摘要:

LATTICE CALCULATIONS ON POINT IMPERFECTIONS IN THE ALKALI HALIDES BY F. G. FUMI AND M. P. TOSI Cattedra di Fisica Teorica dell' UniversitA, Palermo, Italy Received 3rd June, 1 9 5 7 The Born-Mayer model of an alkali halide crystal has been applied by several authors to the calculation of the energies involved in the formation, motion and association of point imperfections. The available calculations concerning vacancies and impurities in NaCl and KCl crystals are critically discussed, and new results are reported for the formation energy of Schottky defects, for the binding energy of vacancy pairs and for the relative heats of solution of NaCl and KCI crystals. 1. INTRODUCTION The alkali halides represent an ideal substance for a quantitative study of point imperfections, both experimentally and theoretically.Experimentally a variety of techniques is available, and theoretically a very simple model has been set up, the Born-Mayer model, which reproduces the properties of the perfect alkali halide crystals to a remarkable degree. The aim of the calculations dis- cussed in this paper is essentially to see how well the Born-Mayer model accounts for the properties of point imperfections in the alkali halide crystals, and how much it helps to progress in their quantitative understanding. The first important attempts to deal theoretically with point imperfections in the alkali halides are due to Jost 1 and Schottky,2 a few years after Frenkel3 had introduced the concept of point imperfection. Jost attempted to evaluate both the energy to form point defects, namely the energy which determines their equi- librium concentration at a given temperature, and the energy to move point defects, which determines their mobility.Jost dealt with vacancies and interstitials, while we shall consider vacancies and impurities, the dominant equilibrium point defects in the alkali halides. 2. THE ENERGY OF FORMATION OF A SCHOTTKY DEFECT Jost noted that the energy Ef of formation of a Schottky defect is equal to the difference of two energies, viz., the lattice energy W,, which is necessary to form a Schottky defect in a rigid, non-polarizable lattice, bound by central forces, and the polarization energy, the polarization of the lattice being caused by the fact that a vacancy is equivalent in its effects at some distance to a point charge of the opposite sign of the ion removed.Jost estimates the polarization energy as the difference in energy of the electric field surrounding a spherical charge of radius R, when the charge is in a medium of dielectric constant K and in vacuu. Thus Jost and Schottky write the formation energy Ef as where r, and r, are the radii of the anion and cation vacancies. If one takes in a 1 * 7 4 8 e 2 ( a> , r0 , first approximation, W, = - 1 - - 9 0 r = r = - r0 92F . G . FUMI AND M. P . TOSI 93 one has El-"" r0 (1.748 (1 - t> - (1 - d ) - 0 . While such a rough estimate cannot obviously give a quantitative value for Ef, it allows one to understand why Ef is so much smaller than W,. The important step forward in making the estimate of Ej.quantitative was taken by Mott and Littleton 4 and consists mainly in a better treatment of the polariza- tion energy.Let us consider for definiteness a positive-ion vacancy. Mott and e2 1 r0 Littleton write the electrostatic part of the work of extraction as MM- + ?e@, where @ is the potential caused in the position of the positive ion extracted by the polarization of the medium, which in the Jost treatment is simply taken equal to - :( 1 - k). They then compute @ by considering the discontinuous nature of the crystal. More precisely, they evaluate the displacement 4ro of the immediate neighbours of the vacancy by an actual balance of the electrostatic and repulsive forces acting on them, and their electronic dipole mero by considering the electro- static field which acts on them, owing to the added charge and to the polar- ization of the medium.In writing down the repulsive force Mott and Littleton consider only the repulsive interaction between nearest neighbours : they determine the constant A of the repulsive potential, A exp (- r/p), through the condition that the crystal is at equilibrium at the observed lattice spacing, and they use for the constant p the value obtained from the measured compressibility of the crystal. For the other ions of the crystal, Mott and Littleton evaluate the electronic and displacement dipoles by treating the crystal as a continuum, and attributing to each species of ion a proportional part of the dipole moment per unit volume or polarization P = (p+ + p-)/2r03. Thus one has rC (2.3) where a+ and cc- are the electronic polarizabilities of the cation and anion, and a is the displacement polarizability, which Mott and Littleton take from the difference between the static and high-frequency dielectric constants of the crystal.@ is then evaluated by appropriate lattice summations ; one finds 6m + 6.3346 M'+ + 4.1977 M' The repulsive contribution to the work of extraction of an ion is computed, instead, by calculating the work to be done against the repulsive forces to put the ion back into the distorted configuration of the lattice. This yields -- (: + i f ) exp (- Y) . r0 It turns out that the Mott procedure to calculate the work of extraction is essentially equivalent to taking this work as equal to the negative average of the potential energy at the position of the ion to be extracted in the initial undistorted configura- tion, and in the final polarized configuration-a physically reasonable assumption.Two relevant criticisms can be made of the Mott approach, namely, that it neglects the elastic distortion of the crystal around the defect, and that it does not consider explicitly the repulsive interaction between non-nearest neighbours.94 CALCULATIONS ON POINT IMPERFECTIONS In Mott’s treatment the nearest neighbours of the vacancy are allowed to displace outwardly because they bear the same charge of the vacancy, and it is reasonable to expect that the surrounding lattice will undergo an outward expansion as a result of these displacements. Mott and Littleton 4 neglected this effect in their calculation, but Brauer 5 has shown that it can be included in Mott’s treatment.Similarly, the repulsion between non-nearest neighbours can be easily included in Mott’s treatment (Bassani and Fumi 6). The values that one finds for the different contributions to the energy to form a Schottky defect in NaCI, by including both these corrections, are listed in table 1 and compared with the values given by the Mott and Littleton approximation and with those found by Brauer,S who included the elastic correction but used TABLE FORMATION ENERGY OF A SCHOITKY DEFECT IN NaCl (IN eV> present work B.M. repulsion H.M. repulsion Mott and Littleton Brauer positive-ion vacancy 8.94 8.94 8.94 8.94 e2 U M - YO j e t - 3.55 - 3.56 - 3.43 - 3.42 repulsive - 0.75 - 0.72 - 0.58 - 0.72 negative-ion vacancy 8.94 8.94 8.94 8.94 e2 aM- r0 - $e@ - 3.04 - 2.87 - 3.17 - 2.90 repulsive - 0.72 - 0.70 - 0.88 - 0.90 formation energy 4.64 + 5.18 4-66 + 5-37 4.93 + 4.89 4-80 + 5.14 - 7.82 = 2.12 - 7.89 = 1.93 - 7.92 5 2 1 1 - 7.91 = 1.91 the Mott and Littleton repulsive potential.We have used two different forms for the repulsive potential, the form originally proposed by Born and Mayer,7,6 and the somewhat more accurate form proposed by Huggins and Mayer : 8 in the latter, only one of the four parameters which enter the repulsive interaction energy between a pair of ions, buexp ( - *) exp (- b). is fixed arbitrarily, namely by, while in the former two parameters, the ionic radii r+ and r-, are fixed more or less arbitrarily equal to the Goldschmidt’s radii.We have also consistently calculated the displacement polarizability a from the assumed repulsive potential.6 For the ionic polarizabilities we have adopted the values reported by Shockley, Tessman and Kahn9 for ions in crystal lattices, and not the Pauling polariza- bilities for free ions ; however, this change is of little consequence for the problem at hand. The different theoretical values for the formation energy of a Schottky defect in NaCl and KCl are collected in table 2, and compared with the available experi- mental values : for KC1 the reported values range from 2-1 eV 10 to 2-4 eV,11 while for NaCl the best value seems to be 2.02 eV.12 These experimental values are probably accurate to about 0-2 eV owing to an uncertainty of about 0-1 eV in the experimental migration energy for the positive-ion vacancy.12 The values 2-14 eV r + rJ TABLE 2.-FORMATION ENERGIES OF SCHOTTKY DEFECTS IN NaCl AND KCI (IN ev) present work B.M.repulsion H.M. repulsion experimental Mott and Littleton Brauer NaCl K C1 1.93 2.1 1 1-91 2.12 2.02 2.14 2.26 2-18 2.2 1 2.1 -2.4F . G. FUMI AND M . P . TOSI 95 for KCl and 1.93 eV for NaCl, quoted as given by the Mott and Littleton ap- proximation, are actually somewhat bigger than the values reported originally by Mott and Littleton (2.08 eV for KCl, 1.86 eV for NaC1). The main difference is due to the fact that Mott and Littleton took for the lattice energy WL the value computed by Mayer and Helmholz,7 which includes the van der Waals and zero-point energies that are neglected in calculating the work of extraction.Of course, Mott and Littleton also used the old values for the electronic charge and lattice parameter. The values found by Brauer 5 have also been slightly modified to eliminate minor inaccuracies in the values used for the lattice energy. It is satisfactory to note that the theoretically more reliable values for the formation energies based on the B.M. and H.M. potentials are in equally good agreement with experiment as the Mott and Littleton and Brauer values. The main point is that the B.M. and H.M. potentials allow a bigger energy gain from the " electrostatic " polarization of the lattice around the negative-ion vacancy, but this gain is balanced by a bigger expenditure of elastic energy. 3. THE ENERGY OF MIGRATION OF A VACANCY To calculate the energy Em of migration of a vacancy, say a positive-ion vacancy, we have to pull out a positive ion from a given crystal configuration (a perfect crystal with one positive-ion vacancy) and put it back, in the saddle-point position, in another crystal configuration (a perfect crystal with two neighbouring positive-ion vacancies).Thus we must evaluate the difference in total energy between the saddle-point and equilibrium configurations, or compute the work to create a positive-ion vacancy next to another one, and the work to introduce a positive ion between them. Jost and Nehlep 1 estimate the migration energy of a positive-ion (or negative- ion) vacancy as the difference in coulombic and repulsive energies between the saddle-point and equilibrium configurations.They write the coulombic energy difference as (-& (3.1) where the first term gives the interaction energy between the moving ion in the saddle-point position and the two neighbouring vacancies, and the other two terms give the interaction energy between the moving ion in its equilibrium posi- tion and the rest of the lattice, which is perfect except for a neighbouring vacancy. Similarly, Jost and Nehlep 1 estimate the repulsive energy difference as [i (5) - 11 f 1.748 e2 = 1.27 e2 - , r0 (3.2) where the first term gives the interaction energy between the moving ion in the saddle-point position and the two nearest ions, and the second term gives the interaction energy between the moving ion in its equilibrium position and its six nearest neighbours. If they allow for relaxation of the two immediate neighbours of the moving ion in the saddle-point position, Jost and Nehlep find that the energy of the configuration decreases by the maximum value (-- 0.9 &/yo) when the ions are displaced outwards by -0.16 ro.This relaxation energy is sufficient to cancel entirely the migration barrier. Clearly this order-of-magnitude estimate cannot give a quantitative answer. It simply indicates that, as one should expect, an important contribution to the migration barrier will be given by the difference in polarization energy between the saddle-point and equilibrium configurations, that Jost and Nehlep did not include. This difference will also bring about a distinction between the positive- and the negative-ion vacancy. This expectation is confirmed by the calculations96 CALCULATIONS ON POINT IMPERFECTIONS of Mott and Littleton,4 who estimated the migration barrier for the positive-ion and the negative-ion vacancies in NaCl, including the polarization energy.The values they find are not zero, but about 3 eV (0.51 eV for the positive ion, 0.56 eV for the negative ion), still too small and far too close to each other. This is likely due to the fact that Mott and Littleton, as Jost and Nehlep, consider only the repulsion between immediate neighbours. It is also probable 13 that the repulsive energy at distances appreciably smaller than the equilibrium distance is sizeably larger than that estimated from the equilibrium form of the repulsive potential.A refined calculation is currently in progress at Palermo. 4. INTERACTION BETWEEN POINT IMPERFECTIONS Theoretical work on the interaction energies- between point imperfections in the alkali halide crystals was first undertaken in America several years ago at the suggestion of Prof. Seitz by Dienes 14 and by Reitz and Gammells and has recently been continued in Italy by Fumi, Bassani and Tosi.6, 16.16~ In this type of work the results given by the Born-Mayer model should not be taken literally, but must be subjected to careful evaluation. Positive-ion vacancies and negative-ion vacancies bear opposite effective charges in the alkali halide lattice, and one is thus led to expect that they will attract each other : the same applies to divalent impurity cations and positive-ion vacancies.One might also expect a priori that the binding energy between a positive-ion vacancy and a negative-ion vacancy, or between a positive-ion vacancy and a divalent impurity cation, at the closest lattice distance of approach will be smaller than their Coulombic interaction energy.16 Indeed, when we create a positive-ion vacancy in the presence of another defect, such as a negative-ion vacancy (or a divalent impurity cation), the polarization of the crystal between the corresponding lattice sites decreases the work of extraction of the positive ion, while the polariza- tion of the crystal outside these sites increases the work of extraction. In the limit of the smallest lattice distance we have only the second contribution. It follows that the binding energy of the two defects at the smallest lattice distance of approach will be smaller than the corresponding Coulombic value, since at sufficiently large separation it must have a Coulombic value.However, it would clearly be impossible by reasoning of this sort to guess either the magnitude of the decrease in binding energy, or the rate at which the binding energy approaches a Coulombic value. The Born-Mayer model allows one to state with confidence that the binding energy of the “ ground state ” of the vacancy pair and of the complex between a divalent impurity cation and a positive-ion vacancy is smaller than the Coulombic value by several tenths of an electron volt, and that the binding energy of these defects approaches quickly a Coulombic value as their separation increases.More specifically, the binding energies of the ground states of the impurity complexes are found to range between 0.32 and 0.45 eV,6 while the Coulombic values are 0.64 eV and 0.69 eV in NaCl and KC1 respectively. Similarly the binding energy of the ground state of the vacancy pair is found to be 0.60 eV in NaCl and 0-72 eV in KC1, against the Coulombic values 0.91 eV and 0.98 eV. The detailed cal- culation of these values for the vacancy pair will be discussed in a separate paper : 16a here, it should only be noted that they are somewhat at variance with the values reported earlier,l6 which were based on Pauling’s polarizabilities and, for KCl, on Dienes’ distortion around the vacancy pair 14 which turned out to be incorrect. Two other results obtained for the impurity complex by the Born-Mayer model 6 seem significant.Indeed the slight increase in the theoretical binding energy of the ground state with the radius of the impurity cation is confirmed quali- tatively by the measurements of Kelting and Witt,ll who find that the ionic con- ductivity of a KC1 crystal containing a given concentration of divalent impurity cations decreases with increasing radius of the impurity cation. Similarly theF. G. F U M I A N D M. P. T O S l 97 measurements by Pick and Weber 17 on the density of KC1 crystals doped with Ca2+ and Sr2t ions confirm the theoretical results6 on the distortion around the impurity and the vacancy, also in their relative magnitude : the contraction around the impurity wins over the dilatation around the vacancy, and the net effect is bigger for Caz.1- ions than for Sr2f ions, It would be illuminating to have similar measurements performed on NaCl crystals doped with Ca2T and Sr2r ions.The specific numerical values for the binding energies of the ground states of the vacancy pair and of the impurity complex should be qualified. They have been obtained neglecting the elastic correction proposed by Brauer,s which would be quite difficult to treat accurately in these cases. However, the fact that the elastic interaction energy between two rigid atomic spheres of radii rl and r2 replacing two atomic spheres of radius ro in an elastic isotropic medium is almost zero for significant differences in radii 18 suggests that this correction ought not to be important.5. SOLUBILITY OF IMPURITIES In recent years Brauer 19 and Lidiard 20 have attempted with little success to calculate the heat of solution of various divalent impurity cations in the alkali halide crystals using the Born-Mayer model. A minor error in Lidiard's calcula- tions was corrected by Tosi,21 who studied also the effect on the heat of solution of the elastic correction proposed by Brauer 5 which had been neglected by Lidiard,Zo and which in this case is of importance as one must deal with the potential energy at the site of the divalent impurity. In most cases the " theoretical" heats of solution are found to be exothermic, while these processes are known to be endothermic.22 The unavoidable difficulty of this type of calculations, which reduces greatly their accuracy,23 is the fact that one must use the experimental lattice energy of the halide salts of the divalent cations since these salts cannot be treated properly by the Born-Mayer model.This difficulty does not arise in the calculation of the heat of solution of some monovalent impurity cations (Tl+ in NaCl, KC1; 5 Naf in KCl, K-: in NaCl), and in effect such calculations give quite reasonable results as table 3 indicates : for the heat of solution of Ti+ ions in NaCl and KCl crystals, Brauer 5 reports values of 0.83 eV and 0.31 eV respec- tively. In his recent review article, Lidiard 22 has fully subscribed to these views. TABLE 3.-HEAT OF SOLUTION OF MONOVALENT CATIONS IN NaCl AND KCl (IN ev) * hi -Y1 Yz X i K+ in NaCl 1 *07 - 7.91 7.09 0-25 Na+ in KC1 - 0.62 - 7-09 7.9 I 0.20 * The heat of solution of a monovalent impurity cation in NaCl at 0" K is given by X i = hi - 71 + y2.Here y1 and y2 are the lattice energies of NaCl and of the chloride salt of the impurity cation, and hi is the energy involved in replacing one Na+ ion in NaCl by an impurity cation. 1 Jost, J. Chem. Physics, 1933,1,466. Jost and Nehlep, Zphysik. Chem. B, 1936,32,1. 3 Frenkel, 2. Physik, 1926, 35, 652. 4 Mott and Littleton, Trans. Faraduy SOC., 1938,34, 485. Schottky, 2. physik. Chern. B, 1935,29, 335. Brauer, 2. Nafurforsch. A, 1952, 7, 372. Bassani and Fumi, I1 Nuovo Citnenfo, 1954, 11, 274. 7 Born and Mayer, 2. Physik, 1932, 75, 1. 75, 19. 8 Huggins and Mayer, J. Chem. Physics, 1933, 1, 643. 9 Tessman, Kahn and Shockley, Physic. Rev., 1953,92, 890. 10 Wagner and Hantelmann, J. Chem. Physics, 1950,18,72. Mayer and Helmholz, 2. Physik, 1932, D98 CALCULATIONS ON POINT IMPEKFEC'TIONS 11 Kelting and Witt, 2. Physik, 1949, 126, 697. Etzel and Maurer, J . Chem. Physics, 1950, 18, 1003. The authors are indebted to Professor Maurer for very informative correspondence on the available experi- mental data on the formation and migration energies of vacancies in the alkali halide crystals. 13 Fumi, Phil. Mag., 1955, 46, 1007. 14 Dienes, J. Chem. Physics, 1948, 16, 620. 15 Reitz and Gammel, J. Chem. Physics, 1951, 19, 894. 16 Bassani and Fumi, Sitppl. Nuovo Cimento, 1955, 1, 114. 16a Tosi and Fumi, to appear in I1 Nuovo Cimento. 17 Pick and Weber, 2. Physik, 1950, 128,409. IsFriedel, Advances in Physics, 1954, 3, 446; appendix, section 1.3. Eshelby, Actu Metallurgica, 1955,3,487. Teltow, Halbleiterproblenze, Band 111, Vieweg, Braunsch- weig, 1956 : section 2.2. 19 Brauer, Z. Naturforsch. A , 1952, 7, 741. 20 Lidiard, Notes of Lectures on Lattice Defects in Ionic Crystals at the University of 21 Tosi, unpublished results quoted by Lidiard, ref. (22). 22 Lidiard, Handbuch der Physik, vol. XX (Springer Verlag, Berlin, 1957), pp. 292-295. 23 Fumi, private correspondence with A. B. Lidiard. Milan, October, 1954, pp. 11-15 ; and unpublished work.

ISSN:0366-9033    

DOI:10.1039/DF9572300092

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DIFFUSION. COEFFICIENTS IN SOLIDS, THEIR MEASUREMENT AND SIGNIFICANCE BY J. CRANK Courtaulds Ltd., Research Laboratory, Maidenhead, Berks Received 21st January, 1957 The standard definitions and methods of measuring diffusion coefficients in solids are briefly reviewed. Assumptions underlying the simple formulae which have been proposed to relate diffusion coefficients measured by different methods are critically discussed in the light of experimental evidence and theoretical considerations which suggest that modification may be necessary. Difficulties in the way of deciding between several alternative mechanisms for diffusion, and of fully understanding how measured diffusion coefficients are related to the atomic jumps constituting diffusion, are formulated. 1. INTRODUCTION The ultimate aim of a theory of diffusion in solids is to interpret measured diffusion rates in terms of lattice parameters and molecular properties with a view to understanding the basic processes involved.Such a theory requires as its foundation accurate and reproducible measurements of diffusion rates in a large number of systems. The value of the experimental results depends largely on their correct phenomenological interpretation and on the separation of the diffusion coefficients into their physically significant parts. Different values can be obtained for diffusion coefficients in one and the same system according to the method of measurement used. Theoretical considerations embodying both thermo- dynamic and mechanistic arguments suggest reasons why this should be so, but critical examination of these arguments casts doubts on some of the underlying assumptions.These doubts are to some extent strengthened by the fact that diffusion coefficients measured by different methods are not always related by the simple formulae proposed. The aim of this paper is not to survey the whole field of the measurement and interpretation of diffusion coefficients in solids, which would be quite superfluous in view of the excellent reviews by le Claire,l but to illustrate the sort of problems of which leading workers are very conscious and are actively exploring at the present time. 2. DEFINITIONS Experimental techniques aim to measure a diffusion coefficient D defined by (1) where c is the concentration of the diffusing species.This definition was prompted by the work of Fourier on heat flow and by early studies of the dif- fusion of low molecular weight solutes in dilute solution. It is based on the con- cept of diffusion in which net transfer results from random molecular motions where any one molecule is as likely to jump to the right as to the left. Certainly as early as Willard Gibbs, however, came the idea that diffusion rates are more fundamentally related to gradients of chemical potential than of concentration. A system is in equiIibrium only when the chemical potential of each component is uniform throughout the system and flow takes place from regions of high to regions of low chemical potential, even though this may sometimes be up a concentration gradient as in up-hill diffusion.2 The use of chemical potential, 99 rate of transfer = J =- Ddcldx,100 DIFFUSION COEFFICIENTS I N SOLIDS however, implies a net force on any one molecule acting in one preferred direction, so that we are extending the meaning of the term diffusion to include a directional flow of molecules under the action of this force as well as a transfer by random motions.If now we write the rate of transfer Ji of atoms of species i as where N is Avogadro’s number, ni is the number of atoms per cm3, Gi is the mobility, defined as the velocity acquired under the action of unit force, and /J-i is chemical potential defined in the usual way, all referring to species i, it is easily shown that the diffusion coefficient defined by (1) is given by where we have written pi = constant + RT log YiNi, (4) in terms of the activity coefficient yi, and Ni is the mole fraction of the ith species.Also Eizi is assumed constant. The diffusion coefficient is thus expressible as the product of a mobility Gi and an activity or thermodynamic term. It is the mobility which is to be related to lattice parameters, jump frequency etc. 3. METHODS OF MEASUREMENT (a) INTER-DIFFUSION EXPERIMENTS The usual procedure for determining the diffusion coefficient of one metal in another, for example, is to place the two metals or alloys in contact and after a suitable time to observe the distribution of chemical composition in a direction perpendicular to the original interface between the two species. The concentration- distance curve is then analysed by the Matano-Boltzmann procedure to obtain a mutual diffusion coefficient which is the same for each metal. But we expect, in general, that different metals will have different diffusion coefficients.This dficulty is resolved by the observation of Smigelskas and Kirkendall3 that in addition to the transfer of A and B molecules in opposite directions by diffusion relative to the lattice, there is a mass flow of the lattice itself, detectable by marker particles. Hence molecules of A are transferred partly by diffusion relative to the lattice and partly by being carried along with the lattice. In a system for which the lattice structure is unchanged by diffusion the net transfers of A and B molecules are equal and opposite, and it is this common transfer that is measured by the mutual diffusion coefficient, D.When these ideas are expressed mathe- matically 4.5 we find, after making certain assumptions, that ( 5 ) D = NADB + NBDA, where DAY D, are the individual diffusion coefficients expressing the transfer of A and B relative to the lattice, while D relates to transfer of A or B relative to the ends of the system not disturbed by diffusion. In order to determine DA and DB separately, we need another observation such as that of the velocity v of the mass flow where (b) TRACER EXPERLMENTS If we put together two alloys of identical composition but with the molecules of one metal in one of the alloys labelled radioactively, we can observe the diffusion of the labelled molecules under conditions of uniform chemical composition.J .CRANK 101 No mass flow occurs and the only gradient of chemical potential arises from the entropy of mixing. If we assume with Darken 4 that the mobility of any chemical species at a given composition is the same whether there is a gradient of chemical composition or not we can show that where DA* is measured by a tracer experiment. Also (C) ANELASTIC METHODS For some systems, the so-called anelastic methods are useful.697 The re- laxation time required for the establishment of a new equilibrium distribution of atoms or atom-pairs following the application of a stress, is measured. This time is related to the mean time of residence of an atom on a particular site and hence to the diffusion coefficient. The measurements are made on a chemically homogeneous material and are free from Kirkendall and chemical potential effects, as are tracer measurements.(d) OTHER METHODS Other methods are based on observations of the steady-state rate of flow of one species through or the rate of evaporation from a plane sheet of a second species. They have been used in lattice systems 8,9 but are more frequently used in solvent+polymer systems and we shall not stress them here. It suffices to say that where dimensional changes of the sheet occur a correction factor is needed before the mutual diffusion coefficient is obtained.5 A promising method of measuring self-diffusion coefficients as used by Kuczynski 10 is based on the sintering of spherical particles to a flat plate of the same material. Diffusion coefficients in ionic crystals can be measured by methods just described.Often the interpretation of the measurements is simpler because the lattice structure is maintained by the slower common ion and mass-flow does not occur. There is, however, the additional requirement that electro-neutrality shall be conserved and the mutual diffusion coefficient in ionic ’systems expresses this condition. Independent measures of the self-diffusion coefficients can be deduced both from this mutual diffusion coefficient and from measurements of electrical conductivity provided the basic mechanisms of diffusion and electrical conduction are the same. If the movement of the common ion is not negligible, the mutual diffusion will be influenced not only by electro-neutrality but by the mass-flow needed to preserve a non-defective lattice.No such situation appears to have been analysed as yet. 4. SOME DOUBTFUL ASSUMPTIONS The main assumptions underlying the so-called Darken relationship 8 are : (i) that changes in lattice parameters are negligible; (ii) that a non-defective lattice is everywhere maintained by complete shrinkage ; (iii) that expansion and shrinkage of the lattice, and diffusion take place only along a direction perpen- dicular to the original interface between the two diffusing species so that diffusion occurs in a region of constant cross-section. The first difficulty can be and has been 11 avoided by measuring distances in numbers of lattice spacings rather than in centimeters, for example. With regard to shrinkage many workers have reported a slight porosity on one side102 DIFFUSION COEFPIClENTS 1N SOLIDS of an interdiffusion system (ref.(l), vol. 4, p. 270). In a brassi-copper system, Ballufi and Alexander 12 estimate that as much as one-third of the zinc leaving the brass is replaced by pores. The suspicion of incomplete shrinkage thus aroused has been confirmed in several ways (ref. (I), vol. 4, p. 271). AS the pores may occupy as much as 30-40 % of the cross-sectional area of the diffusion zone the assumption of a constant, uniform cross-section is doubtful, particularly as we know little about the extent to which diffusion occurs across the pores. Furthermore, changes in lateral dimensions have been observed in several systems.13 In the case of silver and gold, for example, one-third of the total expansion on the gold side of the original interface occurred laterally and produced an increase in cross-section of the diffusion zone.It appears that, in some systems at least,. these effects may be large enough to influence considerably the values of diffusion coefficients deduced from inter-diffusion experiments. There is at least one case in which a reduction of porosity by pressure resulted in a decrease in the diffusion coefficient observed.14 These are essentially phenomenological difficulties. A more fundamental, theoretical difficulty arises from the fact that Darken’s treatment is based on Fick’s law in which the rate of flow of any component depends only on its own gradient. It may be, however, that a more general treatment based on Onsager’s irreversible thermodynamics 15 is needed.The new basic assumption involved is that the rate of flow of any one species in a multi-component system is linearly dependent on the gradients of chemical potential of each species and not simply on its own gradient as in Fick’s law. Thus, cross-terms appear relating the gradient of one component to the rate of transfer of another. If we consider diffusion to occur by a vacancy mechanism the vacancies constitute one of the components of the system. It has been shown (ref. (I), vol. 4, p. 323) that the simple Darken relationship (8) holds only when the vacancies are assumed to be in equilibrium and when the cross-terms between labelled A* atoms and B atoms, and between labelled and unlabelled A atoms in a tracer experiment are negligible; otherwise the relationship (8) needs modification.Theoretical grounds for sus- pecting that cross-terms may not always be negligible are provided by the cor- relation effects, discussed recently.17 It is found that for many atomic jump mechanisms the diffusion coefficient of one species depends not only on the jump frequency of its own atoms but on those of the atoms of the other species present. In this connection, it is interesting to refer to some recent attempts to deduce the cross-term coefficients from inter-diffusion experiments in three component liquid systems.18919 Further doubts arise as to what chemical potential gradient controls diffusion. If, for example, the lattice, and in particular the vacancies, are not in equilibrium as diffusion proceeds there may be little justification for using chemical potentials measured for systems in equilibrium.This difficulty still exists even if irreversible thermodynamics is used. Also we should remember that Onsager’s linear relation- ships are approximations which may hold only for systems not far removed from equilibrium.zo9 21 We have referred to vacancy mechanisms in which diffusion of any atom proceeds by changing place with a neighbouring vacancy or unoccupied site in the lattice. A picture of the Kirkendall mass flow in terms of vacancy diffusion has been proposed,22 the necessary vacancies appearing from and disappearing into dislocations in the lattice or aggregating to form the pores referred to above. There are several diffusion mechanisms in which vacancies are involved in different ways.A quite different type of mechanism is that in which diffusion occurs by the direct interchange of A and B molecules. This is not much in favour at the moment for several reasons. First, since it is likely to occur only between molecules of roughly the same size there will be no mass flow ; secondly, in ionic systems it would not give rise to any transfer of charge since A and B molecules simply carry equal charges in opposite directions ; thirdly, the energies necessaryJ . CRANK 103 for interchange as calculated by Huntington and Seitz 23 are relatively large com- pared with those needed for other mechanisms. However, it is worth noting that discrepancies between diffusion coefficients measured in ionic systems by tracer and by conductivity methods have been found,24 suggesting that the same mechanism does not necessarily account for diffusion and conductivity.Also, relatively few calculations of energies have been made and the conclusions may not be valid for all systems. It seems worth while, therefore, to see what happens to the relationship between tracer and mutual diffusion coefficients if a simple interchange mechanism involving two atoms is operating. By the nature of the interchange process the diffusion coefficients relative to the stationary lattice are the same for both species and each equals the mutual diffusion coefficient which is therefore simply related to the common mobility GAB by the expression In a tracer experiment involving labelled A* atoms and unlabelled A and B atoms, labelled A* atoms can be transferred by interchanging with unlabelled A atoms as well as with B atoms.The self-diffusion coefficient, therefore, includes two terms corresponding to the two mobilities GUS and GA*B, which is the same as GAB. Remembering that there is no gradient of chemical composition it is easy to show that (ref. (l), vol. 1, p. 327) (10) where DA* is the self-diffusion coefficient of species A measured by the tracer experiment. The mutual diffusion coefficient is in this case a fundamentally simpler quan- tity, involving only one mobility, than is a tracer diffusion coefficient which depends on two mobilities. Except by making some simplifying assumption, such that the interchange of A* and A atoms is a negligible occurrence, no simple relation- ship between the mutual and tracer diffusion coefficients can be found because of the mobility GAA+ which cannot be eliminated.Both mobilities may be de- duced, however, from interdiffusion and tracer experiments through (9) and (lo), if diffusion is known to proceed by interchanges. The prevailing idea, therefore, that the result of a tracer experiment is more simply related to a jump frequency than is a mutual diffusion coefficient may need to be modified according to what molecular mechanism is responsible for diffusion. In spite of these apparent sources of error, Darken’s equations are supported by Johnson’s measurements on the gold+silver system (ref. (l), vol. 1, p. 326) and le Claire and Barnes 25 have shown that the marker shift in one set of experi- ments is correctly given by Darken’s eqn.(6). However, in other systems already mentioned, departures from the simple assumptions have been observed, and we can refer in passing to gross failures of the Darken equations in liquidfliquid 20 and in polymer + solvent systems.27 DA+ = GU*kT(NA + NA*) $- GABkTNB, CONCLUSION In general, it appears that the significance of measured diffusion coefficients is not yet clear, if by significance we mean their precise relationship to the atomic jumps which constitute diffusion. First, there is the practical difficulty of obtaining measured diffusion coefficients which are free not only from the phenomenological effects which we have mentioned but also from the effects of gross inhomogeneities in the crystal lattice, of which grain boundaries are prob- ably the most important.The principal difficulties in interpreting satisfactory measurements are those (i) of deciding which of the several postulated atomic mechanisms are responsible for diffusion, remembering that more than one104 DIFFUSION COEFFICIENTS IN SOLIDS mechanism may contribute in a given system; (ii) of establishing the correct random-walk interpretation of diffusion, taking into account possible correlations between successive jumps of one molecule or between two molecules of different species ; (iii) of deciding whether the equations of irreversible thermodynamics, or even more general non-linear ones, should replace the simple Fick law suitably modified for the effects of chemical potential gradients, and, if they should, of measuring the new cross-term coefficients and assessing their physical meaning ; (iv) of discovering whether the crystal lattice is close enough to equilibrium for the usual chemical potentials determined from systems in true equilibrium, to be applied. The successful resolution of these difficulties which are being in- vestigated by several workers will constitute a considerable advance in the study of diffusion in crystalline solids.It is a pleasure to thank Dr. G. S. Park for helpful criticism of this paper. 1 le Claire, Progress in MetaIPlzysics, 1953, vol. I and 4. ZDarken, Trans. Amer. Znst. Min. Met. Eng., 1942, 150, 157. 3 Smigelskas and Kirkendall, Trans. Amer. Znst. Min. Met., Eng., 1947, 171, 130. 4Darken, Trans. Amer. Znst. Min. Met. Eng., 1948, 175, 184. 5 Hartley and Crank, Trans. Faraday Soc., 1949, 45, 801. 6 le Claire, Phil. Mag., 1951, 42, 673. 7 Wert, Physic. Rev., 1950, 79, 601. 8 Harris, Metals Tech., 1947, 14, no. 5. 9Dunn, J. Chem. Soc., 1926, 129, 2973. Gerzricken, Faingold, Ilkevitch and Sakharov, J. Tech. Phys. U.S.S.R., 1940, 10, 786. 10 Kuczynski, J. Metals, 1949, 1, 167. 11 da Silva and Mehl, Trans. Amer. Znst. Min. Met. Eng., 1951, 191, 155. 12 Ballufi and Alexander, J. Appl. Physics, 1952, 23, 1237. 13 Seith and Kottmann, Naturwiss., 1952, 39, 40. 14 Geguzin, Dokl. Akad. Nauk. U.S.S.R., 1956, 106, 839 (through Physic. Abstr.). l5 Onsager, Ann. N. Y. Acad, Sci., 1945, 46, 241. 16 Bardeen and Herring, Imperfections in Nearly Perfect Crystals (Wiley, New York, 17 le Claire and Lidiard, Phil. Mag., 1956, 1, 518. '*Baldwin, Dunlop and Gosting, J. Amer. Chem. Soc., 1955, 77, 5235. DunIop and Gosting, J. Amer. Chem. SOC., 1955, 77, 5238. 19 Fujita and Gosting, J. Amer. Chem. Soc., 1956, 78, 1099. 20 Wirtz, Z. Naturforschutig, 1948, 3a, 380, 672. 21 Ubbelohde, Trans. Faraday Soc., 1937, 33, 599. 22 Seitz, Physic. Rev., 1948, 74, 1513. 23 Huntington and Seitz, Physic. Rev., 1942, 61, 315, 325. 24 Compton, Physic. Rev., 1956, 101, 1209. 25 le Claire and Barnes, J. Metals, 1951, 3, 1060. 26 Johnson and Babb, J. Physic. Chem., 1956,60, 14. 27 Hayes and Park, Trans. Faraday Soc., 1955, 52, 949. 1952), p. 261. McCombie and Lidiard, Physic. Rev., 1956, 101, 1210.

ISSN:0366-9033    

DOI:10.1039/DF9572300099

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年代:1957

数据来源: RSC

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SOME FUNDAMENTAL ASPECTS OF THE MECHANISM OF DIFFUSION IN CRYSTALS BY K. COMPAAN AND Y. HAVEN Philips Research Laboratories, N. V. Philips’ Gloeilampenfabrieken, Eindhoven, Netherlands Received 6th February, 1957 Diffusion data, if available in the form D = DO exp (- E/kT) yield two parameters, whereas for a molecular description more quantities are needed. Some difficulties encountered with this problem are discussed. This problem has kinetic as well as equilib- rium aspects. The equilibrium aspect has been successfully attacked by the theory of lattice defects : it is possible to specify the atoms that are mobile as well as the frequency with which these mobile atoms jump. Recently, a novel kinetic aspect of the mechanism has become of interest, viz., the problem of correlation between the directions of consecutive jumps of an atom.These correlations effectively alter the self-diffusion coefficient in certain mechanisms, but not the diffusion coefficient for drift or the ionic conductivity. The latter aspect has recently been developed to a promising method of disentangling diffusion mechanisms especially in ionic crystals. 1. INTRODUCTION For a discussion of the molecular mechanism of diffusion of particles through crystals often considerably more data are needed, than are given by simple expressions like (1.1) by which relation a diffusion constant is often expressed. Ultimately one wants to know the path along which a particle travels through a crystal, the height of the potential barriers, the jump frequencies of the particles, the concentration of lattice defects, etc.Therefore, often many experimental data are needed and a close analysis will be required. In this paper, some aspects of diffusion in crystals will be discussed, which are connected with the interpretation of experimental data in terms of molecular mechanism. Emphasis will be laid upon a separation of equilibrium and kinetic properties. Although diffusion itself is a kinetic process, it is possible to dis- tinguish also equilibrium aspects. For instance, the concentration of lattice defects is in several diffusion systems an equilibrium property. This partial equilibrium makes it possible to discuss these stages with the conventional thermo- dynamic methods, which simplifies the discussion considerably, a fact which seems not always to have been sufficiently appreciated.To simplify the argument we will make some restrictions. We will confine ourselves to volume diffusion and furthermore restrict the discussion chiefly to tracer diffusion (self-diffusion). By tracer diffusion is understood a diffusion system in thermal equilibrium, except for the fact that some of the atoms are marked (e.g. by radio-activity assuming the isotopes to have equal mass) in part of the system. Thus tracer diffusion is pure “ mixing ”. Fortunately the results can easily be extended to other diffusion systems, for instance, diffusion along crystal boundaries, dislocations, etc., or to the interdiffusion of chemically different substances. 105 D = DO exp (- E/kT)106 MECHANlSM OF DIFFUSION 1 N CRYSTALS 2.TYPES OF DIFFUSION MECHANISMS It has been realized for a long time that lattice defects play an important role in the transport of matter through crystals.1 For some mechanisms of diffusion, however, the influence of lattice defects can be neglected, whereas in other mechanisms lattice defects play an essential role (proportionality between con- centration of lattice defects and diffusion constant). The two groups require a different approach (cf. 3 3 and 4). Therefore, it is appropriate to make the following division, when considering the role of lattice defects in tracer diffusion in simple systems. (The reader is referred to detailed descriptions of the several mechanisms elsewhere?) I. Difusion mechanisms with vanishing role of lattice defects (1) interchange of particles, (2) ring mechanism, (3) interstitial mechanism (for instance, zeolitic diffusion). II.Difusion mechanisms with predominant role of lattice defects (1) interstitial mechanism, (2) indirect interstitial mechanism (inters t i t ialc y) , (3) dumbbell mechanism (Hove 3), (4) vacancy mechanism. This division is not a complete one and must not be taken too rigorously, as the concept of lattice defects is in many cases not a very well defined one (vacancies and interstitial atoms can be cited as examples of lattice defects). The cited mechanisms should be considered as examples of two different aspects of diffusion mechanism. The essential difference can be illustrated with the following example of a dilute solution of atoms A in a parent lattice of atoms B.Let the atoms A be transported through the lattice by an interstitial mechanism. There is an inter- stitial mechanism of group I, if the atoms of A only occupy the interstitial sites. Then, all the atoms A move independently over the interstitial lattice sites. The average jump frequency v* is where V i is the jump frequency when the atom is at an interstitial site. The diffusion of C in a-Fe is an example of this mechanism. There is, however, an interstitial mechanism of group I1 if the atoms A can also replace the atoms B. Then, at a given instant only the atoms at the inter- stitial sites are free to move, whereas the atoms at the lattice sites jump with a much smaller probability. An atom makes in a comparatively short time a number of jumps from one interstitial site to another and then reaches a lattice site where it stays for a comparatively long time. v* = v i (2.1) The average jump frequency v* of an atom A can then be written where n and ( N - n) represent the number of atoms A at interstitial and lattice sites, respectively, and vi and v l the respective jump frequencies.(The second term is very small with respect to the first term.) An example of this case would be the diffusion of C in a-Fe containing Mn, if the C were fixed near the Mn preferentially, the C at interstitial sites then playing the role of " lattice defects ". Henceforth we conclude that for some diffusion mechanisms a factor (n/N), denoting the fraction of " lattice defects ", occurs in the expression for the average jump frequency of an atom, whereas in others this factor does not occur.AsK. COMPAAN AND Y. HAVEN 107 the diffusion constants for interdiffusion depend on the average jump frequency of the atoms, the factor (n/N) appears also in these diffusion constants (cf. fj 3). The absence of an influence of lattice defects in the first group makes an interpretation of the experimental data for group I less complicated than for group 11. In group I, eqn. (1.1) often suffices for the expression of the experimental data, Do being of the order DO = vd2, where v is the vibrational frequency of the atom and d the jump distance. 3. TYPES OF DIFFUSION CONSTANTS Often more than one diffusion coefficient is needed for describing the diffusion experiments with a simple system. Let us again consider a system of two kinds of particles of which the particles of type 1 are the tracers and those of type 2 the parent atoms.Such a system can be described in principle by two independent diffusion coefficients. The one coefficient can be obtained directly from a " drift " experiment and will be called the " drift "-coe%cient ; the other can be obtained from a tracer experiment (" mixing " experiment) and is called the self-diffusion coefficient. A " drift" experiment can be conceived in the following manner. Let there be a vacancy mechanism of diffusion with a concentration gradient of vacancies and let the fraction of tracers everywhere be the same. Then, the drift coefficient can be defined by (3 * 1) where i = il + i2 is the total current and c = c1 + c2 the total concentration of particles.A " tracer " experiment would be the following. Let there again be a vacancy mechanism of diffusion and let the crystal be in thermal equilibrium except for a partial replacement of the parent atoms by tracers (so that the concentration gradient of vacancies is zero). Then, the coefficient for tracer diffusion or the self-diffusion coefficient can be defined by (3.2) where il, 2 are equal diffusion currents il and i2 flowing in opposite directions and (dq, 2/dx) are simultaneously equal concentration gradients dcl/dx and dc2/dx of opposite sign. Eqn. (3.1) and (3.2) can also be written for other diffusion mechanisms for both groups I and 11. It can be shown that in the diffusion mechanisms of group TI the relation between the two different diffusion coefficients of eqn.(3.1) and (3.2) is given by 4 (3.3) where f is a factor which accounts for the correlation between successive jumps of a particle (cf. 9 5) and (n/N) is the fraction of vacancies (for this special case) or the fraction of interstitials (for the interstitial mechanisms). In the diffusion mechanisms of group I, it is not always possible to define a drift coefficient, but when it is possible to define such (or an analogous) coefficient (interstitial mechanism), one finds the relation (3.4) i =- D (dcldx), il, 2 = - D1,2 dc1,2/& , Dl, 2 (tracer diffusion) =f(n/N)D (drift), D1,2 (tracer diffusion) = D (drift). The coefficient D is the coefficient describing the diffusion of lattice defects as follows from the definition (3.1).D is a constant with respect to the con- centrations, if (n/N) is not too large. This coefficient must be used if one wants to describe for instance, the segregation of lattice defects when the temperature of a crystal is suddenly lowered. The drift coefficient is also to be used when external forces are applied, for instance, an electric field in ionic crystals (cf. 3 5). The tracer diffusion D1,2 is to be used in the tracer experiments as described108 MECHANISM OF DIFFUSION I N CRYSTALS above and also in interdiffusion of two different chemical species. Di, 2 is proportional to (n/N) in group I1 and “ constant ” in group I. The factor (n/N), denoting the concentration of lattice defects does not occur in the expressions for the diffusion coefficient in the diffusion mechanisms listed under group I.However, if the diffusion system consists of more components, such a factor may appear. Let us, for instance, consider the system of atoms A with a small concentration of atoms B replacing the atoms A and let there be an ‘nterchange mechanism of diffusion. The concentration of B is supposed to be onstant throughout the system and with the atoms A a tracer experiment has ieen set up. Let a direct interchange of atoms A occur with a much smaller probability than an interchange of an atom A with B. In that case the atoms B play the same role as the vacancies in the vacancy mechanism of group 11, and the factor (n/N), to be used in a “mixing” equation equivalent to eqn. (3.3), denotes now the concentration of atoms B.Thus the mechanisms of group I1 behave as if the systems had one component more than in those of group I. One can, however, consider the vacancies or interstitials as if they were separate components (cf. also $ 3, and then the de- scriptions of the several mechanisms show essentially the same features. One could also say that the drift coefficient of group I1 plays more or less the same role as the mixing coefficient (self-diffusion coefficient) of group I. In an interstitial mechanism with (n/N) denoting the fraction of atoms at interstitial sites, the coefficients D1,2 and D are directly related to v* and V j , respectively, in eqn. (2.2). Here the factor f = 1. When the fraction (n/N) of interstitials becomes very large for foreign atoms, one gets as limiting case that all foreign atoms are at interstitial sites.Therefore, the interstitial mechanism of group I can be considered as a limiting case of the interstitial mechanism of group 11. The above equations can be extended to systems consisting of more com- ponents. Let particles 1 and 2 be different chemical (instead of isotopic) species which diffuse into each other. If both species have different mobilities, “ drift ” and “ tracer diffusion ”, as defined above, are no longer independent quantities. For, if one makes a drift experiment, there will occur concurrently a mixing or demixing process, and if one makes an interdiffusion experiment, there will occur at the same time a drift, resulting (possibly) in a Kirkendall effect. Therefore, the interdiffusion of two chemically different substances requires one additional diffusion coefficient for describing the molecular mechanism.The three coefficients required correspond to the two jump frequencies and the correlation factor (or rather f(n/N)) which is no longer constant, but an intricate function of the jump frequencies and the concentrations. 4. THE ROLE OF LATTICE DEFECTS In the second group of diffusion mechanisms (with a predominant role of lattice defects) it is of value to split off a factor, specifying the concentration of lattice defects, from the diffusion constant and to treat this factor independently. Eqn. (1.1) can be written in the form (4.1) where n/N = CO exp (- Ec/kT) and D = Dd exp (- Ed/kT), respectively, stand for the (fractional) concentration and the “ mobility ” (diffusion constant) of the lattice defects and f is a constant (correlation factor, cf.$5). Usually, f is unity or does not differ much from unity. The mobility of an atom is (n/N) times the mobility of the lattice defects. Dl, 2 and D represent the diffusion co- efficient for mixing and drift, respectively (cf. eqn. (3.1) and (3.2)). Df ,2 = fC0 exp (- Ec/kT)Dd exp (- Ed/kT) Y The constants of eqn. (1.1) and (4.1) are related by E = Ec 4- Ed, (4.2) Do = fcoDd. (4.3)K . COMPAAN AND Y . HAVEN 109 The constant Dd is of the order of Dd m vd2, where v is the vibrational frequency of the atom and d the jump distance. The value of such a separation of factors is that in many cases the concentration of lattice defects is an equilibrium property.The latter factor has often been taken advantage of in the field of ionic conductivity,l but seems to have been neglected in the field of diffusion. An important feature of this equilibrium property is the following. The concentration of lattice defects depends on the composition of the crystal and can be influenced by the addition of " impurities ". We will illustrate this with an example. We consider the diffusion of Na tracers in NaCl containing a small quantity of Ca2f ions, which are supposed to occupy the Na sites, whereas each Ca2+ ion introduces one additional Na vacancy. For convenience we suppose that the Na vacancy is not associated with the Ca2+ ion. The diffusion mechanism is supposed to be a vacancy mechanism, so that the diffusion constant is pro- portional to the concentration of Na vacancies.At higher temperatures there appear also more Na vacancies in the lattice, together with C1 vacancies, because of the thermal equilibrium. These two concentrations must obey a kind of solubility product law, which in its most elementary form reads [Na vacancies][Cl vacancies] = A exp (- EJkT). (4.4) Denoting the concentration of Na vacancies and Ca2f ions by x and a, respectively, then the concentration of C1 vacancies will be (x - a) and eqn. (4.4) can be trans- formed into Several properties of diffusion constants can be illustrated with eqn. (4.5). The experimental value of the energy of activation for diffusion depends in many cases on the composition of a crystal (impurities, etc.) or varies with temperature.It is often the concentration of lattice defects, rather than a change of the jump frequencies of the atoms, that causes such a variation of the experimental constants. Two extreme cases can be distinguished in eqn. (4.9, namely, for high and low temperatures, respectively. At high temperatures, the factor (44a2) exp (- EJkT) is predominant and one finds for the constants of eqn. (4.1), E, = +E,, CO = A* and x = A* exp (- Es/2kT), whereas at low temperatures this factor can be neglected and one finds E, = 0, CO = a and x = a (a is the fractional concen- tration of Ca2+). At intermediate temperatures one will find intermediate values for E, and CO. The value of A* is for several alkali halides of the order of 1-103, whereas a w 10-6-10-3 for usual concentrations.The influence of lattice defects must be determined by carefully designed experi- ments. A powerful method is the addition of different quantities of impurities, as was first proposed by Koch and Wagner 5 for the study of the ionic conductivity in silver halides and also adopted by other authors. Equations like eqn. (4.5) can be derived for several other types of lattice defects. In the field of metallic diffusion, however, this method does not seem to have been applied, as a result of lack of knowledge about the influence of impurities on lattice defects in metals. 5. CORRELATION BETWEEN SUCCESSIVE JUMPS OF A PARTICLE A novel kinetic aspect of diffusion has been discovered by Bardeen and Herring,4~6$7 who noticed that immediately after a jump an atom may have a greater probability to jump backward than to jump forward, so that the next110 MECHANISM OF DIFFUSION I N CRYSTALS step will be correlated with the preceding one.This effect causes the appearance of a correlation factor f in eqn. (3.3), for the correlations between successive jumps affect only the mixing diffusion coefficient, but not the drift coefficient. The cause of this difference is the following. Consider again a vacancy mechanism and suppose that a given atom A has just jumped in the forward direction. The next step of that atom occurs with a greater probability in the backward direction, so that the contribution of atom A to the current iA will be lowered, as compared with the contribution if there were no correlation. At the same time, however, there will be another atom, which has a greater probability to jump in the forward direction.Therefore, the total current i of all atoms (tracers as well as parent atoms) in a given direction will not be affected by the correlations. Another way of reasoning is the following : there is a countercurrent of vacancies, the jumps of which will not be correlated, so that correlation effects cannot influence the drift current. Correlation effects can be expected in mixing experiments of the following type. There must be an interdiffusion of two types of particles (say, tracers and parent atoms), whilst a third type of particle, which may be of uniform concentra- tion, is operative in the mechanism of interchange. This third one may be a vacancy, for instance. The reason is, that with a jump of a tracer a vacancy has also jumped, disturbing a (possible) previous uniform concentration of the vacancies.In following jumps of the vacancy, the equilibrium distribution of this vacancy around the tracer atom will be restored, by which process the tracer will partly be driven back. Direct interchange of a tracer and parent atom (mechanism 1 of group I), however, is of the nature of a drift current and does not show correlation. How- ever, if a third type of particle, say atoms C, whose concentration is very small, would interchange rapidly with the tracer atoms A as well as with the parent atoms, correlation effects would occur here also. The magnitude of the correlation TABLE I.-CORRELATION FACTOR f FOR effects can be calculated.4 It appears that the magnitude of the effect depends on the nature of the crystal lattice as well :::: as on the mechanism of diffusion.In diamond lattice cubic lattice 0.721 table 1 the values of the correlation factor body-centred cubic lattice face-centred cubic lattice 0.781 for a number of lattices have been given.4 hexagonal close packing 0.78 1 Experimentally the determination of the factor f would be possible by a comparison of the “ drift ” and “ mixing ” diffusion coefficients, but in practice the determination of the drift coefficient is too inaccurate to give useful results. The ionic conductivity, however, is directly related to the drift diffusion coefficient by means of the Einstein relation : DIFRISION VIA FREE D = BkT (5.1) where k is Boltzmann’s constant, T is the temperature in OK, and the mobility B (of vacancies for a vacancy mechanism) is given by B = o/ne2.(5.2) Here (T is the contribution to the conductivity of the ions under consideration, n is the concentration of vacancies and e the charge of the ions. From eqn. (3.3), (5.1) and (5.2) we derive f = Dl, 2 (tracer) Ne2/ukT. (5.3) Eqn. (5.3) serves as the basic equation for experimental work. It can be called an extended Einstein relation, but it should be stresses that eqn. (5.3) does not imply a violation of the original fundamental relation of Einstein. For, strictlyK . COMPAAN AND Y . HAVEN 111 speaking, the Einstein relation pertains only to what we have called the drift diffusion coefficient. The application of eqn. (5.3) to the discussion of the mechanism of ionic transport in crystals is now advancing in several laboratories and the results seem rather promising.Some of the earlier results have been discussed in a previous paper.2 6. CONCLUSIONS We draw the following conclusions : 0) (ii) (iii) different types of diffusion constants must be distinguished ; the introduction of a “ drift ” diffusion coefficient and a “ mixing ” diffusion coefficient is helpful for the interpretation of experimental results ; from an experimental point of view, important lines of approach to a molecular interpretation are perhaps the study of (a) the role of lattice defects (impurities) ; (b) a comparison of diffusivity (tracer diffusion) and ionic conductivity (correlation factor). APPENDIX RELATION BETWEEN CONVENTIONAL DIFFUSION COEFFICIENTS AND THE COEFFICIENTS OF 5 3 Often the equations for self-diffusion in a crystal are written in the following form : Here we have used the form for two independent components namely, particles 1 being the tracers and 2 the normal atoms.The independency results from the assumption, that there is still a third “ particle ” (e.g. vacancy) playing a role in the diffusion process. We derive here the connection between the diffusion coefficients of eqn. (A. 1) and the “ drift ” coefficient of 9 3. If we add the two equations of (A. 1) we get: However, the total current cannot depend on dcl/dx or dcz/dx separately, but only on their sum d(c1 + c2)/dx, and therefore we must have (A.2) Let us now consider the case where cl/c2 = K, and thus also dclldx = Kdcz/dx, Dll + 0 2 1 = 0 2 2 + 0 1 2 = D, where D is the “ drift ” coefficient. everywhere in the crystal. From (A. 1) we then get and this will hold true during the whole diffusion process, because there is no reason for isotope separation, assuming a negligible difference in mass. But then also and so112 MECHANISM OF DIFFUSION I N CRYSTALS From (A.2) and (A.3) we find DlZlcl = D21/~2. In the ‘‘ mixing ” experiment we have dclldx = - d ~ / d x , and with this relation, eqn. (A.l) gives (A.4) where Dl, 2 is the “ mixing ” coefficient of 3 3. coefficients of (A.l) and the drift coefficient: From eqn. (A.2), (A.4) and (A.5) we deduce the following relations for the diffusion c1 D + c2 D1,2 c1 + c2 ’ Dll = 1 Jost, Difusion in Solids, Liquids and Gases (Academic Press Inc., New York, 1952). 2 Compaan and Haven, 3rd Int. Conf. Reactivity of Solids (Madrid, 1956). 3 Hove, Physic. Rev., 1956, 102, 915. 4 Compaan and Haven, Trans. Faraday SOC., 1956,52, 786. 5 Koch and Wagner, 2. physik. Chem. B, 1937,38,295. 6 Bardeen and Herring, Imperfections in Nearly Perfect Crystals (John Wiley, New 7 Le Claire and Lidiard, Phil. Mag., 1956, 1, 518. York, 1952).

ISSN:0366-9033    

DOI:10.1039/DF9572300105

出版商:RSC    

年代:1957

数据来源: RSC

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THERMOELECTRIC POWER OF IONIC CRYSTALS BY R. E. HOWARD AND A. B. LIDIARD Clarendon Laboratory, Oxford ; A.E.R.E., Harwell, Berkshire Received 14th January, 1957 The thermoelectric power of an ionic conductor MX fitted with electrodes of metal M is discussed from the point of view of the lattice defects existing in the salt. This power is composed of two parts : (1) a “ homogeneous ” part (coming from the thermal diffusion potential caused by the temperature gradient in the salt) which is related to the “heats of transport ” of the defects, and (2) an “ inhomogeneous ” part (coming from the dependence of M/MX contact potential on temperature) which is related to the en- tropies of formation of the defects. Detailed results are given for the case where MX is a cationic conductor showing Frenkel disorder, and the influence of impurities of type NX;! in MX is discussed.When the lattice disorder is dominated by the presence of impurities the total power contains an important concentration dependent term -(k/e) In [c(l - p ) ] , where c is the molar fraction of impurity and p is the degree of association between impurity N2+ ions and M+ vacancies (k, e are respectively, Boltzmann’s constant and the electronic charge). The power of an impure crystal is found to be un- altered by the establishment of a Soret concentration gradient. Agreement between present theory and the existing, rather scant, data on AgBr is adequate but further experi- ments on AgBr and AgCl are desirable. 1. INTRODUCTION The properties of lattice defects in solid ionic conductors have been extensively studied through electrical conductivity and diffusion measurements.In the present paper it is shown how the thermoelectric power of a crystal MX fitted with metallic electrodes M is also related to the properties of these defects. The experimental arrangement is imagined to be as in fig. 1. We define the thermoelectric potential as the difference in potential V B - VA ; this is opposite to the convention generally employed in connection with metallic thermocouples 1 but of the same sign as that used by de Groot 2 , 3 and Holtan 4 in their thermodynamic treatments of salt and electrolyte thermocells. The potential difference VB - VA is made up of (i) the “ homogeneous ” potential difference Vp - VA due to the temperature difference between P and A; this term is small by comparison with the other terms and will be dropped ; (ii) the contact potential difference VQ - V p z -@(T) ; (iii) the homogeneous potential difference VR - VQ due to the temperature gradient in the salt ; (iv) the contact potential difference Vs - VR = @(T+AT) ; (v) the homogeneous term in the metal electrode and wire, which we shall neglect.The total is VB - VA = (VR - VQ) f @ ( T f AT) - @(T). The quantity ( V B - VA)/AT is the thermoelectric power 8 and is composed of a “ homogeneous ” part Oh, = ( VR - VQ)/AT and a “ heterogeneous ” part Ohet = A@/AT. These sign conventions will be observed throughout.* Owing to the different types of lattice disorder that may exist a general treatment is not possible. We shall therefore deal only with the case where the salt displays predominantly cationic Frenkel disorder, but it will be assumed that this may be modified by the presence of impurity cations of valency differing from M * A recent paper by Patrick and Lawson,s to which we shall refer below, takes - ( VB - V&/AT as the thermoelectric power.113 Our 0 is thus minus their c114 THERMOELECTRIC POWER (aliovalent). This is the type of disorder existing in AgBr 6 and we shall thus be able to apply our equations to the existing data for this compound.5 The prin- cipal results of this theoretical discussion are (i) eqn. (2.6) for Ohorn for a pure crystal (previously given by Patrick and Lawson) in terms of the “ heats of transport ” (qi*, qu*) of the defects, (ii) eqn.(3.10) for the inhomogeneous power of a pure crystal showing the correction to the Wagner7 estimate (square bracket term), (iii) eqn. (3.12) for 8het for an impure crystal, of which the Wagner term may be only a small part, lastly, (iv) eqn. (4.1) for the total power of an impure crystal, applicable whether a Soret con- T T+AT centration gradient is established or not. As would be expected our equations are compatible with the thermodynamic equations developed by HoltanP The agreement with the existing experimental data on the thermoelectric power of AgBr 5 is satisfactory but these data are not very extensive and further experiments, particularly on AgBr with varying CdBr2 contents, would be valuable. Another system for which there already exists a sufficient knowledge of the lattice defects to make thermoelectric studies profitable is AgCl + CdC12.8 1 1 M A 6 FIG.l.-Schematic diagram of the ex- 2. HOMOGENEOUS THERMOELECTRIC POWER perimental arrangement for the measure- ment of the thermoelectric power of an OF AN IONIC CONDUCTOR is necessary to include some results which are not new and this fact will be in- dicated by giving the appropriate reference. We have to deal with a crystal containing two types of intrinsic lattice defects, namely, interstitial ions and lattice vacancies; in addition aliovalent foreign ions may also be present. First, how- ever, we consider the range of intrinsic behaviour in which foreign ions may be neglected. (i) INTRINSIC RANGE In keeping with the approach which treats the system of lattice defects as an “ ideal solution ” we write phenomenological equations for the current densities of interstitials and vacancies as follows : 2 ~ 3 and in which nj and n, are the numbers of interstitials and vacancies per unit volume and qi* and qv* are their “ heats of transport ”.* Di and Du are the diffusion coefficients for interstitials and vacancies which, it is assumed, carry effective charges + e and - e respectively. E is the electric field strength.*They are heats of transport appropriate to the thermodynamic fluxes and forces employed by Prigogine.9 In 9 52 of ref. (3) they are denoted by a double asterisk.l i . E . HOWARD A N D A . B . LlDIAKU 115 Now in the steady state in the absence of an external field the system will be everywhere electrically neutral and there will be no electric current flowing ni == n,, (2.3) j i = j w .(2.4) In addition, since the departures from " thermostatic " equilibrium are always small we can employ the relation (2.5) where N and N' are respectively the numbers of normal and interstitial (cation) sites per unit volume and gF is the Gibbs free energy of formation of a Frenkel defect pair. By use of (2.1)-(2.5) we get an equation for the thermal diffusion field E,5 ninw = NN' exp (- gF/kT), where 4 = Di/Dv is the ratio of mobilities, and is the enthalpy of formation of a Frenkel defect pair. For small AT (fig. 1) the eXPreSSiOn (2.6) gives the homogeneous thermoelectric power, ohom. The quantity 7 (2.7) 4(4i + 3hF) - ( 9 w + 4hF) + + I QM* = is the overall heat of transport for the cations appearing in the thermodynamic treatments of de Groot 2 and Holtan.4 (ii) INTRINSIC RANGE, EXPERIMENTAL Patrick and Lawson5 measured the thermoelectric power of pure AgBr and of AgBr + 0.25 mole % CdBr2 fitted with Ag electrodes.The results are shown in fig. 2. The impure system will be discussed later. In order to analyse these data on the basis of eqn. (2.6) it is necessary to subtract the inhomogeneous power resulting from the dependence of Ag/AgBr contact potential on temperature. Following an earlier estimate by Reinhold and Blachny,7, 10 Patrick and Lawson took 6het =- 140pVldeg. They then found that the remaining power could be satisfactorily analysed on the basis of (smoothed) values for as found pre- viously by Teltow 6 provided that they took qw* =- 0-385 eV, qi* = 0.017 eV, the heat of formation hF for Frenkel defects in AgBr having been taken as 1.27 eV -also from Teltow's work.The negative value for the vacancy is understand- able since the flow of ions (which carry the heat) is opposite to the flow of vacancies. (iii) KINETIC INTERPRETATION OF HEATS OF TRANSPORT The magnitudes and signs of qw* and qi* are qualitatively understandable on the basis of an intuitive generalization by Wirtz 11 of the usual rate process formula116 THERMOELECTRIC POWER for atomic jump frequency. The isothermal expression for the jump frequency of, say, an interstitial ion is 12,139 14 (2.9) where v is the frequency of vibration of the ion in the (thermal) average potential energy field of all the other ions of the lattice and Ag is the Gibbs free energ of activation.Wirtz supposed that Ag may be decomposed into three parts, Agl, Ag2 and Ag3, being energies of activation respectively supplied (i) at the w = Y exp (- Agjkr), Temperature O C FIG. 2.-The thermoelectric power oh,, + ohet as a function of temperature for AgBr fitted with Ag electrodes. The upper curve is for pure AgBr, the lower curve for AgBr + 0.25 mole % CdBrz. Note that the power is negative with the convention of 5 1 (after Patrick and Lawson). initial position of the interstitial-to initiate the jump ; (ii) at the plane of normal atoms between the initial and final positions of the interstitial ion-to move them apart and facilitate the passage of the interstitial through them; (iii) at the final position of the interstitial-to push apart the surrounding ions so that they can accommodate the interstitial.The point of this decomposition is that the com- ponent energies are supplied at slightly different temperatures in an anisothermal system. In place of (2.9) we would thus have for a jump in the same direction as the temperature gradient. equation for the net flow of interstitials, ji, can be derived kinetically. found to have the same form as (2.1) with On this basis an This is qi" = Ahil - Ahi3, (2.11) where Ah is the enthalpy part of the corresponding free energy As. The expression for vacancies is (2.12) qv* =- (Ah,, - &2>. The results (2.11) and (2.12) require the heats of transport to be numerically less than the corresponding heats of activation. The heat of activation for Ag+ interstitials in AgBr from Teltow's results 6 is 0.18 eV ; the value of qi* (2.8) isR .E . HOWARD A N D A . B . LIDIARD 117 only about a tenth of this. Such a result might be expected for an interstitial since a large part of Ahi must be supplied at the midway position whether the ion moves by the direct mechanism or the indirect (" interstitialcy ") mechanism.15 For a vacancy jump Teltow's data give the heat of activation as 0-37 eV which is numerically nearly equal to q,* (2.8) so that here most of the heat of activation appears to be supplied to the ion which jumps. This again is reasonable. Further use has been made of the kinetic approach in connection with the Soret effect in impure systems.16 (iv) RANGE OF IMPURITY CONTROLLED CONDUCTION We imagine now that the ionic conductor is not in its intrinsic temperature range but contains lattice defects other than those introduced thermally.We idealize this situation by supposing that all the defects are introduced by the presence of aliovalent impurities. To be specific we may take AgBr containing a sufficient concentration of CdBr2 that the Ag+ interstitials may be neglected: the two principal defects are now the Ag+ vacancies (effective charge -e) and the substitutionally incorporated Cd2+ ions (effective charge e). However, since the Cd2 ions are on normal lattice sites, they can move only when they are ad- jacent to a vacancy, i.e. when they are " associated." 6 Hence when we consider the flow of impurity ions, attention must be focused on the movement of the (electrically neutral) impurity-vacancy pairs.17 Let nk be the number of impurity- vacancy pairs per unit volume then the current density of pairs (equal to the current density of impurity ions) is (2.13) in which D k is the diffusion coefficient of the pairs (assumed tightly bound) and qk* is their heat of transport.The relation between nw and nk is given by the mass action equation for the association reaction, namely, Nnk -- - 12 exp ((/kT), nW2 (2.14) where 5 is the Gibbs free energy of association. In the true steady state both j, and j k are zero : eqn. (2.2), (2.13) and (2.14) then yield where x = 5 - T(35/3T), is the enthalpy of association. We have written this power as @hom(a) to emphasize that it is established only after a long time since the condition j k = 0 requires the establishment of an impurity concentration gradient (Soret effect).In practice, measurements of the thermoelectric power will be made in a comparatively short time, and there will be no impurity con- centration gradient although there will be a gradient of complexes due to the increased dissociation at the hot end. To find ohom we therefore set j ,= 0 and relate n, and nk through (2.14) ; the result is (2.16) where p 1. nk/(nk -I- n,) is the degree of association. The expression (2.16) may be evaluated for Cd2+ ions in AgBr by using Teltow's results for the association rcaction (x = 0.16 eV and association constants as given in table 2 of ref. (6)) and assuming q,* = - 0-385 eV as found by Patrick and Lawson from their analysis of pure AgBr.At 200" C a Cd2+ concentration of 0.25 mole % gives p = 0.36 and ohom = - 720 pV/deg. The observed total power is - 290 pV/deg : hence118 THERMOELECTRIC POWER the inhomogeneous power in this case must be 430 pV/deg. This is very different from the Wagner estimate - 140 pV/deg. found to be consistent for the pure AgBr, but will be shown to be understandable when we analyse oh& in 0 3 : the inhomogeneous power is strongly dependent on the purity of the crystal. Another important feature of our results for ohet, which we may anticipate here, is the occurrence of a term which just cancels with Xp/eT(l + p ) in (2.16) when no Soret effect is allowed for and with (qk* - x)/2eT in (2.15) when time is allowed for the Soret effect to develop.The total thermoelectric power is the same whether there is sufficient time for a true steady state to be reached or not ; consequently the Soret effect in these systems cannot be studied via the thermoelectric power. The magnitude of the concentration gradient established may be studied directly and is related to qk*. We have accordingly analysed qk* on the basis of the Wirtz kinetic approach by an extension of the method used for studying Dk.17 Since these calculations are not directly relevant to the thermoelectric power we shall not discuss them further in this paper. 3. INHOMOGENEOUS THERMOELECTRIC POWER OF AN IONIC CONDUCTOR We have seen in 8 2 that a consistent analysis of the existing data on pure AgBr can be obtained using the Wagner estimate of the inhomogeneous power: we also saw that the theory of the power of an impure crystal demanded a different value for the inhomogeneous power.In this section we shall examine the in- homogeneous power from the point of view of the theory of lattice defects and show how the Wagner estimate must be corrected. The contact potential between the ionic conductor MX and the metal M may be evaluated formally by setting the electrochemical potential ji of ion M+ in MX equal to the electrochemical potential of ion M+ in M : or, the electrical potential in M relative to that in MX, @(T), is e-1 times p(M+ in MX) - p(M+ in M), where the ps are ordinary chemical potentials. It follows that ehet is 1 c ) e 3T = - -{p(M+ in MX) - p(M+ in M)} = - (s(M+ in MX) - s(M+ in M)}/e, (3.1) where the ss are the corresponding partial entropies.Wagner’s 7 analysis assumes that s(M+ in M) is:almost identical with the total entropy of the metal-the electrons making only a very small contribution on account of their Fermi-Dirac degeneracy. That is s(M+ in M) = C,(M) dT/T. (3.2) F s: Wagner also suggests that s(M+ in MX) is one-half the total entropy of MX, thus s(M+ in MX) = 3 C,(MX) dT/T. (3.3) On the basis of (3.2) and (3.3) one obtains the estimate - 140pV/deg. for AgBrJAg. 10 In order to look critically at the assumption (3.3) we return to the definition of p(M+ in MX) and enquire what is the free energy change on adding one M+ ion to a MX crystal. Since the salt is crystalline the added ion can only be accommodated on surface sites (or dislocation jog sites) or by going into an inter- stitial position or by filling an already existing vacancy.Since the interactions in an ionic solid are primarily electrostatic and nearest neighbour repulsion forces one would expect the addition of one M+ ion (but no X- ion) to the surface toR . E . HOWARD AND A . B . LIDIARD 119 lower the energy of the crystal by about one-half the lattice energy per ion pair. Hence p(M+ in MX)=+g(MX), where g is the free energy per ion pair of the salt. If differentiation of this approximate relation is allowed then (3.3) follows. Thus, if all the ions transferred to or from the metal were removed or accom- modated at surface sites of the salt then one might expect (3.3) to be a good ap- proximation. Under such conditions the surface electrical double layer would have two components, one formed by an excess (or deficiency) of electrons in the metal, the other formed by an excess (or deficiency) of M+ ions on the salt surface. The double [ayer will not, however, be of this type but will be diffuse in a way already described by several authors in other connections.19~ 20921.The reason is that the salt can lower its free energy if the added ions are distributed more randomly in the crystal, since in this way the (configurational) entropy is increased. The contact potential (from which we derive OheJ is then the potential difference between the metal and the interior of the salt beyond the space charge region near the boundary. We thus require the chemical potential of an interstitial ion in the interior of the salt.Let the work required to bring a cation from a state of rest at infinity into a particular (but arbitrary) interstitial position in the crystal (at constant T, P) be gi. Similarly, let g, be the work required to remove a normal cation from the crystal to a state of rest at infinity. The chemical potential of an interstitial ion in a region of the crystal containing ni interstitials per unit volume is then (3.4) This expression is easily obtained as the change of Gibbs free energy on addition of one interstitial. Likewise the change of free energy on creation of one addi- tional-vacancy is (3 * 5 ) In passing, we note that eqn. (2.5) with gF = gi + gv, is obtained from the equation pi = gi + kT In (nJiV’). pv = gv + kT In (n,iN).pi + /-b = 0, interstitial + vacancy -+ 0. corresponding to the quasi-chemical reaction : We may now substitute (3.4) into (3.1) to obtain &t, thus (3.6) 1 3Pi 1 ohet = - - + - s(M+ in M). e 3 T e The explicit form of pi depends upon the impurity content of the crystal. We deal separately with the region of intrinsic conduction where only thermally produced defects need be considered and with the region of impurity controlled conduction. (i) INTRINSIC RANGE Here the numbers of interstitials and vacancies are equal and given by H i = nv = (NN’)$ exp [- (gi + gW)/2kT]. pi = Hgi - gv) + (kT/2) In (NIN’), Hence where Si =- 3gi/3T, s, =- 3gv/3T. In order to bring out the relation of (3.9) to Wagner’s expression we write si = si’ + s(M+/MX), SV = sV’ - s(M+/MX),120 THERMOELECTRIC POWER where s(M+/MX) is the entropy of a M+ ion added to the surface of MX from the standard state at infinity and which may be assumed to be given by (3.3).The quantity si’ is the increase of entropy on taking a surface M+ ion and putting it in an interstitial position; a similar definition is made for s,’. Both sj’ and s,’ originate from the influence of the defects on the vibrations of the crystal. We have therefore 1 k 2e = (- si’ + s ~ ’ ) + - In (NIN’) (3.10) + [i s(M+ in M) - - 1 s(M+/MX)] . e The first two terms constitute the correction to the Wagner estimate. However, for pure AgBr the correction is probably small, because of the consistency of the analysis based on the Wagner estimate. The term (k/2e) In (NIN’) is only - 30 pV/deg. so that si’ and s,’ are thus nearly equal.(ii) IMPURITY CONTROLLED RANGE Here we suppose that the number of thermally produced defects is small and that we have a large number of vacancies introduced by the presence of divalent cations. If the number of substitutionally incorporated divalent cations is Ni per unit volume then n, = Ni(1 - p ) and by (2.5) and (3.4) (3.1 1) Therefore 1 + - (s(M+ in M) - s(M+/MX)). (3.12) In arriving at (3.12) we have assumed that no impurity concentration gradient has been established. For AgBr containing 0.25 mole % CdBr2 we have (cf. $ 2 (iv)) p = 0.36 at 200” C, whence is found to be 465 pV/deg., whereas the Wagner term is - 140 pV/deg. This is consistent with the value 6het = 430 pV/deg. inferred in 2 (iv) if s,’ is about lO-4eV/deg. or about k : such a value is to be expected from the influence of a vacancy on the lattice vibrations.At the present time it is difficult to be more conclusive in our comparison with experiment since the available data are so scanty. We believe that further study of these systems would be valuable, especially studies of thermoelectric power as a function of impurity content (eqn. (2.16) and (3.12)). 4. SUMMARY AND CONCLUSION The explicit expressions which we have obtained for Ohhom and Ohet may be added together to give the total thermoelectric power 8. For an impurity con- trolled system there is some cancellation of terms and we have 1 e + - (s(M+ in M) - s(M+/MX}. (4.1)R. E. HOWARD AND A . B. LIDIARD 121 The concentration dependent term is large and should be detectable experimentally.Although we have not calculated e h e t ( 0 ) it is not difficult to do so and hence to show that ohom( co) + &,et( co) is also given by (4. l), i.e. the Soret effect does not lead to a change in thermoelectric power. It is possible to rewrite our expressions for 0 in such a way as to make contact with the thermodynamic analysis of Holtan.4 We write 0 as the sum of two parts, one of which (Holtan’s “ thermostatic ” part) is (l/e)(s(M-+ in M) - s(MX)) or (l/e)(s(M) - s(MX)) to the approximation that the homogeneous terms in the electrodes can be neglected; s(M) and s(MX) are the entropies (per molecule) of the metal and the salt respectively. The remaining parts are - Q M*/eT (eqn. (2.6) and (2.16)) with (i) in the intrinsic range, (- si’ + sV’)/2e +(k/2e) In (N,”’)+ s(X-/MX)/e, where s(X-/MX) is the entropy of an X- ion added to the surface (ii) in the impurity range (s,‘/e) f- s(X-/MX) + (k/e> In [N/Ni(1 - p ) ] -, Xp/eT(l + p ) .By studying the small amount of disorder which exists in the anion sub-lattice it is easy to show that each of these expressions is just e-1 times the partial entropy of an anion sx. The thermodynamic treatment of this system shows that QM* =- ex* (Holtan 4 3 9) ; hence the non-thermostatic part of our result is (Qx* + Tsx)/eT rsx*/e, which is Holtan’s thermodynamic result for this case (his eqn. (4.22)) ; sx* is called the “ entropy of transfer ”. From a com- parison of his thermodynamic equations with experimental data on a number o salts Holtan concluded that the thermostatic part of the power was often nearly the whole power.In conclusion we suggest the desirability of further measurements of the thermo- electric power of ionic conductors particularly AgBr and AgCl containing Cd2+ ions, as the lattice disorder in these salts is already well studied. The examination of thermoelectric power as a function of impurity concentration should disclose a concentration dependence brought about by the In [c(l - p ) ] term. Our equations can offer no interpretation of this result. 1 Domenicali, Rev. Mod. Physics, 1954, 26, 237. 2 de Groot, L’Efet Soret (North Holland Publishing Co., Amsterdam, 1945). 3 de Groot, Thermodynamics of Irreversible Processes (North Holland Publishing CO., Amsterdam, 1951). 4 Holtan, Electric Potentials in Thermocouples and Thermocells. Thesis (University of Utrecht, 1953) ; alternatively see Proc. K. Akad. Wetensch, B, 1953, 56, 498 and 510: also 1954, 57, 138. 5 Patrick and Lawson, J. Chem. Physics, 1954, 22, 1492. 6 Teltow, Ann. Physik, 1949, 5, 63 and 71. 7 Wagner, Ann. Physik, 1929, 3, 629. * Ebert and Teltow, Ann. Physik., 1955, 15, 268. 9 Prigogine, Etude Thermodynamique des PhCnomtnes Zrrkversibles. Thesis (University of Brussels, 1947). 10 Reinhold and Blachny, 2. Elektrochem., 1933, 39, 290. 11 Wirtz, Physik. Z., 1943, 44, 221. 12 Wert, Physic. Rev., 1950, 79, 601. 13 Seeger, Handbuch der Physik (Springer-Verlag, Berlin, 1955), vol. 7, part 1, p. 383. 14 Lidiard, Handbuch der Physik (Springer-Verlag, Berlin, 1957), vol. 20, 5 15 et seq. 15 Friauf, Physic. Rev., 1957, 105, 843. 16 Howard, to be published. 17 Lidiard, Phil. Mag., 1955, 46, 815 and 1218. 18 Schone, Stasiw and Teltow, Z. physik. Chem., 1951, 197, 145. 19 Verwey and Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, 20 Grimley and Mott, Faraday SOC. Discussions, 1947, 1, 3. 21 Lehovec, J. Chem. Physics, 1953, 21, 1123. Amsterdam, 1948).

ISSN:0366-9033    

DOI:10.1039/DF9572300113

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

DISLOCATIONS AND POINT DEFECTS BY F. C. FRANK H. H. Wills Physical Laboratory, The University, Bristol, 8 Received 4th April, 1957 The principal interactions between dislocations and point defects are : for both foreign and indigenous defects, adsorption, facilitated diffusion, facilitated precipitation ; for indigenous defects, creation and annihilation. Among examples recently discovered are the following. Dislocations in germanium allow the conversion of copper atoms from interstitial to substitutional solution, with remarkable and otherwise astonishing consequences. Processes making a large and continuing departure from equilibrium in the concentra- tion of indigenous point defects cause characteristic helical distortions of dislocations (Amelinckx et al.), or alternatively generate coaxial rings of dislocations (Mitchell et al.).The mechanisms of these processes are discussed. There can be no doubt that the interactions of dislocations and point defects are of much wider importance than can be illustrated in detail by the present experimental evidence. It is probable that they are important in almost every chemical process occurring in the interior of a crystal. This is particularly the case for processes involving the mobility of defects : but even for those that do not, e.g. some photochemical processes at very low temperatures, the effects of dislocations in displacing energy levels, and modifying selection rules by changing the local symmetry, may be of considerable significance. In this connection it is to be noted that a very moderate dislocation density such as 106 cm/cm3 makes some 1014 highly anomalous lattice sites per cm3, immediately adjacent to the dislocation line, and about 1016 in which the symmetry is modified by a strain exceeding 1 %.In badly distorted crystals these figures may be raised by a factor of 105. Since the writer has outlined the principal interactions elsewhere,l a brief summary will suffice here. We recognize (a) ENERGETIC INTERAcnoNs.-The long-range interactions are attributable to the strain field of the dislocation. For example, defects which expand the lattice are attracted into regions where the dislocation strain also expands it. Since all components of dislocation strain alternate in sign with direction around it, there is always some direction along which any kind of defect will be attracted to the dislocation.The leading term in this interaction is a change of potential energy inversely proportional to distance from the dislocation. The short-range inter- action is attributable to the disordered configuration at the dislocation core, which will also as a rule lead to a positive adsorption of defects on the dislocation, just as impurities are usually more soluble in a liquid than in a crystal. A further consequence of the same effects is that since, near the dislocation, the lattice is already strained beyond the Hooke’s law region, and resists further strain less strongly, any process such as the nucleation of a precipitate, which strains the matrix, meets a lower energy barrier here than elsewhere.(b) CHANGES OF MOBILITY. The pressure coefficient of diffusivity, whatever diffusion mechanism operates, is unlikely to be zero. Once again, we have a situation where the dislocation stresses enhance a property at one place and de- press it at another. The effect is mainly one on the activation energy of the process. In mechanism, it is presumably as a rule an increase of the mobility of defects on the dilated side of the dislocation, if it has 122 The depression is unimportant.F . C . FRANK 123 an edge component. Furthermore, lattice vacancies are among the defects which will show a positive adsorption on to dislocation lines (with an additional term in the attraction due to their reduction of elastic moduli). Hence we can commoiib anticipate enhanced diffusion coefficients along dislocation lines, and lower activa- tion energies for diffusion tending to make these effects of particular importance at low temperatures.(c) CREATION AND ANNIHILATION OF DEFECTS.-A dislocation having an edge component may be regarded as the edge (hence the name) of a restricted area of an additional lattice plane inserted into the crystal. The reticular density of the inserted plane is proportional to the sine of the angle between the plane and the Burgers vector of the dislocation. (There is a degree of freedom in the choice of this plane : but the maximum value of this angle is that between the Burgers vector and the dislocation line. For a pure screw dislocation this angle is zero, so that the reticular density of the associated inserted plane becomes zero.) It follows that any dislocation other than a screw dislocation can operate as a source or sink of vacancies and interstitial atoms by moving in some direction other than its glide direction, so as to change the area of the inserted plane. Such motion is known as " climb " in contradistinction from " glide ".It is the only process occurring in the interior of a crystal (free from cavities) by which the number of lattice sites can change. Unlike the interactions (a) and (b) which affect foreign and indigenous point defects alike, this one primarily concerns the indigenous point defects (vacancies, and interstitials which are constituents of the matrix). Under conditions close to equilibrium, say in a long steady anneal, the climb process will allow dislocations to move out of the crystal, or annihilate with each other, reducing its dislocation content : but in conditions far from internal equilib- rium in respect of concentration of point defects, the resulting motions are likely to increase the dislocation content, giving rise to an accelerating reaction.One of the processes by which this can occur (first described by Bardeen and Herring 2, is the analogue of the spiralling or loop formation by steps at the surface in crystal growth, or by gliding dislocations in crystal slip (at a " Frank-Read " source). The condition for this to occur is that the osmotic force per unit length on the dislocation (kT/A) In (c/cg) (where A is the area per lattice site created or destroyed in the plane in which the dislocation moves, c is the concentration and cg the equi- librium concentration of the relevant point defects) can balance the line tension force of order Gb2/r, where G is the rigidity modulus, b a lattice spacing, r the radius of curvature of the dislocation line.These forces are equal when 1- 1 00 GbA N -~ -- b kT I In (dco) I I In (c/co> I in a typical case. Thus departures from equilibrium by a percent or so can suffice to operate the sources if the free lengths of dislocation line are more than about a micron long. Should the crystal be of high perfection, free from dis- locations in regions as large as a millimetre let us say, or alternatively having its dislocations strongly anchored by adsorbed or precipitated impurity, then diffusion to the dislocations or to the free surfaces can fail to regulate the concentration of indigenous point defects to such an extent that 1 In (C/CO) I becomes of an order greater than unity.Then the defects can spontaneously nucleate dislocation loops-an interior process very similar to the nucleation of growth layers on a crystal surface at high supersaturation.& We now proceed to discuss a few special cases which have come to light recently. The role of dislocations in enabling the number of lattice sites to change has become the dominant theme, though all the other properties are involved. COPPER IN GERMANIUM Fuller et aZ.3 found that copper in germanium had the astonishingly high dif- fusion coefficient, for a substitutional solute, of 3-2 x 10-5 cm2/sec, at 700-900" C,124 DISLOCATIONS AND POINT DEFECTS depending little on temperature.Tweet and Gallagher 4 found that this was only the case for germanium of high dislocation content, and that the copper was only found in high concentration within about 20 microns of the dislocation lines, or the free surface, after 15-min diffusion at 710" C . This seemed to suggest that we had here a striking example of diffusion along dislocations while the lattice diffusion coefficient was about 10-8 : but such an explanation required either a quite in- credibly high diffusion coefficient along the dislocation lines, or a quite incredibly high range of influence of the dislocations in enhancing the diffusivity. Van der Maesen and Brenkman 5 suggested that copper in germanium dissolves both substitutionally and interstitially.Finally, Frank and Turnbull 6 proposed an interpretation (since accepted by Fuller 7) which accounts for all of the observa- tions. In equilibrium, the copper is predominantly substitutional, with a solu- bility c,' - 10-7 (atom fraction) but a small amount (ci' - 10-9) is interstitially dissolved. The latter has a high diffusion coefficient, of the order 3 x 10-3 cm2/sec, corresponding to a negligibly small activation energy. The diffusion of substitutional copper, as such, is negligible. The equilibrium vacancy con- centration at this temperature is relatively small, say about 10-10. Then, in highly perfect regions of the crystal, there is a rapid deep penetration, cor- responding to the diffusion coefficient Di - 3 x 10-3 cm2/sec, at a low concentra- tion level, corresponding only to the interstitial solubility.In regions of high dislocation content, on the other hand, the vacancy concentration is replenished by emission from dislocations. Interstitial copper atoms can then become substitutional, maintaining the mass action equilibrium e, = KCic, (where c,, ei, e, are atom fraction concentrations of substitutional copper, inter- stitial copper and vacancies respectively), Then the diffusion coefficient is about 100 times smaller, of the order observed by Fuller, because each copper atom spends only about 1 % of its time in the mobile, interstitial, condition. The low diffusion coefficient, of order 10-8 cm2/sec, implied by the observed distribution of sub- stitutional copper around dislocations and near the free surface is interpreted in a similar way as the diffusion of vacancies in a region saturated with interstitial copper. Most of the time the vacancies are rendered immobile by being occupied by copper atoms.The effective diffusion coefficient (really, of vacancies; ap- parently, of substitutional copper) can be shown to be equal to the self-diffusion coefficient of germanium divided by the solubility of copper, and this quantity (6 X 10-15/10-7) cm2/sec, estimated with a rather long extrapolation, agrees in order of magnitude with what is observed. The mechanism of the diffusion of copper in silicon, which is likewise rapid, is doubtless the same. Since silicon is transparent in the infra-red, this has enabled Dash 8 to observe some extremely interesting patterns of dislocations in silicon, by precipitating copper on them.The precipitation on screw dislocations is generally sparser than on others, but not necessarily absent. Slight changes in treatment determine whether it happens or not. This is observed. HELICAL DISLOCATIONS IN FLUORITE One point was overlooked by the writer in his previous discussion of these matters, until the observations of Amelinckx, Bontinck and Dekeyser 9 9 10 brought it to his attention. A screw dislocation cannot act as a source or sink of vacancies or interstitials : but by any non-uniform glide motion it ceases to be a pure screw dislocation, and then it can. If D is the sideway displacement, the length of the inclined portion, making an angle a with the Burgers vector, is D/sin a.The osmotic force per unit length on the dislocation is proportional to sin ct (through the factor l/A). Hence the condition that the osmotic force can overcome the line tension is approximately I In (C/CO) I > Gb4/kTD - lOb/D, independent of a. (This leads to the useful approximate rule employed by Amelinckx et al. that the conditions governing dislocation climb motions are the same, whatever the orienta- tion of the dislocation, if considered in terms of the configuration projected on toF. C. FRANK 125 a plane normal to the Burgers vector. It is only approximate, because based on Mott and Nabarro's line-tension approximation for the equilibrium of dislocations. A full treatment has to take account of the less important but by no means negligible mutual interactions of dislocations through their stress fields.) The required deviations from the exact screw direction are correspondingly so small as to be unobservable with the microscope if c differs from co by more than about 1 %.If 1 In (C/CO) I is relatively large, thermal fluctuation stresses may cause a sufficient glide out of the screw position. If a small portion in the middle of a line is pulled aside, the two ends of this portion have opposite inclinations tc, and consequently have opposite signs of edge component. They climb in opposite directions. SO the loop takes on a helical twist. The climb has locked it against gliding back into position, but by further climb it can both expand sidewise, and draw out more of the dislocation line at each end from the pure screw position.Each portion so drawn out continues the helical development. By receiving inter- stitial atoms or giving off vacancies, a right-handed screw' dislocation becomes a left-handed helix, and conversely (in the sense of either of the two statements converse to this one). The assumption of Amelinckx et al. was that the dislocation should initially be anchored at its ends in a position somewhat inclined to the pure screw direction. This should produce the result, but seems inessential. The observed helical dis- locations have axes very close to the Burgers vector directions-in fact essentially parallel to these directions within the uncertainty due to their imperfect uni- formity. Actually, they seem remarkably uniform, when one considers that the regular helix is elastically unstable-any loop becoming larger than its neighbours must be expected to grow faster than them, and retard their growth.The processes in which Amelinckx et al. first observed these helical dislocations consisted either in heating CaF2 with sodium or heating CaF2 with surface films of silver in hydrogen. The chemistry of the processes involved is not yet fully clari- fied, but deserves some general discussion. In the first place, one necessary process in all such methods of investigation is the introduction of excess metal (or extraction of halogen, so that excess metal remains) ; the excess metal is required to precipitate on the dislocations and make them visible. In the present case it is in solution at the high temperature in the form of F-centres.The observed helical winding of the dislocations signifies that processes changing the number of lattice sites have occurred. To describe such changes, we must first consider the nature of the CaF2 lattice. It is a face-centred cubic lattice of cations, with its tetrahedral interstices (of which there are two per cation) occupied by anions. The octa- hedral interstices are unoccupied at low temperature. At higher temperatures some anions transfer to octahedral sites (interstitial positions in this case) leaving some anion vacancies in the tetrahedral positions. This transfer does not affect the number of lattice sites, which only changes when cation vacancies are also produced (with a further production of twice as many anion vacancies, to main- tain charge neutrality).This (and its reverse on lowering the temperature) is a process which can occur by dislocation climb. The production of F-centres does not necessarily involve a large change in the number of lattice sites. Halogen atoms may be extracted from the surface, their electrons migrating to anion vacancies. The number of the latter is thus reduced, and is replenished mainly, no doubt, by increasing the number of anions in octahedral sites, and only to a lesser extent by the production of anion and cation vacancies together. If formation of substitutional hydride ions needs to be considered, the same comment applies to this also. The introduction of foreign metal needs special consideration in this case, because that provided is monovalent.The characteristic mono- valent halide structure, that of sodium chloride, stands in a special relationship to that of fluorite. It likewise consists of a face-centred cubic lattice of cations, with anions occupying the octahedral sites, a situation occurring as a high- temperature point defect in CaF2. Thus the conditions exist, in principle, for the126 DISLOCATIONS AND POINT DEFECTS formation of a continuous series of solid solutions passing from the fluorite struc- ture at the one end to the sodium chloride structure at the other, with a progressive decrease in the number of tetrahedral sites, and increase in the number of octa- hedral sites occupied by anions. Keeping the number of anions constant, ex- tracting calcium and substituting monovalent cations, the number of lattice sites increases by one for every calcium ion replaced.The foreign ions are provided at the surface, and must migrate into the interior to intermingle with the rest. They will therefore induce a net flux of cation vacancies from the interior, which must come from climbing dislocations. Alternatively, if cations can migrate inter- stitially, they ultimately enter on to new interior lattice sites formed on dislocations, with simultaneous emission of anion vacancies which migrate ultimately to the surface. It is most probably this process of formation of solid solution which is responsible for the intensive dislocation climb indicated by the helical dislocations. The foregoing discussion illustrates a guiding principle : that in ionic structures, since a lattice point involves both cations and anions, to count lattice points we may count the one or the other as convenient.In the present case, both because they occupy the simpler lattice, and because they are less mobile, it is most con- venient to count cations. GOLD IN ALKALI HALIDES Barber, Harvey and Mitchell11 have introduced a new way of decorating dislocations in alkali halides, by diffusing gold halides into them. The simplest process is to heat together a crystal of sodium chloride and a small piece of gold wire with chlorine sealed in a hard glass tube. In this case there is no excess metal in the ordinary sense, the hot crystal being colourless. On cooling it becomes coloured by spontaneous decomposition of the gold chloride in solid solution.The fine deposit of colloidal gold marks the positions of the dislocations with particularly high resolution. The precipitation occurs when the temperature falls to about 400" C and makes visible that configuration of dislocations which was present at that temperature. We have to conclude that though aurous chloride itself is unstable, the entropy of mixing is sufficient, at high temperature, to stabilize a (very dilute) solid solution of it in an alkali halide. Dislocations are revealed in this way, heating to any temperature greater than 500" C. If the treatment is given for 4 h at 650" C, in the case of NaC1, it also produces some helical dislocations, and very large numbers of coaxial ring systems. These rings, encircling their common [110] axis, lie on narrow double cones, widening out- wards, extending both ways from a central speck near to which they are evidently produced.They appear to be a distinct phenomenon from the coaxial ring systems of Amelinckx et al., which can be interpreted as formed from dislocation helices by combination with another dislocation (which may be the continuation of the same one) : here, on the contrary, there are rare helical systems, which may be suppossed to have formed from the coaxial ring systems by combination with a screw dislocation. Following a discussion with Dr. Mitchell, the best interpretation of these phe- nomena would appear to be that there is intense stress at the central speck (due to chlorine gas pressure from the decomposition of the dissolved gold chloride): that this stress causes some slip making adjacent dislocation loops, the edge com- ponents of which immediately climb, emitting vacancies which diffuse to relieve pressure at the decomposition nucleus.The loop thus becomes a Bardeen- Herring source generating positive loops (those which have the extra plane within the loop). The ring system generated by such a source would normally be a coplanar system of concentric rings, and the elastic interactions between the rings would tend to keep it so: but in the stress field of the central bubble of com- pressed gas the rings are strongly repelled from this bubble, and move away along their glide axis. Then the next ring formed repels the first and pushes it further away along this axis and so on,F. C. FRANK 127 Jones and Mitchell (to be published) have observed some rather similar though generally less regular systems of rings or helices around specks of impurity (probably oxide) in silver bromide heated in bromine. A similar explanation can apply in this case, positive dislocation rings forming near to the oxide speck, deriving their cations from the speck and their anions from surface bromine respectively, while oxygen gas released at the speck causes the stress which drives them away. 1 Frank, Chemistry of Crystal Dislocations, chap. 1, in The Chemistry of the solid State, ed. Garner (London, Buttenvorths, 1956). 2 Bardeen and Herring, Imperfections in Nearly Perfect Crystals, ed. Shockley et al. (New York, Wiley, 1952), p. 277. 20 Frank in Deformation and Flow of Solids ed. Grammel (Report of IUTAM, Madrid Colloquium, 1955 ; Springer, Berlin, 1956), p. 73. 3 Fuller, Struthers, Ditzenberger and Wolfstirn, Physic. Rev., 1954, 93, 1182. 4 Tweet and Gallagher, Physic. Rev., 1956, 103, 828. 5 Van der Maesen and Brenkman, J. Electrochem. SOC., 1955, 102,229. 6 Frank and Turnbull, Physic. Rev., 1956, 104, 617. 7 Fuller and Ditzenberger, J. Appl. Phys., 1957, 28, 40. 8 Dash, J. Appl. Physics, 1956,27, 1193 ; 1956, 27, 1387 (cover photograph). 9 Bontinck and Amelinckx, Phil. Mag., 1957, 2, 94. 10 Amelinckx, Botinck, Dekeyser, and Seitz, Phil. Mug., 1957, 2, 355. l 1 Barber, Harvey and Mitchell, Phil. Mag., in press.

ISSN:0366-9033    

DOI:10.1039/DF9572300122

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年代:1957

数据来源: RSC

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ABNORMAL VALUES OF A AND E IN MIGRATION PROCESSES BY A. R. UBBELOHDE Dep t. of Chemical Engineering, Imperial College, London, S. W.7 Received 13th February, 1957 This note has the object of raising certain problems about migration mechanisms for discussion. No definitive solutions can be presented in most cases. As is well known, standard theories of migration processes, whether in dif- fusion or in ionic conductance, evaluate mobilities in terms of an activation barrier UO and a hopping frequency v. For example, in the discussion presented by Mott and Gurney 1 the diffusion coefficient is written where n/N is the fraction of defect sites, regarded as between successive potential minima, v is the hopping with the frequency of vibration, and the constant independent, a the distance frequency, which is equated C=exp(-TdY) CCV~ dUo allows for the change in barrier as the crystal lattice expands.The ionic mobility v of the same particles in the ideally simple case is governed by the Einstein relation eD =vkT. More information is generally available on conductance, than on diffusion. Furthermore, conductance can be measured after only comparatively small quantitative migration, whereas for analytical reasons diffusion must frequently be allowed to proceed further, with greater consequent lattice distortion, before a measurement can be completed. Much of what follows refers to the first instance to migration in ionic conductance. Various complications in the simple formula have been inferred in the light of varied experimental evidence. Thus plots of the logarithm of ionic conductance against temperature nearly always show a kink, with higher activation energies controlling the migration process near the melting point.It has been variously postulated that the lower portion of such plots, which is found to be dependent on previous history of the sample, arises : through the migration of particles along cracks and channels in the crystal-these are associated with lower activation energies E than for diffusion through the lattice itself. The number of such particles is also smaller, so that small A and E values should occur together ; through the presence of ions of different electrovalency, accompanied by holes, which permit a different migration mechanism with smaller A and E values than through the lattice itself.At higher temperatures the normal lattice migration mechanism predominates for purely algebraic reasons. Again small A and E values tend to occur together ; 128A . R . UBBELOHDE 129 This (iii) through transport numbers changing as the temperature changes. need not require any concurrent trend in A and E. Each of these explanations may apply in appropriate cases. One of the problems for general discussion is how far it is a coincidence that low values of E tend to be associated with low values of A , and conversely, in migration processes in solids. The well-known parallelism between A and E in reaction kinetics in the gas phase arises in part because processes with high F have no chance of being observed under ordinary experimental conditions unless 14 I2 8 i 4 10 2 0 3 0 4 0 5 0 T i m e ( s e c ) FIG.1.-Decay of current at constant voltage in caesium chloride at current densities about 10-6 Ajcm.2 they also have high A values. Does the same explanation hold for solids? If they exist certain processes with low E but abnormally high A values could be observed, whereas it would not be experimentally practicable 7b to observe pro- cesses with high E unless A was also high. General considerations of the theory of rate processes favour a correlation between low A and low E values 7a since if the entropy term exp (AS*/R) has a large +ve value, the process is not very selective energetically, and the activation energy E* in the term exp (- E*/RT) has to approach the value for complete dissociation from any constraints.Values of the entropy term (exp AS*/R) which are small or negative imply much more selective routes for passing from the ground state to the activated complex. Lower activation energies may be feasible provided the route is highly selective. A second problem for general discussion refers to the build-up of an activated state of the solid before a steady-state migration can be established. For example in measurements of ionic conductance, the current flowing as the result of an applied voltage often decays with time, Fig. 1 illustrates results with caesium halides.2 Some kind of activated state of the solid appears to be built up initially. E130 ABNORMAL VALUES OF A AND E and eventually leads to a steady state. In this instance, the effect becomes less pronounced as the temperature rises and eventually at sufficiently high temper- atures any build-up becomes imperceptible.Since the current decays as a result of the applied voltage, the build-up is formally equivalent to the setting-up of a space charge, as a result of charge migration. The micro-structure of this space charge is by no means clear, but it obviously affects the fundamental assumptions made in deriving any simple hopping mechanism of eqn (1). One possibility could be that the rates of hopping of +ve and - ve charge carriers are very different. The distribution of lattice defects must adjust itself to a value in the potential gradient, which differs from the equilibrium value in the absence of a potential gradient, by virtue of the space charge produced.An- other possibility is that the statistical fluctuations which are required to maintain the simple hopping regime (1) do not in fact suffice in crystals, particularly at lower temperatures. Eqn. (1) implies independent micro-fluctuations capable of supplying the activation energy E to surmount the barrier between one site and the next, with a frequency v. The relaxation time of such fluctuations must be small compared with the reciprocal of the hopping frequency, for the simple formula to hold, Consideration of fluctuations in gas-migration phenomena illustrates how the simple regime need not apply for certain transport processes. For example, in the transfer of heat energy in gases, the carriers exhibit a well-known " persistence of velocities " whereby the hotter molecules migrate faster, so that overall heat transfer is greater than in the simple kinetic theory which neglects persistence.In terms of relaxation times, as the molecules travel down a gradient, the relaxa- tion time for maxwellization over all degrees of freedom in any one layer of gas is comparable with the time of movement from ode layer to the next, instead of being much smaller. This leads to the persistence of velocities. With solids, a particle which has just made a hop is endowed with a large amount of activation energy. Unless the relaxation time for dissipation of this energy is small compared with the reciprocal of the hopping frequency a migration phenomenon analogous to the persistence of velocities will apply.The persistence of activation energy after a hop is likely to be much more significant for interstitial mechanisms of diffusion than for hole mechanisms. A unit of structure still retaining the whole or part of its activation energy after a hop will always find interstitial neighbouring sites to hop into, which favours the condition for chain migration, that the time before the next hop must be comparable with or smaller than the relaxation time for the dissipation of activ- ation energy. On the other hand, in a hole mechanism of diffusion a unit of structure after making a hop is most unlikely to find another hole in front of it. There is plenty of time to dissipate the hop energy before the same unit can move forwards. In chain hopping, particular units may move through several lattice positions before they lose their advantage. This can involve thermal segregation effects if a temperature gradient is combined with a diffusion gradient; such effects are well known in fluids but have not been specifically studied in solids to any great extent.Another way of describing this situation is to refer to such migration without complete deactivation after every hop, as a chain mechanism. The principles of detailed balancing require that if a process proceeds by chains, instead of isolated fluctuations, the probability of its initiation must change correspondingly, in order to maintain Boltmann equilibrium.3 A further complication is that in a migration process appreciable departures from Boltzmann equilibrium may be built up, by way of a space charge, or abnormal distribution of lattice defects. Detailed balancing can only be applied in the limit for very small displacements from equilibrium.A third group of problems refers to the trend of A and E values for migrationA . R . UBBELOHDE 131 in solids, as a transition is approached and traversed. Most of the scanty informa- tion in this field refers to ionic conductance measurements, though diffusion measurements by radio-tracer techniques are extending information to non-ionic solids. Perhaps the simplest group of phenomena appears in connection with premelting, and melting in crystals (fig. 2). In the premelting region of ionic crystals such as AgCl or KCNS,4 or caesium salts,2 the steep rise in conductance implies formally very large activation energies.Actually what is happening is b M 2 d - 5 -b -7 22 2 4 26 28 FIG. 2.-Plot of log conductance against 1jT for solid KCNS. that extensive co-operative defects begin to appear in the crystals, so that the steep temperature dependence no longer refers strictly to migration. In such cases the simple hopping eqn. (1) from one lattice point to the next cannot be applied to evaluate the micro-mechanism. In ionic melts, so far as is known the much lower values of E, are associated fairly generally with much lower A , values. E, 11 0 crystal/melt crystal/melt salt NaCl 25.8 1.3 x 104 KCI 23.0 3.1 X 106 KNO3 15.9 5.7 x 1011 KSCN 8.1 300 KHS04 2.2 - 0.2 Values are taken from ref. (4) and ( 5 ) and from Rogers and Ubbelohde, Trans. Faraday Soc., 1950, 46, 1051.Ionic melts must possess a quasi-crystalline structure in which the Sve and - ve charges are predominantly arranged on interpenetrating lattices. The specific heats are not very different from those of the crystals. UO is smaller and dUo/dV much smaller in the more expanded quasi-crystalline melt. Nevertheless the ratio A , crystallA, melt is seen to be far from regular, in relation to the values Eu crystal/E,melt. This can hardly be explained in terms of the simple hopping eqn. (1) and may imply chain migration in solids in cases where the ratio is exceptionally large. Further examples are needed to test this possibility ade- quately.132 ABNORMAL VALUES OF A AND E Just as in premelting phenomena, abnormally large A and E values can arise in the neighbourhood of a transformation in the solid state.This has been attributed to migration along lattice scars and co-operative defects, that arise from the coexistence of sub-regions in a crystalline hybrid.8 Fig. 3 illustrates - 4 -6 -8 b M 0 2 U -10 ~ - I 2 1 t r a n t i t i o n I I I I I I I 1 2 0 t 4 28 32 1 0 4 - T FIG. 3.-Plot of log conductance against 1/T for solid KNO3. the conductance of KNO; around its transformation temperature.5 Again, in the transformation cc-iron + y-iron, a remarkable change in migration parameters takes place, due possibly to migration along lattice scars.6 Reported diffusion coefficients are for a-Fe stable below 910" C , D = 34,000 exp (- 77,200IRT) and for y-Fe, stable above 910" C, D = 0.00104 exp (- 48,OOO/RT). 1 Mott and Gurney, Electronic Processes in Ionic CrystuZs (Clarendon Press, Oxford, 1940). 2 Harpur, Moss and Ubbelohde, Proc. Roy. SOC. A , 1955, 232, 196 ; cf. Brennecke, J. Appl. Physics, 1940,11,202 ; Manning and Bell, Rev. Mod. Physics, 1940,12,215. 3 for a discussion of co-operative or chain fluctuations, cf. Ubbelohde, Trans. Furadav SOC., 1937, 33, 1198, 1203. 4 Plester, Rogers and Ubbelohde, Proc. Roy. SOC. A , 1956, 235, 469. 5 Davis, Rogers and Ubbelohde, Proc. Roy. SOC. A, 1953, 220, 14. 6 Birchenall and Mehl, J. AppZ. Physics, 1948, 19, 217. 7 (a) Ubbelohde, Report Xrne Conseil de Chimie Institut Solvuy, 1956, p. 491 ; 8 UbbeIohde, J. Appl. Physics, 1956, 7, 313. (6) Barrer, Report Xme Conseil de Chimie Institut SoZvay, 1956, p. 490.

ISSN:0366-9033    

DOI:10.1039/DF9572300128

出版商:RSC    

年代:1957

数据来源: RSC

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DIFFUSION OF NICKEL43 IN NICKEL OXIDE (NiO) BY R. LINDNER AND A. AKERSTROM Laboratory for Nuclear Chemistry, Chalmers University of Technology and The Swedish Institute for Silicate Research, Gothenburg, Sweden Received 12th April, 1957 The diffusion of nickel-63 in NiO in form of crystals, metal oxidation layers and sintered pellets has been measured in the temperature range 740-1400" C by the surface decrease method. The temperature function of the self-diffusion coefficient is represented by the equation : D = 1.7 x 10-2 exp (56,00O/RT) cm2 sec-1. Preliminary experiments concerning the self-diffusion of nickel in nickel oxide sintered pellets have previously been published.1 The results and especially the high activation energy found were rather doubtful and it had to be assumed that the short-lived radioactive tracer nickel-65 (half-life 2-56 h) was not suitable for such experiments.Consequently the investigation has been repeated using the long lived nickel-63 (85 years half-life) which, because of its soft P-radiation of 65 kV maximum energy, has to be measured in a flow gas counter. The investigations were initiated by one of us in co-operation with W. J. Moore during a stay at Indiana University (U.S.A.), but it was decided, considering the importance of the investigation for the knowledge of nickel metal oxidation, that two separate and independent sets of experiments should be performed. In spite of practically identical material and equipment the preliminary results ob- tained differed considerably : Moore 2 gave a preliminary value for the activation energy of self-diffusion of 43 kcal whereas our preliminary results 3 yielded a value of 62 kcal/mole.The latter value has been practically confirmed by this in- vestigation. There should be a close connection between the activation energy for self-diffusion of nickel and that for the formation of nickel oxide by metal oxidation, as nickel oxide is considered to be a typical p-conductor and the diffusion of nickel via vacancies should determine both processes. The commonly accepted value for the activation energy of metal oxidation is about 41 kcal/mole as stated by Gulbransen and Andrew.4 In a later report 5 the same authors report a higher activation energy of about 68 kcal for temperatures above 900" C but assume that this value has no real significance as the oxidation process is disturbed by periodical cracking of the oxide layer.EXPERIMENTAL The following nickel oxide preparations were used in the diffusion experiments. Pure nickel oxide crystals, which were practically black, cleaved from a boule obtained by the Verneuil process. The composition of this material was checked by measurements of electric conductivity which extrapolated to room temperature showed values of the order of 10-7 ohm-1 cm-1, which corresponds to a deviation from stoichiometric composition of about 10-6 according to Morin.6 The sintered samples were prepared from nickelous carbonate (Bakers A.R.) which had been heated to about 700" C to decomposition. After pressing, the pellets were fired at 1300" C and obtained a density of 6.2 (85 % of theoretical density).Their electrical conductivity was comparable to that of the crystals mentioned above. Spectroscopically pure nickel foil (Johnson and Matthey), identical with the material used by Gulbransen was oxidized at 1000" C. The radioactive tracer nickel-63 was ob- tained from Harwell and its purity checked by complexing elution from a Dowex 50 ion 133134 DIFFUSION IN NICKEL OXIDE exchange column with EDTA in presence of cobalt carrier. The absorption curve of this tracer and all other measurements were taken in a flowing methane proportional counter of about 40 % efficiency. No deviating absorption characteristics were found if the 90 % argon + 10 % methane " Q "-gas was used. The absorption curve, obtained with thin absorbers of plastic (Pliofilm 0.6 mg cm-2) and aluminium (1.1 mgcm-2) is shown in fig.1 and is practically identical with that ob- tained by Brosi, Borkowski, Conn and Griess.7 As can be seen from the figure, ab- sorption occurs according to two distinct groups of energy corresponding to absorption coefficients of 4950 cm2 g-1 and 1150 cm2 g-1. These values are confirmed by evaluation of the self-absorption curve in nickel metal as given by Brosi, Borkowski, Conn and m9 cm-' FIG. 1.-Absorption curve of Ni63. Griess in which case a composite absorption coefficient of 4200 cm2 g-1 can be stated. This has to be considered when evaluating the diffusion experiments according to the surface decrease method.* For short diffusion times and/or low diffusion temperatures an approximation can be made using only the absorption of the very soft radiation, whereas at high temperatures and/or long diffusion times the absorption of the harder radiation may be used for evaluation.There is, however, a range corresponding to a decrease in surface activity between 40 % and 10 % in which none of these approximations is allowed. Because of this and because of the necessity of finding one equation for evalu- ation for the whole broad temperature range investigated by us, a two-term absorption formula had to be developed which fits the whole range of temperature and times. DIFFUSION EXPERIMENTS The deposition of the radioactive tracer on the diffusion specimen was accomplished by evaporation from a metal foil in waclco and condensation in an apparatus as described by Lindner and Parfitt.9 The metal foil was coated with radioactivity by electrolysis. Because of its higher melting point tungsten was used instead of tantalum which was used in the earlier experiments.A negligible amount of evaporation of foil material may occur, being higher with tantalum. Special experiments with nickel oxide with small additions of tantalum oxide did not show any appreciable effect on the self-diffusion of nickel by added tantalum, within ow limits of error. The radioactive nickel coating (less than 0.1 mg cm-2) was oxidized in air at 500" C during 10 min. During the diffusion run, no evaporation of radioactive nickel oxide occurred as confirmed by an experiment using two coated samples, the active sides of which were separated by a thin platinum ring. No measurable transfer occurred between the coated samples even at temperatures as high as 1100" C.R .LTNDNER AND A. ~ K E R S T R ~ M 135 RESULTS The results are shown in fig. 2. No systematic deviation between the different sets of samples could be found, meaning that diffusion was not affected by the state of the diffusion sample (crystal, sintered pellet or oxidized metal strip). Although there is some scatter the large range in temperature allows sufficient accuracy in determination of the activation energy, which according to the method of least squares is calculated to be 56,040 f 1280 cal/mole. The results of Gulbransen and Andrew for low- and high-temperature oxida- tion of nickel metal are also shown in fig.2 as dashed lines. The values coincide quite well although at lower temperatures the metal oxidation shows a markedly smaller activation energy. 107 T FIG. 2. DISCUSSION The constant value for the activation energy of self-diffusion as obtained from our measurements over a wide range of temperature makes it probable that the true activation energy of self-diffusion has been measured. The activation energy for metal oxidation, however, is markedly higher at higher temperatures and shows a value of 68 kcal, which according to Gulbransen and Andrew is without real physical significance. At temperatures below 1000" C the activation energy of 42 kcal has been commonly accepted, but we are forced to assume that a real difference exists between the activation energies for the two processes in this temperature range.Possibly the diffusion conditions in a growing metal film cannot be completely considered identical to those in a stabilized pure oxide even for p-conductors. A direct experimental investigation on an eventual mobility of oxygen in nickel oxide seems desirable.10 We wish to express our gratitude to Mr. W. Kebler, Speedway Research Laboratory, Linde Air Company, Indianapolis, U S A . , for the nickel oxide single crystals ; and Dr. E. A. Gulbransen, Westinghouse Research Laboratories, Pittsburgh, for the pure nickel metal foil. The support by the Swedish Council for Technical Research and Prof. C. Brossett, Head of the Swedish Institute for Silicate Research is gratefully acknowledged.136 DIFFUSION I N NICKEL OXIDE 1 Lindner and Akerstrom, 2. physik. Chem., 1956, 6, 162. 2 Moore, 6th Meeting Physical Chemistry (Paris 1956), paper 98. 3 Lindner, 3rd Int. Meeting on the Reactivity of Solids (Madrid, 1956). 4 Gulbransen and Andrew, J. Electrochem. Soc., 1954, 101, 128. 5 Gulbransen and Andrew, Westinghouse Scientific Paper, 1956, 60-94602-1 -P9. 6 Morin, Physic. Rev., 1954, 93, 1199. 7 Brosi, Borkowski, Conn and Griess, Physic. Rev., 1951, 81, 391. 8 Steigman, Shockley and Nix, Physic. Rev., 1939, 56, 13. 9 Lindner and Parfitt, J. Chem. Physics, 1957, 26, 182. 10 Ilschner and Pfeiffer, Nuturwiss., 1953, 40, 603.

ISSN:0366-9033    

DOI:10.1039/DF9572300133

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年代:1957

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摘要:

DIFFUSION IN IONIC CRYSTALS AND THE PROCESS OF SINTERING BY W. JOST AND H. J. OEL Institut fur Physikalische Chemie, der Universitat, Gottingen Received 7th Febnmry, 1957 It seems of interest to investigate the connection of sintering and the process of self- diffusion in solids. The " sintering temperature '' of a powder, according to Tammann may be defined as the temperature at which a stirrer sticks in a powder, the temperature of which has been raised slowly. A mean displacement may be calculated from the coefficient of self-diffusion. This is compared with the lattice constant, in order to show whether the self-diffusion is of the right order of magnitude to account for the sticking together of the powder particles. If sintering and self-diffusion have the same underlying mechanism the temperature dependence (i.e.the energy of activation) should be the same for both processes. So far only for some simple metals these two energies of activation have been compared. For compounds, however, there is more than one coefficient of self-diffusion. In ionic crystals, the difference between the coefficients of self-diffusion for the anion and the cation is as a rule, many powers of ten. During the process of sintering the crystal lattice is destroyed in some parts and rebuilt in others. To this end, in an ionic crystal, the same amount of cations and of anions has to change places. Because cations as well as anions have to move, the slower ones are rate determining, as long as the difference in mobility is large. In the processes of tarnishing and ionic conductivity, on the other hand, either cations or anions have to move in this case, the faster ion being rate determining.For these processes the disorder in the cation partial lattice (in ionic crystals the mobility of the cations is mostly much higher than that of the anions) is usually of im- portance, and the anions need not be considered. In sintering, however, self-diffusion of the anions might be involved, and so disorder in the anion partial lattice becomes im- portant. In connection with these considerations the following experiments are being made : measurement of the anion self-diffusion with radiative isotopes (e.g. iodide in AgI), measurements of the velocity of sintering at various temperatures and other experiments which may supply more detailed information on sintering.In AgI it has been shown that cation diffusion is certainly not the rate determining step. Tammann defined as sintering temperature the temperature at which a stirrer moving in a slowly heated powder sticks. One may look for an interrelation of sintering and self-diffusion, for instance in metals, as has been done repeatedly, e.g. by Kuczynski.1 For ionic crystals, there are at least two different diffusion coefficients, often of different order of magnitude, and the process of sintering may be more complicated. We started a n investigation of the rate of sintering and of other relevant properties of the silver halides. Here the mobilities of the cations are rather high and well known, and apparently there is no direct connection between these mobilities and sintering.It may well be that the mobilities of the slower ions determine the rate of sintering. In fig. 1 and 2 we show electric conductivity and specific heat of silver iodide. There are two regions of marked anomalies : between about 120" C and the transi- tion point at 144.6" C , and above 150" C. The lower region can easily be attributed to the process of disordering of the cations (reaching an amount of some percent close to the transition point). The anomalies in the high-temperature region are 137138 PROCESS OF SINTERING less easily explained. It seems certain that they are not due to further increase in cation disorder which is extremely high in cc-silver iodide but only slightly dependent on temperature.T'K 4qo 4$0 sp bQo7?oqQqoo - I - -2- b . F3 -4 - - 5 - -bhl 2b A 22 2tl tb tb I4 1:2 1 $-x lof FIG. 1. & u" "i 2 I 100 200 4 0 tempoc FIG. 2. (after Lieser, Z, physik. Chem., 1955, 2, 238) The mobility of the anions cannot be obtained by measurements of the conductivity and transport numbers (tanion being < 10-3). Consequently, we determined coefficients of self-diffusion by means of 1131 (fig. 3). The results may be represented by D = DO exp (- Q/RT) ; DO = 4.4 x 10-4 cm2 sec-1, Q = 16-2 kcal/mole. Because we cannot give separate values for energy of disorder and energy barrier, we cannot definitely prove that the anomalous specific heat above - 150" C is due to anion disorder. We can, however, choose a reasonable value for the energy of anion disorder which is compatible with both the observed anion mobility and the specific heat.W. JOST AND H . J . OEL 139 4 2:2 2:O b.8 I:b IOYT FIG. 3. FIG. 4. FIG. 5.140 PROCESS OF SINTERING Fig. 4 shows the results of sintering experiments. The ordinate gives the height of powder in a glass tube, as function of time for several temperatures. If sintering is determhed by the mobility of the slower particles, one may try a transformation of the abscissa of fig. 4, using D,t instead of t, as shown in fig. 5 and 6. IOC 90 m & d ao ~ . . . . . . . . . . 1 n3Oc n 23b0c - m 303OC ------ Ip 3bb°C -.-.- FIG. 6. One may further show that observed diffusion coefficients are of a reasonable order of magnitude to account for sintering. If in the stirring experiments a contact time between neighbouring grains of about 10-1 sec is assumed, with a diffusion coefficient of 2 x 10-11cm2sec-1, a depth of penetration of about 50 lattice distances is obtained. 1 Kuczynski, J. Metals, 1949, 1, 169.

ISSN:0366-9033    

DOI:10.1039/DF9572300137

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

POINT DEFECTS PRODUCED BY IRRADIATION AND THEIR ANNEALING IN IONIC AND COVALENT CRYSTALS BY F. P. CLARKE Atomic Energy Research Establishment, Harwell, Berks. Received 8th February, 1957 Atoms are displaced by interaction with energetic particles and point defects are pro- duced. The diffusion of these defects to normal lattice positions may be described mathematically, using as basis a simple physical model, and the description obtained in this way accords with certain experimental data. On this treatment the activation energy for movement of any type of defect is considered to be a discrete quantity. Another method of describing these annealing phenomena is based upon the as- sumption that the rate processes follow a type of equation familiar in gaseous kinetics. It appears that if this description is to accord satisfactorily with certain experiments a distribution of activation energies has to be assumed.These different treatments are contrasted. MODELS FOR THE DISORDER PRODUCED BY HEAVY PARTICLE MOVEMENT THROUGH A LATTICE A heavy charged particle penetrating a solid may lose its energy (i) through inelastic collisions in which there is excitation of the electronic system of the atoms or ions, or (ii) by elastic collisions in which there is momentum transfer to the nuclear mass of the stationary atoms. For high particle energies the losses will be overwhelmingly by excitation ; for lower energies elastic collisions will predominate, but there is not necessarily a sharp cut-off separating these processes, which actually overlap.For a carbon atom moving in diamond for example, ionization losses may be ignored for energies of the moving atom below about 104 eV. The figure will be higher the higher the atomic number of the moving atom, and for potassium in potassium chloride it is about 105 eV. To move an atom from its position in the lattice some minimum energy will be required. Determinations of the minimum energy have not been made for covalent or ionic solids, but the figure is of the order 25 eV. This is based on an estimate by Seitz which has worked out fairly well for metal systems. Hence energy transfer to the nuclear mass which exceeds this minimum value may result in an atomic displacement within the lattice. For uncharged particles such as neutrons, elastic collisions with the nucleus may be made, and at these a considerable proportion of the energy of the particle may be given up.If this energy is greater than -25 eV a displaced atom may result. If the energy is greater than twice this, the knocked-on atom may itself displace other atoms and continue to do so until its energy is used up. With the range of neutron energies met with in a typical pile experiment, the knock-on energy for atoms with atomic numbers greater than 10 is well below the range at which ionization losses might occur, and so only elastic collisions need be con- sidered in interactions between the primary displaced atom and the atoms remaining in the crystal. The maximum energy which a knock-on can lose by non-relativistic elastic collision is given by EM = 4M1M2 2E, where E is the energy of the incident (M1+ M2) atom and hfl its mass.M2 is the mass of the initially-stationary atom. For low energy encounters it is assumed that the collisions are equivalent to those 141142 DEFECTS PRODUCED BY IRRADIATION between hard spheres of a suitably chosen radius.1 In this case collision will impart any energy between Em and zero with equal probability. Hence a two- dimensional representations of the way in which a primary knock-on might dissipate its energy is shown in fig. 1. The incident charged particle makes a collision at A, and a lattice atom is knocked-on to make a further collision at B. Here the atom gives up some random amount of its energy to a second atom, and both proceed to make further collisions. At some collisions, such as C, only sufficient energy is imparted to cause one displacement and an interstitial atom results.In general it is to be expected that the ratio of interstitials to vacancies will be fractional near the point of primary knock-on and greater than unity at the outermost parts of the region of disorder. FIG. 1. interstitial - track of interstitial 0 vacancy - - - track of incident particle. It will be noticed that in the figure the path of a knocked-on atom is shown longer near the point of primary impact. This is because the probability of collision varies inversely with the particle energy, and at around 100 eV or so a particle will make a collision about once every atomic spacing. As the number of incident particles, and hence the number of disordered regions, increases, a point will eventually be reached when the regions of disorder begin to overlap appreciably. When considering the annealing kinetics both of these possibilities have to be considered separately.An alternative model for the distribution of displaced atoms is based on the following arguments. It has been stated that the energy required to force an atom from its place in the lattice to an interstitial position is probably about 25 eV. This energy is required to overcome the potential saddle separating the normal lattice atom and the interstitial. Once over this saddle the atom will drop into a potential trough and an amount of energy, depending on the trough depth, will be released. Alternatively, if an amount of energy less than that required for displacement be given to an atom, this will be given up to the lattice as vibrational energy.In either event energy transfer by elastic collisions may cause a localized rise in temperature. This will operate over a spherical region surrounding the atom of origin, or if a number of atoms in a line are excited in this way a cylindrical hot region will result. These regions have been called spherical and cylindrical thermal spikes.2 Their duration is about 10-12sec and the magnitude of the temperature rise is not considered sufficient to allow general recombination of any vacancies or interstitials in this time. Brinkman 3 has suggested that during the last part of its movement through the lattice, the primary knock-on must be regarded as having a diffusive-typeF .P. CLARKE 143 movement. Thus below some critical energy the remaining kinetic energy of the moving atom will be dissipated collectively over a small volume of the crystal. This may cause a rise above the melting point. The duration of this displacement spike, as Brinkman has called it, is estimated at about 10-11 sec. Any interstitials or vacancies within the melted region must conceptually be replaced by high and low local degrees of atomic disorder. Within the duration of the spike the local strains due to these abnormal concentrations would not persist and so in resolidifi- cation most of the vacancies and interstitials would have disappeared within this region. The size of the region is estimated as under 50 diam. in a tyEical solid. During the initial part of the movement of the knocked-on atom, however, vacancy pairs can still be formed together with thermal spikes.A diagrammatic representation of these ideas (after Brinkman) is shown in fig. 2. On this figure, FIG. 2. Reproduced, with acknowledgment, from J. Appl. Physics, 1954, 25, 962. suggested features of the displacement spike are also shown: (i) the formation of dislocation loops within the displacement spike during the resolidification, and (ii) the formation of small microcrystals of new orientation. Brinkman considers that the main product of the displacement spike should be the dislocation loops, though more rarely an array of such loops could form the boundary of a microcrystal. Lomer and Cottrell4 consider that these loops would disappear rapidly after resolidification.These models have been suggested initially for the metallic state, where a clear-cut means of distinguishing between them has not been attained. As far as covalent and ionic materials are concerned most of the thinking so far has been in terms of point defects or small combinations of point defects. The important point about these models is that on all of them we would expect some vacancy- interstitial pairs to be formed, though in different amounts on each model. The most fruitful approach as far as annealing is concerned is to look for physical property changes which will allow changes in the numbers of these point defects to be followed as functions of temperatures and time. In some ionic solids, notably the alkali halides, point defects may be produced by X-, or y-irradiation, at energies where direct momentum transfer is ruled 0~t.5.2 In such cases there is no question of displacement spikes occurring.Electrons144 DEFECTS PRODUCED B Y IRRADIATlON will produce point defects distributed randomly, provided they are energetic enough to penetrate the crystal. To sum up, with any of the types of radiation considered, point defects will be produced. These may be distributed randomly in the lattice (electron irradi- ation) or in groups (neutron irradiation) as well. In considering models for the annealing processes it will be assumed that any point defects or groups of point defects will not be present in sufficient numbers to interact with each other. For neutron irradiation, if the displacement spike model is valid, we can ignore the groups as far as point defect annealing is concerned, since the spike will itself anneal them.Hence it may be approximated that the distribution of defects is random. (i) DISCRETE ACTIVATION ENERGY tribution of the defects may be divided into ANNEALING PROCESSES On the simple damage models which have so far been considered the dis- (i) those pairs within each others perturbing influence, (ii) those pairs outside this region. Now, interstitials and vacancies will cause the surrounding lattice to become distorted. Fig. 3 shows a two-dimensional representation of the way in which FIG. 3.-The way in which a lattice distortion will assist defect movement. A and B are forced towards V, thus favouring movement of I to normal lattice site B.this distortion might take place in an ionic lattice. The defects perturb the poten- tial barrier to diffusion, so as to favour the mutual annihilation of the defects. In interstitial movement the saddle of the potential barrier is lowered to, say, E, so that the frequency of movement towards the defect under thermal excitation at temperature To K will now be given by v = vo exp (- E/kT), where vo is the lattice frequency. In vacancy movement, the potential barrier for an atom adjoining the vacancy and between it and the interstitial will be lowered, and the frequency of move- ment of this sort of atom into the vacancy will also be increased. This is equiv- alent to a movement of the vacancy towards the interstitial with the same jump frequency. Thus, suppose that the effect of the perturbation is to lower the barrier for vacancy movement from 0-5 eV to 0.4 eV at room temperature, then on the average the vacancy will jump towards the interstitial one hundred times for every jump that it would otherwise have made.Furthermore the nearer theF. P. CLARKE 145 interstitial the vacancy gets the higher will its jump frequency become. Thus it may be assumed (Fletcher and Brown) that the total jump time for a vacancy to become annihilated, if it starts off within the region of distortion, will approximate to the jump time for the initial movement. Stage 1 will comprise those defect pairs within a certain distance of each other corresponding to (i) above, each of which will annihilate on movement.Outside this distorted region the defects will move according to a random walk process. Some will wander back to the distorted regions and recombine. This will comprise stage 2 of the annealing. Others will wander away and may become annihilated at defects other than the ones of origin, or they may become trapped at the surface or at dislocations. These possibilities will comprise stage 3 annealing. We can classify these processes as being unimolecular or bimolecular, or inter- mediary between these two. Stage 1 annealing rate is dependent on the total number of defects per cm3 within the distorted regions and hence we can write dN/dt = - Nv (a unimolecular reaction), where N is the number of close pairs per cm3. In stage 2, dN/dt will be propor- tional to (no.of interstitials per cm-i) x (no. of vacancies one jump away from region of strain around an interstitial) K (density of interstitials) x (density of vacancies x vol. of spherical region containing the first shell of atoms around the region of disorder) Hence the annealing process may be divided into stages. K NI x NV OC N v ~ (a bimolecular reaction), where N, and NV are the densities of interstitials and vacancies remaining after stage 1 process has finished. In stage 3 there will be a steady reduction in the density of vacancies due to capture at dislocations, etc., as well as due to recombination. Hence the process will appear to be intermediate between unimolecular and bimolecular. Mathematical analyses of these stages have been made by Fletcher and Brown 6 for vacancy diffusion, assuming the interstitials to be stationary.Their first analysis assumes an isotropic medium as well as the other physical approximations outlined above. The second analysis deals with a discrete lattice and develops a set of differential equations on a more rigorous basis. The solutions to the equa- tions developed on the analyses may be referred to as the " isotropic lattice " and " discrete lattice " solutions respectively. The physical ideas behind the forming of these differential equations may be summarized. Vacancies in sites which have the same symmetry with respect to the interstitial, have the same probability of escaping or recombining with the interstitial. Adjacent to any site are several others to which the vacancy might jump.The number of these sites and the individual probability for jumping to each is taken into account. The computations involved in this approach are very laborious unless further approximations are made, and for stage 1 annealing there would seem to be some difficulty in determining the relative jump frequencies in the various possible directions. However, by correctly choosing values for the radius of the distorted region around an interstitial a good correspondence can be obtained between the " isotropic lattice " and " discrete lattice " solutions. The isotropic lattice solutions are shown in table 1 for the various annealing stages. One important aspect common to all these solutions is the fact that the annealing dependence is a dependence in the ratio t/T, where Y = 1 / ~ i s already defined.A detailed analysis of annealing data based on this analysis has been made by Brown et aZ.7 for electron bombardment of germanium, using electrical con- ductivity as a measure of the number of defects. The damage here is mainly the production of isolated vacancy interstitial pairs. Fig. 4 and 5 show how this application has been made and how successful it has been.146 DEFECTS PRODUCED BY IRRADIATION TABLE 1 NFO STAGE 3 : NF = 1 + 4nrC2(b>N~o - (3. Total number of vacancies present at time, t = total left due to each annealing stage N = NM+ NL + NF. Ni = number of vacancies originally in ith type of site, T = the appropriate jump time, ri and r, = radii of initial vacancy position, and that of deformed region, measured <b) is the magnitude of the change in radius ri made at each jump in the appropriate stage, from interstitial.averaged over all possible types of jump. '0 0 0 (I, P C 0 +. > U 3 'p c 0 u rc - c .- .- &. 0 0 c .- c u 5 u, Annealing time (min) (4 Annealing curves at digerent temperatures for a group of samples having the same initial conductivity and given the same bombardment. The sample numbers appear above the curves, the annealing temperatures at the right end. FIG. 4. Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 592. From fig. 4(a) an equivalent annealing curve at 220" C is constructed. This is done by superimposing the annealing curves at different temperatures, after adjusting the time-scale proportionality. This can be done since the annealing kinetics are a function of t/r.The composite curve is shown in fig. 4(6). The fitting of various processes is shown in fig. S(a), (b), (c) and (d). It would seem that (d) is the best fit. However, the lattice frequency constant determinedF. P . CLARKE 147 SO 20 I 0 0 I n C 0 u i 9, v W P 0 c4 r c W 4- a - U L L 0 C Y u b 0 c 0 .- c U 0 L t& Equivalent annealing time a t 2 2 0 ' ~ (rain) (b) Composite annealing curve obtained by adjusting the time scales for superposition of the curves of (a) with respect to the curve at 220" C. The arrows above the curves indicate the ten minute annealing points for the original curves. FIG. 4.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1953,92, 592. 8 0 r 7 0. Monomolccu I ar bo 5 0 .3 0 - Equivalent annoo~ing t i m e a t 2 2 0 ' ( m i n ) (4 Theoretical fit of the composite experimental annealing curve of fig. 4(6) with ii bimolecular or single monomolecular recombination process. FIG. 5. Kt.pvoiliicetl, ,r.itli cickno wlerig,neut, from Physic. Rev., 1953, 92, 594.148 DEFECTS PRODUCED BY I R R A D I A T I O N I h c e 9, V Q, a W W 9, 0 Y t c 0 L w Y L c r 9, V + 0 c 0 .- c U 9, LL Equivalent annealing time a t 2 2 0 ' ~ ( m i n ) (4 Theoretical fit of the experimental annealing curve with two monomolecular recombination processes. FIG. 5.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 595. n c 01 V a# Q V m 0 W E c 0 0) , I L Y - w L L t 0, U .c 0 c Equivalent annealing t i m e a t 2 1 0 ~ ~ (min) (c) Theoretical fit of the experimental annealing curve with two monomolecular recombination processes, plus the recombination during liberation.FIG. 5. Reproduced, with acknowledgment, from Physic. Rev., 1953, 92, 595.I CI & c a u Y c E 0 E q u i v a l e n t annealing t i m e a t 2 2 0 " ~ ( r n i n ) (4 Theoretical fit of the experimental annealing curve with the recombination during liberation and bimolecular recombination. FIG. 5.-(After Fletcher and Brown.) Reproduced, with acknowledgment, from Physic. Rev., 1952, 92, 596. from the time constant T is - 1017 sec-1, and the capture radius is about 20 A. The first of these values is highly improbable and the second would seem to be rather large. Curve (c) however, gives a lattice frequency of 1013 sec-1 and a capture radius of 0.9a, where a is the cube edge of the unit cell.For this and other detailed reasons it is considered that this represents the best overall description of the processes. (ii) VARIABLE ACTIVATION ENERGY In order to explain the kinetics of resistance changes in metal films, Vand 8 suggested that the defects responsible for the changes might be distributed in activation energy. Following this concept, Primak 9 has made a detailed mathematical analysis of the kinetics of annealing processes having a spectrum of activation energies. The approach is similar in principle to that used in the exhaustion theory of transient creep.10 It is assumed that the annealing process obeys a differential equation of the kind - (dN/dt) = kqr, (1) where N is the concentration of kinetic processes, t the time and k a frequency factor related to the rate of the process.y is called the order of reaction by analogy with gaseous kinetics. Then if some property is related to N by the relation p = fN and the rate constant may be written as A exp (- E/kT), the equation becomes150 DEFECTS PRODUCED B Y I R R A D l A T l O N For a process having a discrete activation energy this may be integrated to give the propertyp at some time t, in terms of the property at time t = 0, 1 - p = po[l - (1 - y)Bt exp (- E/kT)]' - = Po#,, where 6,, is termed the characteristic annealing function. Thus a plot of log t against 1/T obtained at a fixed p , should be linear. Very often, however, this is not so. If the processes are distributed in activation energy, the measured value of the property P(t) will be the integral of all the discrete processes occurring at the FIQ.&-The course of an anneal. The characteristic annealing function By sweep- ing across the initial activa- tion energy spectrum. temperature of anneal at any instant, within a narrow activation energy range dE Thus the process of anneal may be represented by a function O,,, sweeping across the po against E curve as shown in fig. 6. The point of inflection in the O,, function is given by EO = kT In (Bt) and so as time proceeds, and at any given tempera- ture, the progress of across the initial activation energy spmtrum becomes slower. For low values of y, O,, is steep over the major portion as shown in the figure, and may be approximated by a step function such that P(t) = po(E)dE.1:: Hence it may be shown that 1 dP kT d(ln t)' PO(E0) = - - Hence a plot of po(E0) against EO may be obtained by plotting -!- dP against kT In (Bt). kT d(ln t) An interesting point revealed by this analysis is that very small departures from linearity in the annealing relation between property changes and In t, are most important in determining the activation energy spectrum. Primak and Szymanski's 11 results for vitreous silica illustrate this and are shown in fig. 7. With a discrete activation energy the order of reaction of - 10 would have to be chosen to explain the annealing curves shown. With a distribution in activation energies, however, an order of reaction around one can be chosen to explain them.Of course it is possible that the initial assumption with regard to the rate of the process may be invalid. For example, an annealing process discussed by Kinchin and Pease 12 may be explained with an order of 6. It is not necessary, however, to assume a distribution in activation energy to make this order lower. Kinchin and Pease have shown, from physical considerations similar to those already outlined, that the problem (in a layer lattice) may be treated by applying diffusion equations to the random migration of interstitials to circular sinks assum- ing a discrete activation energy. Marx et al.,l3 Overhauser,l4 and Dienes 15 have analysed isothermal annealing curves on the basis of an activation energy varying due to the presence of other defects.Dienes has emphasized the effect of this on the apparent order of reaction.F. P . CLARKE 151 l o g t ( t i n seconds) (4 Annealing of the radiation-induced changes in the density of an irradiated vitreous silica H, 1000" C. sample. A , 300" C, B, 450" C , C, 550" C, D, 650" C, E, 725" C, F, 800" C , G, 900" C , FIG. 7.-(After Primak.) Reproduced, with acknowledgment, froin Physic. Rev., 1956, 101, 1269. Act,i'vation encr 9 7 (b) The initial distribution of density changes over activation energy from the annealing data above. FIG. 7-(After Primak.) Reproduced, with acknowledgment, from Physic. Rev., 1956, 101, 1271.152 DEFECTS PRODUCED B Y IRRADIATION The suggestion is that an activation energy EO will be modified by the presence of N defects such that it becomes E = EO - uN where, for simplicity, K is a constant, The rate eqn.(1) may then be written dN - = ANyexp(- Eo/kT)exp (orN/kT). dt For a fixed N , the solution of this may be written t = k' exp (Eo/kT) exp (- orN/kT). Hence, although the plot of 1/T against log t should still be linear, the effective activation energy will vary with N. The effect of this on the annealing curve is that a small change of activation energy with N, leads to a large change in the FIG. 8a.-Annealing curves at different temperatures ("C) for a group of samples having the same initial optical absorption at the wavelength of measurement and given the same bombardment. time-scale for any given isothermal anneal. An annealing curve of low order and variable E is very similar to one described by a high y and discrete E.This suggestion was introduced to correct the rate equation dN/dt = ANY exp (-Eo/kT) which assumes an isotropic medium for the whole of the annealing process, The physical concept underlying this suggestion, that the lattice strain near the defect will change the activation energy for diffusion of other defects, is of course the basis of part of Fletcher and Brown's analysis. If it be assumed that a defect will cause a lattice relaxation appreciable over a sphere of radius 5 atomic spacings, then one would expect a general perturbation of all activation energies to exist for defect concentrations in excess of about 1019-1020 per cm3. It is interesting to make an analysis of annealing curves in such a way that some of the alternative approaches can be compared.In the annealing of neutron- caused damage in magnesium oxide, the decay of a particular centre may be fol- lowed by means of the optical absorption arising from an electron trapped at the centre. It is estimated that the absorption in question is caused by about 5 x 1016 defects, which are identified16 as magnesium ion vacancies. The an- nealing curves at various temperatures, and for different specimens, are shown in fig. 8a. Assuming that the annealing can be represented by the type of expression in which t/r is the ultimate variable, then an equivalent annealing curve for 500" C can be composed as shown in fig. 8(6). The first part of this anneal can be ap-F . P . CLARKE 153 proximated to the combination of two unimolecular processes for low annealing times.However, attempts to fit the rest of the curve by a combination of the physical processes described mathematically by Fletcher and Brown were not successful. V - 0 C c 0 0 0 0 V E - 0 0 U 0 LI I - 5 - 0 I 3 4 5 6 7 8 1 0 9 t i m e ( i e c ) (b) FIG. 8b.-Composite annealing curve obtained by adjusting the time scales for super- position of the curves of (a) with respect to the curve at 500°C. The arrows indicate the ten minute annealing points for the original curves. The theoretical fit of this curve 0.1 (1 - e-d2) + 0.6( 1 - e-rjloo) with two monomoIecular recombination processes is shown by the dotted curve. Fig. 9 shows the activation energy spectrum as obtained from the type of analysis developed by Primak.The two unimolecular processes of the former analysis are now replaced by two definite bands. It must be borne in mind that the effect of the interaction of the characteristic annealing function with the initial activation energy spectrum is to make the spectrum which is produced by analysis of the data, broader than it should be. This broadening might be as FIG. 9.-The initial distribution of defects in activation energy, from the annealing data of fig. 8. much as several times kT for steeply rising parts of the spectrum. Hence the width of the first peak might be due to this and other complicating effects, and the band could be interpreted as representing a discrete activation energy. The second band, however, seems too broad to allow of this interpretation.I54 DEFECTS PRODUCED BY IRRADIATION An interpretation to describe the processes by means of an activation energy varying with the number of defects was not made, since it was considered that the number of defects was insufficient to make this approach physically justified, CONCLUSION When certain radiations penetrate solids, atomic displacements may result.These displacements may be distributed randomly or in groups. Once the dis- tribution is known it is possible in principle to set up diffusion equations to describe their subsequent behaviour as the temperature of the solid is raised. Attempts to do this (Fletcher and Brown, Kinchin and Pease) have been successful for particular cases. The solution of these diffusion equations is complicated for any but the simplest cases.Hence approximations have usually to be made which make the final mathematical description of the processes less accurate. The defect concentration is estimated by measuring physical properties (optical, electrical, density, etc.) which vary with this concentration. The exact way in which this variation takes place is usually uncertain and so before the diffusion equation can be set up, approximations have to be made which again may weaken the final mathematical description. Another important factor which may be obtained from physical measurements is the activation energy for diffusion of defects. If one is sure of the type of defect concerned, this activation energy may be estimated theoretically or by separate diffusion experiments. A common procedure, however, has been to assume that the kinetic behaviour will follow a certain type of equation (e.g.dN/dt = ANY exp (- E/kT)) and to deduce the activation energy from this. Largely as a result of using this procedure the concepts of a continuous distribution in activation energy, and an activation energy varying with the number of defects present has arisen. On the first of these concepts it has been demonstrated that, with an assumption about the way a defect with a discrete activation energy would behave, the continuous distribution may be obtained from experimental data. The final picture emerging in this case would seem to depend for its validity on the correctness of assuming a simple equation for the discrete annealing process. The approaches made by Fletcher and Brown, Kinchin and Pease, Primak, Marx et al., Overhauser and Dienes all emphasize that the concept of an order of reaction is meaningless unless it is based on an underlying physical under- standing of the processes involved. The author would like to thank Dr. A. H. Cottrell, F.R.S., and Dr. J. H. 0. Varley for informative discussions and criticisms on the subject matter of this paper. 1 Bohr, K. Danske Vidensk. Selsk. Mat. Fys. Medd., 1948, 18, 8. 2 Seitz and Koehler, Solid-State Physics (Academic Press, New York, 1956), vol. 2, 3 Brinkman, J. Appl. Physics, 1954, 25, 961. 4 Lomer and Cottrell, Phil. Mag., 1955,46, 711. 5 Varley, J. Nuclear Energy, 1954, 1, 130. 6 Fletcher and Brown, Physic. Rev., 1953, 92, 585. 7 Brown, Fletcher and Wright, Physic. Rew., 1953, 92, 591. 8 Vand, Proc. Physic. Soc., 1943, 55, 222. 9 Primak, Physic. Rev., 1955,100, 1677. 10 Cottrell, J. Mech. Physics Solids, 1952, 1, 53. 11 Primak and Szymanski, Physic. Rev., 1956,101, 1268. 12 Kinchin and Pease, Reports Prog. Physics, 1955, 18, 1. 13 Mam, Cooper and Henderson, Physic. Rew., 1952,88, 106. 14 Overhauser, Physic. Rev., 1954, 90, 393 ; 1954, 94, 1551. 15 Dienes, Physic. Rev., 1953, 91, 1283. 16 Clarke, Phil. Mag., 1957 in press. p. 305 et seg.

ISSN:0366-9033    

DOI:10.1039/DF9572300141

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. F. G. Fumi (Universita di Palerrnu) said: I entirely agree with the view of Koehler and Seitz that the available calculations on the energies to form and to move point defects in metals, in particular vacancies, are relatively rudimentary, and thus should not be taken too literally : for one thing, these calculations are based on the free-electron approximation which does not give the correct value for the cohesive energy of a metal such as copper. I should only like to mention that the simple calculations that I did a couple of years ago,* at the suggestion of Prof. Mott, on the formation energy of vacancies in the various alkali and noble metals give results in as good agreement with experiment as the calculation by Huntington 2 for copper, properly corrected.3, 1 I should like to stress that the approximate value that I reported for the forma- tion energy of a vacancy in Au (ca.0.6 eV) 1 should really be interpreted as a lower limit, as I assumed (ref. (l), p. 1012) that the Coulomb correlations between con- duction and core electrons do not contribute in a relevant way, and I used (ref. (1). p. 1013 and table 4) the repulsive energy contribution that I calculated for Cu ( - 0-3 eV), which is bound to be somewhat too large for Au. Thus my theoretical value for Au can be said to agree within its uncertainty with the more recent value given by the quenching experiments of Koehler et aZ.4 (1.00 5 0-05 eV). Similar considerations apply to the values that I reported (ref. (l), table 4) for the formation energy of a vacancy in Ag (ca.0.6 eV), and partly to the value that I reported for Cu (0.9 eV). The best value reported by Huntington 2 for the formation energy of a vacancy in Cu is 1-8 eV; however, Huntington's cal- culation should be modified 3 D 1 and the true predicted value is about l eV, in good agreement with the value that I computed. A second remark which 1 should like to make refers to table 1 of the paper by Koehler and Seitz which essentially coincides with table 1 of my paper.1 I feel that if one discards with the authors the experimental values by Meechan and Eggleston 5 for the formation energy of thermal defects in noble metals since one objects to their method of measurement, one cannot place much faith in the experimental values by MacDonald 6 for the aIkaIi metals, which are obtained by the same method.Prof. R. M. Barrer (Imperial CulEege) said: Si and Ge, with co-ordination numbers of 4 and sizeable lattice atoms, are relatively open crystals. This suggests that they are well suited as media for investigation of interstitial solution and dif- fusion. On this account I wish to direct attention to a mass spectrographic study of diffusion and solubility of H2 and of He in Ge and Si.7 It was possible to study diffusion of H2 and He in Si from 967 to 1207" C and in Ge from 766 to 930" C. Very high diffusion coefficients were found. Thus DHe-Si ranged from 1.70 x 10-4 cm2 sec-1 at 1092" C to 2-17 x 10-4 cm2 sec-1 at 1200" C. Energy barriers were evaluated, and in terms of the Arrhenius equation, with D in cm2 sec-1: DH2-Si = 9.4 x lO--3 exp (- 1l,OOO/RT), & M i = 0.1 1 eXp (- 29,0OO/RT), D H ~ - G ~ = 6 x l o 3 exp (- 16,00O/RT).1 Fumi, Phil. Mag., 1955, 46, 1007. 2 Huntington, Physic. Rev., 1942, 61, 325. 3 Brooks, Impurities and Imperfections (Amer. Soc. Metals, 1955). 4 Bauerle, Klabunde and Koehler, Physic. Rev., 1956, 102, 1182. 5 Meechan and Eggleston, Acta Met., 1954, 2, 680. 6 MacDonald, Bristol Conference on Defects in Solids (Physic. Soc., London, 1955), 7 van Wieringen and Warmoltz, Physica, 1956, 22, 849. p. 383. 155156 GENERAL DISCUSSION In the He-Ge system in particular the pre-exponential constant is very high, and may provide some evidence of persistence of velocity and so long mean free paths, of diffusing helium atoms, as referred to in the paper by Ubbelohde. Prof.F. G. Fumi (Universita di PaZermo) said: I fully agree with Dr. Waddington on the importance of the form adopted for the repulsive potential in lattice cal- culations on point imperfections. This is why Dr. Tosi and I investigated the effect of using the Born-Mayer and Huggins-Mayer forms for the repulsive potential, as opposed to the Mott and Littleton form, on the energy to create a Schottky defect. I feel it would not be particularly valuable to try other forms. Clearly in the calculation of the energy of migration of point imperfections it is especially important to study the effect of the form of the repulsive potential. Prof. R. A. W. Haul (Universitat, Bonn) said: In addition to the experimental methods discussed in Dr. Crank’s paper I would like to mention a method which we have recently 1 used to study diffusion in crystals.Essentially a constant and limited amount of isotopically labelled gas is left in contact with the crystals, e.g. 13CO2 in the case of calcite. Simply by measuring the decrease of isotope concentration in the gas, progress of diffusion in the solid can be followed and by applying the appropriate mathematical solutions, diffusion coefficients can be evaluated. The method has the advantage that only small crystals are required (1-50,~), the surface area of which can be determined, e.g. by krypton adsorption. The method is free from the uncertainties encountered in diffusion experiments with plugs formed by compressing powdered material. Dr. J. Crank (Courtaulds, Ltd.) (communicated) : I should like to thank Prof.Haul for reminding me of his paper. This method is comparable with the one used by Dr. G. S. Park to study diffusion in the isopentane + polyisobutene sys tem.2 Prof. R. M. Barrer (Imperial College) said: In a logical system of naming diffusion mechanisms, care should be taken in the use of the term zeolitic diflusion. In fact this term may not always be synonymous with interstitial diflusion at least as far as the true zeolites are concerned. In these crystals recent crystallographic and other studies have shown that the degree of openness is extremely varied.3 In the most compact crystals the interstitial mechanism may, it is true, be the most correct for migration of water or other molecules. In other members of the family the channel systems are so open that even the idea of localized sites may not be valid, and molecules are clustered quite thickly together as an intra- crystalline fluid throughout the channels. Self-diffusion occurs almost certainly by place interchange, if the crystals are saturated with sorbate molecules (e.g.H20 replacement by D20). When the crystals are not saturated, diffusion down a concentration gradient of sorbate, involving “ holes ” in the intracrystalline fluid, is an appropriate description. Thus, various mechanisms arise in zeolites, and the term zeolitic difliision in reality may involve more than one of these according to circumstances. The term interstitiaf diflzfsion might I suggest be limited to diffusion in a crystal where one atom only is found per interstice, and this atom cannot diffuse unless a vacant interstice is adjacent to it.Dr. L. E. J. Roberts (Harwell) said: It is possible to decide experimentally between the “ direct ” and “ indirect ” or “ interstitialcy ” mechanism for the diffusion of interstitial atoms when a concentration gradient of interstitial atoms can be established using a different isotope to that present in the host lattice, since the isotopic composition of the interstitial atoms will not be the same in the two cases. 1 Haul and Stein, Trairs. Faraday SOC., 1955, 51, 1280. 2 Trans. Faraday Suc., 1957, 53, 107. 3 Barrer, 10th Sulvay Congr. Chem., 1956, p. 21.GENERAL DISCUSSION 157 The results of some isotopic exchange experiments indicate that interstitial anions may diffuse through substances having the fluorite structure by an inter- stitialcy mechanism.It has been demonstrated that interstitial anions can be incorporated in both fluorides and oxides of the fluorite (CaF2) type when some of the cations have a higher valency than the normal lattice valency. We have pre- viously studied in detail the oxidation of U02 at low temperatures (< 200" C) ; oxygen enters interstitial sites with no change of lattice symmetry until the com- position reaches about UO2.25, and the kinetics are those of a simple diffusion process. At temperatures above 300" C , these oxides disproportionate into two cubic phases, U02 and U409; U409 still contains interstitial oxygen and it seems probable that it is an ordered phase having one extra 0 per unit cell of the original U02 structure.The oxidation of solid solutions of U02 in Tho2 follows the same general course, though it becomes progressively more difficult as the percentage of U02 decreases. When the U02 content is less than 50 %, there seems to be no tendency to dis- proportionate into two phases on heating and the systems are to that extent simpler to study than U02 itself. The plan of the exchange experiments was to oxidize at a low temperature with 0 2 enriched in 018, anneal under various conditions and then remove the interstitial oxygen by reducing with CO, and analyse the CO;! for 018. The reduc- tion cannot be quickly carried out below 500" C , and therefore lower annealing temperatures were not considered. The results are shown in the table, expressed THE PERCENTAGE OF LATTICE OXYGEN (Ns %> OF UOz 4- Tho2 CRYSTALS EXCHANGING WITH INTERSTITIAL OXYGEN time N, % mole yo oxidation maximum uoz temp.temp. 100 145" 145 160 25 495 7 370 3 60 3-6 3 65 3 60 1.2 450 500 435" 830 515 495 495 510 51 3 510 506 510 517 1.8 h 2.0 1.2 3-5 1.3 3-75 18 22 38 12 13 36 40 37 9 12 19 15 17 23 3 3 as the percentage of the lattice oxygen that would have to have undergone com- plete exchange with the 0 2 added so as to produce the dilution of 0 1 8 which was observed (N, %). Two trends are clear: (i) exchange was never complete but was always considerable except in the mixed crystal most dilute in UOz, and (ii) in every case the amount of exchange observed was practically independent of annealing time, What exchange occurs does so as quickly as the diffusion process.This is compatible with an interstitialcy mechanism of diffusion. The whole of the lattice oxygen did not take part in the exchange even in times long compared to the time of diffusion through the crystal; this would then mean that every interstitial position is not equaIIy probably occupied, but that diffusion follows a preferred path, from one favourable site to the next. This is not sur- prising for UOa + Tho2 solid solutions, since it is clear that energetically favoured interstitial sites are those nearest the largest number of U atoms. The results on pure U02 must be due to a different cause. The phase division into U02 and U409 must have occurred in every case before reduction com- menced.The results at high temperature would then indicate that further place exchange does not take place once the ordered phase U40g has been formed.158 GENERAL DISCUSSION The fact that the gas recovered on reduction is richer in 0 1 8 than would be ex- pected on the basis of complete exchange may then be due to 018 from the gas phase concentrating on lattice sites in the outer layers of each crystal during the diffusion of interstitial oxygen, and appearing in the gas phase again on reduction by exchanging again with interstitial oxygen by the interstitialcy mechanism. Dr. Y. Haven (Philips Rex. Lab., Eindhoven) said: The value of D = 3 X 10-3 cm2/sec for the diffusion constant of interstitial diffusion of Cu into Ge suggested by Franck is much higher than would be expected if free gas-like diffusion was assumed.From Izl =0.8dz, t = 2.45 x 10-8 cm (free path length), we calculate D = 5 x 10-4 cm2/sec at 800" C . A larger diffusion constant would only be possible if strong correlations between jumps were present, i.e. if teffective was much larger than 2.45 x 10-8cm; or if there was a large activity factor d log y/dN. Neither of these two factors seems to be sufficient to bridge the gap between the two above mentioned values for D, the more so as probably other factors will make D still smaller. Therefore, we conclude that the experimental value D = 3-2 x 10-5 cm2/sec is about the diffusion constant for interstitial diffusion of Cu and possibly only a small fraction of Cu gets substitutionally replaced in the sense of Van der Maesen and Brenkman. Dr.A. Seeger (Stuttgart) said: There is experimental evidence for enhanced diffusion along dislocation lines, which, as mentioned in Prof. Frank's paper, has to be interpreted in terms of the activation energy of diffusion being smaller in the vicinity of the dislocation line than in the bulk of the crystal. Dr. van Bueren mentioned in the discussion that C . W. Berghout in his thesis (Delft, 1956) has treated theoretically the enhanced diffusion along dislocations by considering the change in the energy of migration of vacancies due to the stress field around dislocations calculated from the linear theory of elasticity. We should like to point out that there is another effect that may be of the same order of magnitude as that derivable from first-order elasticity, and that takes account of some of the contributions to the diffusion coefficient which come from the centre of the dislocations.In the usual treatments the last-mentioned contributions are dealt with only qua1 i t at ivel y . The stress-field around a dislocation decreases as the inverse first power of the distance r from the dislocation. The effect of the stress field on the diffusion coefficient (averaged over the polar angle) decreases as r-2, however, since for small strains, first-order effects cancel. The strains due to deviations from linear elasticity vary as the square of the stress components, i.e. as r-2. When averaged over the angle around the dislocation line the effect of these terms on the diffusion coefficient is finite.(There is, for instance, an average dilatation varying with the distance from the dislocation as r-2. In a vacancy mechanism of diffusion it affects both the energy of formation and the energy of migration.) The preceding discussion shows that second-order effects on the diffusion coefficient are formally of the same order of magnitude as first-order effects. Numerically they are presumably smaller than the first-order effects for edge dislocations but larger for screw dislocations. Since the Y-2 law corresponds to a rather slow variation with distance (the integral (27rrdrll.2) diverges) it is seen that the main contributions to the change in the diffusion coefficient due to s"GENERAL DISCUSSION 159 dislocations do not come from the dislocation core proper as is quite frequently implied. For those effects that do originate in the central parts of the dislocation lines the second-order treatment outlined above (supplemented by a suitable cut-off at small r ) should prove more satisfactory than the first-order treatment. Prof.F. C . Frank (Bristol) (communicated) : Dr. Seeger's point regarding the large cross-section of the dislocation for its effect upon diffusion is essentially right, but nevertheless probably not very important. Suppose we express the effect of strain on diffusion coefficient as a power series in the strain components, and consider separately the terms of first, second and higher order. The first- order terms have no net effect, since at opposite equidistant points from the dis- locations there are equal and opposite strains.The total incremental permeation along the direction of the dislocation, due to the second-order terms is, as Dr. Seeger says, logarithmically divergent. This means, however, that the incre- ment between lOA and 1 p from the dislocation line is equal to the increment between 1 ,u and 1 mm which would be equal again to the increment between 1 mm and 1 m if we could have so large a cylinder with only one axial dislocation. Though, in a certain sense, this is an infinite cross-section, the extra permeation is still, for all practical purposes, concentrated in a narrow channel. Finally, we have the higher-order terms. They should only be important within a radius of the order of ten, or a few tens, of Angstroms. The effect of strain on diffusion must be primarily to change the activation energy.Since this affects the diffusion constant exponentially, it may be relatively a very large effect, especially at low temperatures. Hence, in general, and especially at low temperatures, it is essenti- ally right to regard the effect of dislocations on diffusion as concentrated in a narrow channel. Prof. J. S . Koehler (Illinois) said: With regard to Frank's paper, another ex- ample of a process which involves both lattice vacancies and dislocations is the case of steady-state creep at high temperatures. A theory involving climb which seems to fit the observed data has been given by Weertman.1 Dr. P. Riietschi (Electric Storage Battery Co., Philadelphia) (communicated) : The compensation law is a widely observed phenomenon in chemical kinetics.If rate processes of a similar nature are compared one observes that large activation Probability Internuclear distribution NV distance r FIG. 1 energies are connected with large frequency factors and vice versa. To a first approximation the logarithm of the frequency factor is found to increase linearly with increasing activation energy.' 1 3 . Appl. Physics, 1957, 28, 362; 1955, 26, 1213.160 GENERAL DISCUSSION Unfortunately the exact reaction mechanism is unknown for most chemical reactions. The more particles and vibrations are involved in an activation process the more selective becomes the reaction path and the smaller becomes the fre- quency factor and the activation energy. Explanations of the compensation law based on this general idea and using activation entropies and enthalpies and the absolute rate theory are only qualitative and not very transparent.It is, how- ever, possible to give a more satisfactory treatment on the basis of potential energy surfaces and statistics. This can be done relatively simple for the unimolecular decomposition of a diatomic molecule. A potential energy surface for such a reaction is schematically shown in fig. 1. The number of particles Nv in each energy level v decreases exponentially with increasing v according to a Boltzmann distribution. At larger interatomic distances the Morse-type potential energy curve can be approximated by (Ye - V ) = 2Ve exp (-- p(r - re)). Only particles which have acquired the critical energy Ve will undergo the dis- sociation reaction.The number of particles with this necessary energy is given by where N is the total number of oscillators in the system, F(T) is the partition func- tion and AT, is the phase space element. The partition function can be expressed by where 7 is the mean energy per particle and s is the mean entropy per particle. The energy difference Ye - v i s dependent on temperature. The derivative with respect to Tcan now be obtained using formally a linear thermal expansion co- efficient a : Tn combining the given relations one arrives at the following expression for the number of particles possessing a critical energy Ve and the rate constant becomes simply if v denotes a transition frequency such as kT/h. The temperature independent factor preceding the last exponential term depends exponentially on (Ye - Y ) and the empirical relation between the logarithm of the frequency factor and the activation energy would be explained if the term polre does not vary to an appreciable degree for the series of similar rate processes under comparison and if the entropies S of the reacting systems is the same.For more complicated rate processes such as diffusion in solids, it becomes more difficult to picture a model of the activation process. However, using a thermal expansion coefficient and a potential energy surface of an exponential IGENERAL DISCUSSION 161 nature the compensation effect observed for these processes can still be explained. In the diffusion theories of Lazarus and Zener the activation energy was inter- preted in terms of the elastic shear strain energy involved during the jump of the diffusing atom into the imperfection site.Diffusion will become easier as the lattice expands with temperature. As indicated in eqn. (I) of Prof. Ubbelohde's paper, Gurney and Mott have allowed correctly for the decrease in activation energy as the lattice expands with temperature. Using now in addition to a linear thermal expansion coefficient, a potential energy surface of exponential character, the compensation law is obtained. If UO varies exponentially with the interatomic distance, the derivative with respect to the radius (or the volume) eqlials the original function and the frequency factor depends exponentially on UO. The compensa- tion effect for diffusion of solute atoms in metals has recently been explained by Swalin in a similar manner on the basis of the diffusion theory of Lazarus.Prof. R. M. Barrer (Imperial College) said: Suppose in an interstitial diffusion there can be persistence of activation energy, and of velocity, in a diffusing par- ticle, so that it moves in a given direction not just from one interstice to the next, but through n interstices in a line. Then before this behaviour is offered as an explanation of abnormal A factors in the Arrhenius equation D = A exp (- E/RT), the following considerations should be taken into account.1 These considerations show one way in which retention of activation energy by the diffusing particle, may be unable to give persistence of velocity. In a flux rate governed by a chemical potential gradient, the diffusion coefficient may be written as d In a d In 8 D = RTB- where B is the mobility, a the activity of the interstitial particle, and 6 the fraction of interstices occupied.In systems for which the interstitial component is volatile the concentration of interstitial atoms or molecules can, as is normally true, be demonstrated to approach a Langmuir isotherm, d l n a 1 Also in jumps where n = 1, B = Bo(1 - 6) since (1 - 6) is the chance that a given interstice next to the occupied one is vacant, it being thus assumed that no jump can occur into an interstice already occupied. In a continuous sequence of jumps through n interstices B = Bo(1 - 6)n. In this way Barrer and Jost obtained D1 - d2v exp (- E/RT), Dn - (nd)2(1 - 8)"-1 v exp (- E/RT), and for single jumps, and for n-fold jumps with persistence of velocity. the distance between interstices and v the vibration frequency.Then Here d is Dn/Dl - n2(1 - 8)"-1 = &/A1, and one sees that A, > A1 only if n2(1 - 8)n-l > 1. One may conclude that : (i) D, for a multiple interstitial jump mechanism in which activation energy, and so velocity in a given direction, persist over n barriers, is concentration dependent. (ii) If 8 is large, whether or not the interstitial particle retained its activation energy over tinies equal to that needed for multiple jumps, such jumps would in all probability be blocked, and D, and A, would become less important than D1 and A l . 1 cf. Barrer and Jost, Trans. Faraday SOC., 1949,45,928.F162 GENERAL DISCUSSION (iii)If, however, 8 is small D,, and A, could exceed D1 and A1, and n would be governed primarily by the time of persistence of velocity. In this case abnormal A factors in the Arrhenius equation could result. At a first inspection it would seem that multiple jumps of interstitial atoms within a crystal would not be very probable, but that they would be most likely to be found in dilute interstitial solutions such as might arise in interstitial carbides, nitrides or oxides, in hydrogen + transition metal systems, or in diffusion of H2 and He in Ge,l where light atoms are involved. Multiple jumps seem more prob- able still in dilute “ solutions ” of gases in zeolites, but perhaps long mean-free- paths associated with persistence of velocity of the diffusing atoms are most likely in dilute adsorbed films.We have measured surface diffusion coefficients of permanent and inert gases as very dilute adsorbed films in Vycor porous glass and Carbolac carbon plugs.2 The surface diffusion coefficients are of the order 10-3 to 10-4cm2sec-1 at room temperature; the energy barriers E involved are of the order 2000 cal/mole; and the ratios E/AH lie between 0.5 and 0.6, where AH is the jsosteric heat of adsorption. With certain assumptions, the A values suggest mean-free-paths approaching 20 A’. Prof. W. Jost (Gottingen) said: I have two comments on Prof. Ubbelohde’s paper. For a-AgI the empirical formula for cation diffusion is approximately : D = DO exp (- 1130/RT) cm2 sec-1 giving near the melting point, where it still holds, D m DO exp (- 1130/1650) cm2 sec-1.Thus the fraction given by the exponential is so high that an understanding without consideration of correlation seems impossible. As Prof. Barrer has pointed out several years ago, activation may concern not just the moving particle but also its environments. Thus, if before and during a jump all movements have been in phase, they may not be so after the jump, decreasing the probability for a consecutive backward jump. Dr. A. B. Lidiard (Harwell) said: I would like to ask Prof. Ubbelohde why he distinguishes the polarization he describes from the well-known polarization coming from the build-up of a space charge at the electrodes? This is one of the distinguishing features of an ionic conductor and is the reason why a.c.methods and pulsed methods are used to obtain conductivities which are reproducible and independent of the electrode materials. The theory of these polarization effects as a function of applied frequency has been given by Friauf3 and MacDonald.4 Incidentally Friauf’s experiments on AgBr with Ag and Au electrodes indicated that whereas the interstitial Ag+ ion current carriers could discharge easily at the electrodes the Ag+ vacancies could not. This blocking of the Ag+ vacancies has the interesting consequence that the systems AgBr + C a r 2 should show more pronounced polarization effects than pure AgBr, but this does not seem to have been studied systematically yet. The idea of a “ persistence of velocities ” in atom jumps in solids is interesting but is probably difficult to verify.If the idea should apply to interstitialcy migration (where an interstitial atom pushes a normal atom into the next inter- stitial site, itself taking the normal site) then one could tell through simultaneous measurements of conductivity (T and tracer diffusion coefficient DT. The ratio kTciNe2DT (N = number of ions/unit volume) is a pure number, sensitive to the number of ions which move in each elementary act.5 Existing results on 1 van Wieringen and Warmoltz, Physica, 1956,22,849. 2 Barrer and Barrie, Proc. Roy. Soc. A, 1952, 213, 250 ; Barrer and Strachan, Pruc. 3 Friauf, J. Chem. Physics, 1954, 22, 1329. 4 MacDonaId, Physic. Rev., 1953,92,4. 5 McCombie and Lidiard, Physic. Rev., 1956, 101, 1210. Roy. SOC.A , 1955,231,52.GENERAL DISCUSSION 163 AgBr 1 and AgCl2 are almost entirely against the idea that more than two Ag+ ions (the interstitial and the normal) take part in a single interstitialcy jump. However, at high temperatures there is an inconsistency in the results for AgBr which may indicate the onset of a persistence of velocities effect. In considering the kinetics of the proposed mechanism it is useful to recall the steps in the derivation 33 49 5 of the usual exponential formula for jump frequency, v exp (- AG/kT). The number of ions crossing a given plane is calculated from general statistical mechanical formulae (which in turn give the definitions of v and AG), and assuming complete " de-activation " after every jump the atomic mobility can be derived and then used in non-equilibrium calculations.However, if the de-activation is only partial this simple approach cannot be used, e.g. the calculation given in ref. (6), (7) and (8) in the limit of AG --f 0 gives the rate of effusion of a gas but not its rate of diffusion. In favourable cases it is possible that a formal description may be obtained by the use of a relaxation time to de- scribe the transfer of energy between the mobile atoms and the surrounding lattice. Prof. A. R. Ubbelohde (Imperial College, London) (communicated) : Con- cerning the comments by Fumi and Lidiard, the question of persistence of activa- tion energy for migration in a crystal lattice can no doubt be formally represented as a relaxation process. The suggestion that this relaxation time is always of the order l/v, where v is the characteristic lattice frequency, appears to be the basis of the conventional jump equation.This assumption seems likely to be too general to apply to all types of crystal. Activation energy for migration is generally described as " vibrational ", but if many quanta are nominally required to make up the parameter AE the " vibrations " of the activated particle are far from harmonic and AE involves complex processes of accumulation and decay ; persistence times considerably longer than l/v seem possible. Dr. L. E. J. Roberts (Harwell) said: Prof. Ubbelohde has drawn attention to the possibility that a diffusing atom may make multiple "jumps ". Such a process may explain one feature of the low temperature (- 50" to + 50" C) oxidation of uranium dioxide, which follows a logarithmic law and which results in the introduction of interstitial oxygen into the outermost 4-8 atomic layers of the U02 particles.6 The course of the reaction was not affected by long periods of pumping pro- vided the temperature was not raised.The inference is that the distribution of interstitial atoms in the solid did not alter during the annealing period, although a concentration gradient had been set up and the concentration of surface (ad- sorbed) oxygen remained high. This suggests that the reaction can only proceed in this temperature range by a mechanism which allows the entering oxygen atom to move effectively more than one lattice spacing-ither by persistence of motion, as suggested by Prof.Ubbelohde, or by a co-operative movement that has the same result. Prof. A. R. Ubbelohde (Zmperial College) said: Valuable comments and illustrations have been proposed by Lidiard, Barrer, Jost, Young, Roberts and Bloem. I need only add that in my paper the emphasis is on multiple jumps that arise from retention of activation energy for a finite time. It is quite true, as Prof. Jost points out, that only part of this activation energy " belongs " specifically 1 Friauf, Physic. Rev., 1957, 105, 843. 2 Compton and Maurer, J. Physics Chem. Solids, 1956, 1, 191, 3 Wert, Physic. Rev., 1950, 79, 601. 4Seeger, Handbuch der Physik (Springer-Verlag, Berlin, 1959, vol. 7, part 1, 0 12 5 Lidiard, Handbuch der Physik (Springer-Verlag, Berlin, 1957), vol. 20, Q 15 et 6Anderson, Roberts and Harper, J.Chem. Soc., 1955,3946. et seq. seg.164 GENERAL DISCUSSION to the particles undergoing a hop. bouring particles that move sideways will not be as helpful to Retention of activation energy by neigh- further hops in the forward direction, as energy that can be attributed to the hopping particle itself. Clearly in a detailed analysis of persistence of activation energies the whole of a “ hot region’’ around the hopping particle would have to be considered. In the paper, the emphasis is on the general problem of multiple jumps that arise from persistence of activation energies. In specific cases various geometrical considerations may be required to supplement the consideration of finite relaxation times for activation. The example of zeolitic diffusion mentioned by Prof. Barrer, appears to be an instance of surface migration where the geometrical probability of finding neighbouring vacancies is the controlling factor, rather than the persistence of activation energies, In true lattice migration geometrical controlling factors can likewise occur.For example, in diffusion by vacancies, persistence of activation will actually lessen the rate of diffusion since it favours a hop back into the vacancy just created by the movement forward of a particle. The correlations referred to by Dr. Haven appear to be of this nature. In inter- stitial mechanisms the probability of finding the neighbouring interstices occupied may likewise be important in certain cases. In reply to Dr. Lidiard’s question about polarization, presumably any form of space charge that arises from the passage of d.c.current is a form of polarization. The puzzle in the examples referred to is to determine what is the structural origin of a polarization which becomes less pronounced as the temperature is raised. It seems unlikely that actual separation at the electrodes of the metal or negative constituent is involved. Some form of trapping of particles and electrons appears to be involved that reduces the electrical driving force. As the temperature rises the trapping becomes less effective, but much more work is needed to elucidate the problem. In reply to Dr. Seeger’s comment, a correlation between y and UO would express in a specific way the general correlation suggested between large negative entropies of activation and large positive energies of activation.Only really detailed dis- cussion in terms of specific structures can determine whether this correlation is experimentally significant, or whether the accidental requirements of the duration of experimental observations introduce an artificial bias. Prof. R. A. W. Haul (Universitiit Bonn) said: Would it be possible that surface transport could contribute appreciably to the sintering of AgI as is the case in other sintering processes or is only bulk diffusion the determining factor? One could visualize that surface transport may be responsible for the observed rapid initial decrease of the height (fig. 4). Through surface contacts between adjacent particles a consolidated porous system could be formed with slower subsequent changes of the overall volume.Determination of changes in surface area, e.g. by means of the krypton adsorption method (small areas), might give additional information on the progress of sintering. Prof. J. H. de Boer (Geleen) said: The phenomenon of sintering, as used by Prof. Jost, is related to the sticking together of the particles. Other groups of investigators, viz. those who are concerned with adsorbents and with catalysts, when using the word sintering, may mean another phenomenon. They describe the decrease of the surface area (internal surface area in the granules) as a sintering process. Usually this process occurs at lower temperatures than the stickingGENERAL DISCUSSION 165 together of the particles. It may be that in some cases the cause of the phenomena is a similar one, but one may visualize surface diffusion to play a role in the decrease of the surface area by heating.Prof. F. C. Frank (Bristol) said : To Prof. de Boer's two component parts of the phenomenology of sintering-reduction of surface area and sticking-it is necessary to add a third, shrinkage. This is significantly different from the other two, because, whereas they may proceed by surface diffusion processes only, shrinkage necessitates an interior diffusion process (grain boundary diffusion, or " lattice " diffusion). The centres of a pair of quasi-spherical grains in contact with each other can only approach each other if there is a process which removes intervening material. Prof. W. Jost (Gottingen) said: It seems worth while to point out that AgT in many respects has most unusual properties which make it rather improbable that one is dealing with surface effects.The rate of diffusions of the anions, though much smaller than that of the cations, is still to be considered as high. In fact, when extrapolated to the melting point, the values are by about a factor 10 higher than those mentioned by Laurent for alkali halides. In addition, the so-called sintering temperatures of Tamann are much lower, about 0.6 of the absolute melting temperature. As Prof. de Boer rightly has pointed out a marked decrease in surface might occur at temperatures lower than those under consideration. There seem to me three factors pointing to a bulk rather than to a surface diffusion: (i) the absolute magnitude of the diffusion coefficients of the iodide ions which could hardly be explained by surface migration; (ii) its comparatively high temperature coefficient, giving an energy of activation of about 1 eV, and (iii) the fact that we have to explain anomalous specific heats of up to 20 cal/mole above the Dulong-Petit value which cannot be attributed to cation disorder.Prof. Frank showed schematically the region of contact of two spherical grains. In the region of negative curvature the opposite occurs of what we know for small droplets. Surface tension lowers the vapour pressure, the activity, the chemical potentials thus causing a bulk diffusion flow into this region. This is the basis of Kuczynski 1 theory for sintering of metals, who succeeded in cal- culating self-diffusion coefficients from sintering measurements. We do not claim that this must be the explanation, but nevertheless, these observations and con- siderations do exist.Dr. G. Salomon (Delft) (communicated): The stickiness of solids is in part determined by the rate of stress-relaxation in the contact area. A slight pressure may be sufficient to produce a surface contact, but unless elastic stresses are re- leased very quickly, the surfaces will separate as soon as the external force is re- moved, see e.g. the work of Bowden and Rowe. It seems probable that the rate of stress-relaxation has a similar temperature function as self-diffusion. Mr. J. F. Laurent (Universite' de Paris) (communicated): I would point out, in connection with the statement of Prof.W. Jost and Mr. H. Oel, that anionic self-diffusion coefficients are not necessarlily very much smaller than cationic ones. For instance, for monocrystalline potassium halides, we have established 2 in Prof. BCnard's laboratory that the values of the self-diffusion coefficients of ions at few degrees under the melting point are as follows : 01- KI - 2 x 10-9 cm2 sec-1; DE = 5 x 10-9 cm2 sec-1; D:', = 2-5 x 10-9 cm2 sec-1; Dg, = 4 x 10-9 cm2 sec-1; Dgsl = 3 x 10-9 cm2 sec-1; D& = 3 x 10-9 cm2 sec-1. 1 Kuczynski, Physic. Rev., 1949, 75, 1309 ; J. Appl. Physics, 1950, 21, 652, 1224. 2 Laurent and BCnard, Cumpt. rend., 1955, 241, 1204 ; Noyer and Laurent, Compt. rend., 1956,242,3068. Laurent and Bbnard, J.Physics Chem. Solick, in press,166 G E N E R A L DISCUSSION These values show that, in a compound close to its melting point, if the anionic diffusion rate is slightly smaller than the cationic one, it is not negligible.The relative values of anionic and cationic diffusion rates can even be reversed, if we consider the diffusion in polycrystalline aggregates. Our work on ionic self- diffusion in monocrystalline and in polycrystalline samples of potassium halides gave results summarized in fig. 1-3.1 Selfdiffusion in KI Selfdiffusion in KBR 10 20 2 2 24-LogD Selfdiffusion in KCI 18 20 22 24-LOgD FIGS. 1-3. On these curves loglo (self-diffusion coefficient of different ions) are plotted against the reciprocal of the absolute temperature. At the top, results for KI are given on the left, those for KBr on the right, and below those for KC1.For the common cation K+ (dotted line), the diffusion rate in a given compound and at a given temperature is still the same whether the sample is monocrystalline or polycrystalline. On the other hand, for the diffusion of anions, we observe an increasing of the diffusion coefficient, as the grains of the polycrystalline aggre- gate become smaller (the average diameter of grains is written on curves). It is interesting to note that this increase, which becomes larger as we g o from KC1 1 Lament, I.U.P.R.C., July, 1957.G EN E R A Id DISC IJ S S I 0 N 167 to K1, involves no appreciable variation of the activation energy of the diffusion process ; the frequency term DO alone increases. These curves show particularly that at some temperature and for sufficiently small grains, the anionic diffusion rate can be equal or even exceed the cationic one.The very different behaviour of anion and cation in relation to intercrystalline fields does not seem to be caused by the differences of polarity of both ions; as a matter of fact, for ceasium fluoride, the diffusion rate of the cation, Cs+, is largely affected by the presence of intercrystalline surfaces. Therefore, the inter- pretation of sintering phenomena, which take place in polycrystalline aggregates must take into account the diffusion of ions in the zones close to the surface of crystals. Dr. D. A. Young (HarwelE) said: A corollary implicit in Tamann's concept of the stickiness of a microcrystalline particulate specimen is that the mobile species responsible for ionic conduction and the mass transfer adduced in inter-crystalline cohesion can only migrate from one particle to another when the particles are in genuine contact.Thus in particulate specimens such migration can only be detected using simple d.c. or low-frequency a.c. techniques after the samples have been sintered. Recognition of this fact has led to the use of ionic conductivity as a sensitive tool for studying the annealing and sintering of particulate specimens. I wish to stress here that when this method is combined with the simultaneous measurement of thermoelectric power a yet more sensitive and informative tool is obtained. This approach is illustrated by experiments on silver azide. Below 190" C, silver azide crystallizes in a body-centred orthorhombic structure which transforms irreversibly to a denser structure on heating above 190" C.Whilst undecomposed, the crystals do not photoconduct at wavelengths greater than 380 mp and no Hall effect is observed ; thus the conduction in undecom- posed silver azide is ascribed to the motion of ions, On the other hand, once the crystals have partially decomposed (0.1 %), colloidal centres of silver form and the crystals become photoconductive in the visible with a threshold in the near infra-red. This threshold can be correlated with the activation energy for semi-conduction. In this condition the crystals show an n-type Hall effect, from which we conclude that partially decomposed crystals conduct by both ions and electrons.When a freshly-prepared micro-crystalline particulate specimen of silver azide is heated slowly (0-2 deg./min) for the first time to, say, 180" C and then as slowly cooled, the conductivity changes are as shown schematically in fig. 1, provided no decomposition occurs. If, on the other hand, the samples are heated rapidly (2-3 deg/min) to T,, the conductivity at Ta is found to fall isothermally to line ((a) or line (b) as appropriate. Similarly after a rapid rise from T, to T' the con- ductivity is observed to rise isothermally to lines (a) or (b). During this cycle the azide forms a hard cake and it is proposed that T, marks the elimination by sintering (reduction of surface area) of surface conductivity, whereas Tg marks intergranular cohesion, whence line (c) gives the characteristics of bulk ionic con- ductivity. It is not possible to traverse the transition point without sufficient decom- position occurring to convert the azide into an n-type semi-conductor, so in fig.2 the conductivity changes at the transition point of partially decomposed azide are shown. The values of 9 and 6.4 kcal/mole are the E/2-values for semi- conduction in the low- and high-temperature forms respectively. Although it is evident that the behaviour at the transition point cannot simply be explained by rupture of the intergranular contacts during creation of the dense, high-temperature form in case (a), and by retention or rapid reformation of such contacts in case (b) unless the activation energy for the ionic conductivity in the high-temperature form is fortuitously 6.4 kcal/mole, it is nevertheless clear that ionic migration should be detectable above the transition point in case (b).In order to resolve some of these questions the thermoelectric power of a168 GENERAL DISCUSSION lightly cold-worked, partially decomposed, micro-crystalline specimen in the low-temperature form was measured during slow heating from room temperature to above the transition point and back again. The specimen was under pressure and the result obtained is shown in fig. 3. Tt is proposed that the n-type power is due to electrons and the p-type power to mobile interstitial cations. Some loq u b 4 kcal mole -i-- I I I 19OoC I/.,. FIG. 1. FIG. 2. 1 ' 190°C I/T FIG. 3. a. unconfined material. 6. pellets under pressure.intergranular cohesion occurs during the superficial sintering at Tar and is com- pleted rapidly at T'. At the tran- sition point, the intergranular contacts break and the ionic contribution is thereby eliminated, thus giving rise to the leading edge of the n-type peak. However, at this temperature sintering is rapid and the p-type power is quickly restored. On cooling to room temperature, the p-type power is retained. These and other similar corroboratory experiments will be described in greater detail in papers shortly to be published. Dr. A. G. Maddock (Cambridge) said: In connection with Mr. Clarke's dis- cussion of the kinetics of the thermal annealing of radiation damage there is one kinetic pattern that he does not include and which I consider to be especially interesting because it leads to an intelligible molecular model for the annealing process.In a number of annealing processes, as different as the thermal bleaching of irradiated silver phosphate glass,l the recovery of the conductivity of irradiated uranium wires,2 as well as the annealing of many kinds of chemical radiation The ionic power overwhelms the electronic. 1 Davison, Goldblith and Proctor, Nucleonics, 1956, 14, no. 1, 34. ZKonobeevsky, Pravdyuk and Kutaitsev, Proc. Geneva Conf. (Peaceful Uses of Atomic Energy), 1956,7,433.GENERAL DISCUSSION 169 damage,l the isothermal annealing curves commonly reveal that at each temper- ature the fraction of recovery increases at first rapidly with the time of heating but subsequently settles down to a much smaller rate of change (fig.6 of our paper). Thus if the annealing is continued until the rapid change is over, the fractional recovery depends largely on the temperature of annealing and is not greatly influenced by the time of heating; a pseudo-plateau value for the fractional recovery is found. However, longer periods of heating show that annealing is still taking place. Moreover, if a sample, annealed to the pseudo-plateau re- covery at one temperature, is then heated to a higher temperature, it anneals to the pseudo-plateau value characteristic of the higher temperature along a curve similar to the ordinary annealing isotherms. Some time ago Mrs. de Maine and I pointed out2 that this kinetic pattern can be interpreted in terms of the combination of two species that attract one another, according to some inverse power law of their distance apart, by a normal unimolecular process.Thus df'dt = v exp (- E/kT) exp (+ Y/kTx), where f is the fraction of recovery, E is the unimolecular activation energy, Y is some kind of potential and x the distance apart of the fragments. This is indeed the same kind of kinetics as Mott and Cabrera 3 have obtained for the formation of very thin oxide films. In the annealing processes, however, the attractive forces may be electrostatic, or may arise from the stress field in vicinity of the defects, or possibly some other mechanism giving an inverse power law. The second exponential term accounts for the initial fast recovery but as the closer pairs recombine this term soon approaches unity and the first exponential term accounts for the subsequent much slower recovery.f and x are related by a function de- pending on the distribution of fragments. Mr. G. N. Walton (Harwell) said: In connection with Mr. Clarke's paper I should like to mention some work which is being carried out at A.E.R.E., Harwell, by Dr. F. J. Stubbs and myself. This is a study of the emission of fission product gases, krypton and xenon, from fissile material during irradiation by neutrons, i.e. while fission is taking place. We have done this for several materials keeping the neutron flux constant and altering the external temperature so as to vary the temperature of the sample over the range 100-400" C. Radioactive decay of the gases and their precursors occurs during the process of emission, so that a steady emission rate is eventually obtained when the rate of production is balanced by decay and emission.From these rates it is possible to calculate an apparent diffusion coefficient at the different temperatures of the sample. Owing to radio- active decay, gas atoms from deep layers of the sample have insufficient time to get out before decaying, so that the observations, unlike those on stable atoms, only apply to atoms moving in the surface layers of the material. An allowance also has to be made in the diffusion equation for depletion of the surface by atoms which recoil out of the specimen by fission. Plotting our diffusion coefficients against the reciprocal of the temperature gives a value for an apparent activation energy E.For measurements on uranium metal this activation energy is rather lower than estimates of E reported for the diffusion of rare gas atoms in uranium from " out-of-pile " measurements, i.e. on samples after irradiation. apparent activation energy for diffusion eV/atom after irradiation -1 during irradiation - 0.3 The values for the pre-exponential constant in the two cases do not seem to be greatly different. 1 Green, Harbottle and Maddock, Trans. Faraday SOC., 1953, 49, 1413. 2 Maddock and de Maine, Can. J. Chern., 1956,34,275. 3 Cabrera and Mott, Reports Prog. Physics, 1948, 12, 163.1 70 G EN ER A L D I SC LJ S S 1 ON Analysis on the lines of Mr. Clarke’s paper could not account for this irradi- ation effect. In our experiments the energy deposited by fission amounts to about 0.04 eV per atom per min which is comparable to the value of kT (- 0-05 eV per atom).The energy deposited by fission, however, must be distributed over the atoms in a manner quite different from the way heat is normally distributed. For instance, exceedingly steep temperature gradients must be produced inside the material being irradiated and these would be expected to promote the mobility of diffusing atoms. Mr. Clarke’s paper has drawn attention to the effect of a spread in values of the activation energy term E, but for irradiation studies I should like to suggest that attention be given to the effect of a spread in the values of the temperature term T. Mr. F. P . Clarke (Harwell) (communicated) : Concerning Dr. Maddock’s comments, I am grateful to him for describing his molecular model.The rapid nitial rate of recovery is, of course, explicable on the analysis mentioned in my paper. The exact position of the characteristic annealing function relative to the initial activation energy spectrum may be described in terms of the position of the point of inflection of the function. This is given by the expression kT In (Bt). Hence if the temperature is high enough a major part of the damage will be annealed out rapidly; the dependence of the subsequent annealing on the In Bt term will give the pseudo-plateau mentioned by Dr. Maddock. This approach makes no physical assumptions about the nature of the damage apart from saying that it consists of many different types of centre giving altogether a continuously variable activation energy spectrum. Its weakness lies in the fact that it is hard to check independently the truth of the assumption, and my paper showd that a much simpler model could in certain cases explain the ob- servations. This model effectively replaces the potential term in the Maddock- de Maine expression. It assumes that the attraction will lower the potential barrier to diffusion in such a way as to favour recombination of the attracting species, and that once the initial jump has been made, the recombination is certain to occur in a time very short compared with that required for the initial jump. The unknown potential term of the Maddock-de Maine expression appears in this model as an unknown frequency term. The introduction of a potential term should have certain advantages in ex- plaining the pseudo-plateau part of the annealing curve. One would expect such a term to fall off fairly rapidly with distance, but to be strong for say, close defect pairs. Hence for the initial part of the anneal involving close defect pairs the different methods of describing the process might be equally suceessful. For the later stage, however, even though the potential does fall off rapidly with distance, a slight directional perturbation of the potential barrier to diffusion might have time to assert itself appreciably. Concerning Mr. Walton’s comments, I agree that the temperature term is most important in the case of fissile-bearing materials or materials of very low thermal conductivity. The damage will become unevenly distributed during ir- radiation, and on removal from the irradiating source different regions of the crystal will anneal at varying rates. This is as Mr. Walton suggests an important reason for his results not lending themselves to analysis on any of the models T have mentioned. The effect of thermal and displacement spikes on the in-pile annealing cannot in my opinion be covered effectively by any simple addition to the existing models. The simple assumption made in my paper, that such spikes would merely reduce the defect concentration by an unknown amount, is too nai’ve to be quantitatively effective.

ISSN:0366-9033    

DOI:10.1039/DF9572300155

出版商:RSC    

年代:1957

数据来源: RSC