Symmetry properties of eigenvectors

Eckold, G.

doi:10.1107/97809553602060000638

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 281-283

Section 2.1.3.3. Symmetry properties of eigenvectors

G. Eckold^a ^*

^a Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany
Correspondence e-mail: geckold@gwdg.de

2.1.3.3. Symmetry properties of eigenvectors

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In the previous section we used the symmetry properties of the dynamical matrix to derive equation (2.1.3.36). Since the phonon dispersion $[\omega({\bf q},j)]$ is invariant with respect to all symmetry operations $[\{{\bf S}|{\bf v}({\bf S})+{\bf x}(m)\} ]$ of the space group [equation (2.1.3.37)], we conclude that not only is $[{\bf e}({\bf q},j)]$ an eigenvector of the dynamical matrix D(q) but so is the vector $[\boldGamma ^ + ({\bf q},\{{{\bf S}| {{\bf v}({\bf S}) + {\bf x}(m )}}\}\, {\bf e}({{\bf S}\, {\bf q}, j }) ]$ . If the corresponding eigenvalue $[\omega _{{\bf q},j}^2]$ is not degenerate, the (normalized) eigenvectors are uniquely determined except for a phase factor of unit modulus. Hence, the following relation holds: $[\eqalignno{ {\bf e}({{\bf S}\, {\bf q},j}) &= \exp(i\varphi ) \, \boldGamma ({{\bf q},\{{{\bf S}| {{\bf v}({\bf S}) + {\bf x}(m )}}\}}) \, {\bf e}({{\bf q},j}) &\cr &= \exp(i\varphi ) \, \boldGamma ({{\bf S}\, {\bf q},\{{{\bf E}| {{\bf x}(m )}}\}}) \, \boldGamma ({{\bf q},\{{{\bf S}| {{\bf v}({\bf S})}}\}}) \, {\bf e}({{\bf q},j}) &\cr &= \exp(i\varphi ) \, \exp[{- i({\bf S}\, {\bf q}) \, {\bf x}(m )}]\, \boldGamma ({{\bf q},\{{{\bf S}| {{\bf v}({\bf S})}}\}})\,{\bf e}({{\bf q},j}). &\cr&&(2.1.3.40)} ]$ 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 right-hand side of equation (2.1.3.40) independent of $[{\bf x}(m)]$ . Setting the phase angle $[\varphi]$ equal to $[({\bf S}\,{\bf q})\, {\bf x}(m)]$ , we obtain the simple form of the transformation law $[{\bf e}({{\bf S}\, {\bf q},j}) = \boldGamma ({{\bf q},\{{{\bf S}| {{\bf v}({\bf S})}}\}}) \, {\bf e}({{\bf q},j}). \eqno (2.1.3.41) ]$ This choice is, however, not always possible. If there is a symmetry operation $[{\bf S}_{-}]$ that inverts the wavevector, then in addition to equation (2.1.3.40) there is another relation between $[{\bf e}(-{\bf q},j)]$ and $[{\bf e}({\bf q},j) ]$ 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 $[\{{{\bf S}| {{\bf v}({\bf S})+{\bf x}(m)}}\}]$ sends an eigenvector $[{\bf e}({\bf q},j)]$ 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 $[{\bf e}({\bf q},j)]$ at q then among the variety of possible and equivalent orthonormal sets of eigenvectors at $[{\bf S}\,{\bf q} ]$ 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 Brillouin-zone boundary and symmetry operations with $[{\bf S}\,{\bf q}={\bf q}+{\bf g}]$ (where g is a reciprocal-lattice vector): owing to the periodicity $[{\bf e}({\bf q}+{\bf g},j)=]$ $[{\bf e}({\bf q},j)]$ , equation (2.1.3.3), the eigenvectors $[{\bf e}({\bf S}\,{\bf q},j)]$ and $[{\bf e}({\bf q},j) ]$ have to be identical in this case.

If we consider those symmetry operations that leave the wavevector invariant (except for an additional reciprocal-lattice 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 $[G_{o}({\bf q}) ]$ , the vector $[{\bf T}({\bf q},{\bf R}){\bf e}({\bf q},j) ]$ is an eigenvector with respect to the eigenvalue $[\omega _{{\bf q},j}^2 ]$ as well as $[{\bf e}({\bf q},j)]$ : $[{\bf D}({\bf q}) \, \{{{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},j})}\}= \omega _{{\bf q},j}^2 \, \{{{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},j})}\}. \eqno (2.1.3.42) ]$ 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, $[j \to \sigma, \lambda.]$ λ runs from 1 to $[f_{\sigma}]$ if $[f_{\sigma}]$ is the degeneracy of the eigenfrequency $[\omega_{{\bf q},\sigma}]$ . With this notation, equation (2.1.3.42) can be rewritten as $[{\bf D}({\bf q}) \, \{{{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},\sigma \lambda })}\}= \omega _{{\bf q},\sigma }^2 \, \{{{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},\sigma \lambda })}\}. \eqno (2.1.3.42a) ]$ Consequently, the vector $[{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},\sigma \lambda }) ]$ has to be some linear combination of all eigenvectors $[{\bf e}({\bf q},\sigma\lambda')]$ , $[\lambda'=1,\ldots,f_\sigma ]$ , corresponding to the same eigenvalue $[\omega _{{\bf q},\sigma }^2 ]$ , $[{\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},\sigma \lambda }) = \textstyle\sum\limits_{\lambda' = 1}^{f_\sigma }{\tau _{\lambda \lambda ' }^{(\sigma )}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},\sigma \lambda '})}. \eqno (2.1.3.43) ]$ Obviously, the eigenvectors $[{\bf e}({{\bf q},\sigma \lambda }) ]$ $[(\lambda=1,\ldots,f_\sigma)]$ 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 $[G_{o}({\bf q})]$ , the $[f_{\sigma}]$ eigenvectors transform into each other. The corresponding coefficients $[\tau _{\lambda \lambda '}^{(\sigma )}({{\bf q},{\bf R}}) ]$ can be regarded as the elements of a complex ( $[f_{\sigma}\times f_\sigma ]$ ) matrix $[\boldtau ^{(\sigma)}({\bf q},{\bf R})]$ 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 3N-dimensional reducible representation provided by the T matrix operators [cf. equation (2.1.3.21)].

For a given point group $[G_{o}({\bf q})]$ there is only a limited number of irreducible representations. These can be calculated by group-theoretical 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 group-theoretical 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 $[{\bf e}({\bf q},\sigma\lambda) ]$ of the dynamical matrix as columns of a unitary matrix $[{\bf e}({\bf q})]$ in such a way that eigenvectors of the same irreducible representation occupy neighbouring columns: $[{\bf e}({\bf q}) = ({{\bf e}({{\bf q}, {11}})\ldots{\bf e}({{\bf q}, {1f_1 }}){\bf e}({{\bf q}, {21}})\ldots{\bf e}({{\bf q}, {2f_2 }})\ldots}). \eqno (2.1.3.44) ]$ This matrix can now be used for a similarity transformation of the T matrix operators: $[{\bf e}({\bf q})^{- 1} \, {\bf T}({{\bf q},{\bf R}}) \, {\bf e}({\bf q}) = \boldDelta ({{\bf q},{\bf R}}). \eqno (2.1.3.45) ]$ 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 $[\boldDelta ({{\bf q},{\bf R}}) ]$ has block-diagonal form. Moreover, each block on the diagonal consists of the matrix of a particular irreducible multiplier representation: $[\boldDelta ({{\bf q},{\bf R}}) = {\pmatrix{ {\boldtau ^{(1)}({{\bf q},{\bf R}})}& 0 & \,s & 0 & 0 \cr 0 & {\boldtau ^{(2)}({{\bf q},{\bf R}})}& \,s & 0 & 0 \cr \vdots & \vdots & \ddots & \vdots & \vdots \cr 0 & 0 & 0 & \ddots & 0 \cr 0 & 0 & 0 & 0 & \ddots \cr}}. \eqno (2.1.3.46) ]$ The matrix of eigenvectors thus reduces the operators T to block-diagonal 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 $[G_{o}({\bf q}) ]$ . 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 $[\boldDelta ({{\bf q},{\bf R}}) ]$ 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, $[\sigma \to s,a.]$ 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 $[c_{s}]$ denotes the multiplicity of the representation s, then a takes the values $[1,\ldots,c_{s} ]$ . Using this notation, the transformation law for the eigenvectors can be rewritten as $[\displaylines{ {\bf T}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},sa\lambda }) = \textstyle\sum\limits_{\lambda' = 1}^{f_s }{\tau _{\lambda \lambda' }^{(s)}({{\bf q},{\bf R}}) \, {\bf e}({{\bf q},sa\lambda '})}\cr\hbox{for }\lambda=1,\ldots,f_{\rm s}\hbox{ and }a = 1\ldots,c_s. \cr\hfill(2.1.3.47)} ]$ As a well known result from group theory, the multiplicity $[c_{s} ]$ of a particular irreducible multiplier representation s in the decomposition of the reducible 3N-dimensional T-matrix representation can be calculated from the respective characters $[\eqalignno{\chi ({\bf q},{\bf R}) &= \textstyle\sum\limits_{\kappa \alpha }{T_{\kappa \kappa }^{\alpha \alpha }({\bf q},{\bf R})} &\cr &= \textstyle\sum\limits_{\kappa \alpha }{R_{\alpha \alpha }\, \delta (\kappa, F_o (\kappa, {\bf R}))\,\exp[{i{\bf q}({\bf r}_\kappa - {\bf Rr}_\kappa)}}]&\cr &&(2.1.3.48)} ]$ and $[\chi ^{(s)}({\bf q},{\bf R}) = \textstyle\sum\limits_{\lambda = 1}^{f_s }{\tau _{\lambda \lambda }^{(s)}({\bf q},{\bf R})} \eqno (2.1.3.49) ]$ according to $[c_s = (1/{| G |})\textstyle\sum\limits_{\bf R}{\chi ({\bf q},{\bf R})\,\chi ^{(s)*}({\bf q},{\bf R})}. \eqno (2.1.3.50) ]$ The summation index runs over all symmetry elements of the point group $[G_{o}({\bf q})]$ , the order of which is denoted by [|G|] . Hence we are able to predict the number of non-degenerate phonon modes for any of the different irreducible multiplier representations on the basis of group-theoretical considerations. Obviously, there are exactly $[c_{s}\times f_{s}]$ modes with eigenvectors that transform according to the irreducible multiplier representation s. Among these, groups of always $[f_{s}]$ 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 three-dimensional 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.

2.1.3.3.1. Example

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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 [4mm] . It contains all eight symmetry operations and there are five different irreducible representations, denoted $[\tau ^{(1^ +)}]$ , $[\tau^{(1^ -)}]$ , $[\tau ^{(3^ +)}]$ , $[\tau^{(3^ -)}]$ and $[\boldtau ^{(2)} ]$ . The corresponding character table including the reducible representation provided by the T-matrix operators (cf. Section 2.1.3.1.1) has the form shown in Table 2.1.3.3 . The representations $[\tau _{}^{(1^ +)}]$ , $[\tau _{}^{(1^ -)} ]$ , $[\tau _{}^{(3^ +)}]$ and $[\tau _{}^{(3^ -)} ]$ are one-dimensional, and $[\boldtau ^{(2)}]$ is two-dimensional. The upper index, [+] or −, refers to the symmetry with respect to the mirror plane $[m_{x}]$ . According to (2.1.3.50), we may calculate the multiplicities of these irreducible representations in the decomposition of the 30-dimensional T representation. As the result we obtain $[c_{\tau _{}^{(1^ +)}} = 5,\quad c_{\tau _{}^{(1^ -)}} = 3,\quad c_{\tau _{}^{(3^ +)}} = 3,\quad c_{\tau _{}^{(3^ -)}} = 3,\quad c_{\boldtau _{}^{(2)}} = 8. ]$ Hence for the sample structure presented in Section 2.1.3.1.1 we expect to have five phonon modes of symmetry $[\tau _{}^{(1^ +)}]$ , three modes for each of the symmetries $[\tau _{}^{(1^ -)}]$ , $[\tau _{}^{(3^ +)}]$ and $[\tau _{}^{(3^ -)}]$ , and 16 modes of symmetry $[\boldtau ^{(2)}]$ , the latter being divided into pairs of doubly degenerate phonons.

Table 2.1.3.3 | top | pdf |
Character table of the point group [4mm]

	Symmetry operation
	E	$[D_{90}^z ]$	$[D_{180}^z]$	$[D_{270}^z]$	$[m_{x}]$	$[m_{y}]$	$[m_{ [{\bar 110}]}]$	$[m_{ [{110}]}]$
$[\chi _T ]$	30	2	−2	2	2	2	2	2
$[\chi _{\tau _{}^{(1^ +)}} ]$	1	1	1	1	1	1	1	1
$[\chi _{\tau _{}^{(1^ -)}} ]$	1	1	1	1	−1	−1	−1	−1
$[\chi _{\tau _{}^{(3^ +)}}]$	1	−1	1	−1	1	1	−1	−1
$[\chi _{\tau _{}^{(3^ -)}}]$	1	−1	1	−1	−1	−1	1	1
$[\boldtau ^{(2)}]$	$[{\pmatrix{ 1 & 0 \cr 0 & 1 \cr }} ]$	$[{\pmatrix{ i & 0 \cr 0 & {- i} \cr }} ]$	$[{\pmatrix{ {- 1}& 0 \cr 0 & {- 1} \cr }} ]$	$[{\pmatrix{ {- i}& 0 \cr 0 & i \cr }}]$	$[{\pmatrix{ 0 & 1 \cr 1 & 0 \cr }}]$	$[{\pmatrix{ 0 & -1 \cr {-1}& 0 \cr }} ]$	$[{\pmatrix{ 0 & -i \cr i & 0 \cr }} ]$	$[{\pmatrix{ 1 & i \cr -i & 0 \cr }} ]$
$[\chi _{\boldtau _{}^{(2)}} ]$	2	0	−2	0	0	0	0	0