International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 2.1, pp. 294311
Section 2.1.3. Symmetry of lattice vibrations^{a}Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D37077 Göttingen, Germany 
Having presented the basic formulation of lattice dynamics in Section 2.1.2, we will now consider the constraints that arise due to the symmetry of the particular atomic arrangement within a crystal. We shall see in the following how grouptheoretical methods can be used in order:
The theoretical aspects will be illustrated by means of simple examples which may serve as a guide for the application of the formalism to other systems of interest.
The elements of the dynamical matrix as introduced in Section 2.1.2.3 are given byUsing the matrix notation for the submatrices introduced in (2.1.2.22a), this equation reads Since the vector corresponds to a vector of the direct lattice, the righthand side of equation (2.1.3.1) is invariant with respect to changes of the wavevector q by an arbitrary reciprocal lattice vector g. Hence, the elements of the dynamical matrix represent periodic functions within the reciprocal space: The same periodicity can also be assumed for the eigenvalues, or eigenfrequencies, and for the eigenvectors:^{2}Consequently, we can restrict our discussion to wavevectors within the first Brillouin zone.
Owing to the symmetry of the atomic structure, not all of the force constants reflecting the interaction between atoms and are independent. Rather, there are constraints to the elements of the dynamical matrix according to the space group of the crystal. In the following, these constraints will be considered in some detail. Suppose the space group contains a symmetry operation .^{3} When applied to the crystal, this symmetry operation sends atom into another atom and simultaneously atom into . At the same time, the wavevector of a phonon is rotated from into . Hence, the elements of the dynamical matrix that describes the dynamics of the crystal after application of the symmetry operation may be written as or in submatrix notation This submatrix can be related to the corresponding matrix that describes the same dynamical behaviour, but in the unrotated crystal. To this end, we first consider the transformation of the forceconstant matrices under the symmetry operation. Obviously, the interaction between atoms and has to be of the same type as the interaction between and .
Since the potential energy is invariant with respect to symmetry operations, the force constants are related via or in matrix notation
Owing to the symmetry operation, the displacements of atoms and are rotated and transferred to atoms and , respectively (see Fig. 2.1.3.1). Thus, (2.1.3.5) can be rewritten asMoreover, this relation is valid for arbitrary displacements and, hence, the matrices of force constants transform according to or Using the fact that the matrix of rotation S is unitary , the inverse relation is obtained:Hence, the forceconstant submatrices transform like tensors do. One has to bear in mind, however, that the matrices in equation (2.1.3.7b) correspond to different pairs of atoms as illustrated by Fig. 2.1.3.2. Using this result in equation (2.1.3.4a) and remembering the fact that atoms related by a symmetry operation have the same mass, we obtainThe phase factor on the righthand side contains the indices L and of those primitive cells into which the atoms and are sent by the symmetry operation . In general, the phase is not conserved during the transformation and, hence, the sum over cannot simply be replaced by the matrix elements . Rather, we have to consider the phase factor in more detail in order to find the transformation law for the dynamical matrix.
The position vectors of particles and are related viaandConsequently, the vector appearing in the phase factor of equation (2.1.3.8) can be expressed as When inserted into equation (2.1.3.8), the required transformation law for the dynamical matrix is obtained: or or in submatrix notation Again, these are relations between pairs of submatrices of the dynamical matrix. In contrast to the matrices of force constants, however, phase factors have to be considered here. This is because the symmetry operation may send different atoms and located within the same primitive cell (0) into atoms and within different primitive cells L and as illustrated in Fig. 2.1.3.3. Therefore, the product of phase factors in equation (2.1.3.11) is in general different from unity.
Irrespective of the particular primitive cells in which the atoms are located, however, the labels and K of those atoms that are related by a symmetry operation are uniquely determined. Given the label and a particular symmetry operation , the label K may be represented by the functionwhich represents the atom transformation table.^{4} With the definition of unitary transformation matrices we are now able to formulate the transformation law for the dynamical matrix briefly as Obviously, with the help of equation (2.1.3.13), we can allocate a unitary matrix to each symmetry operation. These matrices, however, do not form a representation of the crystal space group in the mathematical sense since the mapping is not a linear one. Rather, we obtain the following transformation matrix for the product of two symmetry operations:The nonlinearity of the mapping is due to the fact that the first matrix on the righthand side of this equation depends on the wavevector rather than on q. If we restrict our considerations to the symmetry operations of the space group G(q) of the wavevector q that leave the wavevector invariant modulo some reciprocallattice vector g(q, S),then equation (2.1.3.13) provides an ordinary (3Ndimensional) representation of this symmetry group.^{5} In the following, we denote the elements of the subgroup G(q) by . The corresponding unitary and Hermitian transformation matrices can be reduced toAccording to equation (2.1.3.14), they commute with the dynamical matrix: This relation contains the symmetry constraints for the dynamical matrix. The independent elements of D(q) may be obtained by application of equation (2.1.3.18) for every operation of the space group of the wavevector.
Another approach to the symmetry reduction of the dynamical matrix is based on grouptheoretical considerations making use of the well known irreducible representations of symmetry groups. It is especially useful for the prediction of the form of eigenvectors and the investigation of degeneracies. Following the treatment of Maradudin & Vosko (1968), we consider the purely rotational elements of the space group G(q) that form the point group of the wavevector . According to equation (2.1.3.17), we associate a matrix operator to each of the elements of G_{o}(q). These matrix operators are uniquely determined by the rotations R and do not depend on the translational parts of the spacegroup operation , as proven by inspection of the individual matrix elements: These T matrices again commute with the dynamical matrix, but in contrast to the matrices they do not provide an ordinary representation of the group G_{o}(q). For the multiplication of two symmetry elements R_{i} and R_{j} the following relation holds: According to equation (2.1.3.16), is a reciprocallattice vector and henceThus, the T matrices provide not a normal but a multiplier representation of the group G_{o}(q). The phase factor on the righthand side of equation (2.1.3.21a) is the complex multiplier characteristic for the (ordered) product of symmetry operations.
For wavevectors within the first Brillouin zone, the reciprocallattice vectors are identically zero (see last footnote^{5}) and the T representation is an ordinary one. The same is true if none of the symmetry elements of G_{o}(q) contains a fractional translation, i.e. for symmorphic space groups. Therefore, multipliers have to be taken into account only if nonsymmorphic space groups and wavevectors on the Brillouinzone boundary are considered.
There are some other restrictions for the dynamical matrix arising from the fact that inverting the wavevector is equivalent to taking the complex conjugate dynamical matrix [c.f. equation (2.1.2.24)]: Hence it is useful to extend our discussion to those symmetry operations that invert the phonon wavevector. Let us assume that the space group of the crystal contains an element with Using equation (2.1.3.14) we obtainIn order to provide a consistent description, we introduce an antiunitary operator which transforms an arbitrary vector into its complex conjugate counterpart Obviously, does not commute with the dynamical matrix but exhibits the following transformation behaviour: On the other hand, we infer from equation (2.1.3.24) thatwhich provide the additional constraints for the dynamical matrix. In component form, this last relation can be written explicitly asif particles () and () are sent into () and () by the symmetry operation , respectively.
If represents the inversion , in particular, then (2.1.3.28) reduces toMoreover, if every atom is itself a centre of inversion (e.g. the NaCl structure) ( and ), the matrix C(q) defined by is a real and symmetric matrix with real eigenvectors for arbitrary wavevectors q.
In terms of group theory we proceed as follows: We add to the space group of the wavevector G(q) the elements of the coset .^{6} This will result in a new space group which we call . If instead of the matrix operator the antiunitary operator is assigned to those symmetry operations that invert the wavevector, then a representation of the whole group is provided. Moreover, all these matrix operators commute with the dynamical matrix.
As before, let us restrict ourselves to the rotational parts of the symmetry operations. The point group of the wavevector is enlarged by the coset yielding the group . In analogy to equation (2.1.3.19), the elements of the left coset will be represented by the matrix operator
The T matrix operators provide a multiplier corepresentation. The multipliers are not uniquely defined as in equation (2.1.3.21a). Rather, the definition depends on the type and the order of the symmetry operations involved. In order to distinguish between the different kinds of symmetry operations, we introduce the following notation:
The multiplication rule is determined by the multipliers Again, this representation reduces to an ordinary representation either for symmorphic space groups [all ] or for wavevectors within the interior of the Brillouin zone.
All matrix operators of the T representation commute with the dynamical matrix. Hence, they may be used for the determination of independent elements of the dynamical matrix as well as for the determination of the form of eigenvectors compatible with the atomic structure.
As an example, we consider a crystal of tetragonal symmetry, space group , with lattice parameters a and c. The primitive cell spanned by the three mutually orthogonal vectors a, b and c contains ten atoms at the positions listed in Table 2.1.3.1 and shown in Fig. 2.1.3.4. Consequently, the dynamical matrix has elements.

The space group contains eight symmetry operations, namely
Obviously, atoms No. 3 to 10 are chemically identical and have the same mass.
For the reduction of the dynamical matrix, we need the function , yielding the label of that atom into which κ is sent by the symmetry operation S. This function can be represented by the atom transformations shown in Table 2.1.3.2. This table displays the labels of atoms κ and K related by a particular symmetry operation and also the relative position of the primitive cells l and L where both atoms are located. This information is needed for the calculation of phase factors in the expression for the matrix operators T. Via the twofold axis, atom 6, for example, is transformed into atom 9 located within the cell which is shifted by the vector .

Let us first consider the case of phonons with infinite wavelengths and, hence, the symmetry reduction of the dynamical matrix at zero wavevector (the Γ point). Here, the point group of the wavevector is equivalent to the point group of the lattice. According to equation (2.1.3.19a), we can immediately write down the transformation matrix for any of these symmetry operations. Using the notationfor the threedimensional vector representation of the symmetry elements, we obtain the T matrix operatorsandSince each of these matrices commutes with the dynamical matrix (, with ), the following relations are obtained for the submatrices: and so on for the other submatrices.
For nonzero wavevectors q along (), the point group contains the identity and the mirror plane only. The respective T matrix operators are the same as for the Γ point: There are, however, symmetry elements that invert the wavevector, namely and . Hence the enlarged group consists of the elements E, , and . Inspection of the atom transformation table yields the remaining matrix operators: andBeing antiunitary, the corresponding inverse operators are^{7}The invariance of the dynamical matrix with respect to the similarity transformation () using any of these operators leads to the following relations for wavevectors along :^{8}andFor the submatrix (and similarly also for and ) we can combine the three relations and obtainHenceObviously, the symmetry considerations lead to a remarkable reduction of the independent elements of the dynamical matrix.
According to equation (2.1.3.3), the phonon dispersion is periodic within the reciprocal space: Moreover, for each symmetry operation of the space group of the crystal, the eigenvalue equation may be written in the form and due to the transformation property of the dynamical matrix, equation (2.1.3.14), Hence is an eigenvalue of the dynamical matrix D(q) at the wavevector q as is itself. If the eigenvalues are not degenerate, i.e. if there is not more than one linear independent eigenvector for each eigenvalue, then If, on the other hand, there are degenerate phonon modes (, ), the most we can strictly infer from equation (2.1.3.36) isWithout any loss of generality, however, it is possible to label the modes at in terms of those modes at wavevector q in such a way that equation (2.1.3.37) remains valid. Hence, we conclude that the phonon dispersion ω(q)^{9} in the threedimensional reciprocal space exhibits the full symmetry of the point group of the crystal, as illustrated in Fig. 2.1.3.5.
Moreover, is an even function of q (Fig. 2.1.3.6). This is always true even if the space group does not contain the inversion, since the dynamical matrix is Hermitian. From the eigenvector equation (2.1.2.20) we have Taking the complex conjugate and remembering that the eigenvalues of a Hermitian matrix are real quantities, Obviously, is an eigenvalue of D(q) just as is. Hence, with the same arguments as above we conclude that
In the previous section we used the symmetry properties of the dynamical matrix to derive equation (2.1.3.36). Since the phonon dispersion is invariant with respect to all symmetry operations of the space group [equation (2.1.3.37)], we conclude that not only is an eigenvector of the dynamical matrix D(q) but so is the vector . If the corresponding eigenvalue is not degenerate, the (normalized) eigenvectors are uniquely determined except for a phase factor of unit modulus. Hence, the following relation holds:The statement giving the atomic displacements as solutions of the equations of motion (cf. Section 2.1.2) was based on Bloch waves, the polarization vector being invariant with respect to lattice translations. It is therefore convenient to choose the arbitrary phase factor in the transformation law for eigenvectors in such a way as to leave the righthand side of equation (2.1.3.40) independent of . Setting the phase angle equal to , we obtain the simple form of the transformation lawThis choice is, however, not always possible. If there is a symmetry operation that inverts the wavevector, then in addition to equation (2.1.3.40) there is another relation between and due to the Hermitian nature of the dynamical matrix. Hence in this case the transformation law may differ from equation (2.1.3.41), as discussed in Section 2.1.3.5.2.
If the dynamical matrix exhibits degenerate eigenvalues for the wavevector q, the most that can be said is that the symmetry operation sends an eigenvector into some linear combination of all those eigenvectors that correspond to the same eigenvalue. Without any loss of generality we may, however, demand that equation (2.1.3.41) remains valid even in this case, since if we would have determined eigenvectors at q then among the variety of possible and equivalent orthonormal sets of eigenvectors at we simply choose that particular one which is given by (2.1.3.41). There is, however, one exception, which applies to wavevectors on the Brillouinzone boundary and symmetry operations with (where g is a reciprocallattice vector): owing to the periodicity , equation (2.1.3.3), the eigenvectors and have to be identical in this case.
If we consider those symmetry operations that leave the wavevector invariant (except for an additional reciprocallattice vector), we are able to obtain special conditions for the eigenvectors themselves. In Section 2.1.3.1 we found that the dynamical matrix commutes with the T matrix operators defined by equation (2.1.3.19a). Hence, if R is an arbitrary element of the point group , the vector is an eigenvector with respect to the eigenvalue as well as : Since eigenvalues may be degenerate, we now replace the index j that labels the 3N different phonon branches by the double index σλ: σ labels all different eigenvalues whereas λ distinguishes those phonons that are degenerate by symmetry,^{10} i.e. that have the same frequency but different eigenvectors,λ runs from 1 to if is the degeneracy of the eigenfrequency . With this notation, equation (2.1.3.42) can be rewritten as Consequently, the vector has to be some linear combination of all eigenvectors , , corresponding to the same eigenvalue ,Obviously, the eigenvectors span a vector space that is invariant with respect to all symmetry operations of the point group of the wavevector. Moreover, this vector space does not contain any proper invariant subspaces and is therefore irreducible. Under the symmetry operations of the group , the eigenvectors transform into each other. The corresponding coefficients can be regarded as the elements of a complex () matrix that induces a unitary irreducible multiplier representation (IMR) of the point group of the wavevector. The complex multiplier is exactly the same as for the 3Ndimensional reducible representation provided by the T matrix operators [cf. equation (2.1.3.21)].
For a given point group there is only a limited number of irreducible representations. These can be calculated by grouptheoretical methods and are tabulated, for example, in the monographs of Kovalev (1965) or Bradley & Cracknell (1972). The multipliers are specific for the individual space groups G(q) and depend merely on the fractional translations v(R) associated with a symmetry element R. It should be noted that for wavevectors within the Brillouin zone and for symmorphic space groups all multipliers are unity and we are left with ordinary irreducible representations. Hence, merely on the basis of grouptheoretical considerations, restrictions for the phonon eigenvectors can be obtained.
A particular phonon can now be characterized by the symmetry of the corresponding eigenvector, i.e. the irreducible multiplier representation (IMR) that describes its transformation behaviour. All degenerate phonons obviously belong to the same IMR. Moreover, phonons with different frequencies may belong to the same IMR. On the other hand, there may also be IMRs to which no phonon belongs at all. For a given crystalline structure it is possible, however, to predict the number of phonons with eigenvectors transforming according to a particular irreducible multiplier representation:
Let us arrange all eigenvectors of the dynamical matrix as columns of a unitary matrix in such a way that eigenvectors of the same irreducible representation occupy neighbouring columns: This matrix can now be used for a similarity transformation of the T matrix operators: Since an eigenvector can never change its symmetry by the multiplication with the T matrix operator and since all eigenvectors are pairwise orthonormal, the resulting matrix has blockdiagonal form. Moreover, each block on the diagonal consists of the matrix of a particular irreducible multiplier representation: The matrix of eigenvectors thus reduces the operators T to blockdiagonal form.
There may be several phonons with different frequencies that belong to the same symmetry (irreducible representation). All purely longitudinally polarized lattice vibrations, irrespective of whether these are acoustic or optic modes, belong to the totally symmetric representation. This is because each vector parallel to q – and in purely longitudinal modes the polarization vectors of each individual atom are parallel to the wavevector – is left invariant by any of the symmetry elements of . A particular irreducible representation may thus appear more than once in the decomposition of the T matrix and, consequently, two or more of the blocks within the matrix may be identical. Therefore, it is convenient to split the index σ that labels the modes of different frequency into two indices s and a, s characterizes the inequivalent irreducible multiplier representations and a is the running index over all modes of the same symmetry but of different frequency. If denotes the multiplicity of the representation s, then a takes the values . Using this notation, the transformation law for the eigenvectors can be rewritten asAs a well known result from group theory, the multiplicity of a particular irreducible multiplier representation s in the decomposition of the reducible 3Ndimensional Tmatrix representation can be calculated from the respective charactersand according to The summation index runs over all symmetry elements of the point group , the order of which is denoted by . Hence we are able to predict the number of nondegenerate phonon modes for any of the different irreducible multiplier representations on the basis of grouptheoretical considerations. Obviously, there are exactly modes with eigenvectors that transform according to the irreducible multiplier representation s. Among these, groups of always phonons have the same frequency. The degeneracy corresponds to the dimensionality of the irreducible representation. The crystallographic space groups give rise to one, two or threedimensional irreducible representations. A maximum of three fundamental lattice vibrations can therefore be degenerate by symmetry, a situation that is observed for some prominent wavevectors within cubic crystals.
Symmetry considerations not only provide a means for a concise labelling of phonons; group theory can also be used to predict the form of eigenvectors that are compatible with the lattice structure. This aspect leads to the concept of symmetry coordinates, which is presented in Section 2.1.3.4.
Let us return to the example presented in Section 2.1.3.1.1. At the Γ point, the point group of the wavevector is identical to the point group of the crystal, namely . It contains all eight symmetry operations and there are five different irreducible representations, denoted , , , and . The corresponding character table including the reducible representation provided by the Tmatrix operators (cf. Section 2.1.3.1.1) has the form shown in Table 2.1.3.3. The representations , , and are onedimensional, and is twodimensional. The upper index, or −, refers to the symmetry with respect to the mirror plane . According to (2.1.3.50), we may calculate the multiplicities of these irreducible representations in the decomposition of the 30dimensional T representation. As the result we obtainHence for the sample structure presented in Section 2.1.3.1.1 we expect to have five phonon modes of symmetry , three modes for each of the symmetries , and , and 16 modes of symmetry , the latter being divided into pairs of doubly degenerate phonons.

So far, we have used the 3N Cartesian coordinates of all atoms within a primitive cell in order to describe the dynamics of the crystal lattice. Within this coordinate system, the elements of the dynamical matrix can be calculated on the basis of specific models for interatomic interactions. The corresponding eigenvectors or normal coordinates are some linear combinations of the Cartesian components. With respect to these normal coordinates, which are specific to each particular crystal, the dynamical matrix has diagonal form and contains the squares of the eigenfrequencies reflecting the interatomic forces.
As shown in Sections 2.1.3.1 and 2.1.3.3, there are constraints for the dynamical matrix due to the symmetry of the crystal lattice and, hence, eigenvectors must obey certain transformation laws. Not all arbitrary linear combinations of the Cartesian coordinates can form an eigenvector. Rather, there are symmetryadapted coordinates or simply symmetry coordinates compatible with a given structure that can be used to predict the general form of eigenvectors without the need for any particular model of interatomic interactions. These symmetry coordinates can be determined on the basis of the irreducible multiplier representations introduced in the previous section.
From the Tmatrix operators and the representation matrices of a particular irreducible multiplier representation we may define another matrix operator with the elements When applied to an arbitrary 3Ndimensional vector Ψ built from the Cartesian coordinates of the individual atoms, this operator yields the particular component of Ψ that transforms according to the irreducible representation s. Hence it acts as a projection operator. Defining a set of vectors by we obtain Using the multiplication rule (2.1.3.21a), it can be shown that the righthand side of this equation reduces towhich can also be written in the formsince if R runs over all symmetry operations of the group the same is true for the product . Comparing this expression with the definitions (2.1.3.51) and (2.1.3.52) we obtainHence the set of vectors span an irreducible vector space and transform into each other in just the same way as the eigenvectors of the dynamical matrix do.
If the corresponding irreducible representation s appears only once in the decomposition of the 3Ndimensional T representation, then the vector space provided by the , , is uniquely determined. Consequently, these basis vectors themselves may be regarded as eigenvectors of the dynamical matrix. In this case, symmetry considerations alone determine the polarization of lattice vibrations irrespective of the particular interatomic interactions.
If, on the other hand, the multiplicity of the representation s is larger than 1, the most that can be inferred is that each of the vectors is some linear combination of the eigenvectors , . By an appropriate choice of the different generating vectors in (2.1.3.52), it is, however, always possible to find a set of pairwise orthogonal vectors that span the same vector space as the eigenvectors . If we repeat this procedure for every irreducible representation s contributing to the T representation, we obtain 3N linearly independent vectors, the symmetry coordinates, that generate a new coordinate system within the 3Ndimensional space of atomic displacements. With respect to this coordinate system the dynamical matrix is reduced to a symmetryadapted blockdiagonal form.
In order to show this, let us denote the matrix elements of the transformed dynamical matrix by (, ) and the components of the symmetry coordinates by (, ). Then the following equation holds, since the dynamical matrix D(q) commutes with the Tmatrix operators and since the symmetry coordinates transform according to (2.1.3.54): Owing to the orthogonality of the irreducible representation, we obtain after summation over all symmetry elements R and division by the order of the groupThis equation proves the blockdiagonal form of the transformed dynamical matrix . Hence, with respect to the symmetry coordinates, the dynamical matrix can be represented by submatrices of dimension that are determined by the individual irreducible representations (s): The elements of the submatrices are given byand must be independent of λ. Obviously, a submatrix may appear once, twice or three times on the diagonal, according to the dimensionality of the respective irreducible representation.
The eigenvectors and eigenvalues of the blockdiagonalized dynamical matrix can be collected from the eigenvectors and eigenvalues of the individual submatrices. Hence, the eigenvectors of correspond to the nondegenerate phonons of symmetry s.
Let us try to find the symmetry coordinates corresponding to our sample structure introduced in Section 2.1.3.1.1 for . Using the irreducible representations displayed in Section 2.1.3.3.1, we write down the projection operator for representation according to equation (2.1.3.51): with the abbreviations
From the results in Section 2.1.3.4, we expect to have five symmetry coordinates corresponding to representation and three for according to the respective multiplicities. Let denote the basis of the 30dimensional space generated by the displacements of the ten atoms in the x, y and z directions, respectively. If we apply the projection operator to the basis vector , we obtain the first symmetry coordinate according to equation (2.1.3.52): In a similar way we may use the basis vectors , , and in order to generate the other symmetry coordinates:(It can easily be shown that all the other basis vectors would lead to linearly dependent symmetry coordinates.)
Any eigenvector of the dynamical matrix corresponding to the irreducible representation is necessarily some linear combination of these five symmetry coordinates. Hence it may be concluded that for all lattice vibrations of this symmetry, the displacements of atoms 1 and 2 can only be along the tetragonal axis. Moreover, the displacements of atoms 3 to 10 have to be identical along z, and pairs of atoms vibrate in opposite directions within the xy plane.
For the representation we obtain the following symmetry coordinates when is applied to , and :
Obviously, none of the corresponding phonons exhibits any displacement of atoms 1 and 2. There is an antiphase motion of pairs of atoms not only within the tetragonal plane but also along the tetragonal z axis.
For the representations we obtain the following projection operators:with
Both representations appear three times in the decomposition of the T representation. Hence, we expect three phonons of each symmetry and also three linearly independent symmetry coordinates. These are generated if the projection operators are applied to the basis vectors , and : Just as for representation , the symmetry coordinates corresponding to representations do not contain any component of atoms 1 and 2. Consequently, all lattice modes of these symmetries leave the atoms on the fourfold axis at their equilibrium positions at rest.
Representation is twodimensional and appears eight times in the decomposition of the T representation. Hence, there are 16 doubly degenerate phonons of this symmetry. According to (2.1.3.51), four projection operators , , and can in principle be constructed, the latter two being, however, equivalent to the former ones:andwith
The projection operator applied to the basis vectors , , , , , , and yields eight symmetry coordinates for eight phonon modes with different eigenfrequencies. Owing to the degeneracy, each of these phonons has a counterpart with the same frequency but with a different linearly independent eigenvector. These new eigenvectors are built from another set of symmetry coordinates, which is generated if the other operator is applied to the same vectors , , , , , , and .The two sets of symmetry coordinates areLooking carefully at these sets of symmetry coordinates, one recognises that both vector spaces are spanned by mutually complex conjugate symmetry coordinates.
Collecting all symmetry coordinates as column vectors within a matrix we finally obtain the matrix shown in Fig. 2.1.3.7. For simplicity, only nonzero elements are displayed. This matrix can be used for the blockdiagonalization of any dynamical matrix that describes the dynamical behaviour of our model crystal.
Whenever two phonon modes of the same wavevector exhibit identical frequencies but linearly independent eigenvectors, these modes are called degenerate. As discussed in the preceding sections, the symmetry of the crystal lattice may cause degeneracies if there are higherdimensional irreducible representations of the point group of the phonon wavevector. Twodimensional representations yield twofold degenerate lattice vibrations, whereas threefold degeneracy may be found for special wavevectors in cubic crystals exhibiting threedimensional irreducible representations. In addition, there are two other reasons for the possible existence of degenerate lattice vibrations: accidental degeneracy and degeneracy due to timereversal invariance of the lattice vibrations. Both phenomena will be described in the following.
The symmetry analysis of lattice vibrations provides a powerful tool not only for the characterization of eigenvectors but also for the presentation of experimental results. In neutron scattering experiments, for example, a series of single phonons may be detected but symmetry determines which of these phonons belong to the same branch, i.e. how the single phonons have to be connected by a dispersion curve. The decision as to which of Figs. 2.1.3.8(a) or (b), which represent the same experimental results as full circles, is the correct one can be made by symmetry arguments only. In Fig. 2.1.3.8(a) the two phonon branches intersect. Thus, there are two degenerate phonons at the single wavevector . From the symmetry point of view, this particular wavevector has no special properties, i.e. the point group is just the same as for neighbouring wavevectors. Hence, the degeneracy cannot be due to symmetry and the respective eigenvectors and are not related by any transformation matrix. As a consequence, the two phonons cannot belong to the same irreducible representation, because otherwise every linear combination would equally well represent a valid eigenvector to the same eigenvalue with the same symmetry. It is, however, highly improbable that the special nature of the interatomic interactions gives rise to this uncertainty of eigenvectors at some wavevector within the Brillouin zone. Rather, it is expected that any infinitesimal change of force constants will favour one particular linear combination which, consequently, must correspond to a phonon of lower frequency. At the same time, there will be another well defined orthogonal eigenvector with slightly higher frequency – just as is represented in Fig. 2.1.3.8(b). Hence, two phonon branches of the same symmetry do not intersect and yield a frequency gap. This phenomenon is sometimes called anticrossing behaviour. It is associated with an eigenvector exchange between the two branches.

(a) Accidental degeneracy of phonons with different symmetry. (b) Anticrossing of phonons with the same symmetry (no degeneracy). 
Accidental degeneracy according to Fig. 2.1.3.8(a), on the other hand, can only be observed if the two phonon branches belong to different irreducible representations. In this case, the eigenvectors are uniquely determined even at since a mixing is forbidden by symmetry.
In Section 2.1.3.3, we considered in some detail the symmetry of phonon eigenvectors with respect to the symmetry operations contained in the point group of the wavevector. We know, however, that those symmetry operations that invert the wavevector give rise to additional constraints for the dynamical matrix. A lattice vibration with wavevector can also be regarded as a wave travelling in the opposite direction on a reversed timescale. Since the classical equations of motion are invariant with respect to time reversal, both phonon eigenvectors, and , are related and additional degeneracies may appear if there are symmetry operations that transform into .
Let A be the rotational part of such a symmetry operation. We have shown in Section 2.1.3.1 that antiunitary matrix operators can be defined that commute with the dynamical matrix. Not only the eigenvectors , , but also are therefore eigenvectors corresponding to the same eigenfrequency. If the latter are linear combinations of the former ones, no new information about degeneracies can be expected. If, on the other hand, is orthogonal to all vectors , it belongs to another vector space corresponding to different indices and . In this case, the eigenfrequencies and have to be identical and additional degeneracies appear that may even relate different irreducible representations s and .
In order to distinguish between the two possibilities, we have to consider the transformation behaviour of the vector . According to the multiplication rule for the Tmatrix operators, (2.1.3.32), we obtainThe operation leaves the wavevector invariant and is therefore an element of the point group . Hence the righthand side can be expressed in terms of the irreducible representation : Obviously, the new eigenvectors transform according to the conjugated representation defined byA being an arbitrary representative of those symmetry operations that invert the wavevector q.
If the irreducible representations and are not equivalent, the eigenvectors and are linearly independent and, hence, the vibrations of the two representations are degenerate. In this case, is called a representation of the third kind. Within the decomposition of the T representation, and always appear in pairs.
If, on the other hand, the two representations are equivalent, there is a matrix with corresponding to a transformation of the coordinate system. It can be shown that the product has the following form: In many cases, A is of order 2, i.e. (for or , for example), and the righthand side of (2.1.3.64) reduces to .
Those representations for which the sign in (2.1.3.64) holds have come to be called representations of the first kind [ ] and those that correspond to the − sign representations of the second kind . Without going into the details of the two cases, we merely wish to present the results that are most relevant for our discussion. For a more detailed presentation see e.g. Maradudin & Vosko (1968).
For representations of the first kind, the eigenvectors and are related by a linear transformation. Hence there are no additional degeneracies due to the timereversal invariance of the equation of motion. Let us consider the special case of a point group whose elements commute with the symmetry operation A. Crystals with a centre of inversion always meet this condition. For wavevectors within the Brillouin zone or for symmorphic space groups all multipliers are unity and . In this case, representations are of the first kind if they are real representations.
For representations of the second kind, and are linearly independent. The corresponding lattice vibrations are degenerate due to timereversal invariance. Consequently, the multiplicity of the representation in the decomposition of the T representation is an even number and pairs of eigenfrequencies are identical: The theory of characters provides us with a rather simple criterion for the distinction between representations of the first, second or third kind. Without any proof, we simply present the result that is particularly important for practical work: Finally, let us consider the special case of a real dynamical matrix. This can be found for crystals in which each atom is a centre of inversion or for special wavevectors on the Brillouinzone boundary or at the Γ point, for example. In this case, the dynamical matrix commutes with the operator that transforms arbitrary vectors into their complexconjugate counterparts. Hence, the vectors are eigenvectors to the eigenvalue as well as itself. Since transforms under the elements of the point group according to the irreducible representation , belongs to the complexconjugate representation. If the two representations are not real ones and therefore not identical, the corresponding lattice vibrations have to be degenerate. Every linear combination of the eigenvectors and is an eigenvector as well. Thus, we are free to choose real eigenvectors, namely: and .
Let us consider the space group . For wavevectors along the hexagonal axis, the point group consists of the three symmetry operations E, and . Being a cyclic group, its irreducible representations are onedimensional (see Table 2.1.3.4). The mirror plane inverts the wavevector and the two threefold rotations are selfconjugated with respect to :If we remember that for symmorphic space groups all multipliers are unity, we obtain the following conjugate representations according to (2.1.3.62): Obviously, and are inequivalent and, hence, pairs of phonons corresponding to representations and , respectively, are degenerate. The two transverse acoustic phonon branches tin particular not only leave the Γ point with the same slope as determined by the elastic stiffness (cf. Section 2.1.2.4 and Chapter 1.3 ) but are strictly identical throughout the whole Brillouin zone.

Another example may illustrate the degeneracy of phonons at special wavevectors where the elements of the dynamical matrix are real quantities. Let us consider the nonsymmorphic space group . For the Γ point (), the onedimensional representations of this cyclic group are collected in Table 2.1.3.5. Obviously, and form a pair of complexconjugated representations as well as and . Therefore, always two lattice vibrations of these symmetries exhibit the same frequencies. The eigenvectors for representation or can be combined with the eigenvectors of corresponding modes of representations or , respectively, to yield real quantities.

For wavevectors within the Brillouin zone along the hexagonal axis, the irreducible representations are the same as for the Γ point. However, the elements of the dynamical matrix are complex and symmetry does not yield any degeneracies. Hence phonons can be distinguished according to the six different representations.
At the Brillouinzone boundary along the hexagonal axis (, the A point), one has to take into account multipliers of the form since the space group is nonsymmorphic. For symmetry operations without fractional translation (, , ) this factor is unity, whereas it equals the complex unit i for the other elements of the point group (, , ). Hence the six irreducible multiplier representations are as shown in Table 2.1.3.6. Now we have three pairs of complexconjugate representations, namely: and ; and ; and and . Again, pairs of phonons of corresponding representations are degenerate. As a consequence, the phonon dispersion curves need not approach the Brillouinzone boundary with a horizontal slope but meet another branch with the opposite slope.

In conclusion, grouptheoretical considerations for wavevectors along the hexagonal axis yield at the centre (Γ point) as well as at the boundary (A point) of the first Brillouin zone pairs of degenerate phonon modes. Both modes belong to complexconjugate representations. This result can be used in order to display the dispersion curves very clearly in an extended zone scheme plotting the phonon branches of different symmetries alternately from Γ to A and from A back to Γ as illustrated in Fig. 2.1.3.9. Here, the phonon dispersion for the roomtemperature phase of KLiSO_{4} is shown as an example. Note that irreducible representations are frequently denoted by the letters A, B, E, T instead of our notation . T and E are reserved for representations that (at the Γ point) are triply and doubly degenerate, respectively. An index or g/u is often used to distinguish representations that are symmetric (gerade) and antisymmetric (ungerade) with respect to a prominent symmetry operation, e.g. a centre of inversion or, in the case of , the twofold axis. The total symmetric representation is always denoted by A. Hence in the preceding example all the representations , , and are Etype representations since they are doubly degenerate at the zone centre due to timereversal degeneracy. Moreover, and are symmetric with respect to . Therefore, the irreducible representations of Fig. 2.1.3.9 can be identified as , , , , and .
It can be seen that all phonon branches cross the zone boundary continuously while changing their symmetry. This behaviour is a direct consequence of the timereversal degeneracy.
In our last example, we recognized that the group of the wavevector consists of the same elements, irrespective of whether the Γ point, the zoneboundary A point or any other wavevector along the hexagonal axis is concerned. This behaviour, however, is the exception rather than the rule. In general, wavevectors on the Brillouinzone boundary exhibit different point groups to wavevectors within the Brillouin zone and the Γ point yields the full point group of the crystal. Obviously, the symmetry of the lattice vibrations changes discontinuously when approaching prominent wavevectors. Phonon branches, on the other hand, represent continuous functions ω(q) within the reciprocal space. Hence the question arises as to how the different irreducible (multiplier) representations associated with one particular phonon branch at different wavevectors are interrelated, i.e. which of the individual representations are compatible.
The solution of this problem is quite simple as long as all of the irreducible multiplier representations are onedimensional. For arbitrary wavevectors within the Brillouin zone, the point group is always a subgroup of as well as of , where represents a wavevector at the Brillouinzone boundary in the direction of q. When leaving a prominent wavevector ( or ), the transformation properties of lattice vibrations with respect to all those symmetry operations that are conserved do not change. Hence, the compatibility relations for onedimensional irreducible multiplier representations can be formulated asThe simple relation (2.1.3.67) does not hold, however, if higherdimensional representations have to be considered at prominent wavevectors. With respect only to the symmetry elements of the subgroup , those representations are not necessarily irreducible. Rather, they may be decomposed into several (up to three) irreducible components . The multiplicities are given by the characters using the following equation: Hence, phonons corresponding to irreducible representations with nonzero multiplicities () will mix at the Γ point (zone boundary) to yield degenerate modes corresponding to the higherdimensional representation.
In conclusion, group theory provides an important tool not only for the labelling of lattice vibrations according to irreducible multiplier representations but also for the assignment of branches at points of degeneracy within the reciprocal space.
To illustrate compatibility relations, let us once more consider the example of space group as introduced in Sections 2.1.3.1.1, 2.1.3.3.1 and 2.1.3.4.1. For wavevectors along we have and there are two irreducible representations, a symmetric one (with respect to ) with and an antisymmetric one with . Remember the representations for the Γ point, shown in Table 2.1.3.7. We immediately recognize that the Γpoint representations and are related to the symmetric representation for nonzero wavevectors along . and , on the other hand, are related to the antisymmetric representation .

The twodimensional representation exhibits the character . When leaving the Γ point along , it therefore splits into the symmetric representation with and the antisymmetric one with . Consequently, there are always pairs of a symmetric and an antisymmetric lattice vibration which degenerate at the Brillouinzone centre and the phonon dispersion along exhibits the principal behaviour as shown in Fig. 2.1.3.10. Here, six modes are displayed which illustrate the six possibilities for relating symmetric and antisymmetric vibrations to the Γpoint representations.
Inelastic neutron scattering is the unique experimental method for the determination of phonons at arbitrary wavevectors. Additional information can be obtained by optical methods, infrared absorption and Raman spectroscopy. For the detection of lattice vibrations, electromagnetic radiation of appropriate frequencies in the THz regime is needed. The corresponding wavelengths are of the order of cm and are therefore very large compared with typical lattice parameters. Consequently, optical spectroscopy is sensitive to longwavelength phonons only, i.e. to Γpoint modes. Moreover, the visibility of lattice vibrations in infrared or Raman experiments is governed by selection rules which, in turn, are determined by the symmetry of the corresponding eigenvectors. We may distinguish infraredactive modes, Ramanactive modes and `silent' modes that are neither infrared nor Ramanactive. Some simple grouptheoretical arguments lead to the criteria for infrared or Raman activity.
Infrared spectroscopy is based on the absorption of electromagnetic radiation by phonons, as shown in Fig. 2.1.3.11. Photons can only be absorbed by those lattice vibrations that are associated with a periodic variation of an electric dipole moment. Since the dipole moment is a vector, it transforms under the symmetry operations of the crystal according to the vector representation which is provided by the ordinary matrices describing the effect of any rotation, mirror plane etc. upon an arbitrary vector of our threedimensional space. It should be noted that the vector representation is in general reducible and can be regarded as the direct product of some irreducible representations. Lattice vibrations can carry an electric dipole moment only if their symmetry is compatible with the symmetry of a vector, i.e. if the corresponding irreducible representation is contained within the vector representation. The multiplicity of a particular irreducible Γpoint representation within the decomposition of the vector representation can be calculated from the respective characters and . Hence we may formulate the criterion for infrared activity as follows: Phonons corresponding to an irreducible representation are infrared active if
(First order) Raman spectroscopy, on the other hand, is based on the scattering of electromagnetic waves by phonons (see Fig. 2.1.3.12). Scattered intensity can only be obtained if the incident wave polarizes the crystal in such a way that it acts as a source for the outgoing wave. This is achieved if the tensor of the polarizability exhibits nonzero elements that relate electric field components in the directions of the incident and scattered waves. Hence, only those lattice vibrations that are associated with a periodic variation of the polarizability tensor can yield (firstorder) Raman intensity. Their symmetry has to be compatible with the symmetry of a tensor, i.e. the corresponding irreducible representation has to be contained within the (reducible) tensor representation . As for infrared activity, we may therefore formulate the criterion for Ramanactive phonons with the help of the characters and : Phonons corresponding to an irreducible representation are Raman active if Without going into details, we note that the tensor representation is the symmetric square of the vector representation and its character may be calculated from the character of , It should be noted that grouptheoretical considerations yield necessary conditions for the visibility of phonons. They cannot predict, however, intensities of active modes since these depend on crystalspecific properties like dipole moments or elements of the polarizability tensor.
As an example, let us once more consider the space group . For , the character table shown in Table 2.1.3.8 summarizes all essential information about irreducible, vector and tensor representations. Obviously, the vector representation consists of the irreducible representations and , the latter being twodimensional. Γpoint phonons corresponding to these two representations are infrared active. All other lattice vibrations cannot be detected by absorption experiments.

Using the multiplicities as calculated from (2.1.3.70), we obtain the decomposition of the tensor representation: Hence phonons corresponding to the representations , , and are Raman active.
All lattice vibrations that belong to the representation are neither infrared nor Raman active. They cannot be detected in (firstorder) optical experiments and are therefore called silent modes.
References
Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids. Oxford: Clarendon Press.Kovalev, O. V. (1965). Irreducible representations of the space groups. New York: Gordon & Breach.
Maradudin, A. A. & Vosko, S. H. (1968). Symmetry properties of the normal vibrations of a crystal. Rev. Mod. Phys. 40, 1–37.