The Journal of
PhysicaIChemistry ~
~~~~~
0 Copyrighl, 1991, by the American Chemical Society
VOLUME 95, NUMBER 6 MARCH 21, 1991
LETTERS Constraints on the Movement of Atoms in the Totally Symmetric Vibrations of Crystal Lattices: Lattice Force Constant Calculations for Sodalites J. A. Creighton,* H. W. Deckman, and J. M. Newsam Exxon Research and Engineering Company, Clinton Township, Route 22 East, Annandale, New Jersey 08801 (Received: June 25, 1990)
Atoms in crystal structures at certain special positions are forbidden by symmetry to move in the zero wavevector totally symmetric lattice vibrations. A rule for identifying these special positions is given, and use is made of the constraint on atom movement to derive equations that give the valence force constants for the aluminosilicateframework of sodalite from Raman spectroscopicdata. The Si+-A1 bending force constant is found to be relatively small, and the equations define a relationship between the totally symmetric frequencies and the Si-O-AI angle, thus providing the basis of a method for determining this angle from experimental Raman data.
Atoms in crystal lattices at special positions which by symmetry have no positional degree of freedom are forbidden to move in the optically allowed (zero wavevector) totally symmetric lattice vibrations. Recognition of this constraint can assist greatly in the interpretation of such totally symmetric modes and in particular can lead to a considerable simplification in the derivation of the equations relating the frequencies of the totally symmetric modes to the lattice force constants and geometries. We illustrate this here by deriving equations that give the valence force constants for the sodalite structure from its totally symmetric vibration frequencies. Atoms at such special positions are constrained to be stationary in the totally symmetric modes of the lattice since any movement of an atom from such a position would perturb the crystal symmetry. Such sites lack any positional degrees of freedom, and the structures that have such sites may be identified by noting that their point symmetry groups are those in which no vectors (none of the translational vectors) are a basis for the totally
symmetric representation. They are thus the sites with symmetries oh, 0,Thq Td, T, S,, D3d, D~drDnh, Cd,Dn ( n = 6,4, 3, 2)1 C3j9 or Ci. Such sites are common in crystals in the cubic and hexagonal systems, but they are found in all systems except monoclinic and triclinic.' Since the site point groups in a given lattice are subgroups of the factor group, it also follows that in crystals showing this property none of the components of the dipole moment vector are a basis for the totally symmetric representation of the factor group, and thus the totally symmetric vibrations of crystals that have such sites are always infrared-forbidden. The totally symmetric vibrations in crystals showing this property are therefore never subject to LO-TO splitting, which is a considerable complication in the full analysis of the lattice dynamics of the infrared-active vibrational modes of crystalline solids.' In order to show the significance of the constraint on the movement of some of the atoms in the totally symmetric modes of such crystals, we use this fact to derive equations for the frequencies of these modes for the sodalite structure, represented
*To whom correspondence should be- addressed. Permanent address: Chemical Laboratories, University of Kent, Canterbury CT2 7NH, U.K.
(1) Decius, J. C.; Hexter, R. M. Molecular Vibrarions in Crysrals; McGraw-Hill: New York, 1977; p 346.
0022-3654191 12095-2099%02.50/0 0 1991 American Chemical Society
2100 The Journal of Physical Chemistry, Vol. 95, No. 6, 1991 by the Substance Na8Si&02&12. This Structure, Space group P43m (=Td4),2consists of an aluminosilicate framework (composition [Si&16024]bcomprised of silicon and aluminum atoms (T atoms) alternating around the vertices of a truncated cuboctahedron ("sodalite cage"), with bridging oxygen atoms between each T atom. The T atoms are shared between adjacent sodalite cages, and the coordination about each T atom by oxygen is close to regular tetrahedral. The framework silicon, aluminum, and oxygen atoms each respectively occupy symmetrically equivalent lattice positions, and one sodalite cage constitutes the primitive unit cell. The anionic framework charge, and the charge associated with the occluded chloride ions, are compensated by nonframework sodium cations. The following discussion treats only the framework vibrations, and the representation of the zero wavevector modes of the framework reduces to 3Al 5A2 8E + 13Tl + 14T2 in the factor group Td.3 In terms of internal coordinates (48 T-O bond length displacements, 24 T U T and 7 2 0-T-O angle displacements) the dynamical matrix is of order 144 with 39 zero roots, and the construction and factoring of this matrix to yield the equations of motion for the three AI modes would clearly be a considerable task. The T atoms are, however, at sites of S4 symmetry and are therefore constrained to be stationary in the A, modes, while the 0 atoms at CI sites have no such constraints on their motion. The three AI modes are therefore simply the vibrations of the set of symmetrically equivalent 0 atoms in three orthogonal directions between fixed T atoms, with the additional requirement for each T 4 - T oscillator that the four 0 atoms around each T atom move symmetricallyin phase about the local S4axis so as to maintain the S, symmetry of the T atom site. The equations that describe these modes are therefore very simple, but more importantly, they may be. derived by considering an equivalent simple "molecular" model. The model is essentially a single T U T oscillator, but in addition to T-O stretching and T-O-T bending, it must also incorporate internal coordinates associated with 0-T-O bending. In order to define the 0-T-0 bending internal coordinates in a way that introduces vibrations corresponding only to AI modes of the sodalite framework, use is made of the fact that in the AI modes of the framework the four 0 atoms-argundeagh-Tatom move symmetrically in phase about the local S4 axis so as to maintain the S4symmetry3fihe T atom site. Therefore, only the two AI (S,) combinationsof the displacements Sa,,in the six (FT-0 angles around each T atom can be involved. Assuming furthermore tetrahedral 0-T-O angles, only one of these is nonzero, namely
+
s
(26a12- 6a13 - 6a14 - ~
-
a 2 6a24 ~
+
+ 2~a,)/fi
where 6a12and 6au refer to displacementsof the 0-T-O angles bisected by the local S4symmetry axis. This coordinate may be replaced at each T atom (but for a normalizing constant) by the displacement 68, of the angle between the local Si axis and one of the T-O bonds. In the model also, therefore, an analogous coordinate W is introduced at each T atom to represent the displacement of the angle between the T-O bond and a spatially fued axis through the T atom that is coincident with the direction that the corresponding S4axis would have in the crystal. To define these axes, dummy atoms X are included in the model, which is thus a nonplanar W-shaped XSiOAlX molecule, in which X and the two T atoms (Si and AI) are assigned to have infinite mass so that they are stationary. The configuration of the model is such that it has C, symmetry if the two T-O bonds are taken to be equivalent, and the internal coordinates are the two T-O bond length displacements 6rl and br2. the two X-T-O bending displacements 68, and M2,and the T+T angle displacement of Sg. The derivation of the C matrix for this molecule by standard methods' is particularly easy on account of the infinite masses (2) Low, J.; Schulz, H.Acta Crysrallogr. 1967, 23, 434. (3) Maroni, V. A. Appl. Spectrosc. 1988,42,481. (4) Wilson, E. B.; Decius, J. C.; Craw, P. C. Molecular Vibrations; McGrsw-Hill: New York, 1955.
Letters of the X and T atoms. The eigenvaluesof GF,for an assumed F matrix, then give the three quantities 4?rzut plus two zero mots, where v, are the three vibration frequencies. Though not required for the model, it is assumed henceforth for simplicity that the S i 4 and Al-O bonds have equal lengths and equal force constants and that the X-Si-O and X-Al-O angles are equal. It is also assumed that the F matrix is diagonal, corresponding to the simplest valence approximation for the potential energy 2 v = fi[(W2+ (br2)21 + h[(W2+ (ae2)21
+f B ( w ) 2
Since the X and T atoms are fixed, the following relationships exist between the coordinates 68' - SO2 = p(6rl - 6r2) cot ( @ / 2 )cos T and
Sg = -p(6rl
+ 6r2) tan ( @ / 2 )
where 7 is the angle by which the X-T bonds are rotated about the T-O bond axis out of the TOT plane (the angle between the normals to the XTO and TOT planes) and p is the reciprocal of the T-O bond length. By transforming to the symmetry coordinates
s1= (6rl + 6 r 2 ) / f i s2= (tie, + se,)/~/?
s3= (6rl - 6 r 2 ) / f i these relationships may be used to remove the two redundant coordinates (68, - 68,) and S@ from the potential energy expression, and the C and F matrices in these coordinates are then g,, = 2P e a 2
(@/a
gI2= gZl= pl.c sin @ cos T 2p2p(1 - COS2 ( @ / 2 )COS2 T)
g22
g33 = 2P sin2 (@/2) fil
= fi + 2Pzfg tan2 f22
f33
@/a
=h
= f, + P% cot2 ( 8 / 2 ) cos2 7
where l.c is the reciprocal atomic mass of oxygen. As will be reported elsewhere? we have evaluated the full C and F matrices for the zero wavevector modes of the infinitely extended sodalite framework and have verified algebraically that the A, blocks obtained by symmetrizing these matrices (with a diagonal F matrix and with or without the assumption of equal S i 4 and AI-O bond lengths, interbond angles, and force constants) are identical with the C and F matrices given by the model. It would be extremely laborious to derive the algebraic form of these matrices by symmetrization of the full sodalite G and F matrices without the assumption of equal S i 4 and AI-O bond lengths and force constants, but use of the model provides a straightforward way of arriving at these algebraic expressions. A singlecrystal Raman study' enables the three AI framework modes of the sodalite Na8Si,gi&OuC12 to be unambiguously assigned to prominent Raman bands at 987, 463, and 263 cm-'. These wavenumbers lie just outside the range of real solutions calculated by use of the C and F matrices given above, but the closest solution, with calculated wavenumbers 987,469, and 253 cm-I, corresponds to the following values of the force constants for bond stretching (f,), OTO angle bending (foro=h/3), and TOT angle bending (f ): f , = 522 J m-2,fOm = 65.5 X lo-" J rad-2,fs = 7.58 X 10-l' J rad-2. Although significance cannot be attached to the precise values of these force constants since there are several approximations implicit in this treatment (the (5) Reference 4, p 143. (6) Creigbton, J. A,; Deckman, H. W.; Newmm, J. M. To be publbhed. (7) Ariai, J.; Smith, S.R. P. J . Phys. C Solid Store Phys. 1981.14. 1193.
2101
J. Phys. Chem. 1991,95,2101-2103 assumption of a harmonic potential function with zero offdiagonal force constants, and neglect of the dynamical effects of the nonframework alkali-metal cations and chloride anions), it is nevertheless notable that& is smaller thanfm and indeed is small on the scale of typical bending force constants. The smallness of& clearly underlies the conformational flexibility of the sodalite framework, in which the T-0-T angle varies from 125O to 139O to 154O on changing the nonframework cation in the series Li, Na, K.8 We believe these to be the first direct experimental data on the force constants in aluminosilicates, and they confirm the results of a molecular mechanics investigation of the sodalite structure in which it was also concluded thatfm
