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

International Tables for Crystallography (2013). Vol. D, ch. 1.11, pp. 279-280

Section 1.11.6.6. Tensor atomic factors (spherical tensor representation)

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:  dmitrien@crys.ras.ru

1.11.6.6. Tensor atomic factors (spherical tensor representation)

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Another representation of the scattering amplitude is widely used in the scientific literature (Hannon et al., 1988[link]; Luo et al., 1993[link]; Carra et al., 1993[link]; Lovesey & Collins, 1996[link]) for the description of resonant multipole transitions. In order to obtain the scattering amplitude and intensity for a resonant process described by some set of spherical tensor components, the tensor that describes the atomic scattering must be contracted by a tensor of the same rank and inversion/time-reversal symmetry which describes the X-ray probe, so that the result would be a scalar. There are well known relations between the components of the atomic factor tensor, both in Cartesian and spherical representations. For the dipole–dipole transition, the resonant scattering amplitude can be written as (Hannon et al., 1988[link]; Collins et al., 2007[link]; Paolasini, 2012[link]; Joly et al., 2012[link])[f^{dd}\sim \textstyle\sum\limits_{jm} e^{\prime *}_je_m D_{jm}=\textstyle\sum\limits_{p=0}^2\textstyle\sum\limits_{q=-p}^p (-1)^{p+q}X_{q}^{(p)}F_{-q}^{(p)},\eqno(1.11.6.50)]where [D_{jm}] are the Cartesian tensor components, [X_q^{(p)}] depends only on the incident and scattered radiation and the polarization vectors, and [F_{-q}^{(p)}] is associated with the tensor properties of the absorbing atom and can be represented in terms of a multipole expansion.

It is convenient to decompose each tensor into its irreducible parts. For example, an [E1E1] tensor containing nine Cartesian components can be represented as a sum of three irreducible tensors with ranks [p=0] (one component), [p=1] (three components) and [p=2] (five components). This decomposition is unique.

For [p=0]:[F_0^{(0)}={\textstyle{1\over 3}}(D_{xx}+D_{yy}+D_{zz}).\eqno(1.11.6.51)]

For [p=1]:[\eqalignno{F_{0}^{(1)}&={\textstyle{1\over 2}}(D_{xy}-D_{yx}),&\cr F_{\pm 1}^{(1)}&=\mp {\textstyle{{1}\over{2\sqrt 2}}}[(D_{yz}-D_{zy}\mp i(D_{xz}-D_{zx})].&(1.11.6.52)}]

For [p=2]:[\eqalignno{F_0^{(2)}&=D_{zz}-F_0^{(0)}, &\cr F_{\pm 1}^{(2)}&=\mp {\textstyle{1\over 2}}\sqrt{{\textstyle{2\over 3}}}[(D_{xz}+D_{zx}\mp i(D_{yz}+D_{zy})],&(1.11.6.53)}][F_{\pm 2}^{(2)}={\textstyle{1\over 6}}[2D_{xx}-2D_{yy}\pm i(D_{xy}+D_{yx})].\eqno(1.11.6.54)]

It follows from (1.11.6.14)[link] that the fourth-rank tensor describing the quadrupole–quadrupole X-ray scattering can also be divided into two parts: the time-reversal part, [Q_{jklm}^{+}], and the non-time-reversal part, [Q_{jklm}^{-}]. Both can be explicitly represented by (1.11.6.3)[link] and (1.11.6.2)[link], in which all these tensors are parity-even. The explicit form of the fourth-rank tensors is suitable for the analysis of possible effects in resonant X-ray absorption and scattering. Nevertheless, sometimes the following representation of the scattering amplitude as a product of spherical tensors is preferable:[f^{qq}={\textstyle{1\over 4}}\textstyle \sum \limits_{ijmn}e^{\prime *}_i e_m k_j^{\prime} k_n Q_{ijmn}= \textstyle \sum \limits_{p=0}^4\textstyle \sum \limits_{q=-p}^p (-1)^{p+q}X_q^{(p)}F_{-q}^{(p)}.\eqno(1.11.6.55)]

Here, the dipole–quadrupole tensor atomic factor given by (1.11.6.10)[link] is represented by a sum over several tensors with different symmetries. All tensors are parity-odd, but the tensors [I_{jml}^{--}] and [I_{jml}^{-+}] are also non-time-reversal. The scattering amplitude corresponding to the dipole–quadrupole resonant X-ray scattering can be represented as[\eqalignno{f^{dq}&={\textstyle{1\over 2}}i\textstyle \sum \limits_{ijm}e^{\prime *}_i e_j(k_mI_{ijm}-k_m^{\prime}I_{jim})&\cr&= \textstyle \sum \limits_{p=1}^3\textstyle \sum \limits_{q=-p}^p (-1)^{p+q}(X_q^{(p)}F_{-q}^{(p)}+\bar X_q^{(p)}\bar F_{-q}^{(p)}).&(1.11.6.56)}]The explicit form of [F_{-q}^{(p)}] can be found in Marri & Carra (2004[link]). Various parts of [F_{-q}^{(p)}] possess different symmetry with respect to the reversal of space [P] and time [T].

The spherical representation of the tensor atomic factor allows one to analyse its various components, as they possess different symmetries with respect to rotations or space and time inversion. For each [p], [F_{-q}^{(p)}] is related to a specific term of the multipole expansion of the system. Multipole expansions of electric and magnetic fields generated by charges and permanent currents are widely used in characterizing the electromagnetic state of a physical system (Berestetskii et al., 1982[link]). The transformation rules for electric and magnetic multipoles of both parities under space inversion and time reversal are of great importance for electromagnetic effects in crystals. The correspondence between the [F_{-q}^{(p)}] and electromagnetic multipoles is shown in Table 1.11.6.2[link]. In this table, the properties of the tensors [F_{-q}^{(p)}] under time reversal and space inversion on one side are identified with multipole terms describing the physical system on the other. In fact, for any given tensor of rank [p=1,2,3,4] there is one electromagnetic multipole of the same rank ([1\to] dipole, [2\to] quadrupole, [3\to] octupole, [4\to] hexadecapole) and with the same [T] and [P] properties. Note that [P]-odd [E1E2] tensors have both [T]-odd (−) and [T]-even (+) terms for any [p], whereas [P]-even tensors (both [E1E1] and [E2E2]) are [T]-odd for odd rank and [T]-even for even rank, respectively (Di Matteo et al., 2005[link]).

Table 1.11.6.2| top | pdf |
Identification of properties under time inversion [T] and space inversion [P] of tensors associated with multipole expansion

After Di Matteo et al. (2005[link]) and Paolasini (2012[link]).

Rank of tensorResonant processTPTypeMultipole
0 E1E1 + + charge monopole
0 E2E2 + + charge monopole
1 E1E1 + magnetic dipole
1 E2E2 + magnetic dipole
1 E1E2 + electric dipole
1 E1E2 polar toroidal dipole
2 E1E1 + + electric quadrupole
2 E2E2 + + electric quadrupole
2 E1E2 + axial toroidal quadrupole
2 E1E2 magnetic quadrupole
3 E2E2 + magnetic octupole
3 E1E2 + electric octupole
3 E1E2 polar toroidal octupole
4 E2E2 + + electric hexadecapole

An important contribution of Luo et al. (1993[link]) and Carra et al. (1993[link]) consisted of expressing the amplitude coefficients in terms of experimentally significant quantities, electron spin and orbital moments. This procedure is valid within the fast-collision approximation, when either the deviation from resonance, [\Delta E = E_c - E_a - \hbar\omega], or the width, [\Gamma], is large compared to the splitting of the excited-state configuration. The approximation is expected to hold for the [L_2] and [L_3] edges of the rare earths and actinides, as well as for the [M_4] and [M_5] edges of the actinides. In this energy regime, the resonant factors can be summed independently, leaving amplitude coefficients that may be written in terms of multipole moment operators, which are themselves single-particle operators summed over the valence electrons in the initial state.

Magnetic scattering has become a powerful method for understanding magnetic structures (Tonnere, 1996[link]; Paolasini, 2012[link]), particularly as it is suitable even for powder samples (Collins et al., 1995[link]). Since the first studies (Gibbs et al., 1988[link]), resonant magnetic X-ray scattering has been observed at various edges of transition metals and rare earths. The studies include magnetics and multiferroics with commensurate and incommensurate modulation (Walker et al., 2009[link]; Kim et al., 2011[link]; Ishii et al., 2006[link]; Partzsch et al., 2012[link]; Lander, 2012[link]; Beale et al., 2012[link]; Lovesey et al., 2012[link]; Mazzoli et al., 2007[link]) as well as multi-k magnetic structures (Bernhoeft et al., 2012[link]), and structures with orbital ordering (Murakami et al., 1998[link]) and higher-order multipoles (Princep et al., 2011[link]). It has also been shown that effects can be measured not only at the edges of magnetic atoms [K edges of transition metals, L edges of rare-earth elements and M edges of actinides (Vettier, 2001[link], 2012[link])], but also at the edges of non-magnetic atoms (Mannix et al., 2001[link]; van Veenendaal, 2003[link]).

Thus, magnetic and non-magnetic resonant X-ray diffraction clearly has the potential to be an important working tool in modern materials research. The advantage of polarized X-rays is their sensitivity to both the local atomic environments of resonant atoms and their partial structures. The knowledge of the local and global crystal symmetries and of the interplay of their effects is therefore of great value for a better understanding of structural, electronic and magnetic features of crystalline condensed matter.

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