.11 0.029 0.3 0.25 0.14 - 0.27 - 0.25 - - - 0.25 0.2 0.29 0.17 - 0.32 - 0.31 - - - - the orientation of the rotational angular momentum of a product molecule.In principle, this orientation is less difficult to study because it is produced by the reaction rather than the experimenter. In practice, the development of the technique was long 10 9 G 05 -I 0 -0 5 0 0.5 1.0 1.5 00 -1 5 detector displacement/mm FIG. 1 .-Deflection profiles for CsBr from Cs+ HBr reaction. Data are shown for three orientations of the deflecting field, where t,h is the angle between the nominal field direction and the plane of the reactant beams. Dashed trapezoid shows undeflected profile observed with deflecting field off; other profiles obtained with deflecting field voltage at 20 kV, buffer field voltage at 2 kV.A positive detector displacement corresponds to deflection towards the high-field region, negative displacement towards the low-field region. delayed by problems involving nonadiabatic changes in the quantization axis. These problems have now been largely solved and, at least for alkali atom reactions, the technique should soon become part of the standard reactive scattering repertoire.118 GENERAL DISCUSSION The experimental method involves deflecting the product molecule (alkali halide) in an inhomogeneous electric field.' Comparison of deflection profiles obtained with the field oriented parallel ($ = 0') and perpendicular ($ = 90") to the plane of the reactant beams reveals whether the product rotational angular momentum has a preferred orientation or pozarization in space.2 When suitably normalized by a simple procedure, the shape of these profiles depends primarily on ( cos2 x), where x is the angle between the rotational angular momentum and the initial relative velocity vector.are given in table 1 and data for the Cs+HBr reaction are shown in fig. 1. The CsBr from Cs+HBr has its rotational momentum J' strongly polarized perpendicular to the initial relative velocity vector V, but all azimuthal orientations of J' are equally likely. In the Ic/ = 0" apparatus mode the deflecting field is nearly parallel to V, the typical J' is approximately perpendicular to the field and the cor- responding projection M' N 0.The rotating molecule is accelerated when the dipole moment points towards the field direction and decelerated when it points away. Since the net Stark effect is unfavourable, deflection occurs towards the low-field side of the inhomogeneous field. In the $ = 90" apparatus mode the field is per- pendicular to V, and the azimuthal symmetry about V gives projections of J' on the field direction which cover the full range, - J' < M' < J', but peak rather sharply at the extremes, M' = +J'. Accordingly, the deflection profile spreads to both the high- and low-field side but shifts somewhat towards high-field because the Stark effect is favourable when IM'/J'( r+. For an intermediate apparatus alignment, $ = 55", the typical situation becomes IM'/J'I 21 5 and the deflection profile becomes essentially the same as that for an isotropic distribution of rotational momenta.In addition to the deflecting field, a buffer field extending from the reaction zone to the deflecting field has an important role. The buffer field is designed to preserve any polarization produced by the reaction. It also allows a normalization procedure which greatly facilitates the data analysis. This employs reduced deflection profiles defined by R(s, $) = [I($, $) - I"($, $)]/Io(s, $), where I(s, $) is the observed intensity as a function of the detector displacement s and the superscript zero refers to '' buffer field off ". These reduced profiles have been shown 2* to be quite insensitive to substantial variations in the deflecting field strength, beam position in the field, collimating slit widths, dipole moment, and distributions of rotational and transla- tional energy.The reduced profiles depend primarily on the coefficient u2 in a Legendre expansion of the distribution of polarization angle, in some cases the u4 coefficient can also be determined from the data. moments of the ~ ( C O S x ) distribution, Results for 10 reactions studied during the past year ~(COS x) = 1 + u2P2(cos x) + a4P4(cos x) + . . ., This yields two (cos2 x) = +(I +$a2) ( C O S ~ x> = $(1+ +a2) - &( 1 -&a4). Table 1 lists these parameters. are : For comparison, the values for sin" x distributions The basic electric deflection technique was developed over 40 years ago by I.Estermann and R. G. J. Fraser. A lucid exposition is given in Fraser's book, Molecular Rays (Cambridge University Press, 1931), edited by Sir Eric Rideal. We have greatly enjoyed Sir Eric's description of these early studies at this Discussion. C. Maltz, N. D. Weinstein and D. R. Herschbach, Mol. Phys., 1972, 24, 133. D, S. Y . Hsu, Ph.D. Thesis (Harvard University, 1973).GENERAL DISCUSSION 119 n = 00 6 4 2 0 (cos2 x> 0 0.1 1 0.14 0.20 0.33 <cos4 x> 0 0.030 0.048 0.086 0.20 The observed polarization is large for the HBr and HI reactions, substantial for Br,, CH,I, and CF31, but almost undetectably small for CCl,, SF,, and SF6. In all cases a2 < 0 and a, > 0, indicating the preferred orientation of J’ is perpendicular to V. As yet, these polarization data have been interpreted only qualitatively or by comparison with simple models.1* We cite two cases to illustrate how the polariz- ation complements other reactive scattering properties.(i) For the direct interaction models which prescribe a repulsive force between the products in the exit channel of an A +BC-+AB + C reaction, the scattering angle and velocity can be calculated by specifying just the total repulsive energy released, whereas the rotational angular momentum requires in addition the repulsive force as a function of separation distance. The situation becomes very simple in the limit of very weak exit AB+C interaction (as assumed in the spectator stripping model) or very small mass of the C atom (mc < mB or mA, as obtains for reactions such as Cs + BrH).The initial collisional angular momentum, which is perpendicular to V, then goes pri- marily into rotation of AB rather than into relative motion of AB and C and thus J’ is strongly polarized perpendicular to V. (ii) For statistical complex models, the product rotational polarization is related to the nature of the coupling in the transition-state for decay of the complex. For “ tight-coupling ”, strong polarization with respect to the direction of the final relative velocity vector V’ is predicted if in the transition-state the effective moment of inertia about the separation axis is small, and weak polarization if that moment of inertia is large. For “ loose-coupling ”, which allows the incipient products to rotate freely in the transition-state, the polarization with respect to V‘ vanishes.Current experiments using a new field configuration are designed to determine the polarization of J’ with respect to both the V and V’ directions. Prof. W . H . Miller (University of CaZ$ornia) said: Contrary to the impression one might like to convey, semi-classical theory cannot accomplish everything. In general, one expects it to be accurate only when the relevant classical trajectory functions (e.g., the classical deflection function O(b) in elastic scattering, the final quantum numbers as a function of their conjugate initial angle variables, n2(q1), for inelastic or reactive scattering, etc.) are sufficiently smooth. If n,(q,) is a highly structured function, as in the example shown by Bosanac, it is clear that the primitive semi-classical formulae will give poor results ; furthermore, no appropriate “ uniformization ” trick is yet known [although see the progress on this topic made by Connor, this Discussion] that will in general salvage the situation.This same complicated structure in the reactive quantum number function has also been seen in collinear H + H2 collisions above the classical threshold for reaction. This shortcoming of the semiclassical theory is of rather minor significance, however, for it has been observed4 that in three-dimensional collision systems (as opposed to artificial collinear ones) the quantum effects present in classically allowed processes (i.e., interference effects) are in most situations quenched, so that an ordinary Monte Carlo trajectory calculation should be completely adequate in such cases.C. Maltz, N. D. Weinstein and D. R. Herschbach, MuZ. Phys., 1972, 24, 133. D S Y Hsu, P h B . Thesis, (Harvard University, 1973). S.-F. Wu and R. A. Marcus, J. Chem. Phys., 1970,53,4026. W. H. Miller, J. Chem. Phys., 1971, 54,5386.120 GENERAL DISCUSSION This would most especially be true when the quantum number function is highly structured, for the interference pattern would then be irregular and thus more suscept- ible to quenching. The general conclusion, therefore, is that under normal conditions a strictly classical treatment is adequate if the process of interest is classically allowed [cf. the example of LaBudde and Bernstein in this Discussion of 0-+2 rotational excitation of H2 by Li+].The semi-classical theory thus makes its most practically important contribution in providing a description of classically forbidden processes (for which strictly classical methods are obviously inapplicable), important examples of which are vibrationally inelastic transitions, reactive tunnelling below the classical threshold for reaction? and non-adiabatic transitions between different potential energy surfaces. Dr. J. N. L. Connor (University of Manchester) (communicated) : I wish to report that it is possible to derive a uniform asymptotic approximation for a one dimen- sional semiclassical S-matrix integral that possesses an arbitrary member of nearly coincident real or complex valued classical trajectories. This uniform approxima- tion generalizes previous results on two and three nearly coincident traje~tories.~'~ The uniform approximation is expressed in terms of a canonical integral and its derivatives.The canonical integral is determined by the number of coalescing trajectories. The constants that appear in the uniform approximation can all be written in terms of quantities that characterize the classical trajectories. The expo- nential functions of primitive semiclassical theory only appear when the trajectories are well separated from each other. Otherwise higher functions appear, which reflect the topological structure of the trajectories. The most important use of the uniform approximation is in the threshold region, as emphasized by M a r c ~ s . ~ As the number of nearly coincident classical trajectories increases? so does the difficulty in evaluating the canonical integral.Nevertheless, the results are still of value, in that they indicate the greatest simplicity that can be hoped for, that retains the uniformity of the original integral in its asymptotic expansion. On the other hand, there may be other aspects of a problem that suggest additional approximations, as has been emphasized by Miller.6 As for the two dimensional integral considered in my paper,' the uniform approxi- mation breaks down when end point contributions are important, when the pre- exponential factor contains a zero, pole or branch point near a classical trajectory and when the phase is very slowly varying (as in a near elastic collision). For these cases, further work is required to derive the appropriate uniform approximation (see the comment of Marcus Prof.R. B. Bernstein, Dr. R. A. LaBudde, Dr. P. J. Kuntz (University of Wisconsin) and Prof. R. D. Levhe (Jerusalem) said: Our recent 3-particle trajectory simulation of the K+CH31 reaction utilized an electron-jump type of potential surface, with a " steric effect '' arising naturally because of the asymmetry of the potential function. J. N. L. Connor, Mof. Phys., 1973 (submitted for publication). J. N. L. Connor, Mol. Phys., 1973, 25, 181. J. N. L. Connor, Mol. Phys., 1973 (in the press). R. A. Marcus, this Discussion. W. H. Miller, this Discussion. J. N. L. Connor, this Discussion. R. A. Marcus, this Discussion. R. A. LaBudde, P. J. Kuntz, R. B. Bernstein and R. D. Levine, Chern.Phys. Letters, 1973,19,7 ; R. A. LaBudde, Ph. D. Thesis (University of Wisconsin, 1973). * J. N. L. Connor and R. A. Marcus, J. Chem. Phys., 1971, 55,5636.GENERAL DISCUSSION 121 Comparison of the results of this calculation with those of Bunker and Goring- Simpson suggests that inclusion of the CH3 internal motion does little to improve agreement with experiment.2 The translational energy dependence of the reactive cross section is somewhat better in the 3-particle calculations ; the product angular and translational energy distributions are very similar in both calculations (both, however, predict the wrong dependence of these distributions on relative collision energy). In the 6-atom calculation the CH3 product is found to be vibrationally cold (at all energies), which is evidence against the importance of the dynamical role of the CH3 group.It appears that the need for a full 6-particle treatment has not yet been demonstrated for the alkali-methyl iodide reactions. In future work it would seem desirable to work on optimizing the 3-particle potential surface before introducing the additional (and computationally costly) complication of the internal structure and dynamics of the CH3 group. Prof. D. L. Bunker (University of CaZgornia) said: It has been remarked at several places in the discussion that in the alkali-methyl iodide reactions, the three methyl hydrogens play a very limited role and might well be omitted from theoretical studies. If we combine the results of the trajectory study we have presented here with others in which CH3 is involved (e.g., T + CH4), we find that the vibrational energiza- tion of product CH3 varies in the expected way with the time scale of the reaction.If the out-of-plane deformation mode has time to relax continuously during the depart- ure of the products, there is no appreciable energy deposition. This is the case for Rb and K+CH31. This does not mean that the steric properties of CH3 do not exert a heavy influence on the correlation of potential surface with scattering properties. We found that angle-dependent Rb-1-C forces and Rb-H repulsions were about equally import- ant. For trajectory studies, we agree that in principle CH3 may probably be replaced by a single entity, but not a spherically symmetric one. Prof. R. M. Harris (Worcester State CoZZege) and Prof.D. R. Herschbach (Harvard University) said : We wish to suggest a simple heuristic explanation for the rather sharp peak which Gersh and Bernstein found in the total cross section for the K + CH31+KI + CH3 reaction. It is useful to consider first two textbook cases, both derived from the venerable model which treats the reactants as point masses subject to a spherically symmetric potential V(R) and assumes reaction occurs for all traject- ories which surmount the centrifugal barrier and reach the “ close collision ” region. In this model, the reaction cross section as a function of the collision energy E is given by a@) = nb;, where b,(E) is the orbiting impact parameter at that collision energy. (i) For a step-barrier potential defined by V(R) = 0 at R> a and V(R) = E at R < a, one finds o,(E) = 0 for E < E and (ii) For an attractive inverse power potential, V(R) = -&(a/R)”, one finds 2 I n for n 3 2 and any E, where the constant C = S(n-2)E.D. L. Bunker and E. A. Goring-Simpson, this Discussion. R, B. Bernstein and A. M. Rulis, this Discussion.122 GENERAL DISCUSSION The observed K + CH31 cross section resembles (i) at low collision energy and (ii) at high energy. This suggests study of a potential having a barrier of height E at R = a and an attractive " chemical well " for R <a. At large distances the potential is qualitatively similar to a Lennard-Jones function and has a " van der Waals well ") as shown by elastic scattering experiments. However, for the model adopted here, this R> a region does not contribute to the reaction cross section.For simplicity we use a four-parameter function, This is designed to give V(a) = E and to extrapolate to an asymptote at V(co) = d. The corresponding reaction cross section is given by d - E = for n = 2. Here E 3 D> d, where D = d++(n-2)(d-E). In the E< D region the cross section is the same as (i) and as E+co it approaches (ii) within a constant factor. The cross section peaks at E = ind and its value there is na2(1 - ~ / d ) ~ ' " . This model gives a good fit to the data, as shown in fig. 1. The values obtained for the barrier height and position, E = 1.0 kcal mol-1 at a = 4.5 A, are consistent with results of an optical model analysis of the K+CH31 elastic scattering,l which found the reaction threshold at V(a) = 1.5 kcal mol-' and a = 4.3 A.In evaluating the a parameter, we took account of the orientation dependence of the reaction probability by introducing a steric factor Po = 0.80, in accord with the optical model analysis and the reactive scattering experiments of Brooks, Beuhler and Bernstein. I I I I I I I - collision energy, Elkcal mol-' FIG. 1.-Total cross section for K+CH,I reaction : curve calculated from model potential with parameters indicated ; points from experiment of Gersh and Bernstein. R. M. Harris and J. F. Wilson, J. Clzem. Phys., 1971, 54, 2088.GENERAL DISCUSSION 123 Reaction cross sections obtained from trajectory calculations may be fitted to the model in the same way. We find the a@) curves obtained in the studies reported at this Discussion by Bunker and by Bernstein can be simulated fairly well by use of suitable V(R) functions.These resemble cuts through the potential hypersurface for collinear K-I-CH3 along a plausible “ reaction path ”, although as yet we have no general prescription for calculating V(R) from the potential surface. Bunker’s cross sections are approximately constant above E - 1 kcal mol-l, in disagreement with experiment. According to our model this occurs because the potential surface chosen drops too abruptly inside the entrance channel barrier. Bernstein’s cross section shows a pronounced peak, qualitatively consistent with fig. 1, but the location (- 14 kcal mol-l), height (- 17 A’) and breadth of the peak differ considerably from the experimental result.The potential surface used provides a switch (“ electron- jump ”) from a covalent to ionic component inside the entrance channel barrier. The drop in a,(E) above its peak occurs because as E increases an increasing number of trajectories cross back from the ionic to the covalent surface. This corresponds, in our model, to trajectories which pass over the entrance channel barrier but are turned back by the centrifugal potential. From our results it appears that relatively minor adjustments of the surface should bring the computed a,(E) closer to experiment. These adjustments include moving the entrance barrier outwards by about 30-40 % and providing a more gentle average descent in the region just inside the barrier. This simple model likewise simulates the cross sections predicted from trajectory calculations for the T+H2 and T+D, reactions by Karplus, Porter, and S h a ~ m a .~ These cross sections have shapes much like fig. 1, but with the ordinate scale multiplied by about and the abscissa scale by roughly 20. Dr. A. Ben-Shaul (Munich) said: The formalism reviewed in the paper of Levine and Bernstein provides a simple means for characterizing product state distributions in chemical reactions. The notion that a number of exoergic reactions resulting with large extent of population inversion can be characterized with the aid of one (1, eqn (31)) or two (Av, OR eqn (32)) temperature-like parameters is most valuable in chemical laser studies. Using equations like eqn (31), (32) of Levine and Bernstein’s paper one can write in a compact form expressions for the necessary lasing conditions, the population inversions and the gain factors in chemical lasers.The main advantage is that of using only one (or two) parameters to characterize all laser transitions instead of different vibrational or rotational temperatures for different transitions. On the basis of these arguments two special cases were recently s t ~ d i e d . ~ These are : (a) completely non-relaxed initial product population, (b) rotationally relaxed but vibrationally non-relaxed population. In this study it was assumed that the initial vib-rotational distributions correspond to a rotational-translational microcanonical equilibrium situation. The rate equa- tions governing the output of the laser were solved (on the basis of the assumption above) for the case of the HF, (F+H,+HF+F) laser, and agreement with some experimental results was found.In this study relaxation effects were not included, but are taken into account in more detailed work which is now in progress. D. L. Bunker and E. A. Goring, Chem. Phys. Letters, 1972, 15, 521. R. A. LaBudde, P. J. Kuntz, R. B. Bernstein and R. D. Levine, Chem. Phys. Letters, 1973, 19, 7. M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Phys., 1966, 45, 3871. A. Ben-Shaul, G. L. Hofacker and K. L. Kompa, J. Chem. Phys., in press. See also Discussion remarks on the paper by Ding et ai.I24 GENERAL DISCUSSION Prof. J. C. Poianyi, Mr. J. L. Schreiber and Dr. J. J. Sloan (University of Toronto) said : In its simplest formulation the Levine-Bernstein theory rests on their obser- vation that in a number of cases the product energy-distribution from chemical reac- tions deviates exponentially from the statistical expectation.For product vibrational distribution, for example, the surprisal (which is a logarithmic measure of the “ devia- tion ” from equilibrium) decreases linearly toward higher vibrational energy V’. In such cases a single parameter, with the dimensions of temperature, provides a useful measure of the extent of the deviation at any specified V’. The situation would seem to be more complicated, however, if the product vibrational-energy distribution is bimodal, since in this case one has the appearance of a deviation from equilibrium which is at first large then small and then once again large before diminishing.energy/kcal mol-’ atomic hydrogen and vibrationally-excited molecular fluorine. FIG. 1 .-Computed product vibrational-energy distribution, k( V’), from the reaction between There is some experimental evidence for what may be a bimodal vibrational energy distribution in the products of the reaction H+Cl,(u> l)-+HCl+C1.2 This is supported by theoretical evidence for a small double-peak in the product vibrational energy distribution from the same reaction. A more conspicuous vibrational double- peak has emerged from a 31) trajectory study that we have made of the reaction H+F,(t, = 4)+HF+F. The potential-energy hypersurface used for this study was an LEPS surface that had been found to give a very good fit to the experimentally-observed product vibrational- di~tribution,~.The results reported here were obtained from a batch of 1700 trajectories, of which 161 were reactive. The relative translational energy in the reagents was fixed at R. D. Levine and R. B. Bernstein, Paper at this Discussion and references therein. A. M. G. Ding, L. J. Kirsch, D. S. Perry, J. C. Polanyi and J. L. Schreiber, Paper at this Discus- sion. 3 C. A. Parr, J. C. Polanyi, W. H. Wong and D. C. Tardy, Discussion remarks in this volume. N. Jonathan, C. M. Melliar-Smith and D. H. Slater, J. Chem, Phys., 1970, 53, 4396; N. Jonathan, S. Okuda and D. Timlin, Ma/. Phys., 1972, 24, 1143. J. C. Polanyi and J. J. Sloan, J. Chem. Phys., 1972, 57, 4988. and also yielded the correct activation energy (2.4 kcal mol-’).GENERAL DISCUSSION 125 T = 2.0 kcal mol-1 and the reagent vibrational energy was held at 9.3 kcal mol-f corresponding to u = 4.The reagent rotational energy was Monte Carlo selected from a 300 K distribution. The resulting product vibrational distribution is shown in fig. 1. A similar double-peaked vibrational distribution was obtained previously in an independent 3D trajectory study, using a different LEPS surface as an approxi- mate model for H+F2(u = l).' for a random sample comprising more than half of the reactive trajectories in the present study showed, without exception, that the F2(u = 4) reagent molecules that formed HF product in the more highly-excited vibrational states, u'> 6, were in the extended phase of vibration (r > re) at the time that the new bond formed, whereas F2 molecules forming product with uf < 6 were in a compressed phase of vibration (r < re).This corresponds to the analysis of a trajectory study of H + Cli (the dagger indicates vibrational excitation) contained in a paper presented at this Discussion.2 The ratio of the times spent by the isolated F2(v = 4) reagent molecule in the extended and the compressed configurations was z(r > r,)/z(r < re) = 2.1. This may be compared with the corresponding ratio of reactive cross-sections from the trajectory study ; o,(d > 6)/o,(u' < 6 ) = 2.6 1- 0.3 (to a 67 % confidence limit). Since the latter ratio is larger than the former, it appears that extended F2, at close HF2 approach, is slightly more reactive than compressed F2.This, together with the substantial anharmonicity of the F,(v = 4) vibration that causes it to spend twice as much time in an extended configuration as in a compressed one, provides an explanation of the observation that the bimodal k ( d ) distribution in fig. 1 has its dominant peak at high U f . * Inspection of the bond-plots and force-plots Prof. R. D. Levine (Jerusalem) (communicated) : Polanyi, Schreiber and Sloan in their comment have outlined the evidence supporting a bimodal products' vibrational energy distribution in exoergic chemical reactions starting with vibrationally excited reactants. Additional support come from the quanta1 (collinear) collision model of Hofacker and Levine. The question raised by Polanyi, Schreiber and Sloan is how to describe the devia- tion from equilibrium when the products' distribution is bimodal.The surprisal of a particular final vibrational state (fraction of energy in energy in vibration fo) has been defined 4- by Here P(f,) is the observed distribution and P"(f,) is the distribution expected on a-priori grounds. To compute P"(f,) we have proposed use of the principles of information theory as applied to statistical mechanics.6 In particular, in the absence of any prior infor- nzation (save for the conservation of energy and other good quantum numbers) we have proposed that all products' quantum states are equally likely. In practice this means that P"(fi,) is the fractional number of products' quantum states which have the vibrational quantum number u, and is a monotonically decreasing function off,.From an information-theory point of view, when one does possess the prior * A . M. G. Ding, L. J. Kirsch, D. S . Perry, J . C. Polanyi and J. L. Schreiber, Paper at this 3G. L. Hofacker and R. D. Levine, Chem. Phys. Letters, 1971, 9, 617. N. C . Blais, Los Alamos Scientific Laboratory, Report LA-4687, 1971. Discussion. R. D. Levine and R. B. Bernstein, This Discussion. R. B. Bernstein and R. D. Levine, J. Chenz. Phys., 1972, 57, 434. E. T. Jaynes in Stahtical Physics (Benjamin, New York, 1963).126 GENERAL DISCUSSJON information that the reactant diatomic is vibrationally excited, one should incorporate it into Po(fu). The general techniques for incorporating such constraints into the distribution have been discussed before 1-4 and the specific application to excited reactants will be published elsewhere.At this point it is sufficient to say that subject to the a-priori information that the reagent diatomic is vibrationally excited, the a- priori distribution is also bimodal. In other words, if Po(.fo) is the unconstrained a priori distribution and P,(fy) is the a priori distribution subject to the known a priori constraints, we have that df") = -~n[Po(fo>l~"(fv)l (2) P"(fv) = Po(fD) expE-s~fy)I. (3) (4) or Here g(fv) is the measure of the a priori information. It is clear that -lnllP~fv>/~"(fo)l = - l ~ ~ ~ ~ f v ) l ~ o ( f v ) l + - ~ ~ [ ~ o ~ f u ) / ~ " ( ~ ) l where the first term is the surprisal of the deviation of P(&) from the (constrained) a priori distribution Po(fy).The temperature parameter 1; ', 1" = - d ln[P(f,?lPO~fv)l/df" ( 5 ) is a characteristic of the potential energy surface and is independent of the reactant's vibrational e~citation.~ On the other hand, -ln[P(~7)/~"(fv>] = 10 &fo +dh) (6) is not necessarily a monotonic function of fy, and does depend on our prior informa- tion. To summarize : The deviation from the unconstrained (or uniform) a priori distribution, need not be a monotonic function off,. However, the deviation of the observed products' vibrational energy from the a priori distribution (including known a priori constraints) should be a monotonic function offv and could be characterized by a temperature-like parameter. Prof. M. Menzinger and Mr. D. J. Wren (University of Toronto) said : To comple- ment the work on total reaction cross-sections (excitation functions a@)) by Bernstein, Bunker and co-workers we wish to report excitation functions for total chemilumin- escence production in the Ba + N20 rea~tion.~ A He or H2 seeded supersonic N20 beam was crossed by a thermal Ba beam and the total light signal was measured by a photomultiplier. Collision energies were computed assuming complete expansion and classical contribution of the N20 rotations.The N,O beam was monitored by a small mass spectrometer. The resulting excitation function is well represented by a 1 / E dependence over the entire energy range 0.1-1.1 eV (CM). This behaviour is reminiscent of a barrier surmounting process (e.g., Langevin- Giomousis-Stevenson) for a Coulomb potential although it is expected that diabatic transitions will play an important role.R. D. Levine and R. B. Bernstein, This Discussion. R. B. Bernstein and R. D. Levine, J. Clzem. Phys., 1972, 57, 434. E. T. Jayties in Statistical Phj%ics (Benjamin, New York, 1963). G. L. Hofacker and R. D. Levine, Chem. Phys. Letters, 1972. 15, 165. ' C. D. Jonah, R. N. Zare and Ch. Oltinger, f. Chem. P ~ J J s . , 1972, 56, 263.GENERAL DISCUSSION 127 Mr. D. S. Perry, Prof. J. C. Polanyi and Dr. C. Woodrow Wilson Jr. (University of Toronto) said : In discussing energy requirements for endothermic reaction use has been made of the detailed information comprised in various “ endothermic triangle plots ”.l, These triangle-plots present contours of equal kendo( V’, R’, T‘), where kendo is the endothermic rate constant, and V’, R’, T’ are the vibrational, rotational and translational energies in the reagents of endothermic reaction.The data were ob- tained by the application of microscopic reversibility 1 * 3* to the corresponding Ice,,( V ’ , R’, T’), obtained in infra-red chemiluminescence studies. Since the energy released in exothermic reaction was (to a good approximation) constant, the endothermic triangle plots gave the effect on the detailed rate constant of enhancing the reagent energy in one degree of freedom at the expense of another. The most striking observation was that enhancement in V’ at the expense of either or both R’ and T’ was capable of increasing the rate of endothermic reaction by orders-of-magnitude. (In the exceptional case Cl+ClH-+CI, +H it was possible to have “ too much ” reagent vibration ; this was termed the “ reverse light-atom anomaly ” 5 ) .In terms of theoretical treatments, such as the Levine-Bernstein approach,6 it becomes important to know the extent to which these data obtained from microscopic reversibility can be trusted in detail. Recorded in the triangle-plots is, for example, information regarding the approximate effect at various levels of reagent vibrational excitation of an increase in reagent rotational or translational energy (as R’ or T’ is increased at constant V’, kendo goes through a maximum whose magnitude, in general, increases with V’). that the kendo were strictly valid only for the reverse (i.e., endothermic) reaction back into the distribution of reagent energy- states which characterised the exothermic reactions. Since the exothermic reactions proceeded at room temperature, the spread of reagent energy-states was small, and one could conveniently characterise the reagent energies for exothermic reaction by their mean values, ( V ) , (R), ( T ) .The rate constants recorded in the endothermic triangle plots could then be symbolised kendo(( V ) ( R)( T)I V’, R‘, T’) (with the initial state to the right of the modulus) ; the endothermic reactions being It was recognised in the original work kendo AB( V’, R’) + C -+A + BC(( V ) , (R)). A more interesting quantity, since it is experimentally much more accessible, would be kendo( V, R, TI V‘, R’, T’). Here, as the reagent energy for endothermic reaction V’, R’, T’, is varied, the product distribution V, R, T ,is at all times the preferred one for endothermic reaction with that particular reagent energy-distribution. It was argued previously that kendo(( Y ) , (R), (T)I V’, R’, T’) recorded in the triangle plots would not differ notably from kendo( V, R, TI V’, R’, T‘), since the endothermic rate constants apply to endothermic reagent-energies only slightly above threshold.The possible variation in product energy-distribution from endothermic reaction was therefore small, and small alterations in V, R, T would not (judging from preliminary trajectory results) markedly alter the detailed rate constants. We wish to report that we have now put this supposition to a systematic test in a trajectory study, and havc confirmed our earlier expectation. K. G. Anlauf, D. H. Maylotte, J. C. Polanyi and R. B. Bernstein, J. Chem. Phys., 1969, 51,5716. J. C. Polanyi and D. C. Tardy, J. Chem. Phys., 1969,51, 5717. R. A. Marcus, J. Chem. Phys., 1970, 53, 604. J . L. Kinsey, J . Chem. Phys., 1971, 54, 1206. J. C. Polanyi, Act. Chem. Res., 1972, 5, 161. R. D. Levine and R. B. Bernstein, Paper in this Discussion and references therein.128 GENERAL DISCUSSION The model potential-energy hypersurface used was that appropriate to the endo- thermic reaction HI+ I-+H + I2 ; endothermicity E, +AH = 0 + 35.5 k a l mol-1 (Ec is the classical barrier height for reaction in the exothermic direction). The masses were mA = m, = nzc (to avoid the light-atom anomaly). In order to simulate our treatment of experimental data ‘ 9 * we first ran a batch of trajectories in the exothermic direction, selecting the reagent energies by Monte Carlo methods from a thermal distribution. Microscopic reversibility was then invoked in order to obtain what we have previously termed kendo(( V ) , (R), (T)I V‘, R’, T’). Next 8 batches of traject- ories were run in the endothermic direction, with different choices of V’, R’, T’, at a fixed V’ + R’ + T’. These yielded kendo( V, R, TI V’, R’, T’). The average discrepancy between these two estimates of the detailed endothermic rate constants was found to differ by less than 10 % of the maximum reactivity. It appears, therefore, that the published endothermic triangle plots, obtained by application of microscopic revers- ibility to the findings for exothermic reaction, can be used as a satisfactory guide to the variation of kendo with vibrational, rotational and translational energy in the reagents of endothermic reaction (at constant V’ + R’ + T’). A full account of this calculation, and related work, is being prepared for publica- tion. We have benefited in this work from helpful discussions with Mr. J. L. Schreiber. K. G. Anlauf, D. H. Maylotte, J. C. Polanyi and R. B. Bernstein, J. Chem.Phys., 1969,51,5716. J. C . Polanyi and D. C. Tardy, J. Chem. Phys., 1969, 51, 5717. D. S. Perry, J. C. Polanyi and C. Woodrow Wilson Jr., Chem. Phys. Letters, 1974 ; Chem. Phys., 1974.

ISSN:0301-7249    

DOI:10.1039/DC9735500113

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

2. ELASTIC SCATTERING Elastic Scattering: Introduction BY J. PETER TOENNIES Max-Planck-Institut fur Stromungsforschung, Giittingen, Germany Receiced 17th June, 1973 In molecular beam scattering experiments it is possible to observe directly changes in the physical properties of atoms and molecules occurring in single collisions between almost any two species. In the general case of a collision in which one of the partners is polyatomic, the following properties may be changed : the direction and magnitude of velocities (translational degrees of freedom), the internal-state quantum numbers (internal degrees of freedom), and the structure (" chemical " degrees of freedom). Depending on the relative collision energy, and the structure of the colliding species, any or all of the above properties may be changed.Collisions in which only the direction (but not the magnitude) of the relative velocity is changed, are denoted elastic collisions. Those in which the internal-state quantum numbers are changed are called inelastic collisions. Those which involve structural changes are called reactive collisions. Elastic scattering, which will be discussed here, is almost invariably the most probable process in low energy collisions. Since the electronic excitation energies of atoms are usually greater than 1 eV, only elastic scattering occurs when two atoms collide at thermal energies. The deflection angle 8, resulting from an elastic collision, is determined by the interatomic force averaged over the collision trajectory. The probability of a given deflection is proportional to the differential cross section da/dw(0) which is one of the quantities measured in a molecular beam scattering experiment. The differential cross section provides direct information on the radial dependence of the intermolecular potential V(R).Within the last ten years all the important elastic cross section features, which can be used to characterize 2nd specify various R-regions of the intermolecular potential, have been observed. Techniques for deriving potential parameters from the measured data are well developed. Although they are in principle applicable to any binary system, the results are most easily interpreted in atom-atom collisions which lead to a lZ-state of the diatom. This state always occurs if one or both of the atoms are in a '5'-state (have closed shells).Then the potential is of the relatively weak van der Waals type. For studying such potentials, elastic scattering has emerged as the best experimental technique. It is now an almost routine laboratory tool (albeit a complicated one) for characterizing van der Waals forces. In this brief introduction we will summarize the methods used in elastic scattering studies and list the important classes of systems which have been studied up to the present time. Finally, we will suggest directions for future work. I n addition we will mention some important recent results. 129 55-E130 ELASTIC SCATTERING A. EXPERIMENTAL METHODS Essentially two types of experiments are performed in elastic scattering studies : (1) measurements of the velocity dependence of the integral cross section and (2) measurements of the angular dependence of the differential cross section.In fig. 1 the basic features of an apparatus used in measurements ofthe integral cross section are shown at the top. The technique is essentially analogous to the measure- ment of an optical absorption coefficient. A velocity-selected molecular beam is passed through a scattering chamber (absorption cell) containing the target gas. The beam intensity, with and without gas in the scattering chamber, is measured by the detector at the far right. From the ratio of the attenuated to unattenuated beam intensity, length of the scattering chamber and the absolute target gas density, it is possible to calculate the absolute integral cross section using Beer's law.In general, measurements of absolute cross sections are difficult to carry out with high precision (see the paper by Bickes et al. in this. volume). Relative cross sections are easier to detector chamber Beam attenuation ==+ 6 (v, ) =+.d (9) f . I 1 1 1 experimentally accessible range '---**--. glories '-._____ '.__---. orbiting 1 I I I 10'2 10" 0 Rm R FIG. 1.-Integral cross section measurements of elastic scattering. A schematic diagram of the apparatus is shown in (a). In (6) the typical velocity dependence of the integral elastic cross section (in arbitrary units) is plotted as a function of the reduced relative velocity. In (c) a typical potential curve is shown, indicating the R-regions probed in the different velocity regions.Note that the glory undulations probe both the well and long range parts of the potential. measure. The useful range of velocities is dictated by the Maxwell-Boltzmann velocity distribution of the source. Since the measured cross sections are averaged over the velocity distribution of the target gas, they have to be deconvoluted to give the integral cross section as a function of the relative velocity g. A typical velocity dependence is shown in the middle of fig. 1, whereg is " reduced "J . PETER TOENNIES 131 by dividing by 2~R,/!i. ( E and R, are defined in terms of the potential curve at the bottom of fig. 1). At low velocities sharp spikes, corresponding to resonances with quasi-bound states due to orbiting, are expected. At intermediate velocities gentle sinusoidal undulations (glory undulations) are observed.They are attributed to a quantum interference between unscattered wave packets and those which, by a compensation of attractive and repulsive forces, are scattered into the forward direction (see fig. 2). At reduced velocities greater than unity the relative energy is so great that the attractive forces no longer significantly affect the trajectory and the integral cross section is determined by the repulsive “ hard core ” of the potential. In this way an approximately universal curve is obtained. INTEGRAL CROSS SECTIONS Symmetry I 2 - unddations (e.g. LHe -‘He): DIFFERENTIAL CROSS SECTIONS n Supernumerary rainbows Syrnmetr) undulations ieg ‘ ~ e - L ~ e ) FIG. 2.-Characteristic trajectories, leading to semiclassical wave interferences.In each case two trajectories are shown, which lead to the same centre of mass scattering angle. Since the action along the two trajectories is different, the phase shifts are also different and an interference results. The symmetry undulations are only possible if the primary beam and target particles are identical. As indicated at the bottom of fig. 1, the orbiting region of a(g) provides information mostly on the long range attractive part of the potential. The glory undulations provide information on the well region and the long range potential. In a first approximation the velocity spacing of the glory maxima is proportional to the product E o R , . ~ The high velocity region relates mainly to the repulsive part of the potential.Thus, the absolute magnitude of the integral cross section and the velocity dependence over a large range of velocities, provides enough information to characterize the entire potential curve. The most comprehensive measurements have been obtained for He-He scattering. This system is discussed in more detail in the next section. In general, the experimentally accessible range of reduced velocities for heavy systems (ml, m,>4) lies in the glory region as indicated in the middle of fig. 1.132 ELASTIC SCATTERING However, under special circumstances such as H-Hg which has a relatively deep well (E = 0.45 eV), the experimentally accessible velocity region may be in the orbiting regime. Thus the first orbiting maxima were recently observed for this system ; H-He and H-Ne lie at the other extreme.Since the well depths are extremely small (2 1 mev) the reduced value of g is usually greater than 1. Unfortunately, with exceptions of symmetric boson systems (e.g., 4He-4He), integral cross section measurements in themselves do not contain enough information to determine uniquely the exact shape of the potential curve. On the other hand, such measurements are relatively easy to carry out and can be used for a rough characterization of the important potential parameters such as E and R,. As dis- cussed in the paper of Fitts and Law in this volume, integral cross sections in the glory region can also be used as a check on the accuracy of a proposed potential model. The significance and uniqueness of potentials, determined by a least squares fit of high velocity integral cross sections, is discussed in the paper by Bickes et al.in this volume. becondary beam source c - = - -7- c ‘ I ) I 8 4 - d 6 d u Scattered intensity - > ab ( 0 ) dd (9) r- dn dw ---__ - - rainbows 1 1 0 Rm R FIG. 3.-Differential cross section measurements of elastic scattering. A schematic diagram of the apparatus is shown in (a). In (6) the typical angular dependence of the differential cross section (in arbitrary units) is plotted as a function of the cente of mass scattering angle. In (c) a typical poten- tial curve is shown, indicating the R-regions, probed in the different angular regions. The essential features of an apparatus used in the measurement of differential cross sections are shown at the top of fig.3. In place of a scattering chamber a well collimated secondary beam is usually used. The scattered primary beam particle intensity is measured by a movable detector which rotates about the scattering volume defined by the intersection of the two beams. The scattered intensity is directly proportional to the differential cross section da/doi(O) in the laboratory system. For comparison with calculated cross sections the measured values have to be trans- formed into the centre of mass system.J . PETER TOENNIES 133 A typical curve of the differential cross section (weighted with sin 8) as a function of angle is shown in the middle of fig. 3. Since absolute differential cross sections are also very difficult to measure (they require calibration of density in the secondary beam) and usually not needed, the ordinate is given in arbitrary units.At small angles the differential cross section depends mostly on the long range attractive part of the potential. Closely spaced interference oscillation (fast oscillations-see fig. 2) are expected and sometimes observed at small angles. The measurement and interpret- ation of such undulations is discussed in the paper by Coggiola et al. The fast oscillations are superimposed on widely spaced oscillations (supernumerary rainbows) which culminate in the main rainbow. The main rainbow occurs at a scattering angle of 0,-2c/Ec, where E,, is the relative energy in the centre of mass system. The fast oscillations and supernumerary rainbows depend on the existence of an attractive potential.The former provide information on Ro and the latter information on the depth E and the shape of the potential curve in the well region. The largest deflection produced by the attractive part of the potential alone is given by OR. Thus at larger angles the fast oscillations quickly disappear. The large angle region is largely determined by the repulsive part of the potential. In general, because of the low scattered beam intensities, differential cross sections are more difficult to measure than integral cross sections. As a result, such measure- ments are usually restricted to a fairly narrow range of beam energies. Furthermore, the observation of several supernumerary rainbows requires that the most probable relative energy is sufficiently low relative to the well depth.The angular distribution, shown in the middle of fig. 3, is thus typical for the heavier systems. The super- numerary rainbows are related to the glory maxima in the integral cross section. The number of supernumerary rainbows observable for any system is approximately equal to the number of glory undulation^.^ This equivalence has to do with the fact that the same interference process is responsible for both phenomena (see fig. 2). The greater difficulties encountered in measuring differential cross sections are rewarded by a more unique determination of the potential curve. Under conditions in which the semiclassical approximation is valid differential cross sections can be " inverted '' to obtain directly a potential curve independent of any particular mathe- matical model of the potential.Techniques for inverting beam scattering data have recently been reviewed by Buck. Under semiclassical conditions the large angle region ( 0 ~ 8 , ) can be inverted using a well-known procedure first suggested by Firsov and based on classical mechanics. The region 8<O, can be inverted using anyone of several procedures, suggested by Buck,6 Klingbeil and Remler.* These methods have in common that uniqueness is only then achieved if the whole inter- ference structure is known, together with some additional information from, for example, large angle scattering, the interference amplitudes, or from the averaged differential cross section. The need for extra experiment21 information has recently been demonstrated by B ~ y l e . ~ who was able to fit supernumerary rainbow data using a number of potentials with very different shapes.Finally, it should be pointed out that although the beam method is probably the best single method for determining intermolecular potentials the accuracy of the results which it provides can be increased if other available information on the system under study is taken into account. Thus, in determining the rare-gas dimer potentials spectroscopic data on the vibrational levels, solid state compressibility coefficients and second virial coefficients have been used to rsfine the potentials obtained from beam experiments. In this wzy it has recently been possible to determine up to nine parameters in the following flexible piecewise Exponential-Spline-Morse-Spline-van der Waals (ESMSV) potential model l o134 ELASTIC SCATTERING Y(R)/& = f ( x ) = A exp(-a(x-1)) O,<x,<xl = exp(al + (x-x1){a2 + (x- x2)[a3 + (x- xl)a4]]) = exp[-2a(x- 1)]-2 exp[-B(x- l)] = bl +(x-x,)(b,+(x-x,)[b,+(x-x,)b,]) x,<x<x4 = Q X - 6 - csx-8 - clox-'o x1 < x < x2 x ~ < x < x ~ x&x< a3 where x = R/Rm and ci = Ci/(cRL).Note that the first term is a Born-Mayer or exponential potential. The second and fourth terms are spline fits, the four constants of which are determined by the requirement of equal potential and slope (force) at the boundaries. The third term is a Morse potential. The constants in the long range potential are sometimes available from theory. Usually the well region is well described by the last three terms. The model is then called an MSV potential.B. SYSTEMS STUDIED Over 200 different combinations of scattering partners have been studied by at least one of the above techniques1 The systems covered in elastic scattering experiments are summarized in table 1 in order of decreasing precision in the potential determina- tion. The first row lists the three groups of systems which have been studied by inversion techniques which yield directly potential curves independent of an assumed model. The rare gas dimers are in the next category since these have been studied TABLE 1. -SURVEY OF SYSTEMS STUDIED IN ORDER OF DECREASING PRECISION OF THE POTENTIAL DETERMINATION type of systems* beam data other data H+-A do/dw(B, E) theoretical C6-values EzzF M-Hg : and u(u) A-A dcr/dw(O, E ) second virial coefficient spectroscopic energy levels - M-A 4 0 A-A daldw(0, E ) absolute u method used no.of to obtain parameters cross section potential model determined inversion none needed - best fit ESMSV 6-9 modified Lennard-Jones potentials and MSV best fit and Buckingham -5 2-3 theoretical C6-values best fit Lennard-Jones and H-A,mol u(u) absolute u Buckingham M-X M- mol da/dw(O, E ) MX-A etc. The bold face symbols represent thelfollowing species : A, rare gas atom : M, alkali atom ; X, halogen atom ; mol, diatomic and polyatomic molecule by combining beam data with other data as mentioned above. The third category consists of two groups of systems which have been carefully studied in beam experi- ments but for which the data were not inverted directly.Finally, in the last category, some systems are listed for which only 2 or 3 parameters of a more restricted potential model (e.g., Lennard-Jones 12 : 6 or Buckingham exp(a : 6)) were deter- mined. For reasons of brevity some systems involving polyatomic molecules have been left out. C. SOME IMPORTANT RESULTS 1. He-He This system has probably been studied more than any other. It is also the simplest closed-shell-closed-shell-atom-atom system and for this reason is best suited for anJ . PETER TOENNIES 135 2 0 4 i: I t- loo'-- I 501 20 50 primary beam velocity/ms-' FIG. 4.-Measured effective integral sattering cross sections for 4He-4He over 4 orders of magnitude in the primary beam energy.14c~ l e The differences in the curves can be attributed almost entirely to the indicated differences in the temperature of the target gas.Experiments from five different groups are shown. 2.0 25 3.0 rlA FIG. 5.-The 4He-4He p~tential.~ The circles denote the potential, obtained by inversion.14h The other potentials are determined by molecular beam scattering data, solid line, 14e dashed line,l4Y crosses 14c and by gaseous properties : dashed dotted line. Several other potentials, derived for this system, are not shown since they cannot be distinguished.136 ELASTIC SCATTERING exact apriori quantum chemical calculation. Recently, configuration interaction (Cl) calculations have succeeded in predicting a potential well E = 0.93 0.03 meV (108 K),12 in good agreement with the experimental vaIue of between 0.888 l 3 and 0.949 meV.O Fig. 4 shows the measured velocity dependence of the integral cross section over a range of lo4 in the centre of mass energy (upper abscissa). The measurements were carried out by 5 groups l4 using the different target temperatures indicated in the curves. The differences in the cross section can be attributed almost entirely to velocity smearing produced by the target gas at the different target temperatures. The undulations observed between ul = 400 and 4000 m-' in some of the measure- ments can be attributed to an interference which can only occur when primary and target gas particles are indistinguishable (see fig. 2). These symmetry undulations contain information on the s-wave phase shift. Using a method suggested by Miller l5 the s-wave phase shifts can be inverted to obtain a model independent potential.Fig 5 shows the inverted potential obtained by Feltgen et aZ.14h together with several other measured potential curves. The agreement, especially with measurements of Gengenbach et aZ.14" who relied on absolute integral cross section measurements, is remarkably good. Cross sections have also been measured at energies up to 2000 eV l6 where, because of the grazing collisions involved, the po- tential is probed up to 20 eV. There is a good agreement with theory over the entire energy range. 2. THE RARE GAS DIMERS As mentioned in the previous section, differential cross sections for the rare gas dimers have been measured with good angular resolution. In the case of the lighter systems only symmetry undulations are seen.These disappear as one goes to the heavier systems Ar-Ar, Kr-Kr and Xe-Xe. For the latter systems, the relative beam energy is low enough with respect to the well depth so that the supernumerary 10 0 % 5. -10 E ka ---- n -2 0 FIG. 6.-Interatomic potentials for the symmetric rare gas dimers. OJ . PETER TOENNIES 137 rainbows become visible. By fitting these results together with spectroscopic and second virial coefficient data Lee and coworkers lo have determined between 6 and 9 of the parameters in the potential model of eqn (1). The potential wells for all the rare gas dimers are shown in fig. 6. With the availability of these precise potentials it is now possible to check the Law of Corresponding States which requires that the reduced potential curves f ( x ) = V(R)/E versus x = R/R, for a given class of systems such as the rare-gas dimers should exactly coincide. Fig.7 shows that the reduced potentials are indeed strikingly similar. There are only small differences of about one percent in x in the repulsive region and in the attractive region near x = 1.4. Despite these small differences, Lee and coworkers l o were not able to find a single reduced potential curve which could fit all of the data available for all of the rare gases. Furthermore they observed that the apparently small differences in the potential curves produce significant differences in calculated values for various observable properties. Lee and coworkers also report that significantly different reduced potentials were derived for the asymmetric He-rare gas pairs and Ne-rare gas pairs. Thus, the “ Law of Corresponding States ” can at best only be considered a crude approximation in these cases.I ! I 1 I I I I I I I I I I x = R/Rm 1.0 1.5 2 .o FIG. 7.-Reduced form of the interatomic potentials for the symmetric rare gas dimers.1° Another result coming out of these studies has to do with the contribution from three-body interactions to the third virial coefficient and solid state properties. With the availability of accurate two-body potentials these can be used to calculate solid properties such as the bulk modulus, zero-pressure energy and nearest neighbour separation. By including only the Axilrod-Teller long range three-body dispersion forces, good agreement with the measured third virial coefficients of Ar and Kr and138 ELASTIC SCATTERING solid state properties of Ne have been achieved.This result suggests that for the rare gases the Axilrod-Teller forces are probably the only important contribution from many-body interactions in the condensed phase. 3. Na-Hg This and other M-Hg systems have been carefully studied by Buck and Pauly, who measured both integral l7 and differential l 8 cross sections with an apparatus having an exceptionally good velocity and angular resolution. Both sets of data were used in a semi-classical inversion procedure to obtain the potential curves directly, without recourse to bulk data which, of course, is not available for these systems. Fig. 8 shows the differential cross sections for Na-Hg at five different relative energies.9: 20 15 10 5 u l o I U .- 2, l5 2 s- .3 5 U .CI 10 a n a z o N 25 20 15 10 5 0 --t------t : 5 10 15 PO 25 30 35 40 e FIG. &-Measured differential cross sections for Na-Hg at five different energies.' With increasing energies the supernumerary rainbows shift to smaller angles. Especially at Ecm = 0.25 and 0.19 eV the fast oscillations are well resolved. In addition to six or seven supernumerary rainbows, the fast oscillations are also cearly discernible. Fig. 9 shows that the inverted potential curves for all five beam energies are virtually identical. The repulsive potential was derived by invertingJ . PETER TOENNIES 139 the differential cross sections measured at 8 ~ 8 , . Also shown in fig.9 is a Lennard- Jones (12 : 6) potential. This and other experimental evidence indicates that the Lennard-Jones (12 : 6) potential is much too " hard " in the repulsive region. The potential curves for the other M-Hg systems are quite similar, but differ some- what in the repulsive region. FIG. 9.-The potential for Na-Hg, obtained from the inversion of supernumerary rainbow data at five different energies (see fig, 8).'* The solid line shows a Lennard-Jones (12 : 6) potential, which has been fitted at the minimum. 4. COMPARATIVE STUDIES Fig. 10 shows a comparison of reduced potentials for three sets of partners with different electronic configurations : He-He, Li-Ar and Na-Hg, for which precise potential curves are now available. As mentioned above, these three systems are representative of the Z electronic-state-diatom potentials formed from atoms of the corresponding groups of the periodic table : VIIIa-VIIIa[closed shell ('S0)-closed shell ('So)] ; Ia-VIIIaCopen shell (2S&-closed shell with 8 electrons ('So)] ; and Ia-IIb[open shell (2S+)-closed shell with 2 electrons (lSO)] respectively.The relatively small differences in the long range potentials in fig. 10 are expected since this part of the potential is largely determined in all cases by the dispersion forces. The large differences in the repulsive region on the other hand are apparently due to the differ- ences in electronic density distributions. These distributions enter directly in the140 ELASTIC SCATTERING calculation of repulsive potentials by methods based on the Thamas-Fermi-Dirac statistical model of the atom.19 Thus it is reasonable to expect different repulsive potential.. for the three systems.1 I He- He 0.9 Li - A r 5.41 K - Hg 52.4 4.9 I 0.5 1.0 RlRm I 5 FIG. 10.-Comparison of reduced potentials for atom-atom systems representative of three different combinations of groups from the periodic table. Large differences are observed in the repulsive potential, which may be due to differences in the electronic charge distributions. D. FUTURE DIRECTIONS The above brief summary of recent results illustrates that eIastic scattering experiments are now capable of providing highly accurate potential data for atom- atom partners producing lX- and 2E-diatom states. In all these cases the partners interact by way of only one potential curve.As pointed out by Bernstein,20 this behaviour will always be the case for the following combinations of atoms in the periodic table : I + 11, I + VITI, I1 + 11, I1 + VIII, VIII + VIII, I1 + VA, and VIII + VA. At the present time reliable data (shown in fig. 10) are available for only three of the possible combinations. With presently available techniques it should be possible to study the other combinations in the near future. For all other combinations of groups in the periodic table the atoms will form diatoms with two or more possible potential curves. A few studies of systems with several potential curves have recently been carried out.21 As an example, fig. 11 shows a measurement of the angular distribution of Naf, formed in inelastic scattering from Na + I.This result is of interest here because essentially similar results would have been obtained if'the elastic Na angular distribu-J . PETER TOENNIES 141 tion had been observed. The measured angular distribution is quite different from that shown in fig. 3 or fig. 8. The differences can be attributed to the fact that the covalent (path 2) and ionic (path 1) potentials are involved in the scattering as illus- trated at the bottom of fig. 1 1 . As the two atoms approach sometimes they interact ecm 10' 15' 20" . O0 so I I fi Na' measured- intensity E,,=13.1 eV -4 "0 100 200 300 400 7 Na' + I - _ _ _ _ - FIG. 11 .-Differential cross section measurements of Na+ ions, produced in the scattering of Na + I at 13.1 eV. The measured angular distribution in (a) is compared with the calculated distribution in (b).7 in the abscissa (bottom) is equal to the product #,,*E,, (in units of eV deg). In (c) the two different potential curves as wll as the different " interaction " paths, leading to ions in the exit channel, are shown. (probability P) by way of the ionic curve. At other tries they interact (probability 1-P) by way of the covalent curve. These two sets of trajectories can produce Naf ions at the same angle. Thus new interferences in addition to those which are present when the encounter is governed by only one potential curve (see fig. 2) can occur. The theoretical angular distribution below the experimental curve was calculated from all available data on the ionic and covalent potential curves.The curve crossing probability was based on the well-known Landau-Zener formula. The agreement is remarkably good and suggests that information on both potential curves can be obtained from this type of double ioterference pattern.142 ELASTIC SCATTERING In general, however, whenever curve crossing is expected to occur, state selection of one or preferably both beams will be necessary in order to be able unambiguously to interpret the results. Experiments of this type have only been carried out in studies of the triplet scattering of the alkali dimers.22 Further work along these lines is extremely important since very little is known about many of the potential curves accessible in such experiments. Another direction for future work is to study the interactions of excited atoms.23 An experiment of this type is presented in the paper by Davidson et al.Use of lasers to excite the atoms appears difficult but feasible. Finally, it should be pointed out that the analysis of elastic scattering experiments in which one or both of the partners is a molecule is still not on a firm theoretical basis. As noted in the paper by Coggiola et al. in this volume, it is customary to attempt an analysis of the data by assuming only a spherical symmetric potential. Such a procedure is probably reliable in first order since elastic scattering is almost invariably much more probable than inelastic scattering. Nevertheless a more rigorous analysis of such systems is needed. This problem has recently been investigated in theoretical studies of He-H,.The potential hypersurface for this system is known from quantum chemical calcula- t i o n ~ . ~ ~ At energies below 0.5 eV, the threshold for vibrational excitation, a rigid rotor potential of the following form is suggested by the calculations. U R ) = Vo(R)E1 +q(R)P2(cos QI where and Vo(R) = MSV potential d R ) = qrep 0<R<2.2A = q(R) 2.2<R<4.5A - qatt 4.5<R<w. - qrep 24 and qattZ5 are known from theory and q(R) is assumed to be a smooth function, joining these two theoretical values. Independent calculations by several different groups 26 show that the integral total cross section (c,,, = Xctz{) obtained in a close-coupling calculation, in which rotational excitation is properly accounted for, is identical to the elastic cross section calculated for Vo(R).This result is a computational confirmation of the Law of Con- servation of Cross Secti~n.~' There is also evidence that the total differential cross section is much the same as the elastic differential cross section calculated for VO(R).26b These calculations have so far only been carried out for Hz molecules initially in the j = 0 state and without including closed channels. Recent calculations show that these closed channels can lead to a break down of the law of conservation of cross section.28 Obviously, such simple laws will also break down for more anisotropic systems, especially if vibrational excitation becomes probable. Substantial experi- mental evidence, that anisotropic and inelastic effects cannot be neglected, comes from the observation of a " washing out " or " quenching " of glory undulations in molecu- lar ~cattering.~~ Thus a considerable theoretical effort is needed before " elastic " scattering experiments on the more complex systems can be analyzed with the same degree of reliability as for atom-atom scattering.f E. SUMMARY Elastic scattering experiments have been performed over a wide range of energies. In the case of He-He, experiments have been reported over seven orders of magnitudeJ . PETER TOENNIES 143 (0.1 meV to 2000 eV). The glory undulations in the velocity dependence of the integral cross section and the main rainbow in the angular dependence of the differ- ential cross section can be measured using a more or less conventional apparatus. These features yield fairly reliable information on E and R, which is roughly inde- pendent of the assumed potential model. A more sophisticated apparatus is needed to resolve the supernumerary rainbows and fast oscillations.The location of several supernumerary rainbows, the spacing of the fast oscillations and some additional data are sufficient to invert the beam data to obtain a potential curve which is essen- tially model-independent. The resulting potentials are good to within a few percent. Equally reliable potential curves may also be obtained from less complete beam data if precise measurements of the equation of state, transport properties, spectroscopic constants or solid state properties are also used. These precise methods are presently restricted to atom-atom interactions in which one of the partners has a closed shell.The author is grateful to U. Buck and H. Pauly for helpful discussions and to John Fenn for critically reading the manuscript. For reviews see : (a) H. Pauly and J. P. Toennies, Adv. At. Mol. Phys., 1968, 1, 195 ; (b) H. Pauly and J. P. Toennies, Methods of Experimental Physics, vol. 7a, p. 227 (Academic, New York, 1968) ; R. B. Bernstein, Adv. Chem. Phys., 1966,10,75 ; ( d ) R. B. Bernstein, Adv. Chem. Phys., 1966, 10, 75 ; ( d ) R. B. Bernstein and J. T. Muckermann, Adv. Chem. Phys., 1967, 12, 389 ; (e) J. P. Toennies in Physical Chemistry an Advanced Treatise, Vol. IV (Academic, New York, 1973), chap. 6. E. F. Greene and E. A. Mason, J. Chem. Phys., 1972,57, 2065. A. Schutte, D. Bassi, F. Tommasini and G.Scoles, Phys. Rev. Letters, 1972, 29, 979. see ref. l(e) and eqn (21) of D. E. Pritchard, J. Chem. Phys., 1972,56, 4206. U. Buck, Rev. Mod. Phys., 1973 to be published U. Buck, J. Chem. Phys., 1971, 54, 1923. E. A. Remler, Phys. Rev. A, 1971, 3, 1949. J. F. Boyle, Mol. Phys., 1971, 22, 993. lo J. M. Farrar, T. P. Schafer and Y . T. Lee, to be published. l 1 For a listing of all systems studied up to Aug. 1970 see ref. (Id) (up to 1967) and R. B. Bernstein, Theoretical Chemistry Institute, The University of Wisconsin, Technical Note Wis-TCl-3976. l 2 (a) P. J. Bertoncini and A. C. Wahl, J . Chem. Phys., 1973,58,1259 ; (6) H. F. Schaefer, D. R. McLaughlin, F. E. Harris and B. J. Alder, Phys. Rev. Letters. 1970, 25, 988. l 3 H. G. Bennewitz, H. Busse, H. D. Dohmann, D. E. Oates and W. Schrader, 2. Phys., 1972,253, 435. l4 (a) H. P. Butz, R. Feltgen, H. Pauly, H. Vehmeyer and R. M. Yealland, 2. Phys., 1971, 247, 60; (b) D. E. Oates and J. G. King, Phys. Rev. Letters, 1971, 26, 735 ; (c) R. Gengenbach, C. Hahn and W. Welz, J. Chern. Phys., to be published ; ( d ) P. E. Siska, J. M. Parson, T. P. Schafer and Y . T. Lee, J. Chem. Phys. 1971,55,5672 ; (e) J. M. Farrar and Y. T. Lee, J. Chem. Phys., 1972, 56, 5801 ; (f) P. Cantini, M. G. Dondi, G. Scoles and F. Torello, J. Chem. Phys., 1972, 56, 1946 ; (9) H. G. Bennewitz, H. Busse, H. D. Dohmann, D. E. Oates and W. Schrader, 2. Phys., 1972, 253, 435 ; (h) R. Feltgen, H. Pauly, F. Torello and H. Vehmeyer, Phys. Rev. Letters, 1973, 30, 820. W. H. Miller, J. Chem. Phys., 1969, 51, 3631. ' (a) R. Klingbeil, J. Chem. Phys., 1972,57, 132 ; (6) R. Klingbeil, J. Chern. Phys., 1972,57,1066. l 6 J. E. Jordan and I. Amdur, J. Chem. Phys., 1967, 46, 165. l7 U. Buck, K. A. Kohler and H. Pauly, 2. Phys., 1971, 244, 180. U. Buck and H. Pauly, J. Chem. Phys., 1971,54, 1929. R. G. Gordon and Y. S. Kim, J. Chem. Phys., 1970,56,3122. 2o R. B. Bernstein, J. Chem. Phys., 1963, 38, 2599. 21 C. A. L. Delvigne and J. Los, Physica, 1973, to be published. 22 (a) D. E. Pritchard, G. M. Carter, F. Y . Chu and D. Kleppner, Phys. Rev. B, 1970,2,1922 ; (b) A. Hoh, H. Oertel and A. Schultz, 2. Phys., 1970, 235, 20. 2 3 H. Haberland, C . H. Chen and Y . T. Lee, Proc. of the 3rd Int. Conf. on Atomic Physics, Boulder, Col., USA, Atomic Physics, 1973, 3, 339.144 ELASTIC SCATTERING 24 (a) M. Krauss and E. H. Mies, J. Chem. Phys., 1965, 42,2703 ; (6) M. D. Gordon and D. *’ G. A. Victor and A. Dalgarno, J. Chem. Phys., 1970,53,1316. 26 (a) W. Eastes and D. Secrest, J. Chem. Phys., 1972,56,640 ; (6) H. Fremerey and J. P. Toennies, 27 R. D. Levine, J. Chem. Phys., 1972,57,1015. 28 J. van de Ree, private communication. 29 R. E. Olson and R. B. Bernstein, J. Chem. Phys., 1968,49,162. Secrest, J. Chem. Phys., 1970,52,120. Chem. Phys., to be published ; (c) R. Gengenbach and P. McGuire, unpublished.

ISSN:0301-7249    

DOI:10.1039/DC9735500129

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Central-Field Intermolecular Potentials from the Differential Elastic Scattering of H&) by other Molecules * BY h O N KUPPERMANN, ROBERT J. GORDON? AND MICHAEL J. COGGIOLAf Arthur Amos Noyes Laboratory of Chemical Physics,§ California Institute of Technology, Pasadena, California 91 109 Received 26th February, 1973 Differential elastic scattering cross sections for the systems Hz + Oz, SF6, NH3, CO, and CH, and for D2+02, SF6, and NH3 have been obtained from crossed beam studies. In all cases, rapid quantum oscillations have been resolved which permit the determination of intermolecular potential parameters if a central-field assumption is adopted. These potentials were found to be independent of both the isotopic form of the hydrogen molecule, and the relative collision energy.As a result of this, and the ability of these sphericd potentials quantitatively to describe the measured scattering, it is concluded that anisotropy effects do not seem to be important in these HZ(Dz) systems. The determination of interatomic and intermo!ecular potentials from molecular beam experiments has received considerable attention over the last few years. Early experiments at high energy with various atomic,l ionicY2 and molecular systems yielded essentially structureless total cross sections. In order to determine the scale of the potential from such data, it is essential to have absolute cross ~ections,~ which require accurate calibration of beam intensities. It has long been recognized that the calibration problem can be avoided if the cross section has structural features that provide an in1 Prnal “ calibration.” Recently, rapid quantum oscillations have been resolved in dif m-rential elastic cross s e c t i o n ~ , ~ - ~ which provide the necessary calibra- tion.The freqaa ncy of such undulations has been related,’* for central-field poten- tials, to the ranp- of the potential according to the approximate expression (1) where A0 is the spacing of the oscillations, p is the reduced mass, v is the relative collision velocity, a is a range parameter for the potential (e.g., the zero of the poten- tial), and A is the de Broglie wavelength. As a result, well resolved rapid oscillations permit the estimation of a independently of the shape and depth of the potential well. A more quantitative fit to experiment of the differential cross sections calculated from an assumed potential permits one to determine more quantitatively this potential for systems subject to central forces.In particular, information about the depth of the attractive well and the steepness of the repulsive part of the potential can be obtained. Partly because of the simplicity of interpreting the experiments for central fields, most of the measurements of quantum oscillations have been for atom-atom scattering. * Work supported in part by the U.S. Atomic Energy Commission, Report Code No. CALT- t present address : Naval Research Laboratory, Washington, D.C. $ Work performed in partial fulfilment of the requirements for the Ph.D. Degree in Chemistry at $ Contribution No. 4650. Ae = (nrz/pva) = (A/20) 767P4-112. the California Institute of Technology.145146 CENTRAL-FIELD INTERMOLECULAR POTENTIALS The first molecular system found to have oscillations given by (1) was D2+N2, reported by Winicur et aL6 In the present study, which is a continuation of their work, we have measured the differential elastic cross sections of D2 and H2 scattered by 02, CO, NH3, CH4, and SF,, in order to obtain information about the corres- ponding intermolecular potentials. The data are discussed from the perspective of a central field approximation. Variation of the relative collision energy and the use of both H2 and D2 with the same scattering partner provide a useful test for the validity of this approximation. Some of the systems reported here have also been studied in total cross section experiments by Butz et al." and by Aquilante et aZ.I2 Information obtained from total and differential cross section measurements on the same systems are mutually complementary.EXPERIMENTAL The crossed molecular beam apparatus is shown schematically in fig. 1 and 2. The main features of the machine are a movable, differentially pumped quadrupole mass spectrometer detector, a differentially pumped supersonic primary beam and a subsonic secondary beam perpendicular to the primary beam, all contained in a bakeable stainless steel 1200 I. main vacuum chamber. The beams intersect the axis of the main chamber at the centre of rotation of the detector. The detector chamber is mounted on a semicircular shaped quadrant arm which pivots about the axis of the main chamber, while the detector is free to move along the rim of the quadrant out of the plane of the beams.Thus the detector can scan both colati- tudinal and longitudinal angles, although only in-plane measurements were made in the present experiments. Pumping in the main chamber is by means of four 6 in. oil diffusion pumps, each having a nominal trapped speed of 12501./s, and a liquid nitrogen cooled titanium sublimation pump, with a calculated speed of 200001./s for air. The primary beam source chamber and buffer are pumped by a 6 in. oil diffusion pump (1250 IJs) and a 6 in. mercury diffusion pump (150 l./s), respectively. FIG. 1 .-Vertical cross section of molecular beam apparatus. N-primary nozzle source, S- skimmer cone, VS-velocity selector, C-primary beam chopper, F-beam flag, CA-secondary beam glass capillary array, IS-electron bombardment ionizer, MF-quadrupole mass filter, EM-electron multiplier, TSP-titanium sublimator pump, OP-Orbion pump, IG-ionization gauge, BV- bellows operated bakeout valve, VP-Pyrex view port.All apertures in the apparatus are circular, with the entrance aperture of the detector housing (0.16 cm diam.) located 8.05 cm away from the intersection of the beams. The exit aperture of the primary beam chamber (0.21 cni diam.) is located 7.9cm away from theA . KUPPERMANN, R . J . GORDON A N D M. J . COGGIOLA 147 scattering centre, and the secondary effusive source (0.16 cm diam.) is 0.5 cm from the centre. The primary beam is formed with the aid of a nozzle-skimmer arrangement giving a measured Mach number of - 15 and an angular FWHM (full width at half maximum) of 1.4".A jacket surrounding the nozzle tube permits one to cool the entire nozzle assembly to liquid nitrogen temperature . \ PRIMARY 1.4* FWHM / .. 2.4' FWHM 0 , 5 CM I I {DETECTOR FIG. 2.-Crossed beam geometry. 0 is the measured laboratory scattering angle. The secondary beam source consists of a glass capillary array attached to the end of a brass tube, which can be tilted out of the plane of the beams by pumping the air out of a stainless steel bellows attached to this source. When the secondary source is tilted, the two beams do not cross, and the background signal intensity can be measured. This procedure is superior to flagging the secondary beam since the latter method tends to modulate the back- ground as well as the signal.The angular width of the secondary beam is 2.4" FWHM. The FWHM cross section of the beam intersection region in the collision plane has the approximate shape of a rectangle 0.17 cm along the direction of the primary beam and 0.22 cm along the direction of the secondary beam. The angular resolution of the detector is approxi- mately 2". The heart of the apparatus is an Extranuclear 324-9 quadrupole mass spectrometer l3 mounted in a bakeable double differentially pumped chamber. The operating pressure in the ionization region, measured with an uncalibrated Bendix miniature ionization tube, is typically 2x Torr with the beams on, whereas in the main chamber, it is about 1 x lod6 Torr under these conditions.To obtain such a large pressure differential, we found it necessary to bake the spectrometer housing and Orbion pump for about 8 h at approximately 200°C whenever the machine was pumped down from atmospheric pressure. The mass spectrometer chamber is equipped with a bellows activated valve 6 cm in diameter which is kept open to the main chamber during the bake-out period in order to accelerate the removal of background gas. Particles entering the mass spectrometer chamber pass successively through a high-effi- ciency electron impact ionizer, a series of electrostatic focusing lenses, and a 23 cm long Paul l4 quadrupole mass filter. Ions are detected by a 14 stage CuBe electron multiplier whose output is amplified by an Extranuclear tuned amplifier followed by a Princeton Applied Research HR-8 phase sensitive detector. The amplified signal is finally converted to digital form by a Raytheon model ADC-24 analog-to-digital converter.The apparatus is interfaced to an SCC-4700 computer, which serves several functions. First, it tilts the secondary beam in (" on " mode) and out (" off" mode) of the scattering plane. Second, the computer periodically samples and averages the amplified signal and subtracts the background from the total intensity. Third, it calculates the standard devia- tions for both " on " and " off" modes. The signal to noise ratio varied from better than 100 at the small scattering angles to a148 CENTRAL-FIELD INTERMOLECULAR POTENTIALS minimum of 10 at the largest one. To correct for long term drift in the signal caused by such factors as fluctuations of beam intensities and gradual build-up of background in the mass spectrometer, a fixed scattering angfe (generally between 3.0" and 5.0") was chosen as a reference angle.After the measrtrement of the signal at each scattering angle, the intensity at the reference angle was remeasured to provide a normalization factor. In this way, individual reIative intensity points were reproducible to within 5 % when remeasured on different days. RESULTS The differential cross sections for the systems H2 + 02, SFt,, CO, NH3, CH4 and D2 + 02, SF6, NH3, were all measured using room temperatdre H2 and D2 beams, with a relative collision energy of approximately 0.06 eV. Measurements of the H2 + SF6 and H2 + NH3 systems were also made using an H2 beam cooled to liquid nitrogen temperature, with a relative energy of approximately 0.02 eV.In addition, the SF6 system was studied using a low temperature beam of para-hydrogen. These experiments scan a wide range in the size, anisotropy and ihitial relative collision energy of the scattering species, and of the corresponding de Broglie wavelengths. The measured differential elastic cross sections are shown in fig. 3 to 7 inclusive, together with the on-line computer determined error bars. The various curves drawn through the measured points were fitted to the data as described below. DETERMINATION OF THE INTERMOLECULAR POTENTIAL In the interpretation of our data we have assumed that the differential elastic cross sections measured are due to the spherically symmetric part of the intermolecular potentials.The reason for this assumption and the tests of its validity are described in the Discussion. In our analysis, a model potential function is assumed and the potential parameters are varied until a least-squares fit of theory to experiment is obtained . In the present analysis we have used a Lennard-Jones (n, 6) potential, where the repulsive exponent n was either fixed at 12 or 20, or was allowed to vary as a fitted parameter. In addition, a Morse-cubic spline-van der Waals (MSV) potential l 5 was used in some systems. The MSV potential is defined by c(exp[ - 2p(r - r,)] - 2 exp[ - p(r - r,,,)]] r < rl 1 - c6r-6 r2 <r The cubic spline function is a set of five cubic polynomials whose coefficients are chosen to smoothly join the inner and outer branches of the potential.The end points were taken such that Y(rl) = -0.75 E , and r2 = rl +0.2 r,. The fitting parameters were E, rm, p and C6. The corresponding differential cross sections were accurately calculated using a partial wave expansion employing both JWKB and high energy eikonal phase shifts, tested against accurate integration of the radial Schrodinger equation to assure the validity of this method. In order to compare the computed cross sections with the data, it is necessary to correct for velocity spread and angular resolution of the apparatus. In trial calculations, we found that the former effect tends to dampen the undulations at CM scattering angles > 15" while the latter damp- ens the small angle scattering to roughly an equal extent.This situation differs from that of Siska et aZ.15 who found that under their experimental conditions with both beams supersonic, the effect of angular resolution was dominant at all scattering angles and that they could lump both corrections into a single effective angular resoIution function. Consequently, the calculated cross sections were transformed to the Y(r) = cubic spline r1 <r,crz20 25 10 15 laboratory angle O/deg FIG. 4 I I I J 5 10 15 20 25 I laboratory angle O/deg FIG. 3 102- I I t I 5 10 15 20 25 I 00 laboratory angle O/deg FIG. 5 H, + N H 3 \ H2 + SF6 I 1 5 I 10 15 20 IO"1 laboratory angle O/deg FIG 6 FIG. 3.-Plot of the product of the scattered intensity I times the sine of the angle 0 against 0 in the laboratory system of reference for H2 + O2 and D2 + 0, collisions.The lower curve has been shifted downwards by one decade. Points are experimental, and curves are theoretical fits. The solid curves are the MSV fits, and the corresponding potentials were used to establish the outer and inner ordinate scales for the Hz + O2 and D2 + O2 results, respectively. The FIG. 4.-Dif€erential scattering results for (room temperature) H2 + SF6 and D2 + SF6 collisions. Explanation FIG. 5.-Differential scattering results for (room temperature) Hz +NH3 and Dz + NH3 collisions. Explanation of FIG. 6.-Low temperature results for H2 + NH, and H2+ SF6 collisions. Explanation of curves is given in fig. 3. The LJ (n, 6) curve for Hz+NH3 was indistinguishable from the MSV curve and was not plotted. Results using upper dashed curve is the LJ (12, 6) fit, and the dotted curves are the LJ (11, 6) fits.of the cvrves is the same as fig. 3. the curves as for fig. 3. The LJ (12, 6) fits were indistinguishable from the LJ (n, 6) ones and were not plotted. para-hydrogen + SF6 were identical to those shown for normal-hydrogen.1 50 CENTRAL -FI ELD INTERMOLECULAR POT ENTI A LS laboratory system and averaged over both the relative collision energy distribution and the detector angular resolution. The potential parameters were fitted to the data by minimizing the weighted sum of squares of the differences between the cross sections calculated as just described and the experimental results, treating the vertical scale coefficient as a fitting para- meter.For the Lennard-Jones potentials with n fixed, the fitted parameters E and B were found using a simple Newton’s method. In the case of the MSV (E, r,, p, c6) and the three parameter Lennard-Jones ( E , B, n) potentials, a general method due to Marquardt l6 was used. In the following sections, the quoted values for the uncertain- ties of the potential parameters are those corresponding to a 95 % confidence level. 10 15 20 25 laboratory angle O/deg FIG. 7.-Differential scattering results for the H2 + CH4 and H2 + CO collisions. Only LJ (12, 6) fits were attempted, and they are shown by the solid curves. All the systems were initially fitted with an LJ (12, 6) potential. The optimum values of E and CT and their 95 % confidence levels are listed in table I together with A, the de Broglie wavelength for each system, and Q the total cross section as calculated from the partial wave expansion.In addition, the results of the LJ (20, 6) and (n, 6) fits are also given in this table. The 02, SF6 and NH3 data were measured with the most accuracy, and hence were chosen for the four parameter MSV fits. The HZ+ CH4 and H2 + CO data were of poorer reproducibility quality and for this reason not submitted to such fits. The corresponding parameters are listed in table 2 together with the values for ;1 and Q. Various calculated differential cross sections are shown in fig. 3 to 7 inclusive. In all cases, the Q thus determined was within 10 % of the value predicted by eqn (1). It is worth emphasizing that while the statistical un- certainties in the fitted potential parameters listed in tables 1 and 2 are often quite small, it does not follow that the “ true ” values of these quantities (e.g., the actual well depth) must lie within the predicted ranges.A .KUPPERMANN, R . J . GORDON A N D M . J . COGGIOLA 151 In fig. 8, 9 and 10 are shown fitted LJ and MSV potentials for the 02, SF6 and NH3 data. In each case, the MSV and LJ potential with fitted repulsive parameter are given for the room temperature H, system, while only the MSV fit is given for the TABLE 1 .-LENNARD-JONES (n, 6) POTENTIAL PARAMETERS AND TOTAL CROSS SECTIONS II 12 13.2 20 12 13.6 20 12 16.2 20 12 16.3 20 12 19.1 20 9.1 12 9.1 12 9.2 12 12 12 CIA 3.38+ 0.03 3.40f0.04 3.46k0.04 3.510.2 3.5f0.3 3.6f0.2 4.05 1 0.06 4.121.0.04 4.151 0.04 4.1 5f 0.08 4.1 8+ 0.05 4.1 4 1 0.04 4.2f 0.4 4.2+ 0.2 4.2f 0.2 3.34f0.07 3.45k0.06 3.34+ 0.09 3.34f 0.08 3.39+ 0.08 3.261 0.07 3.5k0.1 3.7f0.2 e/meV 7.7f 0.9 7.2f 0.9 7.6+ 1.2 7.3k0.6 7.0k0.7 6.7+ 0.9 10.4f0.5 10.4f0.3 10.4k0.8 9.6f 0.3 lO.O* 0.2 l0.5+ 0.2 10.3 1 0.6 10.3 5 0.3 10.3 k0.4 9.6+ 1.2 9.8k 1.4 10.3k0.7 10.3 f 0.8 9.1k0.8 9.1 k 0.7 6.9f 1.5 9.9+ 1.4 )./A 0.84 0.84 0.84 0.61 0.61 0.61 0.81 0.81 0.81 1.52 1.52 1.52 0.58 0.58 0.58 0.87 0.87 1.56 1.56 0.65 0.65 0.84 0.87 Q/A2 2081 15 182f 12 1691.15 270521 251f:20 222+ 30 380+41 361 5 30 335+ 31 326f 3 1 325+ 28 3135 19 3801. 32 331 +_ 30 3345 31 2251.15 260k 18 256k 17 2551 17 2505 21 245k21 2105 18 317+26 corresponding D2 systems. Those potentials not shown, were in general, indisting- uishable from those which were plotted.In the case of the SF6 and NH3 systems, all three potentials are seen to be in very close agreement, while for the O2 systems, the agreement is somewhat poorer. In all cases, however, the potentials overlap through- out the range plotted when the uncertainties in the potential parameters are taken into TABLE 2.-MORSE-SPLINE-VAN DER WAALS (MSV) POTENTIAL PARAMETERS AND TOTAL CROSS SECTIONS system ls!A rm/A E/meV B Cs/eVA6 ?.JA Q/A2 HZ+02 3.3450.05 D Z + 0 2 3.550.2 H2+SF6 4.14f0.02 4.16k0.03 DZ+ SFs 4.2+ 0.2 H2+NH3 3.42k0.05 3.23f0.05 DZ+NH3 3.23k0.05 3.8650.05 4.03 f 0.2 4.63 0.02 4.64k0.03 4.62+ 0.2 3.80+0.05 3.88 f 0.05 3.77+ 0.05 7.2+ 0.6 6.9f 0.9 lO.O+ 0.2 10.21 0.2 10.45 0.8 9.7k0.5 10.2k0.5 9.0k0.8 5.2k0.4 4.8f0.4 6.5+ 0.5 6.3k0.6 6.6f 0.6 4.9+ 0.4 4.8 k 0.4 4.9k0.4 64.8k0.7 0.84 63.1 f0.6 0.61 57.2k0.3 0.81 55.9k0.5 1.52 54.6k0.5 0.58 58.2k0.6 0.87 59.2+ 0.7 1.56 59.810.7 0.65 213f 19 291 2 26 396f 33 325)31 389+ 38 288+21 242 f 20 258-1 18 account.Hence, to within the experimental errors, the potentials for the H2 and D, isotopes are the same for a given scattering partner, and the resulting potential is independent of the mathematical form chosen, and of the de Broglie wavelength. It1 52 CENTRAL-FIELD INTERMOLECULAR POTENTIALS should be noted that agreement of the long range regions of the potentials is expected since both the LJ and MSV forms are chosen to have an r6 dependence and, in addition, the measured scattering is not very sensitive to this region.The range of intermolecular distances sampled in these experiments, and depicted in fig. 8 to 10 inclusive, was approximately estimated by calculating the classical deflection function from the MSV potentials and considering the range of angles in the CM system covered for each system. 30.0 20.0 10.0 % E - n 0.0- -10.0- -20.03,0 H2 + 0 2 MSV - 1 L J ( 13.2-6 1 - - - H, + sF6 MSV - LJ (16.2-6)--- Dz + SFG - MSV s........ - 1 t I 1 I 1 4.0 5.0 6.0 7 ' r l A FIG. 8.-Comparison of the intermolecular potentials over the range of distances sampled for H2+02 ( A = 0.84& and D2+02 ( A = 0.61 A), determined from the data in fig. 3. The solid curve is the H2 + O2 MSV potential, while the dashed curve is the Hz + O2 LJ (n, 6) potential. The dotted curve represents the D2 + O2 MSV potential.FIG. 9.-Comparison of the intermolecular potentials for H2+SF6 ( A = 0.81 A) and D2+SF6 (A = 0.58 A). Explanation of the curves is given in fig. 8. The corresponding curves for H2+SF6 at h = 1.52 A are indistinguishable from those at A = 0.87 A, within plotting accuracy.A . KUPPERMANN, R. J . GORDON AND M. J . COGGIOLA 153 20.0 10.0 5 1 A 2 0.0- -10.0 - - - H2+ N H 3 MSV - LJ (9.1-6)--- D 2 + NH3 MSV ........ -20.01 1 I I I I I 1 3.0 4.0 5.0 6.0 I rlA 3 FIG. 10.-Comparison of the intermolecular potentials for HZ+NH3 ( A = 0.87A) and DZ+NH3 ( A = 0.65 A). Explanation of the curves is given in fig. 8. The corresponding curves for H2 + NH3 at X = 1.56 A are indistinguishable from those at h = 0.87 A, within plotting accuracy.DISCUSSION Ford and Wheeler l1 have shown by semi-classical techniques and for a central- field potential having an overall shape analogous to that of an LJ (12,6) potential that when the deflection function has a relative extremum, interference between the attractive and repulsive branches leads to rapid oscillations superimposed on the broader supranumerary rainbow undulations. In the past, oscillations of the sort reported here have been described qualitatively as resulting from such an interference effect. This description is incorrect for our systems because in the quantum limit, where the de Broglie wavelength becomes comparable to the potential range, the Ford and Wheeler analysis is inapplicable. The breakdown of the semi-classical description is seen in at least two ways.First, we have observed strong undulations at angles considerably larger than the rainbow angle, whereas the semi-classical description predicts that oscillations die out rapidly on the dark side of the rainbow. For example, the LJ (12, 6) fit for the H2 + O2 system predicts a classical rainbow at 15" in the CM, whereas we see strong oscillations out to 25". Indeed, the absence of rainbows both in theory and experiment for these systems shows that the semi- classical approach cannot be used here. Second, accurate quantum mechanical theoretical calculations predict oscillations with a spacing given by eqn (1) for purely repulsive potentials with monotonic deflection functions. The Ford and Wheeler analysis, however, reduces to the classical result whenever the deflection function is single-branched, and no undulations are possible.The oscillations in our systems are more accurately described as a diffraction effect produced at the steep repulsive wall of the p0tentia1.l~ The presence of an attractive well intensifies the diffraction oscillations and can increase their frequency since in this case the appropriate range parameter to use in eqn (1) is r, rather than 0. However, since in most cases the van der Waals minimum occurs at a distance Y, only slightly larger than the zero of the potential, the frequency of the undulations is only slightly affected by the presence of the well. The intermolecular potentials of the systems we have studied are anisotropic ;1 54 CENTRAL-FIELD INTERMOLECULAR POTENTIALS consequently, the interpretation of our data is more complicated than for atom-atom scattering.One approximate way of coping with this difficulty is to separate the potential into a spherical and an anisotropic part. We then assume that the effect of the latter is unimportant due to a combination of rotational averaging and the likelihood that the decrease of the magnitude of this anisotropy with the intermolecu- lar distance, Y, is more rapid than that of the spherically symmetric part, making it already sufficiently small for the distance range sampled by the present experiments. A partial wave expansion can then be used to determine the isotropic part of the potential, as was done in the previous section. Such an analysis, however, is not necessarily correct since the anisotropy may dampen or “ quench ” the oscillations and possibly shift their locations.Rothe and Helbing 2o and Kramer and LeBreton 21 report quenching of the glory undulations in the total scattering cross section of alkali atoms by various large asymmetric molecules. On the other hand, Aquilante et aZ.I2 find no evidence for quenching in the glory scattering of D2 by N2 and several hydrocarbons. Also, Butz et a l l 1 were able to fit the glory undulations in the total cross sections of He, HD and D, scattered by CH4, N2, O,, NO and CO using a spherical Lennard-Jones (12, 6) potential. Only the CO, glories appeared slightly dampened, as compared with their theoretical calculations. Turning to the rainbow maximum, Anlauf et aL2, found that for Ar+N, it was weaker than expected from a Lennard-Jones (n, 6) potential (with best fit obtained for n = 20), and attribute this difference to quenching.Similarly, Cavallini et ~ 1 . ~ ~ compared the rainbow of Ar + N2 with that of Ar + Ar and attributed the dampening of its intensity and the shift of its position to higher angles to anisotropy effects. Tully and Lee,24 after studying the same Ar+N, system, assume that the shift in the rainbow position to larger angles is negligible, but that the quenching is not, and get a slightly deeper well than Anlauf et al. Stolte 2 5 measured the total cross section of Ar+NO with the rotational quantum numbers of NO selected to be J = MJ = 3 andJ = MJ = 4, and found that the anisotropic contribution to the total cross section is less than 1 %.Farrar and Lee 26 have seen rapid quantum oscillations in the differential elastic scattering cross section for the p-H, +p-€3, system, and were able to interpret their data using a central-field assumption. We now consider the theoretical calculations on anisotropy effects on differential elastic cross sections done so far. Cross 27 found in an approximate semi-classical calculation, using a potential with an isotropic part similar to that of K+Kr that anisotropy can significantly quench glory, rainbow and “ rapid ” oscillations. However, Cross’ theory, which is based on the Ford and Wheeler treatment of interference between different branches of the deflection function, is inapplicable to our systems where the undulations are produced to a large extent by diffraction at the steep repulsive wall of the potential.Furthermore, he assumes that the dependence of the isotropic and anisotropic parts of the potential is identical, an assumption subject to question. Finally, the systems treated in the present paper are more highly quantum than that considered by Cross, and the anisotropic effects are expected to be quantitatively different. Wagner and M c K o ~ , ~ ~ in an exact solution of the Schrodinger equation for the scattering of Ar + H,, found no significant quenching or shifting of the rapid quantum undulations. However, their results provide only a lower estimate on these effects since H2 is more isotropic than other diatomic molecules, and rotational transitions, which play an important role in quenching, are less likely for low energy collisions with H,.The range of intermolecular distances sampled in the present experiments, esti- mated by a semi-classical analysis as described in the previous section, and depicted in fig. 8 to 10 inclusive, includes part of the repulsive wall and the minimum in the attractive well. We conclude from the present experiments that in this range, andA . KUPPERMANN, R . J . GORDON AND M . J . COGGIOLA I55 for the hydrogen or deuterium systems considered, effects of anisotropy on the differ- ential cross sections are negligible (within experimental error). This conclusion is based on the following observations. First, the potentials obtained were independent of their assumed mathematical form. Indeed, comparison of the results for the three- parameter LJ (n, 6) potential and the four-parameter MSV potential, as given in fig.8 to 10 inclusive, shows that they are nearly equal, even though their mathematical form in the r-range sampled by the experiments is substantially different. Second, a variety of different secondary scattering partners were studied. We did not find a correlation between the amplitudes of the observed oscillations and the symmetry of the secondary molecule, as would have been expected for significant anisotropy effects. Third, both H2 and D, were scattered by the same secondary molecule. If quenching and angular shifting of the undulations by the anisotropy in the potential were significant, they would be expected to be sensitive to the relative momentum, or wavelength, of the colliding molecules.The fitted potential parameters obtained using the central-field assumption for the two isotopes at the same relative collision energy should as a result be different but, as pointed out at the end of the previous section, these potentials are the same to within the experimental errors. Fourth, the potentials for H2+SF, and H,+NH, were determined at two different relative energies (see tables 1 and 2). The fitted parameters are in excellent agreement with each other, a necessary condition for validity of the central-field assumption. Finally, the p-H2 + SF6 experiments yielded results identical to the n-H, + SF6 scattering at the same relative energy, to within experimental error, thus indicating the insensitivity of the measurements to the distribution of H2 initial rotational states.In summary, 0 5 10 15 20 25 laboratory angle 0 Jdeg FIG. 11 .-Comparison of the differential elastic scattering predicted by total cross section measure- ments with the experimental Hz+02 data from fig. 3. The solid curve represents the LJ (12,6) fit given in table 1. The dotted curve was determined using the LJ (12,6) E and u parameters of Butz et al." given in table 3, while the dashed curve was fitted to the data using the EU product determined by Butz et al.156 CENTRAL-FIELD INTERMOLECULAR POTENTIALS we have found it possible in every case to describe the measured differential elastic scattering cross sections using a spherically symmetric potential which is independent of the de Broglie wavelength 1, of the experiment.Both the position and the amplitude of the rapid oscillations, as well as the overall shape of the cross section are accurately fitted by such spherical potentials ; no effects of anisotropy are observed. Butz et d.ll have fitted a LJ (12,6) potential to their total cross section measure- ments of D2 + 02, D2 +CO and D2 + CH,. The total cross section results yield the product EU, but do not give reliable estimates for the individual parameters. Aqui- lante et aZ.’s results l 2 for D2 +CH, agree with those of Butz et al. To compare the latter’s results with our own, we have used their ECT product values and determined the individual parameters by the Newton’s method described in the previous section. TABLE 3.--COMPARISON WITH LJ ( 1 2 , 6 ) PARAMETERS OBTAINED FROM TOTAL CROSS SECTION MEASUREMENTS system 4 elmeV ref.&SO2 3.3810.03 2.99 3.37 & 0.05 H2+CO 3.5k0.1 3.11 3.41 1 0.4 H2+ CH4 3.7k0.2 2.95 3.52+ 0 . 3 3.6 3.53 0.3 7.710.9 6.3 5.6+ 1.2 6.9+ 1.5 5.7 5.2+ 1 .5 9.9* 1.4 7.4 6.2f 2.0 6.0 6.2+ 2.0 this work ref. (11) (a) this work ref. (1 1) (a) this work ref. (11) (4 ref. (1 2) (a) (a) These valueq were obtained by holding the product EU constant, while allowing u to vary to give a best fit to the experimental data. Based on our previous conclusions that the H, and D, isotopes yield the same scatter- ing potentials, these calculations were done for the H,+O,, €3, +CO and H, +CH, systems in whiah the quantum undulations are more pronounced. The results of these restricted fits are given in table 3, together with the unrestricted ones, as well as those obtained by Butz et al., from their total cross sections..The corresponding differential crow sections are shown in fig. 1 1 for H2 + 02. In addition, our LJ (12, 6) best fit cross section is reproduced for comparison. It is clear that neither the total cross section data, nor the best fit obtained using the constrained product of co give as good an agreement as the unconstrained LJ (12, 6) fit. While this is true for all of the systems compared, it should be noted that the results obtained from the constrained fit are in much better agreement with the differential cross section data than are the predictions from the separate parameters obtained from the total cross sections. This emphasizes the value of total cross section measurements in determining &CT product values, while giving less reliable estimates of the separate parameters.In contrast, differential cross section measurements of the type reported here yield a more accurate description of the intermolecular potential, indicating among other things deviations from the LJ (12, 6) expression, as shown from the H2(D2) + 0, system in In concluding, it should be remarked that the lack of anisotropy effects for the €I,@,)-containing systems described in the present paper, are probably due at least in part to the fact that this molecule is nearly spherical. In addition, rotation21 excita- fig. 3.A . KUPPERMANN, R . J . GORDON A N D M . J . COGGIOLA 157 tioii processes probably result from small orbital angular momenta and manifest themselves at large scattering angles, in a manner determined mainly by the inter- molecular potential at distances shorter than those sampled in the present experiments.One should be extremely cautious in attempting to extend these conclusions to other systems. See for example, J. Amdur and J. B. Jordan, Adu. Chem. Phys., X (Interscience, New York, 1966), pp. 29-74 and references cited therein. A. J. H. Boerboom, H. Van Dop and J. Los, Physica, 1970, 46,458. Yw. N. Belayev, N. V. Kamyshon, V. B. Leonas and A. V. Smeryagin, Entropie, 1969,30,173. The same is true for differential cross sections. See, for example, William H. Miller, J. Chem. Phys., 1969, 51, 3631. U. Buck and H. Pauly, J. Chem. Phys., 1971,54, 1929. Daniel H. Winicur, A. L. Morsund, W. R. Devereaux, L. R. Martin and Aron Kuppermann, J. Chem. Phys., 1970, 52,3299. M. Cavallini, L. Meneghetti, G. Scoles, and M. Yealland, Phys. Reu. Letters, 1970, 24,1469. D. Auerbach, C. Detz, K. Reed and L. Wharton, Abstracts from VII ZCPEAC (North-Holland, Amsterdam, 1971), pp. 541-542. J. M. Farrar and Y. T. Lee, J. Chem. Phys., 1972, 56, 5801, and earlier papers cited therein. See, for example, Richard B. Bernstein, Adu. Chem. Phys., X (Interscience, New York, 1966), p. 100, eqn (VZ.6), with La-L0-(pb/h). H. P. Butz, R. Feltgen, H. Pauly and H. V. Vehmeyer, 2. Phys., 1971, 247, 70. l2 V. Aquilante, G. Liuti, F. Vecchio-Cattivi and G. G. Volpi, Mol. Phys., 1971, 21, 1149. Extranuclear Laboratories, Inc., Pittsburgh, Pennsylvania. l4 W. Paul and H. Steinwedel, 2. Naturforsch., 1953, 8a, 448. Is P. E. Siska, J. M. Parson, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1971,55,5762. D. W. Marquardt, J. SOC. Znd. Appl. Math., 1963, 11,431. l7 Kenneth W. Ford and Johrl A. Wheeler, Ann. Phys., 1959,7, 259. John Adams, Ph.D. Thesis (University of Arkansas, 1969). Robert J. Gordon and Martin Griss, to be published. 2o Erhard W. Rothe and Reinhard K. B. Helbing, J. Chem. Phys., 1970,53,2501. 2 1 H. L. Kramer and P. R. LeBreton, J. Chem. Phys., 1967, 47,3367. 22 K. G. Anlauf, R. Bickes Jr., and R. B. Bernstein, J. Chem. Phys., 1971,54, 3647. 23 M. Cavallini, M. G. Dondi, G. Scoles and U. Valbusa, Chem. Phys. Letters, 1971, 10, 22. 24 F. P. Tully and Y. T. Lee, J. Chem. Phys., 1972, 57,866. 2s Steven Stolte, Ph.D. Thesis (Catholic University of Nijmegea, 1972). 26 J. M. Farrar and Y. T. Lee, J. Chem. Phys., 1972,57, 5492. 27 R. J. Cross, J. Chem. Phys., 1970,52, 5703. 28 A. F. Wagner and V . McKoy, J. Chem. Phys., 1973, 58.

ISSN:0301-7249    

DOI:10.1039/DC9735500145

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Scattering of Metastable Mercury Atoms BY T. A. DAVIDSON, M. A. D. FLUENDY AND K. P. LAWLEY Dept. of Chemistry, University of Edinburgh, Edinburgh EH9 355 Received 10th January, 1973 Differential elastic cross sections for the scattering of the metastable 6 3Pz state of mercury from sodium, potassium and rubidium have been measured at thermal energies. Interference structure is resolved in all cases, suggesting that the atoms interact by a single effective potential in the attractive region probed at these energies. No attenuation of the interference structure is observed so that quenching of the metastable state is not an important process along trajectories sampling only the attractive part of the potential. Collisions between ground state Hg atoms (IS,) and alkali metal atoms (”&) at thermal energies are necessarily elastic and occur under the influence of a single potential.Scattering measurements on these systems have been made by several groups 1-7 and recently Buck and Pauly ’* have reported accurate differential cross section measurements and their inversion to yield potentials. This inversion may be in some degree ambiguous as shown by Boyle lo ; nevertheless the potential between the alkali metals and ground state mercury is now comparatively well established. The corresponding interaction between alkali atoms and the excited states of Hg l 2 is much less well understood. The lowest of such states are the 63P0, 63P1 and 63P, at, respectively, 4.64,4.89 and 5.43 eV above ground. The 63P0 and 63P2 states are metastable with a lifetime 11* l2 - s, comparable to the flight time in the crossed beam apparatus used for the present experiments.The 3P1 state decays after - s. The thermal energy collision of these atoms with other species is now complicated not only by the possibility of electronic energy transfer but also by the fact that a manifold of potentials arises from the separated atoms due to the possible spin pairings (if the partner is not a singlet atom) and mJ states of the Hg atom. TABLE 1.-AE FOR PROCESSES INVOLVING Hg (3P2) AND ALKALI METALS process ionisation of M Na - 0.31 - K Rb 1.11 - 1.27 excitation transfer to 1st excited state of M - 3.33 - 3.82 - 3.87 The photochemistry l3 of these states and in particular of the 3P1 atoms is of great importance and has been widely studied. However, the understanding of the funda- mental processes of electronic energy transfer and their relation to curve crossing must begin with simple systems where a knowledge of the adiabatic potential curves can be obtained.The alkali metal+Hg* system is a useful case for this purpose since both theoretical studies and the unravelling of potential energy curves through magnetic state selection offer the hope of a complete picture. The various processes that can occur are : (a) elastic scattering, (b) intermultiplet transitions AJ, which will be followed by quenching if the 3P1 state is formed, (c) 158T . A . DAVIDSON, M . A . D. FLUENDY AND K . P . LAWLEY 159 Am, transitions within a given J state, (d) ionisation, (e) transfer of electronic excitation to the alkali metal.For the 63P2 state, all these channels are open for the series Na to Cs, though with fairly considerable exoergicities except for cases (a) and (c) ; exoergicity values are given in table 1. In this paper we describe the results of differential cross section measurements on the process (a), elastic scattering accompanied by (c), rn J transitions, i.e., for scattering without change in J value. EXPERIMENTAL The apparatus is shown schematically in fig. I . The two beam sources rotate on a turn- table in front of the fixed detector which is located in a differentially pumped chamber.14 The excited mercury beam is produced by bombarding a ground state Hg beam effusing from a glass capillary array with electrons of controlled energy.The electrons were collimated by a magnetic field and, with an excitation voltage of 10 eV, an electron current - 10 mA excited roughly 1 in lo5 of the Hg atoms resulting in a metastable flux of 101o-lO'l atoms steradian-l s-l. Higher electron energies mainly resulted in increased production of photons as shown by time of flight experiments using a pulsed excitation voltage. After re-collimation (the electron bombardment can produce deflections -2" in the Hg* beam), the metastable atoms crossed a modulated target beam of alkali metal atoms. The scattered atoms entered a UHV chamber (- Torr) via a narrow channel to strike a clean potassium surface plated in situ. Electrons ejected from this surface were focused into a Mullard channel electron multiplier and counted on dual scalers gated in delayed synchronism with the target beam modulation.The maximum counting rate observed was -lo5 Hz while the background count rate for the detector alone (valved off from the main chamber) was 1-0.1 Hz; more typically, when measurements were in progress, the background was 2-3 Hz. Once prepared, the K surface was stable in performance for periods of several months. All the experimental data were recorded on paper tape and processed off line by computer.14 HCJ' OVEN XBEAM WEN ts.p. 1 FIG. 1 .-Apparatus schematic showing electron gun for metastable Hg production and the Auger detector.I 60 SCATTERING OF METASTABLE MERCURY ATOMS The excitation function observed with this equipment is shown in fig. 2 as a plot of main beam signal (normalised by the excitation current) versus excitation voltage.The first maximum with a threshold at -6.0 eV arises from the 6 3Po,l,a states, the other maximum with a threshold at 10 eV observed on a tungsten surface results from photons plus possibly some long lived 303 Hg atoms.lS In the results to be described, 10 eV electrons were used so that only the 63P0,1,2 states need be considered. Of these three possibilities, time of flight experiments showed the photon contribution (from decay of 3P1 Hg) to be only - 10 % of the total signal. Since the scattering cross section for 2537 A photons with alkali metals is very much smaller than for atom-atom collisions, these photons will not be modulated by the cross beam and hence not registered as scattered signal.Calculations l6 for the 3P2 and 3P0 states suggest that the ratio of their cross sections for formation by electron bombardment is roughtly 5 : 1 (i.e., the simple statistical ratio). Their lifetimes are probably similar and since the 3P2 would be more efficiently detected by the K surface, the observed scattering is tentatively attributed solely to this state. Magnetic deflection analysis of the beam to confirm this is in progress. 2 4 6 .8 113 12 14 16 exciter volt age/V FIG. 2.-Apparent excitation function observed for Auger ejection from K and W surfaces as a function of exciter voltage in the source. Because the 3P2 Hg atom flux decays appreciably during transit from scattering centre to detector, the Hg* velocity distribution is considerably distorted from the initial v 2 Maxwellian one out of the exciter.The appropriate velocity distribution a distance L from the source is : where v g is the most probable velocity at source and z is the lifetime. downstream, the most probable velocity v z is given by the appropriate root of the cubic At a distance L From the dimensions of the apparatus (L = 61 cm), the most probable velocity of the Hg* is - 38 % greater than that for a stable species at the same temperature. The relative velocity distribution is not affected so much, the full width at half height being reduced by 10 %. The most probable relative velocity also changes slightly with angle of scattering. Finally, the relative masses and velocities are such that at a given laboratory angle of observation (0) there are two centre of mass angles (x) contributing, leading to fast and slow scatteredT.A . DAVIDSON, M. A . D. FLUENDY A N D K . P . LAWLEY 161 components. However, partly because of the ratio of Jacobians but also because of sub- stantially greater decay, the slow component is <10 % of the fast component and may be neglected. RESULTS AND DISCUSSION Laboratory distribution for HgZk elastically scattered from the alkali metals Na, K and Rb are shown in fig. 3,5 and 7. The corresponding centre of mass angles are also shown on the axes. The results for Rb are rather limited in precision pending further observations. Fig. 4 and 6 show the result of deconvoluting the curves shown in fig. 0' 25 44 angle 75 FIG. 3.-Laboratory differential cross section for Hg 63P2 scattering from sodium.The angles shown below the axis are in the centre of mass system. Velocity 895.0 m s"'. 1 t m t- 0 u M (d X x 1 Y CI Y .d c) $ .- <!I 4 n 0 25 44 75 angle FIG. 4.-Laboratory differential cross section for Hg 63P2 +Na deconvoluted using the main beam profile, data of fig. 3. 55-F162 SCATTERING OF METASTABLE MERCURY ATOMS 3 and 5 using the observed main beam profile as the filter function. The deconvoluted results are rather noisy, but structure partially resolved before deconvolution can now be clearly seen. The laboratory angles are shown above the axis. angle FIG. 5.-Laboratory differential cross section Hg 6 3 P ~ scattering from K. The angles shown below the axis are in the centre of mass system. Velocity 660.0 m s-l. 0 2 0 63 angle FIG.6.-Deconvoluted results for Hg 63Pz+K, same data as fig. 5. The most striking qualitative feature of these results is the presence of strong undulations in the differential cross sections which cover the whole angular range of observation (out to 85" in the centre of mats in some cases) with undiminished ampli- tude. This points to two quite separate conclusions. Firstly, we are seeing the operation of either a single potential or a group of potentials that are very similar inT . A . DAVIDSON, M . A . D . FLUENDY AND K . P . LAWLEY 163 the parts of them covered by the observations. For, if several rather different potentials with similar weights were operating, the net interference structure would be much weakened by superposition of the separate patterns. Furthermore, the presence of strong interference structure means that at least two branches of the deflection function are present.Thus, quenching cannot be removing the inner branch of the deflection function or, if several similar potentials are operating, only one or two at most can be affected in this way otherwise the amplitude of the interference structure would be diminished. FIG. 7.-Laboratory differential cross section for Hg 63P2 scattering from Rb. The angles shown below the axis are in the centre of mass system, Velocity 470.0 m s-l. FIG. &--A family of deflection functions arising from a group of potentials having similar values of u and C, but different values of E. Also shown are the I values of some interfering branches.1 64 SCATTERING OF METASTABLE MERCURY ATOMS Turning to the details of the scattering structure, there is a regular pattern in which two periods may be seen, a high frequency one with an average period of -5.8" (Na) and 3.7" (K) and a lower frequency one of smaller amplitude with a period of 24" and 13" respectively in Na and K. This rather simple structure is similar to that expected from a single potential at collision energies leading to orbiting or a rainbow well beyond the angular observation range.Referring to fig. 8, analysis of the angular periods in the interference structure can be made using the semi-classical relation Ax = 2n/(Z, -Z2) where ZI and Z2 are the orbital momenta values for two interfering branches on the attractive side. The high frequency oscillations would then corres- pond to interference between two attractive branches with deflection x and spaced 2-3 A apart.The low amplitude longer period oscillations similarly arise from regions of the deflection function centred at 2n-x and x, spaced -0.8 A apart, (see fig. 8). A detailed potential fit to this data has not yet been achieved but at least a qualitatively similar cross section can be obtained using the Buck and Pauly potential for the ground state atoms, suggesting that the effective potential of Hg*/M is similar to this. Five molecular states evolve from the atomic pair Hg(3P2)+M(2St) and so the question arises as to why they should all appear so similar. At large separations Hund's case (c) holds (assuming fixed nuclei) and the manifold of states may be classified by their value of Q(= m,(Hg)+m,(M)).In the limit of small separations (Hund's case (a)) the good quantum numbers are A and S. Depending on the order of the molecular states, the following tentative correlations may be made ; molecular mJ ms Q state It is clear that neither the deep lying 2C+ nor the largely repulsive 2Zz correlate with the 3P2 state. Calculations of the interatomic potential in the m,, m, coupling scheme have been made using the limited Hartree-Fock-Slater 2-electron orbitals already calculated for the 3P2 state of Hg.17 These were combined with HFS valence orbital of the alkali metal to form a linear combination of Slater determinants that preserved J and mJ as good quantum numbers. An approximate Hamiltonian using the core potentials (with exchange) of the unperturbed atoms together with the specific electron-electron repulsion terms among the three valence electrons was then used in a first order computation of the energy of each of the five states listed above.Relatively shallow wells ranging from 8 x erg to 13 x erg were found, all much less than the spin-orbit splitting in mercury. The positions of the potential minima were roughly constant at 4 A. The calculated potentials thus bracket the ground state well depth but have rather smaller values of Q. So far, then, we have a picture of five rather similar potentials originating with the 3P2 states, partly as a result of the restrictions of the correlation diagram itself. Turning to the dynamics of the collision, the forces operating depend upon whether mJ in a space fixed system or in a rotating system is a good quantum number.Thus for collisions of large impact parameter the coupling of electronic motion to the inter- atomic motion is weak and the phase shifts depend only slightly on mJ. That is, the adiabatic phase shifts (those calculated assuming mJ a good quantum number) are scrambled. As the impact parameter decreases strong coupling ensues, at first nearT . A . DAVIDSON, M. A . D . FLUENDY AND K . P. LAWLEY 165 the turning point. Finally, for collisions of small impact parameter, mJ in a rotating frame is a good quantum number and the phase shift functions are well separated. The impact parameter at which coupling becomes important is determined by the splitting of the adiabatic potentials.of the 3Pz state on mJ is less than 10 % of the mean value and this presumably means a similarly small range of c6 values. Taking the range of C6 values to be given effect- ively by The dependence of the polarisability A c 6 = 4 A d (3) where A& is the range of well depths quoted above and applying l9 to determine the critical impact parameter for coupling, b,, values N 6 A are obtained with a relative velocity U- 6.6 x 10’ m s-l. At small angles of scattering, the lack of coupling between the Hg* mJ state and the passing atom results in a scrambling of the manifold of interatomic potentials to give one effective potential curve. As b decreases, coupling ensues but the potentials, all belonging to quartet states, remain inherently similar and the interference structure from them coincides. The observa- tion of a weak structure of longer periodicity in o(0) shows that the deflection functions associated with the various adiabatic potentials can not diverge appreciably until near the minimum where the resulting interference structure from each state would be lost (see fig.8) in the averaging over mJ. The observation of quantum structure also sets an upper limit on the size of the quenching cross section since both branches of the deflection function must be present for this structure to be seen. In the absence of a detailed potential fit to the experi- mental data, it is not possible to give a precise value to this limit, but quenching is clearly not an important process for collisions with impact parameters >o and the total quenching cross section can hardly exceed gas kinetic values.No other data on the absolute magnitude of the quenching cross section for Hg 6 3Pz have been reported, though Martin 2o has measured relative total cross sections for the intermultiplet process (d) with a wide range of gases ; preliminary results 21 for excitation of the alkali metals are not incompatible with a cross section of gas kinetic magnitude. We thus adopt the following tentative picture. The authors thank the Science Research Council and N.A.T.O. for financial support and T. A. D. the Carnegie Trust for a scholarship. F. A. Morse and R. B. Bernstein, J. Chem. Phys., 1962,37,2019. F. A. Morse, R. B. Bernstein and H. U. Hostettler, J. Chem Phys, 1962, 36, 1947. H. U. Hostettler and R. B. Bernstein, Phys. Rev. Letters, 1960, 5, 318. E. Hundhausen and H. Pauly, Z. Naturforsch., 1964, 19a, 810. E. Hundhausen and H. Pauly, Z. Phys., 1965,187,305. P. Barwig, U. Buck, E. Hundhausen and H. Pauly, 2. Phys., 1966,196, 343. U. Buck and H. Pauly, J. Chem. Phys., 1969,51, 1662. U. Buck and H. Pauly, J. Chem. Phys., 1971,54, 1929. U. Buck, M. Kick and H. Pauly, Proc. VZZ Znt. Conference on Physics of Electronic and Atomic Collisions (North-Holland, Amsterdam, 1971), p. 543. l o J. F. Boyle, Mol. Phys., 1971, 22, 993. l 1 P. Baltayan and J. C. Pebay-Peyroula, Compt. Rend., 1965,260, 6569. l 3 J. G. Calvert and J. N. Pitts, Photochemistry (Wiley, New York, 1966). E. C. Darwall, M. A. D. Fluendy and K. P. Lawley, Entropie, 1971, 42, 162.166 SCATTERING OF METASTABLE MERCURY ATOMS l4 L. T. Cowley, M. A. D. Fluendy, D. S. Horne and K. P. Lawley, J. Phys. E, (Sci. lnstr.), 1969, l5 M. N. McDermott and W. L. Lichten, Phys. Rev., 1960, 119,134. l6 J. C. McConnell and B. L. Moiseiwitsch, J. Phys. B, 1968, 1,406. l7 E. C. Darwall, M. A. D. Fluendy and K. P. Lawley, Mol. Phys., 1970,19,673. l 8 J. Levine, R. J. Celotta and B. Bederson, Phys. Reu., 1968, 171, 31. l9 M. A. D. Fluendy, I. H. Kerr and K. P. Lawley, to be published. 2o F. J. Van Itallie, L. J. Doemeny and R. M. Martin, J. Chern. Phys., 1972, 56, 3689. 21 R. M. Martin, private communication, 1972. 2, 1021.

ISSN:0301-7249    

DOI:10.1039/DC9735500158

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Scattering Experiments with Fast Hydrogen Atoms Velocity Dependence of the Integral Elastic Cross Section with the Rare Gases in the Energy Range 0.01-1.00 eV BY R. W. BICKES, Jr.?, B. LANTZSCH, J. P. TOENNIES* AND K. WALASCHEWSKI Max-Planck-Institut fur Stromungsforschung, 34 Gottingen, Germany Received 3rd January, 1973 Integral elastic cross sections of hydrogen atoms scattered by rare gases have been measured in the velocity range 1.4-14 x lo5 cm s-' (0.01 < Ecm< 1.00 eV). The experimental data have been fitted to the Lennard-Jones and exp (a,6) potential models and are found to provide information on the attractive and repulsive parts of the potential. The potential well parameters E, Rm and the Born-Mayer repulsive potential parameters A , ac have been determined for all of the rare gases.These c, Rm parameters, in combination with the corresponding rare gas dimer parameters, are shown to be consistent with combining rules. Despite the fact that the H-atom is the simplest of atoms, its interaction potentials with other atoms and molecules are largely unknown. Being an abundant constituent on earth and the most abundant element in outer space these potentials are of funda- mental importance for understanding gas phase reactions,' recombination mechan- isms,2 9 high temperature transport properties and collisional energy transfer, both in the laboratory and in interstellar space.4* Theory only provides reliable values for the coefficient of the leading term in the long range potential.6 The next higher order attractive R-8 term, which is expected to make a large (20-30 %) contri- bution to the well depth, has, however, not yet been calculated.A priori calculations of the short range repulsive potential are only available for H-He ' and H-Ne.8 Work is in progress on H-Ar.' The first measurements of the velocity dependence of the integral cross section for these systems were carried out by Herschbach and coworkers lo* who were able to deduce rough values for the strength parameter (= ER,) for these systems. Recently, more accurate absolute integral cross sections have been measured for Ar, Kr and Xe at H-atom velocities in the range of 1.8 to 6.2 x lo5 cm s-',12 and the results were interpreted in terms of E and Rm. A precision measurement of absolute integral cross sections for H-He has been used in connection with the relative measurements reported here to obtain an accurate potential curve for this system for comparison with theory.' EXPERIMENTAL A schematic drawing of the apparatus is shown in fig.l.14 The hydrogen atom beam is produced by thermal dissociation in a cylindrical tungsten oven, which is heated by electron bombardment to 2500-2600K. At inlet pressures of about 1 Torr the measured degree of dissociation is roughly 0.40. After leaving the oven chamber (at the left in fig. 1) through a conical skimmer, the beam passes through a mechanical Fizeau-type velocity selector (Av/v = 11 %, Av is the FWHM spread ; urnax = 14x lo5 cm s-' at 650 Hz). The velocity t present address : Dept. of Chemistry, University of Waterloo, Waterloo, Ontario, Canada. 1 67168 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS selector is calibrated to 3.2 % by comparing measured and calculated Maxwell-Boltzmann distributions for Hz at low oven pressures (<0.1 Torr).The secondary beam source is a stainless steel many-channel-array (channel diarn. 0.05 mm, channel length 3 mm, overall area = 6.25 x 2 mm,2 2100 channels, 60 % trans- parency).15 Liquid nitrogen cooling is used for He, Ne, Ar, and Kr, whereas Xe is admitted at room temperature. At the inlet pressures of 1-2.5 Torr the effective angular spread (averaged over the array area) is 8" (FWHM).16 An identical secondary beam source is mounted with centre 20mm to one side from the primary beam. An automatic valving system alternates the gas flow through the two sources. This corresponds to turning the secondary beam on and off.The flow through the sources is adjusted so that the rest gas scattering is the same for both secondary beam positions. Thus, the observed beam attenuation can be attributed entirely to the directed component. Diffusion pumps FIG. 1.-Schematic scale diagram of the apparatus. The beam passes from left to right through the velocity selector scattering region and into the electron bombardment detector. The detector consists of an electron bombardment ion source (similar to an UHV Bayard Alpert gauge tube, with the cathode removed and provision made for coaxial ion extraction at one end) and a small magnetic mass spectrometer ( h / m = 10 %, m,,, 21 5). The Ht background gas pressure in the ionisation region is estimated to be 2x lO-L3 Ton.The beam ionisation efficiency is 5 x lo-' A T o r 1 at 4 mA emission current. After amplification by a venetian blind multiplier the pulses from individual ions are counted in fast counters, wbich are read out into DEC PDP/8L mini-computer for on-line evaluation of the integral effective cross section using Beer's law. The H atom beam intensity is about 4x lo4 counts per second at the maximum in the Maxwell-Boltmann distribution (ol = 8200ms-'). The mass one background is typically 250 counts per second. Protons produced by ionisation of the H2 beam component contribute less than 2 % to the mass one signal. For each measured value of the integral cross sections, the computer calculates the standard deviation of the individual cross section.The integral cross section at one velocity is remeasured until this value approaches a constant. This takes on the average about 12 min. The standard deviations of the mean cross sections are typically in the range & 0.8-3 %. They are indicated by error bars in fig. 2. The angular resolving power defining beam geometry is determined by the following slits at the positions L : oven slit : 0.4 mmx 6 mm, L = 0 ; collimation slit : 0.5 mmx 6 mm, L = 27.5 cm (the scattering beam is at L = 34 cm) ; detector enfrance slit : 2 mm x 6 mm, L = 123.5 cm.BICKES, J R . , LANTZSCH, TOENNIES AND WALASCHEWSKI 169 FIG. 2.-The measured effective integral cross sections (in arbitrary units) is plotted as a function of the primary beam velocity.The target beam temperature is 77 K in all cases except Xe (293 K). The curves have been shifted vertically for the sake of clarity. The error bars span the range given by the standard deviation of the mean cross section. The solid line is the best fit calculated curve for the Lennard-Jones (12,6) potential model. RESULTS AND METHODS OF ANALYSIS Fig. 2 shows the measured relative integral cross sections as a function of the primary beam velocity. Note that the measurements span two orders of magnitude in the primary beam energy (0.01 eV to 1 .OO eV), which is nearly the same as the energy in the centre of mass. In all cases, except H-He, the measurements show the velocity dependence expected in the transition region, where both the long range attractive and short170 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS range repulsive parts of the potential influence the integral cross sections.The arrows in fig. 2 indicate the location of the characteristic velocities gc(gc = &&,/ti) which for a Lennard-Jones (12,6) potential characterizes the middle of the transition region. For H-He, gc is about 150ms-', and is well below the range covered in these experiments. Thus, the measured cross section is almost entirely determined by the short range repulsive potential, which is the reason for the small change in the cross section in the investigated velocity range. In addition to the velocity dependence, absolute values are also needed fully to analyse the data. For He, Ne and Ar the absolute values of the cross section were derived from previous work on the systems D-He, D-Ne and H2-Ar.17 The cross sections are : D-He, 38.13 A2 f0.8 % (u, = 7100 m s-l) ; D-Ne, 39.3 A2 3.7 % (v, = 8125 m s-l) and H2-Ar, 77.52A2 +0.65 %(vl = 5215 m s-l).They were corrected for the angular resolving power and converted to values for the H-atoms (assuming identical potentials for H and D atom scattering). The H-Ar cross section was obtained from the ratio oH-Ar/oH2-Ar = 1.23+1 % (u, = 5215 ms-') which was measured in the course of this work. Since highly accurate absolute cross sections were not available for H-Kr and H-Xe, these were estimated by using combining rules as discussed in the next section. The absolute cross sections are summarized in table 1. TABLE ABSOLUTE CROSS SECTIONS (A2) FOR v1 = 5000 m s-' USED IN THE EVALUATION OF POTENTIAL PARAMETERS 00 H-He 39.6 a H-Ne 40.8 a H-Ar 63.0 a H-fi 78.9 H-Xe 105.2 b 0 derived from measurements by Hahn.b estimated using combining rules. As discussed in the last section, it is not possible at the present time uniquely to determine the intermolecular potential from integral cross section measurements. In view of this difficulty, we have in previous work on H-He l 3 and He-H, l 8 and other light systems used all available information (including that from theory) and a many-parameter flexible potential model to obtain the best possible potential. Since accurate integral cross sections and sufficient theoretical data are not yet available for the heavier rare gases, we have departed from this procedure in analyzing these experiments (see also the discussion in the last section).Thus, for comparison with other work l2 the data was first analyzed using the well known two parameter - Lennard-Jones (12, 6) potential : vLJ(12.6)(R) = E[ey2-2ey]* Further analysis was based on the three parameter Slater-Buckingham exp (a,6) potential : A " trial and error " procedure, in which the calculated corrected cross sections are compared with measured or estimated absolute cross sections, was used to deter- mine the best fit potential parameters. For a given potential model and trial para-BICKES, J R . , LANTZSCH, TOENNIES AND WALASCHEWSKI 171 meters, the elastic cross section was calculated by the partial wave method using JWKB phases, which were corrected at low energies using the Rosen and Yennie method.lg This ideal cross section was then corrected for the angular resolution of the apparatus using the approximate differential cross section and analytic geo- metry weighting factor given by von Busch.20 Finally, the corrected cross section was averaged over the secondary beam angular and velocity distributions as well as over the primary beam velocity distribution after velocity selection.22 The largest corrections are of the order of 4-5 % and, since they are reliable to better than 5 %, the overall cross section error (including that from the approximation of the phases) is less than about 0.5 %.Fig. 3 shows the calculated absolute cross section before and after corrections for the system H-He, for which these corrections were largest.1000 2993 soco 10000 vim s-' FIG. 3.-The calculated best fit integral absolute cross section as a function of primary beam velocity for H-He is shown with and without corrections. The calculated cross section without corrections (-.-.-) is compared with that after correction only for the angular resolving power (. . .) and after averaging over all velocity distributions (-). The corrections for the other rare gases are smaller. TheX2-test was then used to determine the goodness of fit. The potential parameters were varied in an automated program 23 until x2 was minimized. Of the order of 120 fit cycles were required to get the final set of potential parameters. POTENTIAL PARAMETERS The potential parameters determined are summarized in tables 2 and 3.Table 2 lists values of the product ER,, obtained using the Lennard-Jones (12,6) potential from the relative velocity dependence, and compares these with previous work. The agreement in the case of the heavier rare gases with the experiments of Aquilanti et al. is reasonably good, although outside the range of error. Our experience indicates that the difference may be due to the inadequacy of the JWKB phases at TABLE 2.-&R, VALUES [meV A] OBTAINED IN THIS AND PREVIOUS WORK. IN ALL CASES, THE LENNARD-JONES (12,6) POTENTIAL WAS USED. THE ERRORS IN OUR VALUES WERE OBTAINED BY COMPARING VARIOUS CALCULATIONS WITH THE MEASURED POINTS Stwalley Aquilanti et al. et al. this work H-He <1.8 - 0.60+35 % H-Ne <15 - 8.9+15 % H--Ar 6.25-25 14.2 16.9+8 % H-Kr 9.4-25 18.4 22.0+5 % H-Xe 12.5-31.2 23.4 26.9+5 %172 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS low energies in the earlier work.The decreasing error with increasing rare gas mass in our work reflects the location of gc with respect to the investigated velocity range. Table 3 summarizes all the potential parameters from both potential models as well as derived Born-Mayer parameters for a purely repulsive potential. Also listed are the X2-values and the number of measurements used in the fit. In all cases except Kr the goodness of fit as given by the x2-value is the same for both potential models. For Kr, the X2-value is larger for the more realistic and flexible TABLE 3.-POTENTIAL PARAMETERS AND X 2 VALUES CALCULATED FOR TWO POTENTIAL MODELS. A AND a* ARE THE DERIVED BORN-MAYER (BM) PARAMETERS.n IS THE NUMBER OF MEASURE- MENTS. THE ERRORS SHOWN ARE THE SUM OF THE ERRORS. system L-J (1 2,6) exp (a, 6) L-J ( 1 2,6) exp (a, 6) :-He 0.16f0.02 0.35f0.04 3.7050.09 3.6950.09 :-Ne 2.82*0.10 2.82f0.10 3.1550.14 3.18*0.15 1-Ar 4.73f0.15 4.80f0.15 3.54*0.13 3.56f0.13 s/meV RmlA :-Kr 5.99f0.20 6.08f0.20 3.67f0.28 3.7030.29 [-Xe 6.85f0.25 6.81 k0.25 3.933Z0.31 3.95f0.32 STATISTICAL AND ESTIMATED SYSTEMATIC U a*"/ A " / x 2 exp (a, 6) A-1 B-M eV B-M L-J( 12,6) exp (a,@ n 13.17 3.569 154.3 18.1 15.7 38 3Z0.95 14.19 4.462 3003 54.2 54.8 53 h l . 0 14.32 4.022 5730 45.2 45.6 49 f l . 0 *l.l 14.14 3.822 6200 73.6 97.3 54 14.28 3.615 7852 96.0 95.0 78 + l . l " The Born-Mayer potential is given by V(R) = A exp (-u*R). The parameters are simple related to those of the exp (a, 6) 52 = &[(a expi: acalq)/AO expi12 where ae,pi and UCalcr are the experimental and calculated corrected cross sections at the otential by the following formula : a* *elocity of.Aaexpr is the standard deviation of the measured points at the velocity ui. = aR,, A = 6~(a-6)-1 exp a. exp ( 4 6 ) potential than for the Lennard-Jones (12,6) potential. A possible explana- tion may be the fact that the absolute cross sections for Kr and Xe were not measured, but estimated by the following procedure based on combining rules. The combining rules of Good and Hope24a and Hudson and McCoubry24b are, respectively : where and I is the ionisation potential. These rules were used to obtain an effective H-H potential from the measured E, Rm values for H-He, H-Ne and H-Ar and those from beam measurements for He-He, Ne-He and Ar-Ar.The values used are summarized in the upper half of table 4. Very good agreement in the two independent determinations of E and R, for H-H is found using the Ne and Ar parameters. The poor agreement in the case of H-He is expected in view of the large uncertainties in its well parameters. The averaged effective H-H potential parameters from Ne and Ar were then used (bottom half of table 4) to estimate the E and R, values for H-Kr and H-Xe using the same combining rules. The products ERm obtained in this way are in excellent agreement (<4 %) with the values obtained directly from the velocity dependence (see table 2). The estimated parameters E and R, also agree well (< 2.5 %) with those obtained in the final analysis given in table 3.The potential para-BICKES, J R . , LANTZSCH, TOENNIES A N D WALASCHEWSKI 173 meters for H-Kr and H-Xe from table 4 were then used to calculate an estimated integral cross section for the Lennard-Jones (1 2,6) potential. These values were inserted into the fit procedure to obtain the final E and R, values in table 3. TABLE 4.-METHOD BASED ON COMBINING RULES USED TO ESTIMATE &(mev) AND &(A) VALUES FOR H-G AND H-Xe. THOSE IN ROMAN THE QUANTITIES IN BOLD FACE ARE MEASURED. ARE DERIVED USING COMBINING RULES. THE LENNARD-JONES (12,6) POTENTIAL HAS BEEN USED THROUGHOUT A He Ne Ar Kr Xe H-A & 0.16 a &n 3.70 Rm 3.15 & 2.82 a & 4.73 a Rm 3.54 & 5.85 Rm 3.63 & 6.82 Rm 3.83 (ERm = 21.2) (cR, = 26.1) A-A H-H 2.06 3.27 2.01 3.72 3.32 average : %%+ 0.05 3.29+ 0.05 23.80 29 2.04 +-{ 4.45 3.29 a this work.-4115 2b 25 30 3:5 40 45 50 RIA FIG. 4.-The best fit potential curves for H-Ne for two model potentials illustrate the model dependence of the experimental results. The curve (1) shows the exp (a,6) potential and the curve (2) shows the L-J (12,6) potential.174 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS The good agreement and internal consistency obtained using these combining rules cannot be evaluated as evidence for their accuracy. Very similar results are also obtained using the simpler combining rules: = +(RmlI+R,,,22) and E~~ = ( E ~ ~ E ~ ~ ) ) . The E and R, values for H-H, however, do seem suitable for making rough guesses of potential parameters for other H-atom systems.Fig. 4 shows the best fit curves for the two potential models for the case of H-Ne. Very good agreement is found, especially in the vicinity of the well region. The largest differences are observed in the extreme repulsive region. It is now well known that the Lennard-Jones potential is seriously in error in this region and that the exp (a,6) potential is more realistic. Fig. 5 shows the best fit exp (a,6) potential curve for all the systems. The solid part of the curves shows the region, which, according to the method discussed in the next section, is probed in these experiments. The dashed portion is thus merely a model-dependent extrapolation. For Ne, Ar, Kr and Xe, the potential curves are all similar and the E and R, values are in the expected order. At small potential energy g H-He is larger than expected.This can be explained by the much smaller polarisability of He compared with the other rare gases (only 2 electrons in the outer shell compared with 8 electrons in the heavier rare gases) so that the C/R6 dispersion terms is much smaller in relation to the repulsive potential. -6 t -*' i5 i'0 2:5 3.0 3'5 +O 45 5.0 25 RIA FIG. 5.-The best fit potential curves for the hydrogen-rare gas partners based on the exp(ct,6) potential model. The solid part of each curve shows the region probed in these experiments. Unfortunately, our E, R, values disagree considerably with those of Aquilanti et al. Probably this has to do with differences in the absolute cross sections.In keeping with physical intuition our R, values for the H-Ar potentials are smaller than those for the Ar-Ar interactions (see table 4), whereas those of Aqilanti et aZ. are larger.BICKES, J R . , LANTZSCH, TOENNIES A N D WALASCHEWSKI 175 DISCUSSION Integral cross section data cannot be directly inverted to obtain a model- independent potential curve,3o 31 Direct inversion and a unique determination of the potential curve is only possible to a high degree of reliability using differential cross section data with several supernumerary rainbow extrema. 32 Since the necessary extrema can only be observed for heavy partners satisfying the requirement B = $R; 2 500, it would appear that the determination of potentials for hydrogen atom interactions for which B-2-50 always involves some uncertainties.Two aspects of this problem will be briefly discussed here : a quantitative estimate of the potential region probed in these experiments and the significance of the potential curves. Intuitively, the potential region probed by integral cross section measurements should be related to the local expectation value of the potential, which is defined by R++AR R - 3AR (V(R’))R = J J V(R’)$(R;O)$*(R;e)R’2 dR’ (3) where $(R) is the total scattering wave function 1 which is calculated for the best fit potential V(R). Fig. 6a and 6b show typical calculated curves of (V(R’)), for two extreme primary beam velocities for the system H-Ar. Also indicated in each curve is the location of the weighted mean value of the classical turning points Rlmln for each partial wave, E R .(21+1) sin2 ql ~ ( 2 1 + l)sin2 y f l ’ ( R . ) = I l m l n man where Rlmin is the classical turning point for partial wave 2 and (22+ 1) sin2 vl is the partial cross section (ql is the scattering phase shift). It is seen that (Rmin) agrees well with the location of the greatest extremum in the (V(R’))R curves. Thus, at the low velocity (fig. 6a), the integral cross section probes mostly the attractive potential and the largest contribution comes from R 2: 3.5 A, which is just the location of the potential well. The repulsive potential beyond the classically forbidden region is also probed to a small extent. At high velocities (fig. 6b), however, only the repulsive potential in the vicinity of R E 2.5A is probed.To obtain a quantitative estimate of the range probed we have arbitrarily set the borderline at the point beyond which the potential contributes less than 15 % (see fig. 6a and 6b). According to this criterion the region of the H-Ar potential probed lies in the range 2.3 AS RS4.8 A. to calculate the cross section contributions from various R-regions. These calculations indicate that the contributions from the repulsive and attractive parts of the potential partly compensate each other as suggested by the areas under the corresponding portions of the (V(R’)), curves. This compensation effect is borne out further by calculations based on the exp (a,6) potential with variable potential parameters. In these calculations, E was varied away from the best values and for each new value of e the best values of R, and a A more exact analysis is possible by using Calogero’s variable phase176 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS were redetermined.This procedure was repeated until the confidence limit per- centage point corresponding to the x2 value was equal to about 70 % (corresponding to a 1 CT error in a normal distribution). These potential parameters define one of the dashed curves in fig. 7. E was then varied in the other direction to obtain the second dashed curve. Thus the true curve is expected to lie between these two limiting curves with a confidence of about 70 %. It is gratifying to see that the 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 RIA (4 -10 I 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 RIA (b) FIG.6.-The calculated local expectation value of the potential ( V(R’)>R is shown as a function of the interatomic distance for H-Ar . <Rmin), the weighted mean vaIue of the classical turning point, is also shown. Fig. 6a is for v1 = 1700 m s-’ and fig. 66 is for ol = 13 OOO m s-’. bo is given by u = nb;. The calcdations are based on the Lennard-Jones (12,6) potential.BICKES, J R . , LANTZSCH, TOENNIES AND WALASCHEWSKI 177 recent SCF calculations by Bondybey et aL8 agree with our results within these limits. Further changes in the potential parameters beyond these curves lead to large changes in the confidence levels. Thus these limiting curves provide a rough estimate of the uncertainties in the potential deterrninati~n.~~ - I - - 15 20 25 30 35 4.0 4 5 50 RIA FIG.7.-The curves 2 and 3 indicate extreme potentials with larger x2 values than the best fit (see text). Curve 1 shows the best fit (minimum in x2). All curves have been fitted to the experimental data for H-Ne using the exp (a,6) potential. Curve 4 has recently been calculated by Bondybey et a1.' In all studies of this type there appears to be one point on the repulsive potential at which the best fit and extreme curves all cross. Apparently this point is deter- mined best in the experiments. The physical significance of this singular point is not yet fully understood and is presently under investigation. We thank W. Bauer for his help with the measurements and calculations, R. Feltgen for lending us his program for calculating averaged integral cross sections and R.Gengenbach for helpful discussions. The construction of the apparatus was financed by the Deutsche Forschungsgemeinschaft. C. Rebick and J. Dubrin, J. Chem. Phys., 1970, 53,2079. V. H. Shui and J. P. Appleton, J. Chem. Phys., 1971,55, 3126. R. T. Pack, R. L. Snow and W. D. Smith, J. Chem. Phys., 1972,55,926. L. Spitzer, Jr., Difuse Matter in Space, (Interscience, New York, 1968). J. Schafer and E. Trefftz, 2. Naturforsch., 1970, 25a, 863. G. Starkschall, J. Chem. Phys., 1972, 56, 5728. ' G. Das and S . Ray, Phys. Rev. Letters, 1970, 24, 1391. ' V. Bondybey and P. K. Person and H. F. Schaefer, 111, J. Chem. Phys., 1972, 57, 1123. G. Das and A. C. Wahl, private communication, 1972. 1967,46,2172. l o M. A. D. Fluendy, R. M. Martin, E.E. Muschlitz, Jr. and D. Herschbach, J. Chem. P/zYs.,178 SCATTERING EXPERIMENTS WITH FAST HYDROGEN ATOMS l1 W. C. Stwalley, A. Niehaus and D. R. Herschbach, J. Chem. Phys., 1969,51,2287. l2 V. Aquilanti, G. Liuti, F. Vecchio-Cattivi and G. G. Volpi, Chem. Phys. Letters, 1972,15, ?05. l 3 R. Gengenbach, Ch. Hahn and J. P. Toennies, Phys. Reu. Letters, 1972,7,98. l4 Details of the apparatus will be described in the dissertation by B. Lantzsch, 1973, Gottingen. l6 D. R. Olander and V. Kruger, J. Appl. Phys., 1970,41,2769. l7 Ch. Hahn, Max-Planck-Institut fur Stromungsforschung, Report No. 118/1972. l8 R. Gengenbach and Ch. Hahn, Chem. Phys. Letters, 1972,15,604 ; R. Gengenbach, J. Strunck l9 M. Rosen and D. R. Yennie, J. Math. Phys., 1964,5,1505. 2o F. von Busch, 2.Phys., 1966, 193, 412. 21 K. Walaschewski, Max-Planck-Institut fiir Stromungsforschung, Report No. 106/1971. 22 The primary beam distribution is given byf(u,) = I(vl), T(vl), where I(vl) is the Maxwell- Boltzmann distribution and T(vl) is the velocity selector transmission curve. Neglect of Z(ul) leads to significant errors at very high or very low velocities. 23 Minuit, CERN Computer Library D 506. 24 (a) R. J . Good and Ch. J. Hope, J. Chem. Phys., 1970,52,540. 2 5 M. J. Cantini, M. G. Dondi, G. Scoles and F. Torello, J. Chem. Phys., 1971, 56, 1946. 26 P. E. Siska, J. M. Parson, T. P. Schaefer and Y. T. Lee, J. Chem. Phys., 1971,55,5762. 27 J. M. Parson, P. E. Siska and Y. T. Lee, J. ChemPhys., 1972,56,1511. 28 M. Cavillini, M. G. Dondi, G. Scoles and U. Valbusa, Entropie, 1971, 42,136. 29 T. P. Schafer, private communication, 1972. 30 H. Pauly and J. P. Toennies in Methods of Experimental Physics (Academic Press, New York, 31 R. Diiren, G. P. Raabe and Ch. Schlier, 2. Phys., 1968, 214,410. 32 U. Buck, J. Chem. Phys., 1971,54, 1923. Manufactured by Brunswick Corp. Chicago, Ill. and J. P. Toennies, J. Chem. Phys., 1971,54, 1830. (b) G. H. Hudson and J. C. McCoubrey, Trans. Faraday SOC., 1960,56,761. 1968). F. CaIogero, Variable Phase Approach to Potential Scattering (Academic Press, New York, 1967). 34 The systematic errors in the absolute cross sections are the largest source of experimental error. To study their effect, a series of potential curves were calculated from the H-Ne data using different assumed values of the absolute cross section. In all cases the curves were found to be virtually unchanged and only horizontally shifted along the R-direction. The change in R, was in agreement with that expected from the simple formula mR&, namely dRm/R, N +do/., and in agreement with the high energy approximation (H. Pauly and J. P. Toennies, Adv. At. Mol. Phys., 1965, 1, 302).

ISSN:0301-7249    

DOI:10.1039/DC9735500167

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

Characterization of the Intermolecular Potential Well from Elastic Molecular Beam Scattering Data BY DONALD D. FITTS AND MARTHA LIAO LAW Dept. of Chemistry, University of Pennsylvania, Philadelphia, Pa. Received 29th December, 1972 The spacings of the glory extrema in the velocity dependence of the total elastic cross sections for atom-atom scattering are analyzed in terms of several recently-proposed analytical forms for the argon-argon and neon-neon pair potentials of intermolecular force. In order to subject these new potential forms to more stringent tests, we compare our calculations with experimental data on argon-argon scattering. Our findings are that a series of potentials developed by Barker and his collaborators to fit macroscopic properties of argon are reasonably consistent with the positions of the glory extrema, and that the so-called MSV potentials proposed by Parson, Siska and Lee to fit their differential scattering cross-section measurements on argon are somewhat less consistent. Elastic molecular-beam scattering data are now widely used to study the form of the intermolecular pair potential, Buck has outlined a procedure for inverting within the semiclassical WKB approximation the differential elastic scattering cross section at various energies to obtain directly the orientation-averaged pair potential V(r) over a range of interparticle separations r which includes the minimum in the potential well.In order to obtain a unique potential, this procedure requires some additional scattering data including the total elastic cross section.have applied this inversion to their high-resolution measurements for Na-Hg scattering with gratifying success. However, at the present time, for most systems sufficient molecular-beam scattering data are not available for such an inversion, so that one must consider procedures which utilize only incomplete data for studying Particularly useful observations in this connection are the extrema or “ glory undulations ” in the velocity dependence of the total cross section Q(u) for atom-atom scattering. These glory extrema occur when the pair potential V(r) has a minimum containing one or more bound states. The relationship between the spacing of these oscillations and the potential V(r) has been studied by Bernstein and O’B~ien.~ Their analysis relates the depth E of the attractive potential well and the interparticle separation r, at that minimum to an experimental “ glory observable” Z by the relation ~ r , = tiGI, where R is Planck‘s constant divided by 2n; and the “ glory constant ” G is dependent on the shape of the potential selected for the interpretation of the experimental data, but is independent of E and r,.Although it is well-established through applications to thermodynamic properties and to gaseous transport coefficients that the two- and three-parameter potential forms, such as the Lennard-Jones 12-6, are poor representations of the true pair potential for argon-argon and for neon-neon interactions, molecular beam data, including glory extrema, are often analyzed in terms of such restrictive analytical forms.Recently, several new, and presumably more accurate, representations of the argon-argon pair potential have been proposed in order to account quantitatively for the macroscopic properties of argon. Moreover, argon-argon and neon-neon Buck and Pauly V(r>. 179180 ELASTIC MOLECULAR BEAM SCATTERING DATA potentials have been developed to reproduce the corresponding experimental differential scattering cross sections. It is our purpose here to use these more realistic estimates for the potential form in the analysis of the glory extrema in the total elastic scattering cross sections. POTENTIAL FORMS The analytical forms for the intermolecular pair potentials V(r) that we wish to consider are all of the type where p is the reduced separation r/rm and u(p) is the shape of the potential function.The integrals which we need to compute in the analysis of the glory extrema depend on this shape factor u(p) rather than on the complete potential V(r). Attempts to account quantitatively for the thermodynamic properties of gaseous, liquid, and solid argon, for the transport properties of gaseous argon at low pressures, and for molecular-beam scattering and spectroscopic data on argon pairs have led to a series of proposed analytical forms for the true (as opposed to the effective) argon-argon pair potential V(r). Thus, Barker and Pompe (BP) and Bobetic and Barker (BB) proposed for the shape factor V(r) = w(r/rm) = EU(P), (1) M 2 Barker, Fisher and Watts or equivalently, where the subscripts indicate the respective authors of the potential forms.Fisher, and Watts potentials. Smith (BFW) then refined the potential to the form VBFw(r) = 0.75 VBB(r) + 0.25 vBp(r), UBFW(P) = 0.740 183 uBB(0.999 52 p) + 0.259 861 t&p( f .001 38 p), (3) (4) Barker, have conveniently tabulated the various parameters for all three In their analysis of the absorption spectrum of diatomic argon,8 Maitland and (MS) proposed a modification of the Bobetic-Barker potential : uMs(p) = uBB(p) + 0.025 exp [ - 50(p - 1 .33)2]. (5) This form is very close to the BFW shape factor in eqn (4). Parson, Siska and Lee lo have measured the differential cross section for argon- argon scattering at E = 10.0 x 10-14 erg and used this data along with experimental second-virial coefficients, the vibrational energy-level spacings for Ar2, and a theo- retical calculation of the dispersion energy to determine the parameters in a Morse- spline-van der Waals (MSV) potential shape, 2 4 m d P ) = exP [-2P(P-1)-2 exP [-p(P-1)1, for o<p<pl, = - C6pp6- c8p-8, for p2 < p < 00.} (6) = bl f (P - P 1)ib2 + (P - P2)Ib3 + (p - Pl)b41) 9 for P1 < P < P27 They proposed two different sets of values for the parameters /3, b l , b2, b3, b4, c69 and c,, giving two potential shapes which they call MSV I1 and MSV 111. Hanley et aZ.I2 have calculated the viscosity coefficients and second-virial co- efficients for argon with both the BFW and MSV I11 potentials. 1 They found that both potentials give results in reasonable agreement with experiment and that these experiments do not favour one potential form over the other.D.D. FITTS AND M . L . LAW 181 For neon, Siska, Parson, Schafer, and Lee l 3 evaluated the parameters in the MSV potential shape, eqn (6), to fit their measurements of the differential cross sections for neon-neon scattering at E = 9.83 x and 3.90 x erg. Barker l4 used this potential form to calculate the second-virial coefficients, viscosity coefficients, and thermal diffusion factor for gaseous neon. The calculations agree with the macroscopic data essentially within experimental error. With each of the potential shapes discussed here (BP, BB, BFW, MS, MSV 11, MSV 111 for argon and MSV for neon), there is a corresponding pair of values for E and r,, listed in table 2 for later use, as determined by the respective authors of the potential forms.These values for E and r, are not required in the analysis of the glory extrema; it is the purpose of the analysis when coupled with experiment to determine these parameters. If the values obtained from the glory extrema agree with the previously determined values, we may regard this as additional evidence in favour of the proposed potential. GLORY EXTREMA and Bernstein and Muckerman l 6 have reviewed the quantum- mechanical theory of elastic scattering relating to the glory extrema in the total cross section. A semiclassical treatment relates the indices N of the extrema to the maximum value, for that relative velocity uN, of the scattering phase shift ~ I , ( P ~ ) , The index N (N = 1, 1.5, 2, 2.5, . . .) is an integer for maxima in Q(u) and a half- integer for minima.Within the semiclassical analysis of scattering, the phase shift in the JWKB approximation may be expanded in a power series in 1/E to give (8) The wave number k, the relative velocity u, and the relative kinetic energy E are related by where p is the reduced mass of the interacting pair of atoms. The coefficients yl, y2, . . . depend only on the reduced classical turning point rc/rm = pc and on the potential shape u(p). The first two coefficients are l 7 Bernstein Um(UN) = ( N - %)n. (7) q/krm = E ~ ~ / E + E ~ ~ ~ / E ~ + E ~ ~ ~ / E ~ + . , . Rk = p~ = (2pE)*, (9) where x = rJr. Thus, from eqn (7)-(9)$ if the experimentally determined values of uN or EN are plotted as N - 2 against l / u N , the resulting curve should pass through the origin with initial slope Z equal to ~r,,,/tiG, where G is the glory constant of Bernstein and Mucker- man,16 pc being the value of pc for the glory trajectory (zero deflection).G = ZPYl(PG), (12) Alternatively, the182 ELASTIC MOLECULAR BEAM SCATTERING DATA data may be plotted as (N-$)o, against l/E, in which case the curve intercepts the ordinate axis at I = &r,/kG and has an initial slope S, equal to 2&2r,,32(&-)/7th. In the semiclassical theory of scattering, the maximum in the phase shift occurs when the classical scattering angle O(b,E) in the centre-of-mass coordinate system is zero for finite b (or rc), where b is the impact parameter.16 At sufficiently high energies, the classical scattering angle is given by l9 O(b, E ) N 8E-l [u(pc)-ti(pc/x)~(1-x2)-3 dx.(13) J: Therefore, in order to find pG, the value of pc at maximum phase shift, for a given potential shape u(p), we set the integral in eqn (13) equal to zero. The integral was evaluated for trial values of pc and the root pc of the equation was determined by an iterative inverse linear interpolation method known as the Secant Method.20 The potential-shape-dependent integrals Y1(PG) and y2(pG) were then evaluated and the glory constant G calculated from eqn (12). For potential forms with many parameters, these integrations are best performed numerically on a high-speed computer. Thus, the integrals in eqn (lo), (1 l), and (13) were evaluated by an Nth-order Gaussian quadrature formula,21 where N is an odd integer. The computer routine increased N from an initial value stepwise until two successive evaluations of an integral agreed within one part in 10'.The inte- grands diverge at both limits of integration, which occasionally resulted in computer overflow. Therefore, the lower limit was increased by 1 x 10-15 and the upper limit decreased by the same amount in the Gaussian quadrature integration procedure. In test calculations, this change in limits made no difference in the accuracy of the results. RESULTS AND DISCUSSION The calculated values of pG and the corresponding constants G, yl(pc), and YZ(PG) are given in table 1 for the potential shapes which we are studying. For comparison purposes, the corresponding results for the Lennard-Jones (LJ) 12-6 potential shape are also given.potential shape LJ 12-6 BPa BB b BFW C MS MSV I1 MSV 111 MSVf TABLE I .-CALCULATED GLORY INTEGRALS PG G Yl(P,) 0.947 13 3.726 2 0.421 56 argon 0.946 72 4.005 1 0.392 20 0.949 62 3.890 5 0.403 75 0.948 94 3.930 6 0.399 63 0.949 41 3.935 1 0.399 18 0.950 29 4.011 9 0.391 54 0.946 83 3.969 6 0.395 70 neon 0.951 98 4.082 3 0.384 78 YdPG) -0.166 44 -0.166 92 -0.167 32 -0.167 21 -0.167 47 -0.166 50 -0.166 21 -0.167 31 a J. A. Barker and A. Pompe, Austral. J. Chem., 1968,21,1683 ; b J. A. Barker and M. V. Bobetic, Phys. Rev. B., 1970,2,4169 ; C J. A. Barker, R. A. Fisher and R. 0. Watts, Mol. Phys., 1971, 21, 657 ; d G. C. Maitland and E. B. Smith, Mol. Phys., 1971,22, 861 ; e J. M. Parson, P. E. Siska and Y. T. Lee, J. Chem. Phys., 1972,56,1511; f P.E. Siska, J. M. Parson, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1971,55,5762.D. D . FITTS AND M. L . LAW 183 The glory constants G for the argon potentials are close in value as should be expected since their shapes are similar. The Lennard-Jones G differs significantly from any of the glory constants for these more realistic potentials. The coefficient 72(pG) of the second-order term is more constant than is yl(pG) for all these potentials, consistent with the original claim of Bernstein and O’Brien that Y&G) is nearly independent of the potential shape u(p). TABLE 2.-PARAMETERS OF THE PAIR POTENTIALS FOR ARGON er,k- lKA from glory extrema previous values a potential a E k - 1IK rmlA ermk- 1IKA Z = 1 840m/s Z = 1 8 80m/s BP 147.70 3.7560 554.76 563 575 BB 140.235 3.7630 527.70 547 559 BFW 142.095 3.7612 534.45 552 564 MS 142.5 3.75 534 553 565 MSV I1 144.4 3.715 536.4 564 577 MSV I11 140.75 3.760 529.2 55s 570 a references same as in table 1.Bernstein and O’Brien have suggested that both E and r, for an assumed potential shape be obtained from the experimentally determined glory undulations by making use of both the first-order term yl(pG) and the second-order term y2(pc) : ~ r , is determined from the intercept Z and ~ ~ r , from the initial slope Sl of a plot of ( N - Q)uN against 1/E. Buck 22 and Bernstein and L a B ~ d d e , ~ ~ however, have demonstrated that even small uncertainties in the experimental data will result in appreciable error in ~ ~ r , . We think it best, therefore, to use the second-order term Y2(pG) only to improve the fitting of the experimental data so as to ensure a more reliable value for Z and hence for &rm.At the present time, there are no glory data for neon-neon scattering. However, the total elastic cross sections Q(u) with resolved glory undulations for argon scattered by argon have been measured by Baratz and Andres 24 and by B r e d e ~ o u t . ~ ~ The velocities V, of the glory extrema for N = 1, 1.5, 2, 2.5 and 3 as determined by Bredewout 25 agree within 0.1 to 4.5 % with the corresponding values read from the graphical presentation of Baratz and and re^.^^ We evaluated the intercept I and the initial slope S1 by fitting Bredewout’s values of vN to the equation (N-Q)u, = Z+SIEN-l according to the method of least squares and found Z = 1840 m/s and S , = - 1.23 x A similar fit to the equation (N-8) vN = I+SIEN-l + S2EN-2 gives Z = 1880 m/s and S1 = - 3.13 x 10-l8 J m/s.Using these two values for I, we calculated the products EY, for the potential shapes that we are investigating and compare them in table 2 with the values obtained by the original authors of these potentials. This comparison may be regarded as an independent test of these potential forms, since total cross sections were not used by the original authors in their selection of parameter values. We note that the products ~ r , for the Barker-type potentials (BP, BB, BFW, MS) are in somewhat better agreement with the previously-determined values than are those for the MSV potentials. We also note that the discrepancy between the products ~ r , as determined from the glory extrema and as determined from macroscopic properties for a given Barker-type potential is comparable to the maximum discrepancy between Bredewout’s data and Baratz and Andres’ data.Thus, present indications are that the Barker-type potentials are reasonably consistent with the positions of the glory extrema, and that the MSV potentials are somewhat less consistent. J m/s.184 ELASTIC MOLECULAR BEAM SCATTERING DATA This work was supported by the Materials Research Laboratory Program of the National Science Foundation through the Laboratory for Research on the Structure o€ Matter at the University of Pennsylvania. * U. Buck, J. Chem. Phys., 1971,54, 1923. U. Buck and H. Pauly, J. Chem. Phys., 1971,54,1929.R. B. Bernstein and T. J. P. O’Brien, Disc. Faraday SOC., 1965, 40, 35; J. Chem. Phys., 1967, 46, 1208. D. D. Fitts, Ann. Rev. Phys. Chem., 1966, 17, 59. J. A. Barker and A. Pompe, Austral. J. Chem., 1968,21,1683. J. A. Barker and M . W. Bobetic, Phys. Reo. B, 1970, 2, 4169 ; see also J. A. Barker, M. V. Bobetic and A. Pompe, Mol. Phys., 1971, 20, 347 and J. A. Barker, M. L. Klein and M. V. Bobetic, Phys. Rev., B, 1970, 2,4176. ’ J. A. Barker, R. A. Fisher and R. 0. Watts, Mul. Phys., 1971,21,657. * Y. Tanaka and K. Yoshino, J. Chem. Phys., 1970,53,2012. G. C. Maitland and E. B. Smith, Mol. Phys., 1971,22,861. G. Starkschall and R. G. Gordon, J. Chem. Phys., 1971,54,663. lo J. M. Parson, P. E. Siska and Y. T. Lee, J. Chem. Phys., 1972,56,1511. l2 H. J. M. Hanley, J. A. Barker, J. M. Parson, Y. T. Lee and M. Klein, Mol. Phys., 1972,24, 1 1. l 3 P. E. Siska, J. M. Parson, T. P. Schafer, and Y. T. Lee, J. Chem. Phys., 1971,55,5762. l4 J. A. Barker, Chem. Phys. Letters, 1972,14,242. l5 R. B. Bernstein, Adv. Chem. Phys., 1966, 10, 75. l6 R. B. Bernstein and J. T. Muckerman, Adv. Chem. Phys., 1967,12,389. l7 F. T. Smith, J. Chem. Phys., 1965,42,2419. l8 E. F. Greene and E. A. Mason, J. Chem. Phys., 1972,57,2065. l9 E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, New York, 1938), p. 119. 2o A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965), p. 323-329. 21 A. H. Stroud and D. Secret, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, 22 U . Buck, J. Chem. Phys., 1972,57,578. 23 R. B. Bernstein and R. A. LaBudde, 1972, Report WIS-TCI-469 ; J. Chem. Phys., 1973, 58, 24 B. Baratz and R. P. Andres, J. Chem. Phys., 1970,52,6145. 2 5 J. W. Bredewout, private communication ; unpublished data of J. W. Bredewout and C. J. N. N. J., 1966). 1109. van den Meijdenberg.

ISSN:0301-7249    

DOI:10.1039/DC9735500179

出版商:RSC    

年代:1973

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. U. Buck, Dr. H. 0. Hoppe, Dr. F. Huisken and Prof. H. Pauly (Gottingen) said : We would like to give an example for a direct inversion of differential cross section data to the potential mentioned in the Introduction by Prof. J. P. Toennies. We have measured the differential cross section for the system Li-Hg in the energy range of 30-50 x 10-14 erg. There is a special reason to study this system. Early attempts with relatively restricted input data have given potential parameters which vary over wide ranges. Furthermore a potential derived by Olson from glory scattering data and rapid oscillations from the other alkali mercury potentials obtained by inver~ion.~ In order to resolve the rainbow structure, an experimental arrangement has been used which provides a velocity resolution of about 3 %.Both beams have been analysed by self calibrating velocity selectors so that the absolute value of the relative velocity is determined within 1 %. Fig. 1 shows the experimental results. The quantity I(0) which is proportional to the differential cross section, multiplied by sine, is plotted versus the deflection differs significantly 8 - - 6 - a c .- V I - h m W CI 4 - 2 - I ‘1 FIG. 1.-Measured differential cross section for Li on Hg in centre of mass coordinates for E = 48.9 x 10-14 erg. U. Buck, J. Chem. Phys., 1971, 54, 1923. R. Olson, J. Chem. Phys., 1968, 49,4499. W. Stwalley, J . Chem. Phys., 1971, 55, 170. U. Buck and H. Pauly, J. Chem. Phys., 1971, 54, 1929; U. Buck, M. Kick and H. Pauly, J. Chem. Phys., 1972,56, 3391.185186 GENERAL DISCUSSION angle 8 in the centre of mass system. Four rainbow extrema with well resolved rapid oscillations over the whole angular range are displayed. These data, taken at six different energies, have now been used to obtain the potential by direct inversion in a unique way. In the first step the deflection function is constructed using the following input data : the positions of the rainbow extrema, the positions of the rapid oscillation, the maximum phase shifts from the glory scattering (which agree very well with our extrapolated values from the supernumerary rainbows), the backward scattering near the first rainbow maximum, the calculated van der Waals constant. In the second step, the potential is determined by direct integration over the weighted deflection function.2 The results are shown in fig.2. The potential resulting from each of the different energies is the same to within the experimental error. The minimum distance r, and the corresponding well depth are 3.00A and 16.9 x erg, respectively. FIG. 2.--PotentiaIs for LiHg obtained by the inversion of differential cross section measured at different energies. Our accurate results confirm the potential of Olson. This means that the potential of Li-Hg differs significantly from the other alkali-mercury potentials in contrast to the behaviour, for instance, of the rare gas potentials. Both size parameters and the form are different. Another interesting feature is that the contribution of the C s r 8 term for Li-Hg is small compared with the other systems.Dr. R. Gengenbach, Prof. J. P. Toennies, Dr. W. Welz and Dr. G. Wolf (Gottingen) said : As pointed out in the paper by Bickes et a!. the value for the well depth E for the W. C. Stwalley and K. L. Gamer, J. Chem. Phys., 1968, 48, 5555. U. Buck, J. Chem. Phys., 1971,54, 1923.GENERAL DISCUSSION 187 system H-He reported there is not very precise because of the relatively large beam energies used in these experiments. In a computational study of the effect of various &-values on the low velocity (< 1000 m/s) integral cross section a Ramsauer-Townsend effect was found to occur for a certain range of eva1ues.l To measure the low velocity integral cross section we have developed a liquid N, cooled r.f. discharge source to produce a " cold " H-atom beam. With this source we have been able to extend the velocity range down to 400 m/s (I&,,, = 0.75 mev).A liquid He cooled ( 5 K) scattering chamber was used to reduce the target motion. Fig. 1 shows the measured cross section for H-He and H,-He. The decrease in the H-He cross section at low velocities provides the first clear evidence for a Ramsauer-Townsend effect in atom-atom scattering., It is not observed for H,-He, where the measured points are well fitted by the calculated curve based on a potential reported earlier.3 '60 N 5 . - tz EcmlmeV I 2 3 5 10 20 30 50 (H2) I I I I I I 1 2 5 10 . 2'0 3 o ( H ) \ H -He ---- H,-He 80 '\ - 40 L I I I I ! I , I I LOO 600 800 1000 2000 3000 vl/m s-' FIG. 1 .-Measured integral cross sections for H-He (a) and H,-He (+) at 5 K target temperatuie as a function of the primary beam velocity.Comparison is made to calculated effective cross sections based on the potential model given in ref. (3) for H2-He and in ref. (1) for H-He with E = 0.46 meV. The horizontal bars at the bottom indicate the full half width of the distribution of relative velocities. - The H-He measurements can be fitted by two MSV potentials with identical repulsive regions but different wells characterized by E = 0.46 and 0.69 meV. The system D-He, for which a minimum only occurs for E = 0.46, has also been studied. A minimum was observed so that all the measurements are consistent only with E = 0.46. Prof. V. Aquilanti, Prof. G. Liuti, Dr. F. Vecchio-Cattivi and Prof. G. G. Volpi (Perugia) said: Some time ago, we measured the absolute total cross sections for elastic scattering of H atoms with Ar, Kr and Xe,4 in a velocity range also covered R.Gengenbach, Ch. Hahn and J. P. Toennies, Phys. Rev. A, 1973, 7, 98. A Ramsauer-Townsend effect has also recently been observed for He4-He4 by R. Feltgen, H. Pauly, F. Torello and H. Vehmeyer, Phys. Rev. Letters, to be published. V. Aquilanti, G. Liuti, F. Vecchio-Cattivi and G. G. Volpi, Chern. Phys. Letrers, 1972, 15, 305. ' R. Gengenbach and Ch. Hahn, Chem. Phys. Letters, 1972, 15,604.188 GENERAL DISCUSSION by the more extensive measurements of the Gottingen gr0up.I In this range, given a potential model, the velocity dependence of cross sections yields the product EY,, while their absolute values allow the estimate of r, : our analysis in terms of potentials of the Lennard-Jones family leads to apparently good fits of the data, although it fails to reproduce the known long range behaviour.Therefore it is not surprising that Toennies' analysis in terms of a potential which is similar to a Lennard-Jones (12,6), via the ad hoc assumption of proper long range behaviour by a set of combining 35 3.0 f I I I I 1000 2000 velocity/m s-' FIG. 1 .-Dependence of cross section Q on velocity for O,-Ar and 0-Ar collisions, as against log 0. The solid lines have been calculated for a Lennard-Jones (12,6) potential with the parameters shown and for a scattering chamber temperature of 77 K. rules, yields a good fit to the velocity dependence of moss sections but predicts absolute values in disagreement with our measured ones. Notwithstanding these gross drawbacks of simple model potentials in describing these systems, we must realize that the experiments with such unstable species as hydrogen atoms appear to be still in a stage which prevents the use of more sophisticated analysis.Introduction of more flexible potential forms, with additional parameters, must presumably wait for improvements in the molecular beam techniques. The present situation resembles closely that of experiments with alkali atoms a dozen years ago : of course, differ- ential measurements and/or detection of quantum effects would help considerably. R W. Bickes, Jr., B. Lantzsch, J. P. Taennies and K. Walaschewski, this Discussion.GENERAL DISCUSSION 189 In this connection, we present in fig.1 some results on the scattering of oxygen atoms and molecules by argon. The apparatus is the same as described earlier ; mass spectrometric detection allowed the simultaneous measurement of cross section for 0 and 02, and the relative scale of the data plotted in the figure for the two systems is therefore accurate. Although the potential parameters given in the figure, as well as the calibration of the ordinate scale, rely on a preliminary analysis of the data, the glory undulations superimposed on a v-2/5 dependence on velocity are clearly seen. Dr. K. P. Lawley (Edinburgh University) said: We have just had three examples of elastic scattering involving atoms that are not in an S state; O(3P2,1,0) (Volpi), I(2Ps) (Lee) and Hg*(3P2) (Davidson et al.).In such cases several potentials evolve from the separated atoms if J and m, are not selected. In molecular spectroscopic terms these states can be labelled by their A and R values. If, as in the case of heavier atoms, the J state is known, the number of molecular states that are adiabatically accessible is much reduced and they probably have a common A value. If the atomic partner is in a IS state there will be 2J- 1 such potential curves correlating with a given J state. It is therefore interesting that in all three examples quoted above structure was observed in either the total or differential cross section. The simplest deduction is that the potentials of states that differ only in R-and apparently, in the case of 0, in A-are rather similar.In the case of scattering from excited states (Hg*) there is however, the possibility that selective quenching might be responsible for some simplification of the scattering pattern. One must in any case remember that in these thermal energy experiments the repulsive (i.e., negative) branch of the potential is not really being sampled and it is just this region that a larger splitting of the manifold of potentials might be expected. Dr. J. F. Ogilvie and Mr. R. W. Davis (Memorial University of Newfoundland) said : We have been able to deduce some generalisations about the nature of the potential energy function, in the region of the energy minimum, between very weakly bound atoms. We have transformed other published potentials, derived from molecular beam and spectroscopic experiments and from thermal and transport properties, into the Dunham form : (UlD,) = abx2[1+ aid].Here U = potential energy, D, = depth of potential well, x = (R-Re)/&, R = instantaneous internuclear separation, and Re = separation at the energy minimum ; a; and a, ( i > 1) are parameters adjusted to fit the data for the potential function, from whatever source. This Dunham function can be used to express a real potential, smoothly varying in the region of interest, to the required accuracy in a given range of X by employing sufficient terms in the summation. discovered useful generalisations for strongly bound diatomic molecules in their ground states. For hydrides XH and non-hydrides XY, a1 and a, in the Dunham function were found to be relatively constant within these classes of molecules. We add two further classes M2 and A2 which consist of alkaline- i = 1 Calder and Ruedenberg ' V. Aquilanti, G. Liuti, E. Luzzatti, F. Vecchio-Cattivi and G. G. Volpi, 2. phys. Chem., 1972, 79,200. J. L. Dunham, Phys. ReLl., 1932, 41, 721. G. V. Calder and K. Ruedenberg, J. Chem. Phys., 1968,49, 5399.1 90 GENERAL DISCUSSION earth dimers Mg, and Ca,, and noble-gas dimers He,, Ne,, Ar, and Kr2, respectively. For these molecules, 92 < D, < 1 1 OOO J mol-l. The values are listed in the table : XH XY Mz Az - 01 2.4+ 0.1 3.2k0.1 4.9+ 0.2 7& 0.5 a2 3.8+_0.2 6.4+ 0.75 18 26+4 The A2 molecules are formed from closed-shell atoms ; therefore the values obtained for them may be appropriate and transferable to the non-directional part of the interaction between ordinary non-polar molecules. We believe these new values may find application in the study of intermolecular forces, in particular possibly in the construction of empirical potential surfaces for elastic and inelastic collisions. Fur- ther details of the results will be published separately.

ISSN:0301-7249    

DOI:10.1039/DC9735500185

出版商:RSC    

年代:1973

数据来源: RSC