Intermolecular Forces and Orientational Phase Transitions in Molecular Crystals BY ALFRED HULLER Institut fur Festkorperforschung, Julich, West Germany Received 1 1 th December, 1979 The conditions under which one expects orientational phase transitions are discussed in the introduction. Then we consider the interactions between molecules and construct a Hamiltonian which is compared with the spin Hamiltonians that are in common use in the field of critical phe- nomena. A definition of the order parameter is given with special attention to the ordering of higher multipoles and with a discussion of its experimental determination. Finally the oppor- tunities of obtaining information about critical exponents are considered, but the prospect is rather unpromising. 1. INTRODUCTION In this paper we consider crystals which consist of molecules, polyatomic ions or side groups.These units will be denoted by the common name " molecules " and the crystals they form by the name " molecular crystals ". Methane (CH,) is an example of a proper molecular crystal, whereas in the ammonium halides (e.g., NHZCl-) the " molecule " is an ammonium ion, and in dimethylacetylene the " molecule " is a methyl group. If the molecules are tightly bound units, one may distinguish between the high-frequency internal modes and the low-frequency external m0des.l For our purposes the internal modes may be neglected, whereby the number of degrees of freedom per molecule is reduced to three translational and one, two or three rotational degrees of freedom: one rotation angle for a side group like CH,, two for a linear molecule like N2 and three for a three-dimensional molecule like CH,.Depending on the question as to whether the translational and rotational degrees of freedom are ordered or disordered, we distinguish four main classes of condensed molecular phases : (1) isotropic liquids with translational and rotational disorder, (2) liquid crystals which are rotationally ordered but translationally disordered, (3) orientationally disordered crystals (formerly also called plastic crystals) with trans- lational order and (4) proper crystals (brittle crystals) where both translational and rotational degrees of freedom are ordered. We are concerned with the orientationally disordered phases (3) and transitions towards complete order. Order may be achieved in several stages.Then we observe several orientational phase transitions with a stepwise reduction of the orientational disorder. Orientationally disordered phases are characteristic of crystals consisting of globu- lar molecules.2 The globular shape is evident for cage-like molecules such as ada- mantane or the carbonanes. Small and highly symmetric molecules such as methane may also be considered to be spherical. Electron density contours in the outer parts of the electron shell (where the overlap between neighbours occurs) are veryA . HULLER 67 closely spherical. The weak non-isotropic interactions mainly originate in the relatively weak multipole-multipole interactions. The same argument holds true for hydrogen, but in addition to that the quantum-mechanical uncertainty principle prevents a sharp localization of the molecular axis, thereby amplifying the globular appearance of H,.For globular molecules the angular average of the intermolecular forces is much stronger than the variation of these forces with respect to rotations of the molecules at a fixed distance. The radical part of the potential is responsible for the translational order, i.e., for the formation of a regular lattice of the molecular centres of mass. The angle-dependent part of the potential is responsible for the orientational ordering of the molecules. Consequently, for globular molecules orientational order sets in at lower temperatures than translational order. One observes the phenomenon of orientational disorder in crystals (ODIC). Unfortunately, in most of the phase transitions between crystal phases of different orientational order, the centre of mass structure also changes.In hydrogen, for example, the orientationally disordered phase is h.c.p. whereas the ordered phase is f.c.c. The simultaneous transformation of the centre-of-mass structure distorts the picture of a rotational phase transition. Therefore, the few examples where the symmetry of the centre-of-mass structure is not changed at the phase transition are of crucial importance. Our discussion will be based on the 1/11 phase transition in solid methane and on the II/IV transition in the ammonium halides (see fig. 1). In both cases the centre of mass structure is cubic above and below the phase transition.I 2 1 -- 2 O - 1 160 200 240 T / K FIG. 1 .-Generalized Stevenson plot for the p-T phase diagram in the ammonium halides which has been introduced by Press et al.14 The roman numbers denote the three phases which differ in the orientational order of the ammonium ions: I1 is disordered, IV is ordered in a parallel way (ferro), and in I11 there is antiparallel order (anti-ferro). TCP denotes the tricritical point. For pressures higher than the tricritical pressure the II/IV phase transition is continuous. All other lines denote first-order transitions. 2. INTERMOLECULAR FORCES The angle-dependent intermolecular forces are of the same nature as the central forces between atoms. They arise from the interactions between the permanent electrostatic monopole and multipole moments (electrostatic forces), the interactions between mutually induced multipoles (dispersion forces) and the overlap between68 INTERMOLECULAR FORCES A N D ORIENTATIONAL PHASE TRANSITIONS the electron shells (valence forces).With the exception of the interaction between two hydrogen molecule~,~ the intermolecular forces cannot be calculated from first principles. Therefore, with the exception mentioned, all the potential functions found in the literature, as e.g., atom-atom potentials4 and the Kihara core m0de1,~ are phenomenological expressions containing a number of adjustable parameters. Here we only discuss the expansion of the intermolecular potential V(Rij, mi, mj) between two molecules i and j in a twofold multipole series : 6 , V(Rjj, mi, CO~) zzz 2 A ~ ~ ; m , n , ( R i j ) D ~ A ( m i ) D ~ ; ~ , ( ~ j ) ; (2.1) 1 , m , n .l',m',n' Rij denotes the vector distance between the centres of mass of the two molecules, mi collectively denotes the set of Euler angles (ti, T i , q i ) which define the orientation of molecule i in space. The functions DgA(m) are a complete set of orthonormal func- tions in the Euler angle space, e.g., the Wigner functions. Then 0 I I < co and -1 rn, n I 1 where I stands for the total angular momentum, rn and n are projec- tions of 1. The coefficients Aii',m,,,(Ri j ) are phenomenological functions of Ri which comprise the effect of all three types of interactions. For purely electrostatic forces the distance dependence is [Rij1-'-''-'. In many cases it is advantageous to replace DgA(co) by a complete and orthonormal set of symmetry-adapted functions, e.g., the cubic rotator functions UF;(co) where I is still an angular momentum, but p and v do not have a physical interpretation. High molecular symmetry and a corresponding choice of the proper basis set Uh$(m) greatly reduces the number of non-vanishing coefficients in expansion (2.1).Let us consider expansion (2.1) for two neutral methane molecules with electrostatic interactions only. Modifi- cations due to dispersion and valence forces will be discussed afterwards. The lowest order non-vanishing terms are of order (I, 1') = (3, 3). The octopole moment ( I =I= 3) is the lowest-order multipole moment of a neutral tetrahedral molecule. Therefore there are no terms of order 1 = 0, 1 and 2.Terms of order 1 = 0 are not excluded by symmetry, but they are absent for neutral molecules. The functions A:?,;),, (R) have been calculated by James and Keenan : (2.3) I = A:?,;?, (R) = 2 ~,,P*(R/R) where Z3 is the octopole moment of CH4. The dimensionless matrix Dfi,&, only de- pends on the direction of R. Eqn (2.2) with terms of order (I, Z') = (3,4), (4,4) etc., neglected constitutes the James and Keenan' model for methane. If valence and dispersion forces are important the l/R7 dependence of the octopole term is changed. The new R dependence of the potential cannot be calculated but it can be determined from tunnelling experiments under pressure. This has recently been demonstrated for (NH4+),SnCl; -.* An experiment on CH4 is under way.' More important than the different R-dependence of the coefficients Ai;,';l,vp is the fact that now all symmetry- allowed coefficients are non-vanishing, as e.g., the octopole-monopole interaction As U ~ ~ ) ( c o j ) = 1 is a constant V(390) does not depend on the orientation of molecule j .Summing eqn (2.4) over all neighbours j of molecule i one obtains a contribution to the crystal field experienced by molecule i:A . HULLER 69 where the angular brackets denote an average over the intermolecular distances. The crystal field persists into the disordered phases. It is distinguished from the molecular field V,(co,,) which is obtained from a summation of the interactions which depend on the orientation of both partners. The octopole-octopole term then reads : where the angular brackets denote an average over molecular distances and over w j the orientations of the neighbours of molecule i.In disordered phases the angular average vanishes; therefore the molecular field is zero. Our discussion of V, and V, has been based on the molecular-field approximation which neglects correlations of the intermolecular distances and of the orientations of the molecule j with the orientation of molecule i. The crystal field in methane seems to be rather weak.’O This can be understood in terms of the nearly spherical shape of the outer parts of the electron shell and the charge neutrality of the molecules. The eight nearest- neighbour Cl- ions of a NHZ tetrahedron create a strong crystal field with pronounced minima. The strong crystal field leaves the molecule a choice between two different orientations (more precisely between two groups of 12 equivalent orientations each).Around the two equilibrium orientations, given by Euler angles CU, = (O,O,O) and m2 = (O,n/2,0) the molecule performs small librations with 8” amplitude.” The ammonium-ammonium interaction also contains an (I, Z‘) = (3,O)-term which contributes to the crystal field. The (3, 3)-term (2.2) is the same as in CH4. The electrostatic approximation should be excellent as the NHZ ions do not overlap. For the two equilibrium orientations allowed by the crystal field six of the seven functions U$)(w) are zero. For the libra- tional ground state we obtain <Ui”,(w)) = sc with 5 = 0.893 for NHZ and < = 0.908 for N D p s = & l for wavefunctions centred around col and w2.The octopole- octopole interaction [eqn (2.2)] in the librational ground state thus reduces to : This is different in NHZCl-. Ui:)(co) is equal to 1 for co, and equal to - 1 for co2. For the nearest-neighbour distance R i j = a(1, 0,O) one finds’ Ill,, = -6 and which shows that the orientational ordering in NHZC1- can be described in terms of a pseudo-spin Ising model with a nearest-neighbour ferromagnetic exchange constant J = 6c21$/R7. This is, however, not the whole story: apart from direct interactions to more distant neighbours there are indirect interactions mediated by the halide ions. There are two contributions to the effect, namely the spin-phonon coupling12 and the coupling via the electronic polarizabilities l3 of the halide ions.The indirect interactions favour the antiferromagnetic ordering which is found in NHZ Br- and NH,+I- for zero and moderate pressures. All these interactions contain a factor c2. Therefore phase transitions in the deuterated ammonium halides should generally occur at temperatures which are higher by a factor (0.908/0.893)2 = 1.034 than in protonated samples. This is indeed true for disordered to ferro-ordered transition with transition temperatures of 242 and 249 K for NH4Cl and ND4Cl, respectively. At the disordered to anti-ferro-ordered transition the isotope effect is reversed. In deuterated samples the transition is 30 K lower (!) than in protonated s a m p l e ~ . ~ ~ * l ~ It is the opinion of the author that this reversed isotope effect is not understood on a70 INTERMOLECULAR FORCES AND ORIENTATIONAL PHASE TRANSITIONS microscopic basis, but that there are two approaches towards an understanding. The first one starts from the generalized Stevenson plot', for the ammonium halides (see fig.1). One finds that the replacement of Br- in NHZBr- by Cl - corresponds to the application of an external pressure of 1.8 kbar. Deuteration corresponds to a pressure of 1.3 kbar both in NHZBr- and in NHZCl-. A shift of the zero-pressure line on deuteration not only explains the different sign of the isotope effect for the transition into the ferro-ordered and antiferro-ordered phases, but also its size which depends on the slope of the transition line in thep-Tdiagram. The problem with this explanation is that we do not understand on a microscopic basis why deuteration should correspond to the application of a pressure of 1.3 kbar.The lattice constant change on deuteration is Aa/a z 0.0005, an effect which can be achieved by an externally applied pressure of 0.25 kbar. The second explanation of the isotope effect relies on the difference in the hydrogen bond with protons and deuterons.16 The reasons for a reduction of the hydrogen bonding on deuteration are dynamic in nature. The theory of hydrogen bonding is, however, not developed to the point where a quantitative comparison is possible. In molecular crystals the interactions typically fall off like l/Rn with n = 5, 7 etc. This is in sharp contrast to magnetic systems where the exchange interaction falls off much more rapidly with distance.In many orientational phase transitions, interactions with distant neighbours are important, and if these interactions are com- peting as in the ammonium halides they lead to a complicated phase diagram and thus may influence the critical behaviour. We have given a detailed discussion of the octo- pole-octopole interaction which is responsible for the ordering in CH, and NH: C1-, two important examples for rotational phase transitions. In the case of dumb- bell molecules as, e.g., H2 or N2 the interaction is a little less complicated than eqn (2.2). It retains, however, its basic quality-the tensorial character. The orienta- tion of a dumb-bell is defined by two angles, the polar coordinates 8 and q. Therefore, the interaction is expanded into a double series of functions (the spherical harmonics) defined in that space.The quadrupole moment has only five components, so the 7 x 7 matrix D,,.(R/R) is replaced by a 5 x 5 matrix. Apart from these fine details the tensorial character of the interaction implies that D,,. depends on the direction of the intermolecular distance. In this respect the interaction is radically different from the isotropic exchange interaction in a Heisenberg ferromagnet. The inter- action not only depends on the relative orientation of the two molecules, but on the orientation of the molecules with respect to R i j . Therefore the phase transition in molecular crystals cannot be mapped on a Heisenberg model. Contrary to the Heisen- berg case the transverse excitations (librations) in the ordered phase always have a gap.This is true even for the case of vanishing crystal field which is almost ideally realized in solid hydrogen. On the other hand, for strong crystal fields (NHZCl-, adamantane, K+CN-, etc.) an Ising model represents the phase-transition properties almost ideally. 3. THE ORDER PARAMETER ITS DEFINITION In the Ising limit (strong crystal field) the definition of the order parameter q is obvious. For a ferromagnetically ordered phase we define where <si) is the expectation value of the ith pseudo-spin. In cases with weak crystalA . HULLER 71 fields where we cannot construct a pseudo-spin Ising model such a definition is im- possible. We first claim that a purely rotational phase transition (i.e., without rearrangement of the centre-of-mass structure) can only take place when the molecular symmetry is lower than the symmetry of the centre-of-mass structure.This statement will be discussed afterwards. Methane molecules with tetrahedral symmetry on a cubic (f.c.c.) lattice are an example. The hamiltonian [kinetic energy plus the interaction given in eqn (2.2)] then has the full symmetry of the lattice; in our example this symmetry is cubic. In the high-temperature phase the symmetry of the hamiltonian is not broken by the orientational order of the molecules. If we expandf(co) the orientational distribution function (also called a probability distribution function p.d.f.) for a molecule into a series of symmetry-adapted functions (adapted to the symmetry of the lattice) only those functions are allowed which are simultaneously invariant under the symmetry operations of the molecule and of the lattice.For the CH4 example such an expansion reads: 7 ~ 1 7 co 21 4- 1 the cubic rotator functions U$&(co) have been introduced in Section 2. There are (21 + 1) functions of order 1 but only very few are allowed functions. In our example the non-vanishing coefficients are: AiO,), Ai:), A::), etc. For vanishing crystal field f(o) is a constant, then A::', A::), . . . are all zero. The functions Uii)(co), Uit)(co), . . . modulate the orientational probability density f (co), their coefficients Ai;), Ai:), . . . depend on the strength of the crystal field and for finite crystal fields they are non-zero at all temperatures. When the cubic symmetry of the Hamiltonian is broken, the symmetry at a methane site is spontaneously reduced from cubic to tetrahedral.Then the functions Uii)(co), Ui:)(co), . . . become symmetry-allowed. Their coefficients A!;), Ai;), . . . are zero above the phase-transition temperature and non-zero below. Normally the lowest order coefficient A$;) is identified with the order parameter q. The new definition of q is valid both for weak and strong crystal fields. For strong crystal fields it is identical with the former definition by eqn (3.1). When the molecular symmetry and the site symmetry are identical, spontaneous symmetry breaking is not possible. EXPERIMENTAL DETERMINATION OF THE ORDER PARAMETER In a hypothetical example of polar molecules that undergo a phase transition into a phase with parallel orientation of the dipoles, one would observe a macroscopic electric moment which is proportional to the order parameter.If higher-order multipoles such as quadrupoles, octopoles, . . . order, no macroscopic moment is observed. The same is true for antiparallel dipole order. Therefore, apart from the hypothetical case of parallel dipole orientation, there is no means of observing the order parameter macroscopically. Neutron and X-ray diffraction is the only possi- bility for its determination. A diffraction experiment determines S(Q) = J dcoS(Q, co) which is the Fourier transform of the equal-time density-density correlation function (p(rl, t)p(rz, t ) ) . More precisely stated, p(rl, t ) is the density of the scattering power, i.e., the scattering length density if one considers the scattering of neutrons by nuclei, the electronic spin density in the case of magnetic neutron scattering and the electron density in the case of X-ray scattering.72 INTERMOLECULAR FORCES AND ORIENTATIONAL PHASE TRANSITIONS We divide the density into its contributions from the N molecules in the crystal: Then the correlation function contains terms of the type (pi(rl, t)pj(r2, t ) ) which we separate (pl(r1, t ) ~ j ( r 2 , t>> = <pi(rd><pj(rJ> + <bi(ri, t ) - pi(r)I[~j(~z, t ) - ~j(r)l>* (3.4) The first term comes from the time-independent single-molecule distribution (it leads to Bragg diffraction, the second term contains the correlations); it is responsible for the diffuse intensity between the Bragg points.The Bragg intensity is connected with the probability distribution of scattering centres in a coordinate system fixed in the crystal. To relate (p,(r)) with the expansion (3.2) which defines the order parameter, we introduce a second (primed) coordinate system fixed in the molecule which rotates with it. If the molecular structure is known, b(r‘), the scattering length density in the primed coordinate system is also known, and can be expanded in terms of sym- metry-adapted surface harmonics : b(r’) = C brrm~(r‘)K1,,*(8‘, 9’) (3.5) I’m’ with known coefficients b,.,,(r’). In our examples the proper set of functions are the cubic harmonics. K31- x’y’z’ and K41 - xt4 + f 4 + zr4 - 3/5 are the lowest-order functions with tetrahedral and cubic symmetry, respectively.Using the definition of the cubic rotator functions the scattering-length density in the crystal frame for a given orientation co of the mole- cule. b(r) is averaged with the orientational p.d.f. from eqn (3.2) to yield: (p(r)> = I d(?f(co,)b(r) = C alm(r)K1m(6 9) (3.8) on follows from the orthogonality relations of the cubic rotator functions. The coeffi- cients alm(r) can be measured in a diffraction experiment; they determine the order parameter. If AC) is the order parameter then ~~31(r) is zero above T, and finite below. Its temperature dependence, i.e., the critical exponent p, can be determined from the intensity of the Bragg points (with h, k, I all odd) to which it contributes. The diffuse scattering caused by the second term in eqn (3.4) can be used to obtain information about correlations, in particular about critical correlations in the vicinity of a phase transition.The intensity of the critical scattering is related to the magni- tude of the fluctuations, i.e., to the critical exponent y. The width of the scattering in Q-space is inversely proportional to the correlation length <, i.e., to the critical ex- ponent v. Information about the critical slowing-down is obtained from the energy depend- ence of the critical scattering, that is to say from a determination of S(Q, co).I8A . HULLER 73 4. CRITICAL PHENOMENA Towards the end of the preceding Section, methods to determine critical exponents for orientational phase transitions in molecular crystals have been discussed.The presentation has perhaps been misleading. It may have left the impression that there is a wealth of examples of continuous phase transitions accompanied by critical phenomena. The phase transitions in molecular crystals are almost exclusively discontinuous phase transitions. The use of critical laws like AC) w [(T, - T)/T,]/p is mi~leading,l~-~~ since one obtains spurious values of the critical exponents (e.g., p = 1/6) which have nothing to do with critical fluctuations or low dimensionality. The arguments why a critical exponent p should not be defined at a discontinuous phase transition have been given in ref. (22) andwill not be repeated here. So far the only example where a continuous orientational phase transition has definitely been found is the II/III phase transition of NH,+Cl- and ND,+Cl- under pressure.Here measurements of critical exponents are very difficult. This is seen from the two determinations 23 of /3 (for p = const and for T = const) which differ by 30%. The phase transitions in molecular crystals are not interesting from the point of view of critical phenomena. These systems do not belong to the domain where scaling laws or universality can be tested. The study of phase transitions in molecular crystals is important for a better understanding of intermolecular forces. Very little is known about these forces, which are the basis for all properties of condensed molecular phases. Phase dia- grams are a very sensitive test for the angle dependence of these forces. For the description of the phase transitions themselves a molecular-field theory is mostly sufficient.A very elegant and extremely successful example of such a calcula- tion is the James and Keenan’ theory of solid CH,. The reason for the applicability of molecular are-field theory to slightly discontinuous phase transitions is the fact that the incipient phase transitions are interrupted by the discontinuity before the fluctuations become divergent. The non-divergent fluctuations that occur may be treated within the Ornstein-Zernicke approximation. This is not the case. G. Venkataraman and V. C . Sahni, Rev. Mod. Phys., 1970, 42, 409. J. Timmermans, J. Phys. Chem. Sol., 1961, 18, 1 . I. Silvera, Rev. Mod. Phys., to be published. A. I. Kitaigorodosky, Molecular Crystals and Molecules (Academic Press, New York, 1973). T. B. MacRury and W. A. Steele, J. Chern. Phys., 1976,64, 1288 and references therein. J. 0. Hirschfelder, C. F. Curtis, and R. B. 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