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

International Tables for Crystallography (2013). Vol. D. ch. 1.11, pp. 275-276

Section Tensor atomic factors: internal symmetry

V. E. Dmitrienko,a* A. Kirfelb and E. N. Ovchinnikovac

aA. V. Shubnikov Institute of Crystallography, Leninsky pr. 59, Moscow 119333, Russia,bSteinmann Institut der Universität Bonn, Poppelsdorfer Schloss, Bonn, D-53115, Germany, and cFaculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
Correspondence e-mail: Tensor atomic factors: internal symmetry

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Different types of tensors transform under the action of the extended orthogonal group (Sirotin & Shaskolskaya, 1982[link]) as[A_{i_1^{\prime}\ldots i_n^{\prime}}=\gamma r_{i_1^{\prime}k_1}\ldots r_{i_n^{\prime}k_n} A_{k_1\ldots k_n} ,\eqno(]where the coefficients [\gamma=\pm 1] depend on the kind of tensor (see Table[link]) and [r_{{i_1^{\prime}k_1}}] are coefficients describing proper rotations.

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Coefficients [\gamma] corresponding to various kinds of tensor symmetry with respect to space inversion [{\bar 1}], rotations [R], and time reversal [1^{\prime}]

Tensor typeExampleTransformation type
[R][{\bar 1}R][1^{\prime}R][\bar 1^{\prime}R]
Even Strain 1 1 1 1
Electric Electric field 1 −1 1 −1
Magnetic Magnetic field 1 1 −1 −1
Magnetoelectric Toroidal moment 1 −1 −1 1

Various parts of the resonant scattering factor ([link] possess different kinds of symmetry with respect to: (1) space inversion [{\bar 1}] or parity, (2) rotations [R] and (3) time reversal [1^{\prime}]. Both dipole–dipole and quadrupole–quadrupole terms are parity-even, whereas the dipole–quadrupole term is parity-odd. Thus, dipole–quadrupole events can exist only for atoms at positions without inversion symmetry.

It is convenient to separate the time-reversible and time-non-reversible terms in the contributions to the atomic tensor factor ([link]. The dipole–dipole contribution to the resonant atomic factor can be represented as a sum of an isotropic, a symmetric and an antisymmetric part, written as (Blume, 1994[link])[D_{jk}=D_0^{\rm res}\delta_{jk}+D_{jk}^{+}+D_{jk}^{-},\eqno(]where [D_0^{\rm res}=(1/3)({\rm Tr} D)],[\eqalignno{D_{jk}^{+}&={{1\over 2}}(D_{jk}+D_{kj})-{{1\over 3}}({\rm Tr} D)\delta_{jk}&\cr&= {{1\over 4}}\sum\limits_{a,c}{{m\omega_{ca}^{3}}\over{\hbar\omega}}(p_a^{\prime}+p_{\bar a}^{\prime})(\langle a|R_{j}|c\rangle \langle c|R_{k}|a\rangle+\langle a|R_{k}|c\rangle \langle c|R_{j}|a\rangle&\cr&&(}]and[\eqalignno{D_{jk}^{-}&={{1\over 2}}(D_{jk}^{-}-D_{kj}^{-})&\cr&={{1\over 4}}\sum_{a,c}{{m\omega_{ca}^{3}}\over{\hbar\omega}}(p_a^{\prime}-p_{\bar a}^{\prime})(\langle a|R_{j}|c\rangle \langle c|R_{k}|a\rangle-\langle a|R_{k}|c\rangle \langle c|R_{j}|a\rangle, &\cr&&(}][p_a^{\prime}=p_a/[\omega-\omega_{ca}-i\Gamma/(2 \hbar)]] and [p_{\bar a}^{\prime}=p_{\bar a}/[\omega-\omega_{c\bar a}-i\Gamma/(2 \hbar)]]; [p_{\bar a}] means the probability of the time-reversed state [|\bar a\rangle]. If, for example, [|a\rangle] has a magnetic quantum number m, then [|\bar a\rangle] has a magnetic quantum number [-m].

In non-magnetic crystals, the probability of states with [\pm m] is the same, so that [p_{\bar a}=p_a] and [\langle \bar a|R_{j}^{s}|\bar c\rangle=\langle c|R_{k}^{s}|a\rangle ]; in this case [D_{jk}] is symmetric under permutation of the the indices.

Similarly, the dipole–quadrupole atomic factor can be represented as (Blume, 1994[link])[\eqalignno{f^{dq}_{jk}&= {i\over 2}\sum_{ac}p_{a}{{m\omega_{ca}^{3}}\over{\hbar\omega}} \times \{\langle a|R_{j}|c\rangle \langle c|R_{k}R_{l}|a\rangle k_{l}&\cr&\quad-\langle a|R_{j}R_{l}|c\rangle \langle c|R_{k}|a\rangle k'_{l}\}&(\cr&={i\over 8}\sum_{ac} {{m\omega_{ca}^{3}}\over{\hbar\omega}} \{I^{++}_{jkl}(k_{l}-k_{l}^{\prime}) +I^{--}_{jkl}(k_{l}-k_{l}^{\prime})&\cr&\quad+I^{-+}_{jkl}(k_{l}+k_{l}')+I^{+-}_{jkl}(k_{l}+k_{l}^{\prime})\},&(}]where[\eqalignno{I_{jkl}^{\mu\nu}&=\textstyle\sum\limits_{ac}(p^{\prime}_a+\mu p^{\prime}_{\bar a}) \{\langle a|R_{j}|c\rangle \langle c|R_{k}R_{l}|a\rangle k_{l}&\cr&\quad+\nu\langle a|R_{j}R_{l}|c\rangle \langle c|R_{k}|a\rangle k'_{l}\} &(}]with [\mu,\nu=\pm 1]. In ([link] the first plus ([\mu=1]) corresponds to the non-magnetic case (time reversal) and the minus ([\mu=-1]) corresponds to the time-non-reversal magnetic term, while the second [\pm] corresponds to the symmetric and antisymmetric parts of the atomic factor. We see that [I^{--}_{jkl}(k_{l}-k_{l}')] can contribute only to scattering, while [I^{-+}_{jkl}(k_{l}+k_{l}^{\prime})] can contribute to both resonant scattering and resonant X-ray propagation. The latter term is a source of the so-called magnetochiral dichroism, first observed in Cr2O3 (Goulon et al., 2002[link], 2003[link]), and it can be associated with a toroidal moment in a medium possessing magnetoelectric properties. The symmetry properties of magnetoelectic tensors are described well by Sirotin & Shaskolskaya (1982[link]), Nye (1985[link]) and Cracknell (1975[link]). Which magnetoelectric properties can be studied using X-ray scattering are widely discussed by Marri & Carra (2004[link]), Matsubara et al. (2005[link]), Arima et al. (2005[link]) and Lovesey et al. (2007[link]).

It follows from ([link] and ([link] that [I_{jkl}=I_{jlk}] and the dipole–quadrupole term can be represented as a sum of the symmetric [I_{jkl}^{+}=I_{kjl}^{+}] and antisymmetric [I_{jkl}^{-}=-I_{kjl}^{-}] parts. From the physical point of view, it is useful to separate the dipole–quadrupole term into [I_{jkl}^{+}] and [I_{jkl}^{-}], because only [I_{jkl}^{-}] works in conventional optics where [{\bf k}'={\bf k}]. The dipole–quadrupole terms are due to the hybridization of excited electronic states with different spacial parities, i.e. only for atomic sites without an inversion centre.

The pure quadrupole–quadrupole term in the tensor atomic factor is equal to[f_{jk}^{qq}={\textstyle{1\over 4}}Q_{jlkm}k_l^{\prime}k_m\eqno(]with the fourth-rank tensor [Q_{jklm}] given by[Q_{jlkm}=\sum_{ac} p_a {{m\omega_{ca}^{3}}\over{\hbar\omega}} {{\langle a| R_{j}R_{l}| c\rangle\langle c| R_{k}R_{m}| a\rangle}\over {\omega-\omega_{ca}-i({\Gamma}/{2\hbar})}}.\eqno(]

This fourth-rank tensor [Q_{ijkm}] has the following symmetries:[Q_{jlkm}=Q_{ljkm}=Q_{jlmk}.\eqno(]

We can define[Q_{jlkm}=Q_{jlkm}^{+}+Q^{-}_{jlkm} \eqno(]with [Q^{\pm}_{jlkm}=\pm Q_{kmjl}], where[\eqalignno{Q^{\pm}_{jlkm}&={\textstyle{1\over 4}}\textstyle\sum\limits_a(p_a^{\prime}\pm p_{\bar a}^{\prime}) (\langle a| R_{j}R_{l}| c\rangle\langle c| R_{k}R_{m}| a\rangle&\cr&\quad \pm\langle a| R_{k}R_{m}| c\rangle\langle c| R_{j}R_{l}| a\rangle). &(}]We see that [Q_{jlkm}^-] vanishes in time-reversal invariant systems, which is true for non-magnetic structures.


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