Artigo Porto Normal Mode Determination in Crystals

SERGIO P O R T 0 COMMEMORATIVE ISSUE

Normal Mode Determination in Crystals ~~

D. L. Rousseau Bell Telephone Laboratories, Murray Hill, New Jersey 07974, USA

R. P. Bauman Department of Physics, University of Alabama in Birmingham, Birmingham, Alabama 35294, USA

S. P. S. Porto Instituto de Fisica, University of Campinas, Campinas, Brasil

~~

The group theoretical methods by which the symmetries of normal modes in crystals may he determined are outlined, and a series of tables are presented to facilitate rapid determination of the selection rules for vibrational transitions. Emphasis is placed 3n the method of nuclear site group analysis in which the number of infrared and Raman active modes of each symmetry may he obtained without detailed analysis of the symmetry elements in the crystallographic unit cell or the construction of correlation tables. By using the tables presented here for most cases identification of the crystallographic space group is sufficient information to allow determination of the vibrational mode selection rules by inspection. Several examples are included in which crystals are analyzed by each of the methods.

INTRODUCTION

The current interest in phonon spectroscopy, which has been stimulated by the development of the laser for Raman scattering and the steady advance of inter- ferometric techniques in the far-IR, has made the rapid determination of selection rules for vibrational transitions most important. However, owing to the wide diversity of molecular and ionic crystals which may be studied, a variety of techniques are necessary to treat each problem appropriately. Analysis of the internal motions of a crystal relies heavily upon the symmetry properties of the crystal, which are best treated on the basis of group theory. Although the theory is not quite as simple as for molecules, the methods are quite similar. The determination of the number of vibrational modes, their symmetries, and hence the selection rules for IR absorption or Raman scattering, may be carried out in the same general manner as for free molecules,''2 bear- ing in mind a few differences to be discussed and the absence of rotational and translational freedom for the unit cell. The purpose of this paper is to present the basic group theoretical methods which may be employed to determine crystal selection rules, to illustrate the methods with examples, and to present a set of tables organized to facilitate rapid vibrational selection rule determination. It must be emphasized that the paper is intended to enable the researcher to quickly and con- veniently determine the selection rules from a minimal amount of crystallographic data. Keeping this in mind proofs and justifications of the techniques described have been omitted so that the methodology would not become lost. Ample references have been included so that the interested reader may, if he wishes, consult the original works and study the mathematical foundations of the methods.

There are several methods by which the unit cell may be analyzed and the selection rules determined. Some of these methods consider only k = 0 p h ~ n o n s ~ . ~ . ~ (i.e. near the Brillouin zone center) and others consider points throughout the Brillouin zone.6 Generally the first order phonon spectrum (near k = O), exhibits the greatest intensity and is the object of the vast majority of investigations. For this reason only k = 0 techniques will be discussed in this paper. There are three techniques of selection rule determination at the zone center. First is the factor group analysis m e t h ~ d , ~ developed by Bhagavantum and Venkatarayudu, in which the symmetry properties of the crystal are determined by studying the effect of each symmetry operation in the factor group on each type of atom in the unit cell.

The second method is one which we wish to label as molecular site group analysis as presented by Halford4 and H ~ r n i g . ~ In this technique the symmetry properties are determined first for the unit of interest (i.e. a molecule or an ionic grouping in the unit cell) as an isolated species. These results are then analyzed in terms of the site symmetry and finally in terms of the factor group symmetry. Winston and Halford7 demonstrated that this method is equivalent to the factor group analysis method. Detailed discussions of this technique have been made recently by Fateley etal.' and by Ferraro and Ziomek.'

A third technique was developed by Mathieu to treat molecules.'" In this paper this last method, which we term nuclear site group analysis, is extended to crystals and is presented in conjunction with a set of tables to greatly ease the burden of selection rule determination. This method is simply a generalization of the molecular site group analysis in that a site symmetry analysis is carried out on every atom in the unit By determining the site symmetry of each atom in this way a set of tables may be constructed to allow symmetry and selection rule determination without the construction of a

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correlation diagram." Such a set of tables (Tables B) are included here.

Space groups, point groups and factor groups

The symmetry of an infinite crystal in its equilibrium configuration may be described by a space group. This is a group of symmetry operations, which obey the mathematical criteria for a group, and which, through their application on the crystal, either leave each atom unchanged or carry it into an identical atom. An infinite set of translational symmetry operations in addition to point group symmetry operations constitute a space group. (A point group consists of a set of symmetry elements all of which pass through a single point.) Rather than using the infinite space group to determine the symmetry properties of a crystal, it has been found to be an excellent spectroscopic approximation to consider only a single unit cell of the crystal and treat the infinite set of translations between unit cells as identity operations. The atoms within a unit cell are related by the familiar point group symmetry operations in addition to screw axes and glide planes. (A screw axis symmetry operation is a rotation followed by a translation along the axial direction. A glide plane operation is a reflection across the plane followed by a translation along the plane.) By thus separating out the translations, a finite group remains, which is a factor group of the space group and which describes the symmetry of a unit cell. It has been termed a unit cell group or a factor group and is isomorphous with one of the 32 crystallographic point groups. The crystal class, as defined originally by the analysis of the external symmetries of the crystal, determines which of the 32 point groups is applicable to a particular crystal.

The factor groups differ, however, from the point groups in several important ways.

1.

2.

3.

4.

There need be no point within the unit cell that is left invariant by all the symmetry operations, and therefore it is not necessary that rotational axes and symmetry planes meet at a common point. A point is considered to be invariant if it is either left in place by a particular symmetry operation or is carried over to a position, in an adjacent unit cell, which it could have reached by a simple translation of one lattice unit. That is, if it goes to a corresponding point in another unit cell, or does not move, the point is called invariant; but if it goes to another position within the same unit cell, or to a position in an adjacent cell corresponding to such other position in the original cell, it is not considered to be invariant under the operation in question. As pointed out previously, some of the symmetry operations will include translations by fractions of a unit cell length (screw axes and glide planes). These are taken as equivalent to, that is they map into the same class as, the operations without the translational components. In particular, screw axes may be substituted for, or mixed with, pure rotational axes, and glide planes are similarly taken as equivalent to mirror planes. There may be more than one set of operations falling within the same class. For example, there may be

more than one center of symmetry, or where one C2 axis is present in the point group, there may be several (e.g. four) C2 axes in the unit cell group, all equivalent by symmetry, or there may be several C, axes plus several two-fold screw axes, all these operations mapping onto the simple C2 axis of the point group.

It is through the multiplicity of possibilities described above that the 32 point groups give rise to 230 unit cell groups. These factor groups are normally labelled either by the Schoenfliess symbol in which the point group of the crystal class (e.g. C2") is given a numerical right superscript to designate the factor group (e.g. C;,); or they are labelled by the Hermann-Mauguin symbols in which the crystal class symbol (e.g. Pmm2) is modified to indicate explicitly the substitution of a screw axis or a glide plane (e.g. Pmc2,). A detailed description of the labelling systems may be found in Vol. I of The Inter- national Tables For X-ray Crystallography.12 Spectros- copists generally use the Schoenfliess symbols, while crystallographers find the Hermann-Mauguin symbols more convenient. Both symbols are used in the Tables presented in this paper.

Site symmetry

Within the unit cell there are sets of points in which each member of the set has an identical environment. These points are designated as sites and the symmetry operations associated with one of these points define a group (site group) which is a sub-group of the factor group and is isomorphous with one of the 32 point groups permit- ted in crystal structures. Each member of the set shows the complete symmetry of the appropriate site group for that set. All the points within a set may be generated by applying the operations of the factor group to one of the points. The number of equivalent points, n, in a set is equal to the order of the factor group, H, divided by the order of the site group, h, i.e. n = H/h. For example, in the factor group D3d, there are 12 equivalent sites in a set having C1 symmetry (n = 12/1), and 3 points in a set with C2,, symmetry (n = 12/4). From symmetry considerations the position of a point of given site symmetry may be precisely fixed within the unit cell, or in cases of the C,,,, C,,, and C, site groups, it may have from one to three degrees of freedom since in these groups the sites may lie anywhere along an axis, along a plane, or within the volume of the unit cell. Therefore in unit cells which contain these site groups of indeterminate position, an infinite number of sets of such sites may exist within the cell. For example in a unit cell belonging to the C:, factor group there are four sets of C4,, sites each of which contains only one equivalent point, two sets of C2h sites with two equivalent points within each set, an infinite number of sets of each C2 and C, sites each of which has four equivalent points, and an infinite number of sets of C1 site symmetry with eight points per set. These 'special' positions are those the atoms may occupy when distributed in the unit cell. The symmetry of the sites then defines the symmetry of the local environment of the atom occupying the site, and since it is very useful to have at hand when determining selection rules, the symmetry of all the sites of each factor group have been tabulated in Tables A.

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NORMAL MODE DETERMINATION IN CRYSTALS

SYMMETRIES AND SELECTION RULES OF VIBRATIONAL MODES

Factor group analysis

The determination of the selection rules by direct use of the factor group is a method analogous to the standard technique for free molecules as long as the differences between the point group and the factor group are kept in mind. Symmetry analysis is made by applying all of the symmetry operations of the factor group to each atom in the unit cell, and reducing the representation thereby obtained in order to determine the number of normal modes belonging to each irreducible representation.

Consider a unit cell containing N atoms. Under a certain factor group symmetry operation, R, there will be some number of atoms U ( R ) that are left invariant (under the definition previously given). Each such atom will contribute xT(R) to the trace in the ith class of the reducible representation of the vibrational displacement matrix. xT(R) is the trace under the operation R for a vector displacement, and for Cartesian coordinates may be given by

XT(R) = *1+2 cos & (1) the plus sign corresponding to a proper rotation (not involving a reflection) and the minus sign arising for an improper rotation (i.e. an inversion, reflection or rotation-reflection). The angle & is the angle of rotation for the symmetry operation, R. Therefore the total contribution from all the atoms to a reducible representation (I') of each symmetry operation R is determined by x(R) where

x ( R ) = U(R)xT(R) ( 2 ) For a point group of m classes, one now has m numbers x(R) . These are the characters of the reducible representation of the motions of the nuclei of the unit cell under consideration. This representation may be reduced by the standard formula

or since within each class k, the x j (R) are the same, Eqn (3a) may be written as

In Eqns (3a) and (3b), nj is the number of times the irreducible representation of symmetry species rj appears in the reducible representation and is therefore the number of lattice modes of symmetry r j ; H is the order of the point group; xj (R) is the character of the irreducible representation, Ti; X ( & ) and xi(&) are the characters of the reducible and irreducible representations, respectively, of the kth class of the point group containing g k elements (symmetry operations).

The selection rules for modes having the symmetry of the various rj's are determined by the character tables of the point groups, so that when the number of modes, nj, belonging to each rj has been found, the number of spectroscopically active modes is known. In this manner the correct number of lattice modes may be determined,

within the Born-von Karman approximation, and excellent agreement with experimental observations are obtained. In determining the values of U ( R ) there are two features of the unit cell which may give rise to ambiguities.

It will be apparent that the symmetry operations that include translations by fractional lengths of the unit cell, i.e. the screw axes and glide planes, cannot leave any point in the primitive unit cell invariant. Special consideration must be given, however, to the non- primitive centered structures. A screw or glide operation will move a point to another point within the same centered unit cell, or to a non-corresponding point within an adjacent cell. However, to find the number of distinct lattice modes we must choose as our unit cell the primitive cell. For the purposes of screw and glide operations, it is therefore necessary to identify a priori which points within the centered unit cell correspond to identical points within different primitive cells, e.g. a corner and a center point. The interchange of such points, then, cor- responds to invariance when determining U ( R ) . The numbers U ( R ) will be two or four times larger for the centered unit cell than for the primitive cell simply because the larger cell contains more atoms. In determining the values of U ( R ) for a given class it is usually sufficient to consider the operations of a given class together. However, in some instances the nuclei are invariant to only one member of a class at a time. In other words, one symmetry operation may leave a certain nucleus invariant and interchange the remaining equivalent nuclei in the set. Another member of the same class will leave a different nucleus of the set invariant and interchange the remaining members of the set. If this is the case, in evaluating U ( R ) the total number of equivalent nuclei under all of the operations in the class should be divided by the number of operations in the class, or else a notation made as a reminder not to multiply by the number of operations in the class (gk) when applying Eqn (3b) to reduce the representation.

A more detailed Dicture of the svmmetrv modes of the crystal may be obtained by independently considering each set of equivalent, and of course identical, nuclei within the unit cell. The same procedure as outlined above is carried out for each of these smaller sets of numbers U'(R). In this way one arrives at sets of irreducible representations corresponding to each set of nuclei. This can greatly facilitate the detailed writing of symmetry coordinates and also the qualitative determination of the type of symmetry modes that are appropriate to each irreducible representation.

When there are distinguishable groupings within the unit cell, which may be molecular (e.g. benzene, H2 or Ss) or ionicspecies (e.g. Cog- or NO;), it is often helpful to calculate separately the internal vibrational modes of these groups and the external modes (often referred to by others in the literature as the lattice modes) in which the molecular group moves as a unit. The molecular mode frequencies are generally higher than the external lattice mode frequencies, and agree well, when comparisons are possible, with frequencies observed in gaseous or liquid samples. The external lattice modes

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appear solely as a consequence of the crystalline structure. The separation of internal molecular and external lattice modes may be easily performed.

The external lattice modes are of two types, which may be described as translations and rotations of the molecular groupings. (The translations exist even for single atoms or icins; the rotations apply only for diatomic or polyatomic groupings.) The acoustic lattice modes may be considered as translatory modes of the unit cell as a whole.

The numbers of internal and external modes may be calculated by finding the characters of the representations for the several types of motions. For all possible motions, the character is U(R)xT(R), as given previously. The representation of the acoustic modes has the character xT(R). Translatory lattice modes (including the three degrees of freedom corresponding to acoustic modes) give rise to the character [ UM(R) + UA(R)]XT(R), in which UM(R) is the number of polyatomic molecular units left in place (though possibly rotated) by the operation R and UA(R) is the number of monatomic particles, not included in any of the molecular groupings, left in place by the operation R. Similarly, the character of the representation of rotatory lattice modes is UMxrot(R), where XrOt(R) is the character, under the symmetry operation R , of a general r ~ t a t i o n , ’ ~ and may be written as

Xrot(R) = * X T ( R ) = f (* 1 + 2 cos f$) = 1 *2 cos f$ (4) Here we adopt the usual convention in which the upper sign is for a symmetry operation that is a proper rotation and the lower sign is for a symmetry operation that is an improper rotation (involving a reflection).

The total number of external lattice modes, including the acoustic modes, is therefore

[LTM(R)-k UA(R)IXT(R)+ UM(R)Xrot(R) ( 5 ) and the number of internal molecular modes is found by subtracting this number of external modes from the total number of modes. That is, the character of the representation for the internal molecular modes is

Nuclear site group analysis

The methods of site analysis provide a completely different procedure for obtaining similar symmetry information. Coinsider a factor group, G, of order H. As stated previously a site group, g, which must be a sub- group of G will have order h and n equivalent points of g site symmetry where n = H/h. Each class of sub-group g will map onto one or more of the irreducible representations of the group G. Therefore motions of nuclei determined for their site positions, may be correlated into the factor to obtain the complete description of all the nuclei in the unit cell.

The number of lattice modes arising from each equivalent set of nuclei, their symmetries, and their selection rules may be obtained by the following procedure. Assume the ith set of nuclei rests on sites of point symmetry g, within a unit cell of symmetry corresponding to the point group G. In the sub-group g, one, two or three of the irreducible respresentations will correspond

to the behavior of the three components of a vector. These are simply the irreducible representations of g, said to have allowed IR activity. The irreducible representations of G onto which the IR active irreducible representations of g, map, give the symmetries (under G) of the lattice modes arising from the ith set of nuclei, occupying the sites gi.

There are certain generalizations that simplify this mapping procedure. For example, sites of symmetry C1 give rise to three modes belonging to each non- degenerate symmetry species of the point group of the unit cell, to six modes belonging to each doubly degenerate species, and to nine modes belonging to each triply degenerate species. When two vector components transform under the same symmetry species of g,, the number of modes for the unit cell is similarly doubled. A convenient check of the mapping operation is given by the condition that for each site symmetry, g,, the number of lattice modes (number of non-degenerate plus twice number of doubly degenerate plus three times number of triply degenerate frequencies) is three times the number of nuclei occupying those sites (i.e. 3N degrees of freedom). In the mapping process it is necessary to distinguish between irreducible representations of dimension two and the ‘separably degenerate’, or reducible, representations of dimension two; the latter are not to be considered as degenerate in the restricted sense of this paragraph.

The technique to use is the following: the space group appropriate for the crystal under investigation must be known as must be the number of nuclei of each type within the unit cell. Then one must consult the tables presented in this paper for that space group to determine which site symmetries are compatible with it (Tables A). Frequently the number of nuclei of a certain type within the unit cell may be placed only on a single set of sites compatible with the space group. If more than one set of sites are possible for any set of nuclei then a more detailed analysis of the crystal must be made to determine the symmetry. This may often be most easily accomplished by comparing the X-ray crystallographic data with the map of the unit cell given in Ref. 12. Having established the site symmetry of each kind of nucleus, only the mapping procedure remains, in which irreducible representations of the three vector components are mapped onto the irreducible representations of the factor group and the total number of lattice vibrations is thereby determined. Tables of this information (Tables B) are included in this paper and will be discussed subsequently.

Molecular site group analysis

If distinguishable molecular or ionic groups exist in the unit cell methods analogous to those discussed above may be applied and the ‘external’ motions of these species may be readily determined. Consider a molecu- Iar grouping, M, which has symmetry described by the point group G M and which is located in a unit cell described by the point group G. The molecule M must be centered on one of the lattice sites, which we may describe as a site of symmetry gi, one of a set of n equivalent sites in the unit cell. The group gi then must be a sub-group of both G and GM. Exactly analogous



considerations of the groups gi and G as for the nuclear site group analysis case give the translational lattice modes (including the acoustic modes) attributable to the n equivalent molecules. (These data are presented in Tables B.) By classifying the rotational degrees of freedom in sub-group gi, and by then correlating this information with the factor group G, the librational lattice modes may be obtained in a similar fashion. (This information for non-linear polyatomic molecules is tabulated in Tables C.)

A more thorough analysis may be made, though slightly more time consuming, in which all the motions of the free molecule (or ion) are considered and the effects on these motions of the site symmetry and the unit cell symmetry are determined. An isolated molecule M of symmetry G M may have q vibrational degrees of freedom in addition to three rotational and three translational degrees of freedom. When the molecule is placed on some site of symmetry gi of equal or lower symmetry than GM, a correlation may be made between the irreducible representations.of G M and those of gi. In this way the vibrational, rotational and translational degrees of freedom may be followed so to speak from its isolated free molecule symmetry to its restricted site symmetry. Since the symmetry of the site is always equal to or lower than that of the free molecule (gi is a sub-group of GM) vibrational modes which had been inactive to IR absorption or Raman scattering may become active due to the lower symmetry. The irreducible representations of the site group are then correlated with those of the unit cell group to determine the final number of lattice modes. A set of correlation tables are included in this paper in Tables D. The effect of going from the site group to the unit cell group is to cause splittings of the free molecule frequencies. This splitting is often referred to as correlation field splitting and results from the interaction between the n molecules in the unit cell.

Selection rule breakdown in Raman scattering

Generally, the group theoretically determined selection rules hold very well. Occasionally, experimental difficulty arises in both IR absorption and Raman scattering in which apparent violations occur due to depolarization effects owing to anisotropy and optical rotation in the crystal. When appropriate technique^'^ are applied, these problems may be overcome usually and the predicted behavior observed. However there are two situations, unique to Raman scattering, which can exhibit selection rule breakdown. These are effects caused by incident laser resonances with real electronic transitions (resonance Raman scattering) and splitting due to long-range electrostatic forces. In both of these cases the first order phonon spectrum is modified. We do not discuss the problem of multiple phonon scattering in which k = 0 selection rules are no longer valid and points throughout the entire Brillouin zone must be considered.

It is usually assumed that the polarizability tensor for phonons is symmetric, and this appears to be a very good approximation when the exciting frequency for the Raman spectrum is far from an electronic absorption band. If proximity to an absorption band does produce

asymmetry, the selection rules will be modified, by addition of allowed transitions that were previously forbidden. In particular, in listing the transformation properties of the components of the polarizability tensor, as in Tables E, entries appear of the form aXy. These entries arise from the sum axY+ayx. The difference term, ax,, - ayx, which is assumed to be of zero magnitude, will usually have different transformation properties. Under conditions of resonance excitation such modes may become allowed and therefore may appear in the Raman spectrum.

The difference terms for off-diagonal tensor components are not listed as such in Tables E although they have precisely the transformation properties of rotations. R, transforms as aXy -ayx, R, transforms as ayz -azy, and R, transforms as a,, -a,,. The transformation properties of these rotations are explicitly given in Tables E. In addition to thc observation of asymmetric components in the Raman tensor other selection rule breakdowns due to properties of the resonant excited state have been r e p ~ r t e d . ' ~

When Raman active phonons are simultaneously IR active they carry a polarization which results in a frequency splitting between the transverse and longitudinal waves. This splittin is reflected by the Lydanne- Sachs-Teller relationship in which the static dielectric constant so is related to the high-frequency dielectric constant sm by

1E

(7)

for a crystal in which there exists only one optical polar phonon. This phenomenon has been well characterized and has been found to give rise to three anomalies in the Raman s p e c t r ~ m . ' ~

1. 'Extra' lines appear due to the removal of the transverse and longitudinal degeneracy.

2. In various orientations these lines are found to shift in frequency. The frequency shifting results from the variation in the strength of the coupling caused by the short-range and the long-range forces in the crystal. To predict this shifting properly it is necessary to consider the magnitude bf these forces and to consider carefully the geometrical arrangements and thereby determine which kind of phonon (transverse or longitudinal) is being observed.

3. The intensities vary as a function of orientation in a manner not predictable by simple group theory. To interpret the intensities properly, again it is necessary to consider the magnitude of the crystal forces and the geometrical arrangement. By drawing momen- tum conservation diagrams" one can easily predict the frequency shifts and intensities such that the spectra may be well understood. In the construction of the diagrams it is necessary to know the macro- scopic polarization associated with each phonon. Loudon" has discussed this and gives a set of tables of this information. In Tables E of this paper this information is also included in the following way: the polar phonons are those in which there is allowed IR activity. This is designated by the polarization directions X , Y or Z. If the phonon is Raman active the polarizability tensor components are written in the correct order such that a one to one correspondence

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may be made between the tensor component and the associated polarization direction. For example, consider Table 19E (C3,). An A 1 phonon is polarized in the 2 direction. Therefore the tensor components a,,, ayy and azz are all polarized in the 2 direction. The polarization for an E phonon reads (X , Y) and the two sets of degenerate Raman tensor components associated with this read (xx-yy, xy), and (yz, xz). Therefore, the x polarization is associated with aXx, ayy and aYz (the first term within each of the sets of parentheses). Similarly the a,? and ax= phonons are polarized in the Y direction.

TABULAR PRESENTATION OF SYMMETRY INFORMATION

Whichever calculational procedure is employed, the information of direct interest is determined by the site symmetry and the unit cell point group. Therefore, since there are only 32 point groups that can describe the symmetry of crystals, and only 230 finite space groups for which the sites must be determined, it is possible to present all of the necessary information in the moderate number of tables, given below.

At the heading of each set of tables is given the Schoenfliess notation of the point group (followed by the Hermann-Mauguin symbol for Tables A). Tables A present the finite space groups by number, with the Schoenfliess and Hermann-Mauguin symbol of each, followed by the total number of possible site symmetries for that space group. The number of equivalent sites of the given symmetry is given in parentheses immediately after the point group symbol for the site symmetry, and the letter, or letters (Wyckoff notation), identifying the set or sets of sites of the same symmetry are given preceding the point group symbol. For example, the entry m(f+ e)C; (4), among the sites for the finite space group D:d (P4b2), indicates that there may be four equivalent nuclei occupying the sites designated ‘ f ’ on C, axes parallel to the z, or c, axis of the crystal, and that there may be also (or instead) four equivalent nuclei occupying the sites designated *e’ , also on C2 axes parallel to the z axis. Although most site symmetries define a set of points, and can therefore be occupied by only one set of nuclei in predetermined positions, the C, and C,, sites (Cl, C2, C3, C4 and Cg, and C, = Ctu, C2,, G U , and Go) are not unique and thus can be occupied by more than one set of nuclei. In other words there exists an infinite number of possible sets of the Cg sites discussed above. This is indicated by infinity symbol preceding the Wyckoff letter notation. In this same unit cell group (Dzd) we find from the table the entry ( b + a)S4(2) which indicates that there are only two sets of S4 sites ( b and a ) each of which have two equivalent points per set.

The number of equivalent sites in any set is given by the order of the point group divided by the order of the site symmetry sub-group for the primitive (symbol P ) and primitive rhombohedral (symbol R) lattices, but the number is greater for the centered lattices. The side- centered lattices (symbol A, B or C) have twice as many points, the body-centered lattices (symbol I) have twice as many points, and the face-centered lattices (symbol

F ) have four times as many points as the primitive lattice structures. Also, the rhombohedral lattices may be represented by a hexagonal unit cell, which will have three times the volume and thus contain three times as many points as the rhombohedral cell.

For many of the point groups there are non- equivalent symmetry elements of the same type and therefore non-equivalent sites of the same symmetry. In most of these point groups it is sufficient to indicate the ‘diagonal’ elements, and the site point groups derived from them by primes, or to add superscripts giving the axes lying in or parallel to the symmetry elements. Where there is possible ambiguity of labeling, the method of labeling is indicated below the table.

Tables B, for each point group, list each possible site symmetry for the finite space groups corresponding to that point group, and give the number of lattice modes, of each symmetry, produced by a set of nuclei on such sites. Tables C are a mapping, onto the unit cell point group, of the site symmetry point group irreducible representations that correspond to pure rotations. Tables D are a set of correlation tables giving the correlation map for all subgroups of each point group.

Tables E are a set of character tables of the point groups, and also contains the character of a general vector, XT defined in Cartesian coordinates. Symmetry properties of translational vectors, and hence the elec- tric dipole absorption selection rules, are given, along with the symmetry properties of the rotations. The acoustic modes belong to the representations which contain the translation vectors. In the last column the symmetry properties of the polarizability tensor components are listed, which provide Raman scattering selection rules.

Examples of calculations and use of the tables

In the examples which follow selection rules are determined for several crystals by one or more of the methods discussed here. However, each crystal is analyzed by nuclear site group analysis to demonstrate the facility of this method and the convenience of the tables.

Sodium chloride is a well-known example of a cubic face-centered structure containing four sodium ions and four chloride ions per (non-primitive) unit cell. Since the space group describing the crystal is Og, we find from Table 32A that only Oh sites can accommodate two sets of four equivalent points in each. The lattice modes can now be read directly from Table 32B. Each set of nuclei (sets a and b ) on sites of symmetry Oh give rise to one triply degenerate mode of symmetry Flu. Thus NaCl has two triply-degenerate lattice modes, both of symmetry Flu. From Table 32E we find that these are IR active and Raman inactive. One is an acoustic branch and the other is an optical branch.

Another well-known face-centered cubic crystal is diamond in which there are 8 carbon atoms in the 0; unit cell. From Table 32A we find that these 8 equivalent carbon atoms must lie on Td sites (on either set a or b) . From Table 32B a set of Td sites gives rise to two lattice modes, one Flu and one F2g mode. The Flu is an acoustic mode and the F2g is a Raman active optical mode with off -diagonal non-zero tensor components.

Barium titanate, BaTi03, is also cubic (0;) in its high-temperature phase, but changes to tetragonal (Citi)



Cb" 4

( 0 ) ( b )

Figure 1. Unit cellsfor cubic (a) and tetragonal (b) barium titanate. In the cubic phase the Ba2+ ions are at the cube corners, the 0'- ions are at the face centers and the Ti4+ ions are at the body centers. In the tetragonal phase the vectors represent the relative directions that the atoms are displaced with respect to the cubic phase. Adapted from Ref. 21.

at about 120 "C. These two structures, each containing one formula unit per primitive cell, are illustrated in Fig. 1. It should be noted that the three oxygen atoms in the unit cell are equivalent in the cubic phase; but in the tetragonal phase the atom which lies along the unique axis is distinguishable from the other two. We may readily determine the symmetries of the lattice modes for these two structures by the method of nuclear site analysis (i.e. direct use of the tables).

For the cubic phase it may be readily determined by consulting Table 32A that the barium and the titanium atoms (one of each per unit cell) must occupy oh sites and the 3 oxygen atoms must lie on D 4 h sites. From Table 32B each nucleus on an O h site will contribute an Flu symmetry mode and the three oxygen atoms will contribute 2FlU + F2u, giving a total of 4FlU + F2u. The acoustic mode is Flu as indicated in Table 32E, so the vibrational modes have the symmetry species 3F1, + F2u, of which the Flu modes are IR active and the FZu mode is silent.

To determine the symmetry of the lattice modes of the tetragonal phase (Ci"), it is necessary first to consult Table 13A, in which it is found that only C4u sites will accommodate a single atomic species within the unit cell. These sites are labelled a and b but we must remember that there are an infinite number of C4u sites within the unit cell. Therefore the Ba, the Ti and the unique oxygen all must occupy C4u sites. The remaining two oxygens must lie on C2, sites. Consulting Table 13B the lattice modes from the Ba, Ti and unique oxygen are 3A + 3E. Those from the remaining oxygens are A1 + B1+2E giving a total of.4A + B1 + 5E. Table 13E shows that the acoustic modes are A l + E and, therefore, there are 3A1 optical modes which are simultaneously IR and Raman active, one B1 mode which is Raman active, and 4 E modes which are also IR and Raman active. The IR active modes split into transverse and longitudinal waves. This will have no influence on the IR absorption spectrum since only transverse phonons can interact with the IR radiation. However, Raman scattering may be observed from both transverse and longitudinal waves and therefore twice the number of Al and E modes predicted by group theory may be found in the experiment.

t i t ! . A!. t i

Figure 2. Symmetry operations of the cubic phase, 0:' of barium titanate. This figure was adapted from Ref. 20 and uses the standard notation for the various symmetry

It is also illustrative to determine the selection rules for the cubic phase of BaTi03 by factor group analysis. The symmetry operations'' for 0; are drawn in Fig. 2. (These operations should be compared to the unit cell in Fig. l(a).) The first step in the analysis is to determine the value of U ( R ) for the barium, titanium and oxygen atoms. The first two lines in Table 1, giving Us,(R) and UTi(R), are straightforward. Each Ba atom is on a site of oh symmetry and is therefore invariant to precisely the symmetry operations listed, and similarly for the Ti atoms. Note, however, that the symmetry elements are not necessarily the identical ones for Ti as for Ba. For example, the C4 axes passing through Ti atoms do not pass through Ba atoms and vice versa.

Table 1. Factor group analysis of BaTiO, in its cubic phase

Oh : i 6S4 3uh 8Ss 6 r d Us. : 1 1 1 1 1 1 1 1 1 1 U T i : 1 1 1 1 1 1 1 1 1 1 Uo: 3 1 3 0 1 3 1 3 0 1 UBaTiOa: 5 3 5 2 3 5 3 5 2 3 X T : 3 1 - 1 0 - 1 - 3 - 1 1 0 1

E 6C4 3 C i 8C3 6Cz

3 1 - 1 0 - 1 - 3 - 1 1 0 1

9 1 - 3 0 - 1 - 9 - 1 3 0 1



The method of counting invariant oxygen atoms is less obvious. There are three oxygen atoms per unit cell, so three are invariant under the identity operation E. Also, each is on a center of symmetry, each is on an intersection of three mutually perpendicular planes of symmetry (perpendicular to the unit cell edges), and each is on an intersection of three mutually perpendicular CZ axes parallel to the unit cell edges (of which only one is actually also a C4 axis). There are, under Oh symmetry, necessarily three mutually perpendicular C4 axes (with operations C4 and C:). Each oxygen atom ( D 4 h symmetry) is on only one C4 axis, and each C, axis leaves only one of the three oxygen atoms invariant. This applies as well to the three distinct S4 axes. Similarly, there are necessarily six Cz axes, which pass diagonally through the facles of the unit cell (and an equivalent set passes through the center parallel to the faces), but each oxygen atom is on two and only two of these six axes. Furthermore, each of these Cz operations leaves only one oxygen atom invariant, so U0(6Cz)=l . The diagonal planes of symmetry, 6 u d , similarly leave one oxygen atom invariant. No oxygen atoms are on C3 or S6 axes.

We next obtain the representations of the Ba, Ti and 0 motions b y multiplying the U ( R ) values by XT

obtained from Table 32E. Reduction of the representation for Ba and Ti motions is trivial; the representation is clearly the irreducible representation Flu. The representation of the oxygen motions is reducible to the sum of irreducible representations, 2F1, +F2,.

Calcium tungstate (CaW04) contains four formula units per body centered unit cell of space group C,", (141/a). The modes for this rather complicated unit cell may be readily found by consulting Table 11A. Since there are four Ca atoms, four W atoms, and sixteen identical oxygen atoms it is immediately seen by inspection of Table I1A that the calcium and the tungsten atoms each must occupy S4 sites, and the sixteen oxygen atoms must be on C1 sites. From Table 11B the vibrational modes of the crystal are then 3A, + 3A, + 38, + 3B, + 3E, + 3E, for contributions from the oxygen atoms and 2A, + 2B, + 2E, + 2E, for the contributions both from the calcium and tungsten atoms. The total number of optical branches are 3A, + 4A, + 5 B, + 3B, + 5E, + 4€, while the acoustic branches are A, + E,. We also find from Table 11E that the A,, B, and E, modes are Raman active and the A, and E, modes are IR active.

More information about the details of these vibrational modes may be obtained from the tables. First the tungstate ion may be treated independently. In other words consider a unit cell composed of two particles in each formula unit-a calcium ion and a tungstate ion. Each must occupy S4 sites, and from Table 11B this gives 2A,, t 2B, + 2Eg + 2E, modes. A, + I:, are acoustical modes 50 there remain A, -I- 2B, + 2E, + E,, modes which are optical modes of translational character. By consulting Table 11C we may also assign A, + B, + E, + E, as librationill modes resulting from oscillations of the tungstate ion. No librational modes may be found of course from the Ca ion since it has no rotational degrees of freedom. It should be pointed out that since E, and E, modes arise from both translational and librational motions, the normal coordinate describing these degrees of freedom will be a combination of those symmetry

coordinates describing the librational and translational motions and may not therefore be designated as pure librational or pure translational modes. The total number of 'external' motions of CaW04 are thus A, + A, + 2B, + B, + 3E, + 2E,. By comparing with the previous result for the total number of modes, the 'internal' motions of the tungstate ion are 2A, + 3A, + 2B,+2B,+2Eg+2E,.

A more thorough analysis of these internal motions may be obtained if the method of molecular site group analysis is used. The isolated tungstate ion W0;- has T d

symmetry. By standard techniques* the vibrational motions of this ion may be designated by the following: AI(YI) , E(vz), Fl (rotation), 3F2 (translation, v3, v4). These representations of T d specify the motions of the W0:- ion, and the designations in parenthesis after each representation are the standard Herzberg nota- tions. It is now most convenient to construct a correlation table by consulting Table D. In Fig. 3 each of the T d representations are listed for the tungstate ion. Since the ion occupies an S4 site, the effect of the lowering of the symmetry is determined by forming the correlation between the Td and the S4 representations. The contribution of the calcium ion, also on an S4 site, is only of a translational nature. By then correlating the site symmetry representations to those of the unit cell, the effect of having the several ions within the unit cell interacting with each other is taken into account. To determine the nature of each lattice mode it is now necessary to consider its origin. The 3A, and the 3B, modes each may be divided into a librational lattice mode and a mode derived from each V I and VZ. The 5B, and the 5A, modes each contain two translational lattice modes and modes derived from VZ, v3 and v4. The 5E, and the 5E, modes each result from two translational modes, one librational mode, and v3 and v4. It should be noted that an A, translational mode and an E, translational mode form the acoustic branches of the crystal. Of the internal modes, the mode v1 should appear only in the Raman effect and as a single line. In the vz region there should appear two lines in the Raman spectrum (A, + B,) and one in IR absorption (A,). There should be two lines for both v 3 and v4 in both the Raman spectrum (B, +E,) and in the IR spectrum (Atd +EL,). The magnitude of the splitting between any pair of these modes is determined by the forces in the crystal and may

1

Free Unit

species sym. syrn. sym. modes Ionic ion Site cell Vibrational

Ca" - s4 c4,

A / \ (trans, "3, @4)3F2 f , , - 5 f o ( v 3 , v4, trans, lib, acoustic)

Figure 3. Correlation table fo r CaW04. The translational (trans) and rotational (rot) degrees of f reedom of the free molecule become translational (trans) and librational (lib) lattice modes in the crystal.



I += (a) (b)

Figure 4. (a) Body-centered unit cell (non-primitive) of calcium tungstate” with space group cih. (b) Symmetry operations for Cih. This is from Ref. 12 and used the standard notation for symmetry operations.”

not be readily predicted. In many instances such splitting is very small and therefore may not be observed.

Since CaW04 has a body-centered crystal structure it also serves as an illustrative example to solve by the method of factor group analysis. Preliminary consideration of the crystal with the tables greatly facilitates the process, however. Remembering that we previously determined that oxygen atoms are on C1 sites and the calcium and tungsten atoms are on S4 sites, we may immediately state which symmetry operations will interchange the atoms in the unit cell and may therefore exclude them from our consideration. (Any symmetry operation not included in that group of operations defined by the site symmetry will make a zero contribution to the number of unshifted atoms U ( R ) . )

The oxygen atoms will then only make a contribution to U ( R ) under the identity operation, while the Ca and W atoms of symmetry S4 will have contributions under E, C2, S: and S4. The values of U ( R ) for the oxygens are trivial, and are listed in Table 2. To determine the values for the Ca and W atoms it becomes necessary to consider the unit cell and the symmetry operations depicted in Fig. 4. The C2 axis and the C2 screw axis both map into the same C2 class in the C4,, point group. Consideration of either will give the correct result. However, if the C, screw axis is used it is necessary to identify the corner position as being identical to the center position and the lower side positions as being identical to the upper side positions in a primitive cell. Since equivalent C2 axes

Table 2. Factor group analysis of CaWO, (face-centered,

c 4 h : E C, C: C: i S, S: U h

uca : 4 0 0 4 0 4 4 0 u w : 4 0 0 4 0 4 4 0

XT: 3 1 + I -1 -3 -1 -1 +1

non -primitive unit cell)

uo4: 1 6 0 0 0 0 0 0 0 UCaWOa : 2 4 0 0 8 0 8 8 0

12 0 0 -4 0 -4 -4 0 UWXT UO&T : 4 8 0 0 0 0 0 0 0 UCaWO&T: 72 0 0 -8 0 -8 -8 0

Tea= Tw= 2A, + 2Bg + 2Eg + 2Eu. ro4 = 6Ag + 6A, + 6 4 + 6B, + 6Eg + 615,. rcawo4 = 6Ag + 10A, + 10Bg + 6B, + 10Eg + 1 OE,. rcawo4(primitive) = 3Ag + 5A, + 5Bg + 38, + 5Eg + 56,.

pass through all calcium and tungsten atoms, to determine the number of unshifted atoms under this operation, the counting may be done by considering all of the C2 axes in turn, observing only those atoms being on the axes. Alternatively only a single C2 axis may be considered and its effect on all of the atoms in the unit cell determined. By either technique all the tungsten and all the calcium atoms in the unit cell remain in identity positions under the C2 symmetry operation.

The S4 axes are coincident with the C, axes previously discussed. The corresponding planes associated with the S4 axes pass through the tungsten and the calcium atoms lying on the axis under consideration. Again one has a choice of considering each of the S4 axes in the unit cell and only its effect on those atoms being on it, or only one and its effect on all the atoms in the unit cell. If only one axis is used it is again necessary to identify the equivalent positions based on a primitive cell analysis. The remaining results in Table 2 are found by standard techniques. Twice the correct number of lattice modes are found by this method because the centered unit cell is doubly primitive.

Naphthalene and diphenyl belong to ,a large class of compounds which crystallize in the C 2 h space group with two molecules per unit cell. The ‘external’ lattice modes of these crystals may be readily determined by use of the tables. Since there are two molecules per unit cell, by consulting Table 5A we find that each molecule must occupy a Ci site. Therefore we may read off the translational degrees of freedom from Table 5B (3A, + 3B,) are the librational modes from 5C(3A, +3B,) . The acoustic modes are A , + 2B,. Thus the external lattice modes are 3A, + 3B, + 2 A , + B , of which the A, and B, are Raman active and the A , and B, are IR active.

CONCLUSION

In this paper three methods of selection rule determination of k = 0 phonons have been described. The primary purpose of this presentation was to organize these techniques in a unified manner and to illustrate them in light of the tables which have been included. It is appropriate at this time to discuss some of the advan- tages and disadvantages of each method.

The most complicated method and the one which requires the most thorough understanding of the unit cell is that of factor group analysis. To determine the selection rules of the lattice modes in this way a map of the unit cell must be available and careful consideration must be made of the effect of each symmetry operation on every nucleus in the unit cell. However, if symmetry coordinates are to be written explicitly it is necessary to select generating coordinates (for example, a generating coordinate might be a Cartesian vector located on a specific nucleus), and apply each of the operations of the factor group on it to generate the symmetry coordinates for each irreducible representation. The entire procedure also consists of various orthogonalization methods to obtain meaningful symmetry coordinates. If such a detailed analysis of the symmetry modes is desired it is best to initiate the study by determining the selection rules by the method of factor group analysis.

@ Heyden & Son Ltd, 1981 JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981 26’1

D. L. ROUSSEAU. R. P. BAUMAN AND S. P. S. PORT0

If selection rules are to be determined for a crystal which has distinct molecular or ionic groupings the method of molecular site group analysis should be used. Information will then be obtained showing how each isolated internal motion will be affected by placing it in the unit cell. This is extremely important information to have at hand when trying to interpret IR or Raman spectra of such crystals.

All crystals may be analyzed readily by the method of nuclear site group analysis by using the tables presented in this paper. Furthermore, this method may be applied to free molecules equally well by determining the sym-

metry of each nucleus (site symmetry) and then determining the irreducible representations by inspection from Tables B for the point group of the molecule. This method should be extremely useful for both molecules and crystals due to its speed and simplicity. While it may be desirable to use one of the other methods it is often convenient to determine the selection rules by this method first since it is less subject to ambiguities and human error than are the other methods. In any event it is an important and easy technique that all spectro- scopists should have at hand when studying vibrational motions.

REFERENCES

1. G. Herzberg, Molecular Spectra and Mo!ecular Structure: I1 Infra-Red and Raman Spectra of Polyatomic Molecules, D. Van Nostrand Company, Inc., Princeton, N.J. (1945).

2. E. B, Wilson Jr, J. 5. Decuis and P. C.. Cross, Molecular Vibrations, McGraw-Hill Book Company, Inc., New York (1955)

3. S. Bhagavanturn and T. Venkatarayudu, Proc. lndian Acad. Sci. Sect A 9.224 (1939); S. Bhagavantum, Proc. lndian Acad. Sci. SecF. A13, 543 (1941); S. Bhagavantum and T. Venka- tarayudu, Theory of Groups and Its Application to Physical Problems, Audhra University, Waltair, India (1962).

4. R. S. Halford, J . Chem. Phys. 14, 8 (1946). 5. 5, F. Hornig, ,J. Chem. Phys. 16, 1063 (1948). 6. A. A. Maradudin and S. H. Vosko, Rev Mod. Phys. 40, 1

(19681; A. t. Warren, Rev. Mod. Phys. 40, 38 (1968); G. Venkataraman and V. C. Sahni, Rev. Mod. Phys. 42, 409 (1970).

7. H. Winston and R. S. Halford, J. Chem. Phys. 17,607 (1949). 8. W. G. Fatelev, N. T. McDevitt and F. F. Bentley, Appl. Spec-

trosc. 25, 155 (1971); W. G. Fateley, F. R. Dollish, N. T. Pi/lcDevitt a m F. F. Bentley, Infrared and Raman Selection Rules for Mderular and Lattice Vibratiom : The Correlation Method, 'j\lile~i-lrirerscience, New York (1972).

9. J. R. Ferraro #and J. S. Ziomek, Introductory Group Theory and Its Application to Molecular Structure, Plenum Press, New Yoik (1975i.

30. J.-P. Mathleu Spectres De Vibration et Symetrie, Herman et cie, Paris ;1945),

11. D. M. Adams and D. C. Newton, J. Chem. SOC. A 1970,2822. 1 2. International Tables for X-ray Crystallography. Inter nat i on a I

Union of Crystallography, Kynock Press, Birmingham, UK (1 952).

13. J. E. Rosenthal and G. M. Murphy, Rev. Mod. Phys. 8, 317 (1936).

14. J. F. Scott and S. P. S. Porto, Phys. Rev. 161,903 (1967); S. P. S. Porto, J. A. Giordmaine and T. C. Darnen, Phys. Rev. 147, 608 (1966).

15. P. F. Williams and S. P. S. Porto, Phys. Rev. B 8, 1782 (1973). 16. R. H. Lydanne, R. G. Sachs and E. Teller, Phys. Rev, 59, 673

(1941). 17. H. Poulet, Ann. Phys. Paris 10, 908 (1955). 18. C. A. Arguello, D. L. Rousseau and S. P. S. Porto, Phys. Rev.

181, 1351 (1969). 19. R. Loudon, Adv. Phys. 13,423 (1964). 20. lnternationale Tabellen m e Bestimmung von Kristallstruk-

21. C. Kittel. Introduction to Solid State Physics, p. 404. Wiley.

22. H. Strunz, Mineralogische Tabellen, Akademische Verlags-

tunen I, p. 354. Gebruder Borntraeger, Berlin (1935).

New York (1966).

gesellschaft, Leipzig (1957).

Received 29 August 1980


TABLES -I___

Ordering within each set of tables

Table Crystal syste:?'

Triclinic 1 L

Monociin:c 3 4 5

6 7 8

0 rt ho r ho m h ic

Tetragonal 3 10

Point Group symbo s Srhoenfliess Herrrann-Mauguin

c, 1 C, 1

CZ 2 c s m c2 h 2/m

0 2 1 VI 222 c2 " mrn2 (2mm, mrn) DZh ( v h ) mmm(2lm 21m 2 fm)

c4 4 4

-

Table Point Group symbols Crystal system no Schoenfliess Hermann-Mauquin

18 0 3 32 19 c3v 3m 20 DBd 3m(32/m)

22 C3h(S3) 6i3irn) 23 cfih 61m 24 D6 622(62) 25 C6" 6mm 26 D3h 6m >! 27 D6h 6/mrn(6/m 2 lm 2 fm)

Hexagonal 21 c6 6

Rhombohedra1 16 C, 3 (Trigonal; '7 s6(@3#1 3


Tables A. All sites for each of the 32 crystallographic point groups

2A Ci (i) Space Group Sites

c: (P i ) .[iCl(2)] + (h+g+f+etd+c+b+a)ci(l)

C: (Bllb) -CaC1(4 )I I 5A C2,(2/m)

space Group S i t e s

1

JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981 263 @ Heyden & Son Ltd, 1981


TABLES 7A-9A ALL SITES FOR EACH POINT GROUP

8A D2h (m) continued Soace



ALL SITES FOR EACH POINT GROUP

10A S4 (4) Space Group S i t e s

s: (pi) -[hCl(4)] + -[ (g+f+e)C2(2)1 + (d+C+b+a)Sh(l)

1 2 A D4 (422)

SDaCe

TABLES 10A-15A

1 j A C4v(4mm) continued

l4A Ded(42m) space Group S i t e s



TABLES 15A-20A 15A D,,,(&/mmm) continued

-11

SDace

," Space


ALL SITES FOR EACH POINT GROUP 21A c ~ ( 6 )

Space



ALL SITES FOR EACH POINT GROUP

32A Oh(mjll l) space Group S i t e s



Site Representatisns

Cl 3A1 + 3A2 + 3B1 + 3U2

A1 + A;, + 2B1 + 2B2

C:z(ocz) 2A1 + A2 + 2B1 + B2

C?(<') 2A1 + + B1 + 2B2

CP Y

CP

A1 + B1 + B2

i

Tables B. Irreducible representations that result from occupying each of the sites within each space group. These give the translational lattice modes

1 B C. D2h

Site Representations I 3A

2 8 Ci

Site Representations

I 3Ag + Xu I c1

3%

3B C2


3A' + 3A"

2 A ' + A"

5B '2h

Site Representatims 1 3Ag + 3Au + 3Bg + 3B, C1

ci >AU + 3Bu

A + AU + 2B + 2BU

2A + AU + R + 2BU

C2

cs


3Ag + 3Au + 3B + 3BlU+ 3B

A + A, + B + Blu+ 2B2g + 2B2u+ 2B + 2BjU

A + A, + 2B + 2B1,+ B + B2,+ 2B3g+ 2BjU

A + A, + 2B + 2Blu+ 2B + 2B2,+ B%+ B3,

PA + A, + 2BLg + Blu + B + 2B2,+ B + 2B3,

2 A + % + B + 2Blu+ 2B + B2,+ B + 2B3,

2A + A, + Blg + 2Blu+ B + 2B2u+ 2B + B3,

3% + 3Blu+ 3B2,+ 3B3,

Ag + Blu+ B + B2,+ B + Bju

Ag + B + Blu+ B2u+ B3g+ B3, 1g

A + B + Blu+ B + B2,+ Bju

B + Blu+ B + B2,+ B + B3,

A,+ Blu+ 2B2,+ 2B3,

A, + 2Blu+ B2u+ 2B3,

A, + 2Blu+ 2B2,+ B3,

+ 3B2,+ 3Bx+ 3B3, C1 18

c:(c;) 1g 3g

C;(C;t 18 2g

C,"(C,X) 18 2g

cy( ow)

CiZ( axz )

c y (YZ )

2g %

16 2g %

a3 3s

Ci

C~y(C~+axz+~z)

Cz,($+ow+d")

C&(C~oxy+oxz)

2g 3g

g 1g 2g

D2 1g 2g k

CZh( c;+ow)

CZh( c p y

c;,(c;+oYz)

D2 h BlU+ B2u+ B3U

3A + 3B + 3E

A + B + Z E

A + E


3A + 3B + 3E C1

A + B + 2 E I B + E 1

A + 2B1 + 2B2 + B 118 Cllh

Representations

3Ag+ 3Au+ 3Bg+ 3Bu+ 3 E g + 3EU

3Au+ 3Bu+ 3Eu

A + A, + B + B, + 2E + 2EU

2A + A, + 2B + Bu + E + 2E,

A, + BU + 2EU

A + A u + E + E u

A U + B + E C E , g g

Au + Eu


D. L. ROUSSEAU, R. P. BAUMAN AND S. P. S . PORT0

TABLES 12B-196

1 2 B D4

Site Representstions

/A. + 3A2 + 3Bl + /B2 + 6E

dl + A2 + B1 + B2 + 4 E

A1 C 2A2 + B1 + 2B2 + 3 E

C1

C;(C;)

C2 (C,)

c;(C;) A1 + ?A2 + 2B1 + B2 + 3E

D2(C;+2C2)

D;(C>ZC;)

A2 + B2 + 2E

A2 + B1 + 2E

A1 + A + 2 E c4

D4 A2 + E

REPRESENTATIONS WHEN EACH SITE IN EACH SPACE GROUP OCCUPIED

15B D,,, continued

13B C u V

site Representations

3A. + /A2 + 3B1 + 3B2 + 6E C1

c2 A1 + A:, + B1 + B2 + 4 E

C:(ov) ZA., + AT + 2B1 + B2 + 3E

cz(od) ZA., + A;, + B1 + 2BP + 3E

c 2 y v ( ~ . z ) + B1 + 2E

C g v ( o , )

c4

Al + B2 + 2E

A1 + A2 + 2E

A1 + E c4 Y

14B D2d


/AA1 + XP + 3B1 + 3B2 + 6E

A1 + 42 + B1+B2 + 4E

A1 + 2% + B1 + 2B2f jE

PA1 + $ + B1 + 2B2 + 3E

A2 + B2 + 2 E

C 1

c:(C;)

c2 (c,)

cs

D2

C 2 Y

s4

A1 + B:, + 2 E

B1 + B- + 2E

B2 + E '2d

4

Representations

) A ~ $ >Alu+ 3 A + 3AA2u + 3B + 5BlU+ 3BZK + 3BZu+ 6Eg+ 6EU

3AIU+ ?Apu+ 3BIu+ 3BZu+ 6EU

A1$ Alu + A:?$ AZU+ B 1.s + Blu+ Bpg+ Elzu+ 4Eg+ 4Eu

2g 16

C;(Cz)

C 2 ( C 2 ) A + Alu+ 2 A + 2A2u+ B + Blu+ 2B2&i 2BZU+ 3Eg+ 3EU

c j ( C ; ) A + Alu+ 2 A + >AAzu+ 28 + 2BIu+ B2g+ B2U+ 3Eg+ 3EU

c:(oh) 7 A + AIu+ PA + A2u+ 2B + Blu+ 2B + BZu+ 2 E + 4 E U

c:(av) ?Alg+ Alu+ A + 2AZu+ 2B + Blu+ B i- 2BZu+ 3E + 3Eu

C 2 ( o d ) ?A i- Alu+ A + 2%u+ B + 2Blu+ 2B2$ B2u+ /Eg+ 3EU

C : h ( C 3 6 h ) Alu+ A + Blu+ B2u+ 4 %

C2h(C7+ a v )

C i h (C;+O,)

k3 2 K 1.s

1.s 2.s 1.3

l e 2K 1.s 2 K

2.s 1.s 2.s

1 K 2.s 1.s

2u

Alu+- PAzu+ Blu+ 2BPU+ Xu

Alu+ 2A2u+ 7Blu+ B2"+ 3EU


D;(C:+ZC;) A + A2U+ B + Blu+ 2 E + 2EU

C ; y ( C ~ + 2 0 V ) A + AZU+ B + 2EK+ 2Eu

c ~ v ( c ~ + 2 0 d ) A + Aau+ Blu+ B + 2Eg+ 2EU

C ( C + o + a ) A + A2$ %u+ B + B + BzU+ E + 2EU

Cjy(C;+ah+od) A + A + A2u+ B + B l g B + E + 2Eu

2.5 1 K

1.5 1.5

1.5 2.5

2 v 2 h d l g 1 K 2g

1.s 2.s 1.5 2.s g

Dzh(C;+'c2) B>u+ 2Eu

D j , ( C 2 2 C ; ) A2u+ Blu+ 2Eu

A + A + A + AZu+ 2E t 2EU

AIU+ A2u+ B + B + Z E t 2Eu

AlU+ A + 2EU

c4 18 lU 2E

54 1.3 2.s .s

'4 h 2u

D 4 %g+ Azu+ Eg+ Eu

c4 " A I K + A2"+ E + Eu

D i d (C;+2C;+oV) AZu+ B, + E + EU

D Z d ( C 2 2 C 2 + a d ) AZu+ B + E + Eu

dK K

1.s .s

'4 h A 2 U + EU

3 A + 3E

A + E

17B S,

Site Representations -

C 1

'i 3Au+ 3Eu

?A + 3AU+ 3Eg+ 3EU

A + Au+ E + Eu

' 6 A u + EU -

18B D,


C 1 >A1 + 3%' 6E

c2 A1 + PA2 + 3E

A1+ Ap+ 2 E

A2 + E D3

- 19B C3"

site Representations -

C 1 )Al+ >A2+ 6 E

cs 2A1+ A2+ 3E

A1 + Ag + 2E

A x + E

c3

5" -




25B C6"


3A1+ 3A2+ 3B1+ 3B2+ 6E1+ 6E2

A1+ ++ 2B1+ 2B2+ 4E1+ 2E2

2Al+ ++ 2Bl+ B2+ 3El+ 3E2

2Al+ A2+ B1+ 2B2+ >El+ 3E2

C1

C2

c:(o,)

c: ( a d )

C2 v A1+ Bl+ B2+ 2%+ E2

A1+ A2+ B1+ B2+ 2E1+ 2E2

c;"(a") A 1 1 + B + El+ E2

Cdjv(ad) Al+ B2+ El+ E2

'6

c3

A1 + A2 + 2E1

'6" A1 + E l 1

TABLES 208-278

2OB Djd I


C1

ci 'AlU+ >Azu + 6EU

3Alg + ?Alu i 3%g+ 3AZu+ 6Eg + 6EU

A1$ Alu + 2A

2A + AIU+ A + 2AZu+ 3E + 3Eu

+ 2AZu+ 3E + 3EU C2 2g

c5 Ig 2g

'2 h Alu+ 3EU

A + AIU+ A + AZu+ 2E + 2EU

Alu+ A 2u + 2EU

A

c3 1g 28

'6

+ AZu+ E + EU D3 2g

c3. 1g

D3d %u+ EU

A + AZu+ E + EU

21B C6


C1 3A + 3B + 3E1 + 3E2

A + 2B + 2E1 + E2

A + B + E 1 + E 2

~ !: A + E 1 1 3A* + 3A" + 3Ev + 3E"

2 A ' + A" + 2E' + E"

A' + A" + E' + E"

A" + E ' I '3h

23B a6h

Site c' Representations

C1

Ci 3Au+ 3Bu+ Xlu+ 3EZu

A + Au+ 2B + 2BU+ 2E

2A + Au+ B + 2BU+ E

)A + TAU+ 3B + 3BU+ 3Elg+ 3Elu+ 3EZg+ 3E2u

+ 2Elu+ E + E2,

+ EZu

C2 1g 2g

CS 1g 263 + 2Elu+ 2E

'Ph Au+ 2Bu+ 2Elu+ EZu

c3

'6 AU + BU + ElU+ E2u

A + A, + Elg + Elu '6

2& AU + B + Elu+ E

3h

'6h 4, + ElU

24B D6

Site Representations I 3A1+ 3A2+ 3B1+ 3B2 + 6E1 + 6E2

Al+ A?+ 2B1+ 2B2+ &El+ 2E2

A1+ 2A2+ B1+ ?B2+ 3E1+ 3E2

Al+ 2A2+ 2Bl+ B2+ 3E1+ 3E2

C1

c:(c;)

C2(C2)

c;(c;)

I D2 %+ B1+ % + 2El+ %


24B DF continued


A + A2+ B1+ B2+ 2E1+2E2 c 3

D3(3C2) A2+ B2+ El+ E2

Dj( 3C;) A2+ B1+ El+ E2

'6 A 1 2 + A + '2E1

D6 A2 + El

27B Dsh

Representations

3A + 3AlU+ 3$g+ 3A2u+ 3Blg+ 3BlU+ 3B2$ 3BZu+ 6Elg

+ 6~~~ + 6E2g + 6 4 u

3Alu+ 3A2,,+ 3Blu+ 3B2"+ 6Elu+ 6EZu

A + A + A + A + 2B + 2Blu+ 2BBng+ 2BZU+ 4Elg+

Site

C1 18

ci

C;(C;) 1g 1u 2g 2u 1g

4Elu+ 2 q g + 2 q U

A + Alu+ 2% + 2A2,,+ B + Blu+ 2B + 2BZu+ 'El$ C2(C2) 1g 1g 2g

3% 3E2g+ 3%

163 2g 1g 2g 1g

)Elu+ 3E2g+ 3E2u

A + Alu+ 2A + 2A2,,+ 2B + 2Blu+ B + B2u+ 3E + C;(c;)

2A + Alu+ 2% + AZu+ B + 2Blu+ B + 2BZu+ 'Elf C:(ah) 1g 18 28

JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981 271

TABLES 276-318

A + E + 3F c3

T F



278 D6h continued


&El,,+ 4E + 2%,,

PA1$ A + A + 2A7u+ B + 2BlU+ 7B2$ B2U+ 3E +

3ElU+ >EZg+ 3EZU

7 A + Alu+ A + 2%U+ 2B + Blu+ B + 2B2u+ >El$

2E

c;(u,) lU 2g 1g 16%

18 2-3 2g

3ElU+ 3 4 $ 3E2U

c&(c2ah) AIU+ A7u+ 7Blu+ 2BZU+ 4ElU+ 2EZu

C2h(Cz+ud) hiU+ 2AzU+ BIU+'B;,u+ 3Ezu

c;h(C$+uv) AIU+ 2A 2u +alU+ BZU+ >EIU+ 3EZu

AZg+ +u+ B + BIU+ B + €Izu+ 2E + 2ElU+ E + 9 16 2 g 2g Ezu

Cz (Cz+u +u ) A + A + B + Blu+ B + B2U+ 2E + 2ElU+ E + EZU

c (c +a +a ) A + A + A7u+ BlU+ B + BZu+ E + 2ElU+ 2E7g+ Eau

c' (cy+o +a ) A + A + A + B + BIU+ B2u+ E i 2Elu+ TE + EZU

2v 2 Y d l g 2u lg 78 1s 2s

2v 2 h v lg 2g 2g It3

2 v 2 h d l g 2g 2u lg 2 g

A + Blu+ BZu+ 2EIu+ E2,

A + A + A + A2u+ B + BIU+ B- + BZu+ 2E1$ 2Elu+

2E + 2E7U

A + A + Blu+ 2ElU+ 2EZu

A + A*,,+ B + BZu+ Elg+ %u+ EZg+ E2,

A + A + B + B + E + E I U + E + E Z U

A + Aau+ BIU+ B + E + Elu+ E + E2,,

9 + AZu+ B + B2u+ E + Elu+ E + EpU

D2 h 2u

1g lU 2g I& c g

7 g

'6 lU 2u

",( 3C* ) 28 2g

Dj( 3C; ) 2g 7u 1g lU 1g 2g

c;"("" ) 1u 2 g I& 2g

I& 164 I& 2g

Dxd(3C2+303) Azu+ B2u+ Elu+ EzU

Djd(3C;+3a,) 4 2u + Blu+ E lU + EgU

4 + Alu+ A + A2,,+ 2E + 2Elu '6 1u 26 1g

'3h hlu+ 1g 2.s 262

'6h 1"

B + B + 2Elu+ 2E

E + 2Elu

D6 2g E 1g + El" E +

Alg+ A + E + Elu '6" 2" 1.3

D3h(3c7+30,) BZg+ ElU+ Epg

Djh(3C$+3ud) A2"+ B 1g + Elu+ E 2g

D6 h A2"+ E1u

~


I 3 A + 3E + 9F C1 I

29B T,


3A + >Au+ 3Eg+ 3EU+ 9Fg+ 9Fu C1

Ci 3AU+ 3EU+ 9Fu

A + Au+ E + Eu+ 5F$ 5Fu c2

PA + A + 2E + Eu+ 4Fg+ 5Fu cs g u g

'2h Au+ Eu+ 5FU

D2 3Fu+ 3FU

A + E + 2F + 3Fu c2 Y g g k!

D2 h 3FU

c3 A + Au+ E + EU+ 3F + 3FU

' 6 Au+ Eu+ 3Fu

T F + FU

'h FU

)B 0


3A1+ 3A2+ 6 3 + 9F1+ 9F2

A1+ A2+ 2E + 5F1+ 5F2

A1+ 7A2+ 3E + 5F1+ 4F2

3F1 + 3F2

A2+ E + 3Fl+ 7F7

A1+ E + 3Fl + 2F2

C1

C7@)

C$(C*)

D2 ( 3 4 )

D;(Cg+2c2)

c4

D4 2F1 + F2

A1 + A2 + 2E + 3F1 + 3F2

A2 + E + 7F1 + F2 D3

T F + F2

0 F1

318 Td


>A1+ 3A2+ 6E + 9F1+ YF2

A1+ A2+ 2E + 5F1 + 5F2

2A1+ A2 + )E + 4F1 + iF2

c1

C2

cs

D7 3F1 + 3F2

A1 + E + 2F1 + 3F2

A2 + E + 2F1 + 3F2

C2"

s4

D7 d 3 F ~

A1 + A2 + 2E + 3F1 + 3F2

Al + E + F1 + 2F2 5" T F1 + F2

Td F2




Oh


3A + 3Alu+ 3A + 3APu+ 6Eg+ 6Eu+ 9Flg+ 9FlU+ 9Fzg 1g 28

+ gFzU

3Alu+ 3APu+ 6EU+ 9FlU+ 9FZu

A + AIU+ AZg+ %u+ BEg+ 2EU+ 5F18+5Flu+ 5F2$ 5F2u

A + Alu+ 2% + 2AZu+ 3E + 3Eu+ 5F18+ 5Flu+ 4F + 4FP,

2A + Alu+ PA + A2u+ 4Eg+ 2EU+ 4F + 5Flu+ 4F + 5F21

2A + Alu+ A + 2%u+ 3Eg+ 3Eu+ 4F + 5Flu+ 5F2g+ 4F2,

A l U + A2,,+ 2EU+ 5Flu+ 5F2u

Alu+ 2%,+ 3Eu+ 5Flu+ 4FZu

3Flg+ 3FlU+ 3PZg+ 3FZu

A2$ %u+ Eg+ Eu+ 3F + )Flu+ 2F + 2FZu

A1$ kzg+ 23 + 2F + 3FlU+ 2F + >FpU

A + A2,,+ E + EU+ 2F + >Flu+ 3F + 2F2U

A + A + AZu+ 2 E + Eu+ 2F1$ 3FlU+ 2F + 2F2u

16

1s 2B

1g 28 265

1g 263 18

16 2 6

1g 2g

1g 18 28

1B 2 B 267

"lU+ 3 F ~ u

D;,($+Z$u$2~& AZU+ Eu+ 3FlU+ 2FZu

A + A + E + Eu+ 3F + 3Flu+ 2F + 2F2u

Alu+ A + E + Eu+ 2F + 3FlU+ 3F + 2FZu

A + Eu+ 3Flu+ 2FZu

2F + 2Flu+ F + F2,,

A + E + F + 2Flu+ F + F2u

F 1g + 2Flu+ 2F 28 + F2u

A + E + F + 2Flu+ F + F2u

c4 1g 1u g 1 B 2s

s4 2g B 16 28

'4, lU

4 h3 26

c4 Y 1g g 1.s 2g

DZd(X:+od)

D (C2+2C2+0,,) b d 4 2.3 8 1g 2g

'4 h 2FlU + F2u

A + A + A + A2u+ 2E + 2EU+ 3F + 3Flu+ 3F2$ 3F2,,

Alu+ AZu+ 2Eu+ ?Flu+ 3FZu

A + A2u+ E + Eu+ 2F + 2Flu+ F + F2,,

A + A2u+ E + Eu+ F + 2Flu+ 2F + F2u

A + E + 2F + F2,

1g lU 2g 1g

'6

D 3 2g 1g 2g

C3V 1.3 1g 2B

'3d 2u u 1u

T F + Flu+ F + FZu

Th Flu + F2u

0 F1g+ Flu

Td Flu + Fa3

Oh Flu

1g 2g

TABLE 328


D. L.. ROUSSEAU, R. P. BAUMAN AND S . P. S. PORT0

Tables C. Librational lattice modes of polyatomic units when placed on each of the possible sites in the unit cell

Re resentations

zc c:

3c c2

mepresentntions

:A + 3B I

4c cs - Hepresentations

I_ Site

Representations

!A$ ?AAu+ 3Bg+ ?Bu

!Ag + 3Bg i~ "2 A + Au+ 2B + 2BU

A + 2%+ 2B + Bu

A i Z B

I cs

g u


C 1 3A + 3Au+ 3B1$ 3BlU+ 3B2$ 3B2u+ 3BJg+ 3B3u

A + Au+ B + Blu+ 2B + 2B2"+ 2B t 2BJu

A + Au+ 2B + 2Blu+ B b BzU+ 2B + 2BgU

A + Au+ 2Blf 2Blu+ 2B + 2B2u+ B + B

A + 2Au+ B + 2Blu+ 2B + BZu+ 2B + B

c;(c:) 1g 2.3 2

$(C:) 16 2 g 36

C,X(C,x) 2g 3e 3u

C:Y(qXY) 16 2b3 33 3u

C;"(OXz) 1g 2g

C Y ( 0 Y Z )

A + 2Au+ 2 8 + Blu+ B C 2B2,i+ 2B3$ BTu

A + 2Au+ 2B + Blu+ 2B + Bzu+ B + 2BjU 18 2R 36

c i )Ag+ 3B1$ 7B2f 3BBjg

C ~ v ( C ~ + o x z + o y z ) Au+ B + B + B2u+ B3g+ BJu

C:b(C:+~xy+~z) A,,+ B + BlU+ B + B3g+ BJu

C ~ r ( C ~ : t , v + o x z ) A"+ B + Blu+ Bzg+ BzU+ BJg

16 26

16 2 g

16

D2 B1gf B l U + B2g + B2u + 379 + B3u

c;h(C~+nw)

C?&(C;+O~') A + 2B + B + 2B

C 2 x h ( C 3 d " ) As+ 2B + 2Bpg+ B 16

D2 h B 16 + B 2 g + B 36

A + B + 2 8 + 2Bx 6 16 26

6 16 2 6 x

x

3A + 3B + 3E

A + B+2E

A + E

1oc Sh


c1

c2 A + B + P E

3A + 3B + 3 E

A + E I s4

11c Cirh


)A + 3Au+ 3Bg+ 3Bu+ 3Egt 3Eu C1

'i Xg+ ?Bg+ 3Ep

A + Au+ B + BU+ 2E + 2 E U

A + ?Au+ B + 2Bu+ 2 E +

A + g g B + 2E R

A + Au+ E + Eu

c2

c s E,,

c2 ti

c4

s4 A + Bu+ E + EU

A + E '4h 6 6




C 1 3A + 3 E

A + E c3

c

TABLES 12C-19C

rC Dllh continued

Representations Site

D : , ( C 2 2 C 2 )

D;(C:+2C;)

C ~ y ( C ~ + 2 a y )

C&(Cz+20d)

CZv(C2%fud) Alu+ A + AZu+ Blu+ B + BZu+ 2 E + Eu

C~y(C;+oh+od) + Blu+ B2U+ 2 E + Eu

A2g+ ASu+ Bag+ B2u+ 2E6- isu

A + A + B + Blu+ ZEg+ 2Eu

Alu+ A2$ Blu+ B + 2E + 2Eu 2g g

Alu+ A + B + B2u+ 2Eg+ 2Eu 2 g 1g

2g 2g

2E 2U 1g

Alu+ qZE+ ASu+ B 1g

A + B + 2 E D2 h( C 2 2 C:, ) 2.3 26 g

Dih( c;+zc; ) A + B + Z E 213 1E 8

A 1g + A 1u + A 2g + A2u+ 2E + 2Eu

A + A + Blu+ BZu+ 2E + 2EU

A + A + 2 E

c4

s4 1g 2g

'4h lg 2E g

A + A + E + E U D4 2g 2u g

C4" AlU+ %g+ Eg+ EU

D i d ( Cg+2Ci+oy) 28

D 2 d ( C ~ 2 C 2 + ~ d ) 2g

D4 h A2g + E g

A + B2,+ E + EU

+ Blu+ E + Eu A

LIBRATIONAL LATTICE MODES


C 1 3Al+ 3A2+ 3El+ 3B2+ 6 E

CX(C:)

C2(C2)

C g c i )

D 2 ( C 2 2 C 2 ) %+ B2+ 2 E

D$(C$+PCb) %+ B1+ 2 E

A1+ A + 2 E

A:, + E

A1+ A2+ B1+ B2+ 4 E

A1+ 2A2+ B1+ 2B2+ 3E

A1+ 2A2+ 2B1+ B2+ 3E

c4

D4

13c c q y


C 1 3Al+ 3%+ 3B1+ 3B2+ 6 E

A1+ %+ B1+ B2+ 4 E

Al+ ++ B1+ 2B2+ 3 E

A1+ 2%+ 2Bl+ B2+ 3E

C2

C:(O,)

c:('Td)

C2yJ'T") A2+ B2+ 2E

C:"( Qa) ++ B1+ 2 E

A1+ %+ 2E c4

c4 " % + E

D2d


>Al+ 3A2+ 3B1+ 3B2+ 6 E

Al+ A2+ B1+ B2+ 4 E

A1+ 2A2+ B1+ 2B2+ 3 E

A1+ 2A2+ 2B1+ B2+ 3 E

C 1

C;(C;)

C Z ( C 2 )

cs

D2 A2+ Be+ 2E

c 2 " A2+ B1+ 2E

s4 A1+ A2+ 2E

D2 d A:, + E

1 7 C Sg


C l

C. 3Ag+ 3Eg

3A + 3Au+ 3Eg+ 3Eu

A + Au+ E + Eu C 3

'6 g g a + E

Site

>A1+ 3A2+ 6 E

A1+ A2+ 2E

19c cJv


C l >Al+ >AAp+ 6 3

c A1+ 2A2+ 3E

A ~ + a*+ ZE

a2 + E 5"

0 Heyden & Son Lttf, 1981 JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981 275

TABLES 20C-27C

c:(uh)

cI(a,)

C2"

c3

A i + 2A; + A; + 2A; + 2E' + 4E"

A i + 2A: + 2AS + A; + 3E' + 3E"

A; + A; + A; + E l + 2E"

A i + A; + A; + A; + P E ' + 2E"

A; + A; + E ' + E"

A; + A; + E' + E"

D3

c3"

cm A; + A$ + 2E"

A; + E" '3h

D. I-. ROIJSSEAU, R. P. BAUMAN AND S. P. S. PORT0 LIBRATIONAL LAlTlCE MODES

&

2oc Dx


3A + 3AlU+ >A2$ 3 A 2 b 6Eg+ 6Eu C 1 1g

3Alg+ 3A + 6Eg

Alg+ "\lu+ 2AZg+ 2A2u+ 3Eg+ 3Eu

A + 2Alu+ 2% + AzU+ >Eg+ 3Eu

C l 2g

C 2

C S 1g

'2h 1g ?g

'3 1g lU 2g

' 6 Ale;+ %g+ 2 E g

A + 2A + 3Eg

A + A + A + A2u+ 2Eg+ 2 E U

24c D6 continued

Slte Representations

D2 A2+ B1+ B2+ 2E1+ E2

A1+ A2+ El$ B2+ 2E1+ 2%

A + B2+ El+ E2

A2+ B1+ 54 E2

c3

D~(3c2)

D y 3c; 1

'6 A1+ A2+ 2E1

' 6 A2 + E l

A2g+ %u+ Eg+ Eu

3A1+ 3A2+ 3B1+ 3B2+ 6E1+ 6E2

Al+ %+ 2B1+ 2B2+ 45+ 2%

%g + E g

Representations

3A + 3 B + 3%+ 3%

A + 2B + 2%+ E2

A + B + E l + %

1

I

22c c,.


3A1+ 3A" + 3 E ' + 3E"

A' + 2A" + E ' + 2E"

A' + A" + E ' + E"

A' + E"

23C C6h


3Ag+ 3Au+ 3Bg+ 3BU+ 3E1$ 3ElU+ 3E2$ 3EpU C1

Ci ?Ag+ 3Bg+ 3Elg+ 3EZg

A + AU+ 2B + 2Bu+ 2E + 2ElU+ Em + Epu

A + 2Au+ 2B + Bu+ 2 E + Elu+ E + 2EZu

A + 2B + 2 E

C2 1g d.3

CS 18 2 8

'2h g g 1g

c3 1g 2g

' 6 g g 1g 2 8

'6 u 14

+ EZg

A + Au+ B + BU+ E

A + B + E + E

+ El,,+ E + E2,

I A t A + E + Elu

A1+ 2A2+ B1+ 2B2+ >El+ 3%

Al+ 2A2+ 2B1+ B2+ 3q+ 3%

%+ B1+ B2+ 2 5 + E,,

A1+ %+ B1+ B2+ 2E1+ 2E2

% + B 2 + 5 + 4

A2+ B1+ El+ %

A1 + % + 2E1

A2 + El

26C D,

A + B + E + Epu 1 '3h 8 u 1g

2 7 C D6h

Representations

3Alg+ 3Alu+ 3 5 g + + 6Elg+ 6Elu+ 6 E

3Blg+ "Iu+ 3B2$ TBzU

+ 6%u iAg+ 26

24C D6

Site Hepresentations

?A1+ PA,+ 3B1+ 3B2+ 6E1 + 6E2 Cl

C:(C;)

C2(C2)

A,+ A2+ 2Bl+ 2B2+ kEl+ 2 E 2

Al+ 2A2+ B1+ 2B2+ 3EL+ 3E2

CgC;) Al+.:2A2+ 2B1+ B2+ 3El+ 3E2

276 JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981

3A1$ %g+ 3 B + 3 B + 6 E + 6E2g

A + AIU+ A2$ +u+ 2 B + 2Blu+ 2 B + 2B2,

C i 1g 2g 1g

1g 1g 2 %

45g+ "lU+ 'sg+ 2%u

+ 3E1u+ 3%g+ 3%

3Elu+ 3%' 3E2u

A + A + 2% + B + Blu+ 2B2g+ 2B2u+ 3Elg C2(C2) 163 lU B 1g

C p ; ) 1g 2g 1g 25 15 A + Alu+ 2A + 2%,,+ 2 B + 2Blu+ B + B2,+ 3 E +




3A + 3E + 9F

A + E + 5F

C 1

c2

D2 3P

c3 A + E + 3 F

T F i

LIBRATIONAL LATTICE MODES Dr. continued

~~


A + 2Alut A 2B + 2$u+ 2B 18 + Blu+ 2B 2g + Bau+ 4Elg 1g

'9u+ 2%' "u

+ TElU+ 3%g+ YU

3Elu+ 3%g + 3%

Alg+ %&+ 2Blg+ 2B2$ 4E1g+ 2%g

16 A + 2Alu+ 2$g+ %u+ 2Blg+ Blu+ BZ8+ 2BZu+ 3E 1g

A 1g + 'Alu+ 2Aag+ %u+ B 1g + 2Blu+ 2B 2s + BZu+ 3E18

A 163 + 2+$ Blg+2BZg+ 3Elg+ 3%g

"I&+ 2%g+%g+ B2&+ 3E1g+ 3 E ~ g

%g+ %u+ Big+ BlU+ Bzg+ B2u+ 2Elf 2 E l b %g+ %I

A + A +B + Blu+ B + BZu+ 2 8 + 2ElU+ E2$ EZu

Alu+ %&+ %u+ B + B + BZu+ 2Elg+ Elu+ E 2g + 2EZu

Alu+ %E+ APu+ B + Blu+ B + 2% + El,,+ %g+ 2%u

A + B + BZg+ 2E +

A + Alu+ AZg+ %u+ Blg+ Blu+ B 28 + BZu+ 2E1$ 2Elu

lU 2g 1g 2B 18

1g 2g

1E 2g g

2g 1g 1g %

2%g+ 2%

1g

A + A + B + BZg+ 2E + 2E

A + A + B + BZu+ Elgt Elu+ 4$ EZu

18 2g I E 1E 28

2g 2u 2g

A 2g + Azu' Blf BlU+ pg+ pu+ %g+ E2u

AIU+ A + BZu+ E 1g + Elu+ E2g+ E2u

Alu+ A + B + B + E + EIU+ E + EaU

A + B + E + E Z g

A + B + E + E 2 g

A

A + A + Blu+ BZu+ 2% + 2EZu

+ B 2g 1g

2g lU 2g 1g 2g

28 2g 1g

2g I g 1g

I E

18 2g

1g 2g 1.8

+ Alu+ AZg+ AZU+ 2Elg+ 2Elu

A + A + 2 E

%g+ %u+ El&+ E l u

Alu+ A + E + El,, 2g 18

A + BZu+ E + E2U 2g 1g

A 2g + BlU+ El$ E2u

A + E 1g

TABLES 27C31C

Th


3A + ?Au+ 3Eg+ 3Eu+ 9Fg+ 9FU

3A + 3E + 9 F

A + Au+ E + EU+ 5F + 5FU

A + PAU+ E + 2Eu+ 5F + 4Fu

C 1

C i g g g

C2

ca

'2h A + Eg+ 5Fg

D 2 3Fg + 3Fu

c2v

'2 h 3Fg

c3

Au+ Eu+ 3F + 2Fu

A + Au+ E + Eu+ >Fg+ 3Fu

'6 Ag+ Eg+ Pg

T F + FU

Th Pg

)C 0


C 1

C2($

C& (C2)

3Al+ "2' 6E + 9F1+ 9F2

A1+ %+ PE + 5P1+ 5F2

A1 + 2%+ 3E + 5F1+ 4F2

D 2 ( X $ 3Fl + 3F2

D&(Cg+2C2)

c4

%+ E + 3F1+ 2F2

A1+ E + 3F1+ 2F2

D 4 'PI + F2

c3 A1+ ++ PE + 3Fl+ 3F2

%+ E + 2P1 + P2 D 3

T F1 + FP

F1 C

31c Ta


C 1

C2

cs

D2 3Fl+ 3F2

c2v

s4

'2 d 2Fl + F2

3Al+ yL2+ 6E + 9F1+ 9 4

A1+ %+ 2E + 5F1+ 5 4

A1+ 2%+ 3E + 5F1+ 4F2

%+ E + 3F1+ 2F2

A1+ E + 3F1+ 2F2

A1+ A2+ 2E + 3P1+ 3F2

%+ E + 2F1+ F2

c3

C 3 V

T p1 + PP

Td p1



TABLE 32C LIBRATIONAL LATTICE MODES

32c Oh


3~ + 3 ~ ~ ~ + 3 ~ ~ ~ + 3%,,+ 6Eg+ 6Eu + 9Flg+ 9Fl,,+ 9F2g C 1 1g

9F2u

ci 3A 1g + ,A2$ CEg+ 9F1$ 9FZg

A + A ~ " + A + AZu+ 2E + 2EU+ 5Flg+ 5Flu+ 5F + 5F2u

A + Alu+ 2 A + 2AZu+ 33 + 3Eu+ 5P + 5Flu+ 4F;g+4F2u

A '+ 2Alu+ A + 2AZu+ 2E + 4EU+ 5F + 4Flu+ 5F + 4%

A + 2A + 2A + AZu+ 3E + 3Eu+ 5F + 4Flu+4F2$ 5F2"

A + A + 2E + 5F + 5FZg

A 1g + 2+g+ 3Eg+ 5E 18 + &Fag

3F1$ 3Flu+ 3F2$ 3FZu

$&+ A2u+ Eg+ Eu+ 3F1$ 3FlU+ 2F 28 + 2F2u

Alu+ AZu+ 2Eu+ 3F + 2Flu+ 3F + 2P2u

Alu+ A + E + Eu+ 3F + 2Flu+ 2F + 3F2u

Alu+ Aag+ A2u+ E + 2Eu+ 3F + 2Flu+ 2F + 2F2u

C2($ 1g 2g 28

C i ( C 2 ) 16 2g 1.3

c:(uh) 1g 2% I& 2g

C:(Ud) 1g l U 2g 1g

Cnh(C:+uh)

Cib(C2+od)

D2 ( 3Cf )

D;(C:+2C2)

C&(C2+20,)

C:y(C:+2~d)

C ' ( C +a +o )

D 2 h ( X E + % h ) Ylg+ 3F2g

1.3 26 I&

h3 2g

2g g 1g 23

2 v 2 h d 1g 2g

D;h(C~+2C;pn2u,,) +&+ Eg+ 3Flg+ 2F 26

A + A + E + Eu+ 3Flg+ 3Flu+ 2F' + 2F2u

A + A2u+ E + Eu+ 3P + 2Flu+ 2F + 3F2u

A + E + 3 F + 2 F

c4 1-6 lU g 263

$4 1g 193 28

'4 h g 26

2F + 'Flu+ F + FzU

AIU+ Eu+ 2F + Flu+ F + F2u

ZF 1.3 + Flu+

D4 1s 2g

c4 " 1.3 a3

D2d(3C:+od) 2g + 2 F p ~

D ' (C2+2C2+sh) A2u+ Eu+ 2F 18 + Flu+

2F + Fag

A +

A + A + 2E + 3F + 3Fpg

A,, g+ %u+ E + Eu+ PF 1g + 2Flu+ FPg+ F2"

Alu+ A + Eg+ Q+ 2F + Flu+ F 2g + 2F2,,

A + E + 2F + FPg

2g + F2u 2d 4

D4 h 1g

c3 1g

' 6 1g 28 8

D3

C3" 2g I&

Djd 2g 8 1g

T F + Flu+ F 2 R + F2=

Th 1g 2g

0 Flg + F l u

Td F1g + Fzu

%&+ A2u+ 2Eg+ 2Eu+ 3F1$ 3FlU+ 3F 26 + 3 F a

1g

F + F

Oh



Tables D. Correlation tables relating the irreducible representations of each point group to the irreducible representations of each of its subgroups

C2" c1 c2 c,xz(a;~) C F ( U 7 )

A1 A A A ' A '

A:, A A A " A"

B1 A B A ' A"

B2 A B A " A '

1 4 2 2 2 i

1D C.

B I A A B B

B 2 A B A B

~~

A, A AU A A"

D~~ c1 c;(cX) c:(cz) c:(cg) ciY(exY) c:'(ox2) c~~(&'~) ci

A A A A A A ' A ' A ' A

A, A A A A A" A" A" AU

A g B1g A A

B B A" A ' A ' 4, B l U A A

A g B 2 g A

A B A ' A" A ' AU B2U A

B A A" A" A ' A g B 3 g A

B A A ' A ' A" A" B3u B

B B A ' A" A"

A B A" A ' A"

1 8 4 4 4 4 4 4 4 i

A AU B A '

l I 4 2 2 2

20 C.

6D 4

B 3 A B B A

4 2 2 2


D . L. ROUSSEAU. R. P. BAUMAN AND S. P. S. PORT0

'4h

A g

A"

El3

BU

Eg

Ell

- '1 '1 ' 2 '5 '2h '4 '4

g A A A AK A A ' A

A AU A A" A, A B

A Ag A A ' Ag B B

A A, A A" A, B A

2A 2Au 28 2A" 2Bg E E

2A 2Au 2B 2A ' 2Bu E E

Ag

A U

EK

E"

1

2). 2~ A+B A+B BZ+B3 B2++B, E

8- h i i 2 1 2 1

A A A

A AU A

?A 7Ag

?A PAU E

b 3 2

E

D4

A 1

14.

B1

B2

C?" s4

A 1

B1 A 2

A2

B1 A 1

A A A A A ' A

A A B A" A

A A A A A " B

A A a A ' B

Z A 2B A+B A'+A" B2+B, B1+B2 E

c;(c;) C2(C2) C$(C&) (CE+2C2) (C;+2C$) c4 cl-

A A A A

81 B1 A A A B B

A A A B A B 1

A B 81 A A B A I

15D-see opposite

D3d

Alg

Alu

Eg

Eu 1

16D C,

1 '1 '1 '7 's '2h 'j 4 CJ"

g *l A 1 A Ag A A ' kg A

AU A A U A 1 A2

g A 2 A2

A A,, A A"

A Ag B A" Bg A A

A AU B A ' BU A AU A2 A1

?A 2Ag A+B A ' + A ' A +B

?A ?AU A+B A'+"' A$Bu E EU E

U U EK E

E

1 2 6 t b 3 a ? ? 2

CORRELATION OF EACH POINT GROUP TO ITS SUBGROUPS

17D SL

E ZA ?B A'+A" A'+A" B1+B2 B1+B2 E

18D D,

1 E i 2 A A+B E

1 i 2

21D C,

A A A A

B A B A

?A 2B

1 2

A" A"

2A 2A '




15D

D&h

A l g

A l "

A2g

A2u

B1g

B l U

B2g

Bzu

Eg

Eu

'fh '2h '$h c 1 c i c g ( c f ) c z ( c 2 ) C b ( c i ) c:("h) c:(av) c:(ud) ( c 2 u h ) (c2+av) (';+Od)

A A g Ag Ag

Al l

Bg

BU

Ag B g

BU

BS A8

A, B" AU

A' A' A ' A A Ag A

A AU A

A Ag A

A Au A

A Ag A

A A" A

A Ag A

A A" A

A A A" A" A" A, AU

A ' A" A" Ag Bg

A' A ' A' A, Bu

A' A ' A" Ag

A" A" A' AU 4,

A' A" A' Ag

A" A ' A"

B B

B B

A B

A B

B A

B A

2Bg Ag+Bg A +B g g 2A PAg 2B A+B A+B 2A" A'+A" A'+A"

2A 2Au 2B A+B A+B 2A' A'+A" A'+A" 2Bu A"+ B, Au+Bu

1 1 6 8 8 8 8 8 8 8 4 4 4

l 5 D D,,, c m t i n u e d

Eu E Eu E E E E

TABLES 15D


D. I>. ROUSSEAU, R. P. BAUMAN AND S. P. S. PORT0

EZg

EZu

1

TABLES 23D-260

2A 2Ag 2A 2A ' 2Ag E Eg 4 E'

2A 2Au ZA 2A" 2Au E EU E:, E"

1 2 6 6 6 3 4 2 2 2


A2

B1

B2

El

E2

1

2 3D '6h

'6h I '1 'i ' 2 's 'Ph ' 3 '6 '6 '3h

A2 A

A2

A 1

A A B B B1 A A2

A B A B B2 A A1

A B B A B j A A2

2 A 2 B A+B A+B B2+Bj E E E El

2A 2A A+B A+B A+B1 E E E E2

1 2 6 6 6 3 4 2 2 2

A Ag A A ' Ag

A AU A A" Au A AU A A"

A Ag B A" Bg A Ag B A"

A AU B A ' Bu A AU B A'

A Ag A A '

2A 2Ag 2 B 2A" 2Bg E Eg El E"

2A 2AU 2B 2A ' 2Bu E Eu 5 E'

c6v

A1

+

B1

B2

El

Eg

'1 cp cI(Ov) '2" c3 ';v(OV) ' 6

A A A' A ' A1 A A1 A1 A

A A A"

A B A ' A"

A" A2 A A2 A2 A

B1 A A 1 A2 A B A" A ' B2 A A2 A1 B

2A 2B A'+A" A'+A" B1+B2 E E E El

2A 2 A A'+A" A'+A" A1+% E E E E 2

D-y,

A i

A;

";

E'

E'

1

c1 '2 c:(oh) c:("v) C ~ V '3 Dj '3" '3h

A A A ' A ' A 1 A A 1 A 1 A '

A" A" A2 A A 1 A2 A A A"

A B A ' A" B Z A % % A '

A B A" A ' B1 A Az A 1 A"

2A A+B 2 A ' A'+A" A1+B2 E E E E'

2 A A+B 2A" A'+A" %+El E E E E"

12 6 6 6 3 4 2 2 2

282 JOURNAL. OF RAMAN SPECTROSCOPY, VOL. 10, 1981 @ Heyden & Son Ltd, 198 1


E l g

ElU

E2g


2Bg A +B g g

2Ag 28 A+B A+B 2A" A'+"' Ar+A"

2Au 2B A+B A+B 2A ' A'+"' A'+"' 2BU AU+Bu

2Ag 2A A+B A+B 2A I A'+"' A1+A" 2Ag Ag+Bg

2A

2A

2A

TABLES 27D

A Ag A

A AU A

A%

AU A

Ag

A AU

Ag

A AU

A A

A

A

A

A A

A A

B B

B B

A B

A B

B A

B A

A '

A"

A '

A"

A"

A '

A"

A '

A g A ' A ' A

A" A" A U AU

Ag Bg

A ' A ' AU BU

Bg A g

A' A" BU AU

A" A"

A" A '

Bg A ' A" B

g

A" A ' BU BU

27D Dc. cant inved

1% /I E I

A 1 A 1 A l g A 1g A A ' A g Ai Ai Ai A2 A2 Alu Alu A A" Au A1 A2 A;

A2 A2 AZg Apg A A ' Ag A2 A2 A$

A 1 A 1 %u A 2 u A A" AU A2 A1 A;

A2 A,, Alg Azg B A" Bg B1 B2 A!

A1 A:, Alu Agu B A s 8, Bl B1 A i

A 1 A2 A 2g Alg B A" Bg B2 B1 A:

A2 A1 AZu Alu B A ' B, B2 B2 A$

E E Eg El E" Elg El El E" 53

E E EU EU El E' Elu El El E'

E E E6 E2 E' E2g 4 E2 Eg

Ai A;

A6

A;

A;

Ad

A;

Ai E'

E'

E'

E EU E* 4u E2 E2 E" E" E2u Eu 1 1 4 4 4 4 2 2 2 2 2 2 2


D.

As

A,

Eg

E,

Fg

TABLES 280-32D

28D T

A g A A 1 A A A Ag A A' A A

A A, A A" A2 A" A A u A

2A 2Ag 2 A 2A' 2Ag 2A 2 A 1 E

2A 2AU 2A 2A" 2% 2A 2% 2A E E" E

6

A A l l

E E8

2 A

3A 3Ag A+2B A'+2A" Ag+2Bg B1+B2+B3 "2cB1+B2 Blg+B2g+Bjg A+E A K g +E

F

L. ROIJSSEAU, R. P. BAUMAN AND S. P. S. PORT0

?A A+2B B1+B2+B3 A+E


F,

1

29D T,

3A 3A, A+2B 2A'+A" AU+2Bu B1+B2+B5 A1+Bl+B2 Biu+B2u+B3u A+E A u + E ~

24 12 12 1 2 6 6 6 3 8 Ll 2

0

A1

E

F1

F2 1

D i C1 C2(C:) CQ(C2) D 2 ( 3 C f ) (Cg+2C2) C h D4 C? D3 T

A A A A A1 A A 1 A

% A A B A 81 B Bl A A2 A

A A

2A 2A A+B 2A A+B1 A+B A1+B1 E E E

3A A+PB A+2B B1+B2+B, Bl+B2+B, A+E ++E A+E A2+E F

3A A+2B 2A+B Bl+B2+B3 A+B2+B, W E B2+E A+E A1+E F

24 12 12 6 6 6 3 b h 2

A

A

6 6 1 2 4 12 12 6 3

*d c1 c2 C S D2 c2 " '4 %d '3 '3"

A A1 A A 1 A1 A A A '

A 2 A A A " A 2 A 2 A 6 B1 A

E 2A 2A A'+"' ZA A1+A2 A+B A1+B1 5

F1 jA A+2B A'+2A" Bl+B2+5., A2+B1+B2 A+E A2+E

Fp 3A A+2B 2 A ' t A " B1+B2+B3 A1+BL+B2 IKE B2+E

- n '2h '2h D '

oh c1 c1 c2(c$) c;(C2) C:(oh) ci(qd) (Cf+ah) (C2+od) D 2 ( 3 C z ) (Cc+7C2) -

A A g A g

A ' A ' A A A 1 5 A Ag A

A1U A AU A A A" A" AU AU A A

B A B 1 A g B g A 2 g A A g A

A 2 u A A" A a A' A ' A" BU A BI

i

A' A"

A +B A+B1

Eu P A PAU 2 A A+!3 PA" A'+A" 2 A U AU+B" 2 A A+B1

FIK 1 >A >AK A+2B A+.?B A'+EA" A ' + 2 A A +PB Ag+2Bg Bl+B2+Bj B1+B2+B3

Flu 1 3A >Au A+PB A+.'B 2A'+A" ?A'"'' Au+2Bu Au+2Bu B1+B2+BJ B1+B2+B

FTg 1 3A >Ag A+% 2A-B A'+2A" P A ' + A ' A +PB 2Ag+Bg B1+B2+BJ A+B2+B3

FEU 1 3A 3AU A+PB 7A-B 2 A ' + A ' A ' + 2 A Au+2BU 2AU+BU B1+B2+B3 A+B2+B3

2% K g " Eg 2A 2Ag ?A A+% P A ' A'+A'

g e

3

- 1 1 i! 2,I 211 2L 24 24 i 2 12 12 1 2




F2u

Oh c o n t i n u e d

( c 2 + 2 0 h ) (c$+2od) (c2+ah+od) ( 3 4 + 3 ' h ) (c:++2c2+oh+ad) c4 s4

A 1 A1 A 1 A g Ag

A2 A2 A2 AU AU A B

A 1 A2 B 1 Ag B 1 g

A2 A 1 B 2 AL B 1 u B A

c;" CZ" c; " D2 h %h

A A

B B

2% %+A2 2Ag A +B A+B A i B

2% A1+% %+B2 2% %+A" A+B A+B

g 1g

A2+B1+B2 A2+B1+B2 A2+B1+B2 B 1g +B 2g +B x 1g +B 2g +B 3g A+E A+E

A1+B1+B2 A1+B1+B2 A1+B1+B2 Blu+B2u+B3u Blu+B2u+B3u A+E B t E

Blg+B2g+B3g A +B +B A2+B1+B2 Al+Bl+B2 Al+%+B2 g 2g 3g

A1+B1+B2 A2+B1+B2 Al+%+Bl BlU+%U+B3U AU+%+B3U W E A+E I , 1 2 12 12 6 6 12 12

Bu+Eu B2+E B1+E A2+E A1+E BZu+Eu A+E Au+Eu

A Bg B1 B 1 B 1 B2 B 1 g Ag

BU B 1 B2 A 1 A2 B l U AU A

A +B Ag+Bg A1+B1 A1+B1 A1+B1 A1+B2 E 18 1g

Au+Bu A1+B1 A2+B2 %+A2 AlU+BlU EU

A + E g g

Ag+Eg A2+E A2+E A2+E A2+E AZg+Eg A+E

A +EU A2+E A1+E B2+E B2+E AZu+Eu A+E Au+Eu

A + E g g

Bg+Eg B2+E B2+E B2+E B1+E B2$Eg A+E

1 6 6 6 6 6 3 16 8

T 0 Oh D3 '3" D3d Th Td

A A l g A 1 A 1 A l g Ag A 1 A 1

A2 A l U A AU A 1 A2 A 1 u A 1

A A2 A2

A 2 U A2 A1 A 2 u A AU A2 A 1

E g E g E g E

E E EU E EU E E EU

F1.a A2+E A2+E A2$Eg F F g F 1 F1

Flu A2+E A1+E AZu+Eu F F U F1 F2

F 2 g A1+E A1+E Alg+Eg F Fg F2 F2

F2U Al+E A2+E AlU+Eu F FU F 2 F 1

A2 A2 A 2 g Ag

E E E E

1 8 8 4 4 2 2 2

TABLES 32D



c2’ i axY ax’ oYc

1 1 1 1 -1 -1 -1 -1

Blu j 1 1 -1 -1 -1 -1 1 1

1 -1 1 -1 1 -1

1 -1 -1 1 -1 1

Tables E. Character tables.

1E

S e l e c t i o n R U ~ S

~

RZ axy

TZ

‘ Ry axz

TY

7 E

YT 5 -1 1 1

RE

3E

S e l e c t i o n R u l e s I

X T i 3 1 1 - 1

6E 10E

R * I 1 1 -1 -1 1



XT

TABLES 11 E-17E

1 1 E

I I 3 1 -1 -1 -1

-1 -1 axx- ayy'axy

C4"/ E 2C4' C2' 20" 2od

A1 1 1 1 1

A2 1 1 1 -1 -1

B1 1 -1 1 1 -1

B2 1 -1 1 -1 1

E 2 0 -2 0 0

YT 3 1 - 1 1 1

13E

Select ion Rules

l T Z a:,+

RZ

axx -

axY

(T~.T~);(R~,R~) (a;,, a;,)

CHARACTER TABLES

1 2 E

I E ZC,,' c2' 2c2 zc21i Select ion Rules I D4 I I I

14E

YT 3 -1 -1 -1 I

E 2C4 Cgz 2C2 2CZ1 i 254 oh 20" 2od

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 -1 -1 -1 -1 -1

1 1 1 -1 -1 1 1 1 -1 -1

1 1 1 -1 -1 -1 -1 -1 1 1

1 - 1 1 1 - 1 1 - 1 1 1 - 1

1 -1 1 1 -1 -1 1 -1 -1 1

1 -1 1 -1 1 1 -1 1 -1. 1

1 -1 1 -1 1 -1 1 -1 1 -1

2 0 - 2 0 0 2 0 - 2 0 0

2 0 - 2 0 0 - 2 0 2 0 0

17E

Select ion Rules

S6 E C3 Cj2 i S6 S65

Ag

' Select ion Rules

1 1 1 1 1 1 R Z



TABLES 18E-23E

; %

CHARACTER TABLES

19E

20E 21E

2% *1g

*1u

*2“

E

Eu

Select ion Rules E :‘Jj X:, i 2S6 ’Od

l i l 1 1 1

1 1 1 -1 -1 -1

1 1 - 1 1 1 . 1

1 1 -1 -1 -1 1

? -1 3 2 -1 O

22E

Selection Rules

I

I

?Ti/> e = e

Select ion R u l e s

1 1

1 1

-1 -1

-1 -1

- e * - C

- P - c *

-r.* -E

- E - C *

c * E

C C *

EX E

E E *

1

1

1

1

C

E *

E

c *

0

E*

C

E*

1 1

1 1

1 -1

1 -1

E * -1

E -1

c * -1 E -1

e * 1

€ 1

C * 1

e l

1

-1

-1

1

-1

-1

1 1

1

1

-1

-1

1 1 1 1 1

-1 -1 -1 -1 -1

1 1 -1 -1 1

-1 -1 1 1 -1

P E* -E* -E 1 - E

-E* -E E - c - E x .* :* ::I

C * C ’ :i 1

E e * e * E

-C -E -1

- E -c* - C -1

3 2 2 0

@ Heyden & Son Ltd, 1981 288 JOURNAL OF RAMAN SPECTROSCOPY, VOL. 10, 1981


Dsh

Alg

Alu

Aeg

ApU

Blg

El"

B2g

Bpu

Elg

Elu

EZg

EZu

CHARACTER TABLES

E 2Cg' 2 C 3 C2' 3 C 2 3C2' i 2 S j 2S6 ah 3ad j a v

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 -1 -1 -1 -1 -1 -1

1 1 1 1 -1 -1 1 1 1 1 -1 -1

1 1 1 1 -1 -1 -1 -1 -1 -1 1 1

1 -1 1 -1 1 -1 1 -1 1 -1 1 -1

1 -1 1 -I 1 -1 -1 1 -1 1 -1 1

1 -1 1 -1 -1 1 1 -1 1 -1 -1 1

1 -1 1 -1 -1 1 -1 1 -1 1 1 -1

2 1 - 1 - 2 0 0 2 1 - 1 4 0 0

2 1 -1 -7 0 0 -2 -1 1 2 0 0

2 -1 -1 2 0 0 2 -1 -1 2 0 0

2 -1 -1 2 0 0 -2 1 1 -2 0 0

24E

25E

1- 26E

27E

E 2Cg 2C3 C2' 3C2 3 C 2 '

1 1 1 1 1 1

1 1 1 1 - 1 - 1

1 -1 1 -1 1 -1

1 -1 1 -1 -1 1

2 1 - 1 -2 0 0

2 - 1 - 1 2 0 0

3 2 0 -1 -1 -1

E 2C6 2 5 C2 >av >ad

1 1 1 1 1 1

1 1 1 1 - 1 - 1

1 -1 1 -1 1 -1

1 -1 1 -1 -1 1

2 1 - 1 - 2 0 0

2 - 1 - 1 2 0 0

3 2 0 - 1 1 1

TABLES 24E-27E

Selection Rules I "xx + "yy="zz

9 0 - 1 1 - 2 I

E 2C3 3C2 ah 2 S j >av

1 1 1 1 1 1

1 1 1 -1 -1 -1

1 1 - 1 1 1 - 1

1 1 -1 -1 -1 1

2 - 1 0 2 - 1 0

2 - 1 0 - 2 1 0

Selection Rules

Selection Rules

@ Heyden & Son Ltd, 1981 JOURNAL OF RAMAN SPECTROSCOPY, VO1. 10, 1981 289


TABLES 28E-32E CHARACTER TABLES 2 RE

E UC,, 45' 3C2 1 U S 6 4S6' >oh

1 1 1 1 1 1 1 1

1 1 1 1 -1 -1 -1 -1

0 -1

30E

A1 I A2

1 1 1 1 1

1 --1 1 1 -1 I

E 6s4 j c p 8C3 6ad Select ion Rules

1 1 1 1 1 1 axx+ a + a

YY z z

I I 1 -1 1

I 3 -1 -1 0 1

Select ion Rules

a + a xx yy+ azz

290 JOURNAL. OF RAMAN SPECTROSCOPY, VOL. 10, 1981 @ Heyden & Son Ltd, 1981

Documents

Artigo Porto Normal Mode Determination in Crystals