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

International Tables for Crystallography (2006). Vol. D, ch. 2.2, pp. 307-310

Section 2.2.15. Electric field gradient tensor

K. Schwarza*

aInstitut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.15. Electric field gradient tensor

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2.2.15.1. Introduction

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The study of hyperfine interactions is a powerful way to characterize different atomic sites in a given sample. There are many experimental techniques, such as Mössbauer spectroscopy, nuclear magnetic and nuclear quadrupole resonance (NMR and NQR), perturbed angular correlations (PAC) measurements etc., which access hyperfine parameters in fundamentally different ways. Hyperfine parameters describe the interaction of a nucleus with the electric and magnetic fields created by the chemical environment of the corresponding atom. Hence the resulting level splitting of the nucleus is determined by the product of a nuclear and an extra-nuclear quantity. In the case of quadrupole interactions, the nuclear quantity is the nuclear quadrupole moment (Q) that interacts with the electric field gradient (EFG) produced by the charges outside the nucleus. For a review see, for example, Kaufmann & Vianden (1979[link]).

The EFG tensor is defined by the second derivative of the electrostatic potential V with respect to the Cartesian coordinates [x_{i}], i = 1, 2, 3, taken at the nuclear site n, [\Phi_{ij}={{\partial^{2}V}\over{\partial x_{i}\,\,\partial x_{j}}}\bigg|_{n}-{{1}\over{3}}\delta_{ij}\nabla^{2}\bigg|_{n},\eqno(2.2.15.1)]where the second term is included to make it a traceless tensor. This is more appropriate, since there is no interaction of a nuclear quadrupole and a potential caused by s electrons. From a theoretical point of view it is more convenient to use the spherical tensor notation because electrostatic potentials (the negative of the potential energy of the electron) and the charge densities are usually given as expansions in terms of spherical harmonics. In this way one automatically deals with traceless tensors (for further details see Herzig, 1985[link]).

The analysis of experimental results faces two obstacles: (i) The nuclear quadrupole moments (Pyykkö, 1992[link]) are often known only with a large uncertainty, as this is still an active research field of nuclear physics. (ii) EFGs depend very sensitively on the anisotropy of the charge density close to the nucleus, and thus pose a severe challenge to electronic structure methods, since an accuracy of the density in the per cent range is required.

In the absence of a better tool, a simple point-charge model was used in combination with so-called Sternheimer (anti-) shielding factors in order to interpret the experimental results. However, these early model calculations depended on empirical parameters, were not very reliable and often showed large deviations from experimental values.

In their pioneering work, Blaha et al. (1985[link]) showed that the LAPW method was able to calculate EFGs in solids accurately without empirical parameters. Since then, this method has been applied to a large variety of systems (Schwarz & Blaha, 1992[link]) from insulators (Blaha et al., 1985[link]), metals (Blaha et al., 1988[link]) and superconductors (Schwarz et al., 1990[link]) to minerals (Winkler et al., 1996[link]).

Several other electronic structure methods have been applied to the calculation of EFGs in solids, for example the LMTO method for periodic (Methfessl & Frota-Pessoa, 1990[link]) or non-periodic (Petrilli & Frota-Pessoa, 1990[link]) systems, the KKR method (Akai et al., 1990[link]), the DVM (discrete variational method; Ellis et al., 1983[link]), the PAW method (Petrilli et al., 1998[link]) and others (Meyer et al., 1995[link]). These methods achieve different degrees of accuracy and are more or less suitable for different classes of systems.

As pointed out above, measured EFGs have an intrinsic uncertainty related to the accuracy with which the nuclear quadrupole moment is known. On the other hand, the quadrupole moment can be obtained by comparing experimental hyperfine splittings with very accurate electronic structure calculations. This has recently been done by Dufek et al. (1995a[link]) to determine the quadrupole moment of 57Fe. Hence the calculation of accurate EFGs is to date an active and challenging research field.

2.2.15.2. EFG conversion formulas

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The nuclear quadrupole interaction (NQI) represents the interaction of Q (the nuclear quadrupole moment) with the electric field gradient (EFG) created by the charges surrounding the nucleus, as described above. Here we briefly summarize the main ideas (following Petrilli et al., 1998[link]) and provide conversions between experimental NQI splittings and electric field gradients.

Let us consider a nucleus in a state with nuclear spin quantum number [I>1/2] with the corresponding nuclear quadrupole moment [Q_{i,j}=({1}/{{e}})\textstyle\int {\rm d}^{3}r\rho_{n}(r)r_{i}r_{j}], where [\rho_{n}(r)] is the nuclear charge density around point [{\bf r}] and e is the proton's charge. The interaction of this Q with an electric field gradient tensor [V_{i,j}], [H=e\textstyle\sum\limits_{i,j}Q_{i,j}V_{i,j},\eqno(2.2.15.2)]splits the energy levels [E_{Q}] for different magnetic spin quantum numbers [m_{I}=I,I-1,\ldots,-I] of the nucleus according to [E_{Q}={{eQV_{zz}[3m_{I}^{2}-I(I+1)](1+\eta^{2}/3)^{1/2}}\over{4I(2I-1)}} \eqno(2.2.15.3)]in first order of [V_{i,j}], where Q represents the largest component of the nuclear quadrupole moment tensor in the state characterized by [m_{I}=I]. (Note that the quantum-mechanical expectation value of the charge distribution in an angular momentum eigenstate is cylindrical, which renders the expectation value of the remaining two components with half the value and opposite sign.) The conventional choice is [|V_{zz}|>|V_{yy}|\geq|V_{xx}|]. Hence, [V_{zz}] is the principal component (largest eigenvalue) of the electric field gradient tensor and the asymmetry parameter [\eta] is defined by the remaining two eigenvalues [V_{xx},V_{yy}] through [\eta={{|(V_{xx}-V_{yy})|}\over{|V_{zz}|}}.\eqno(2.2.15.4)](2.2.15.3)[link] shows that the electric quadrupole interaction splits the ([2I+1])-fold degenerate energy levels of a nuclear state with spin quantum number I ([I>1/2]) into I doubly degenerate substates (and one singly degenerate state for integer I). Experiments determine the energy difference [\Delta] between the levels, which is called the quadrupole splitting. The remaining degeneracy can be lifted further using magnetic fields.

Next we illustrate these definitions for 57Fe, which is the most common probe nucleus in Mössbauer spectroscopy measurements and thus deserves special attention. For this probe, the nuclear transition occurs between the [I=3/2] excited state and [I=1/2] ground state, with a 14.4 KeV [\gamma] radiation emission. The quadrupole splitting between the [m_{I} =\pm(1/2)] and the [m_{I}=\pm(3/2)] state can be obtained by exploiting the Doppler shift of the [\gamma] radiation of the vibrating sample. [\Delta={{V_{zz}eQ(1+\eta^{2}/3)^{1/2}}\over{2}}.\eqno(2.2.15.5)]For systems in which the 57Fe nucleus has a crystalline environment with axial symmetry (a threefold or fourfold rotation axis), the asymmetry parameter [\eta] is zero and [\Delta] is given directly by [\Delta={{V_{zz}eQ}\over{2}}.\eqno(2.2.15.6)]As [\eta] can never be greater than unity, the difference between the values of [\Delta] given by equation (2.2.15.5)[link] and equation (2.2.15.6)[link] cannot be more than about 15%. In the remainder of this section we simplify the expressions, as is often done, by assuming that [\eta=0]. As Mössbauer experiments exploit the Doppler shift of the [\gamma] radiation, the splitting is expressed in terms of the velocity between sample and detector. The quadrupole splitting can be obtained from the velocity, which we denote here by [\Delta_{v}], by [\Delta={{{{E_{\gamma}}}\over{{c}}}}\Delta_{v},\eqno(2.2.15.7)]where c = 2.9979245580 × 108 m s−1 is the speed of light and Eγ = 14.41 × 103 eV is the energy of the emitted [\gamma] radiation of the 57Fe nucleus.

Finally, we still need to know the nuclear quadrupole moment Q of the Fe nucleus itself. Despite its utmost importance, its value has been heavily debated. Recently, however, Dufek et al. (1995b[link]) have determined the value Q = 0.16 b for 57Fe (1 b = 10−28 m2) by comparing for fifteen different compounds theoretical [V_{zz}] values, which were obtained using the linearized augmented plane wave (LAPW) method, with the measured quadrupole splitting at the Fe site.

Now we relate the electric field gradient [V_{zz}] to the Doppler velocity via [\Delta_{v}={{{{eQc}}\over{{2E_{\gamma}}}}}V_{zz}.\eqno(2.2.15.8)]In the special case of the 57Fe nucleus, we obtain [\eqalignno{V_{zz}\,\,[10^{21}\,\,{\rm V}\,\,{\rm m}^{-2}]&=10^{4}{{2E_{\gamma}\,\,[{\rm eV}]}\over{c\,\, [{\rm m}\,\,{\rm s}^{-1}]Q\,\, [{\rm b}]}}\Delta _{v}\,\,[{\rm mm}\,\,{\rm s}^{-1}]&\cr&\approx 6\Delta_{v}\,\, [{\rm mm}\,\,{\rm s}^{-1}].&(2.2.15.9)}]EFGs can also be obtained by techniques like NMR or NQR, where a convenient measure of the strength of the quadrupole interaction is expressed as a frequency [\nu_{q}], related to [V_{zz}] by [\nu_{q}={{3eQV_{zz}}\over{2hI(2I-1)}}.\eqno(2.2.15.10)]The value [V_{zz}] can then be calculated from the frequency in MHz by [V_{zz}\,\,[10^{21}\,\,{\rm V}\,\,{\rm m}^{-2}]=0.02771{{I(2I-1)}\over{Q\,\,[{\rm b}]}}\nu_{q}\,\,[{\rm MHz}],\eqno(2.2.15.11)]where (h/e) = 4.1356692 × 10−15 V Hz−1. The principal component [V_{zz}] is also often denoted as [eq=V_{zz}].

In the literature, two conflicting definitions of [\nu_{q}] are in use. One is given by (2.2.15.10)[link], and the other, defined as [\nu_{q}\,\,[{\rm Hz}]={{e^{2}qQ}\over{2h}},\eqno(2.2.15.12)]differs from the first by a factor of 2 and assumes the value [I=3/2]. Finally, the definition of [q=V_{zz}/e] has been introduced here. In order to avoid confusion, we will refer here only to the definition given by (2.2.15.10)[link]. Furthermore, we also adopt the same sign convention for [V_{zz}] as Schwarz et al. (1990[link]) because it has been found to be consistent with the majority of experimental results.

2.2.15.3. Theoretical approach

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Since the EFG is a ground-state property that is uniquely determined by the charge density distribution (of electrons and nuclei), it can be calculated within DFT without further approximations. Here we describe the basic formalism to calculate EFGs with the LAPW method (see Section 2.2.12[link]). In the LAPW method, the unit cell is divided into non-overlapping atomic spheres and an interstitial region. Inside each sphere the charge density (and analogously the potential) is written as radial functions [\rho_{LM}(r)] times crystal harmonics (2.2.13.4)[link] and in the interstitial region as Fourier series: [\rho(r)=\left\{ \matrix{\textstyle \sum\limits_{LM}\rho_{LM}(r)K_{LM}(\hat{r})\hfill & \hbox{inside sphere}\hfill\cr \textstyle\sum\limits_{K}\rho_{K}\exp({iKr})\hfill & \hbox{outside sphere}\hfill}\right. \eqno(2.2.15.13)]The charge density coefficients [\rho_{LM}(r)] can be obtained from the wavefunctions (KS orbitals) by (in shorthand notation) [\rho_{LM}(r)=\textstyle\sum\limits_{E_{k}^{j} \,\lt\, E_{F}}\textstyle\sum\limits_{\ell m}\textstyle\sum\limits_{\ell^{\prime}m^{\prime} }R_{\ell m}(r)R_{\ell^{\prime}m^{\prime}}(r)G_{L\ell\ell^{\prime}}^{Mmm^{\prime}},\eqno(2.2.15.14)]where [G_{L\ell\ell^{\prime}}^{Mmm^{\prime}}]are Gaunt numbers (integrals over three spherical harmonics) and [R_{\ell m}(r)] denote the LAPW radial functions [see (2.2.12.1)[link]] of the occupied states [E_{k}^{j}] below the Fermi energy [E_{F}]. The dependence on the energy bands in [R_{\ell m}(r)] has been omitted in order to simplify the notation.

For a given charge density, the Coulomb potential is obtained numerically by solving Poisson's equation in form of a boundary-value problem using a method proposed by Weinert (1981[link]). This yields the Coulomb potential coefficients [v_{LM}(r)] in analogy to (2.2.15.13)[link] [see also (2.2.12.5)[link]]. The most important contribution to the EFG comes from a region close to the nucleus of interest, where only the [L=2] terms are needed (Herzig, 1985[link]). In the limit [r\rightarrow0] (the position of the nucleus), the asymptotic form of the potential [r^{L}v_{LM}K_{LM}] can be used and this procedure yields (Schwarz et al., 1990[link]) for [L=2]: [\eqalignno{V_{2M} & =-C_{2M}\int_{0}^{R}{{\rho_{2M}(r)}\over{r^{3}}}r^{2}\,\,{\rm d}r+C_{2M}\int _{0}^{R}{{\rho_{2M}(r)}\over{r}}\left({{r}\over{R}}\right)^{5}\,\,{\rm d}r&\cr&\quad +5{{C_{2M}}\over{R^{2}}}\sum_{K}V(K)j_{2}(KR)K_{2M}(K),&(2.2.15.15)}]with [C_{2M}=2\sqrt{{4\pi}/{5}}], [C_{22}=\sqrt{{3}/{4}}C_{20}] and the spherical Bessel function [j_{2}]. The first term in (2.2.15.15)[link] (called the valence EFG) corresponds to the integral over the respective atomic sphere (with radius R). The second and third terms in (2.2.15.15)[link] (called the lattice EFG) arise from the boundary-value problem and from the charge distribution outside the sphere considered. Note that our definition of the lattice EFG differs from that based on the point-charge model (Kaufmann & Vianden, 1979[link]). With these definitions the tensor components are given as [\eqalignno{V_{xx} & =C\left[V_{22+}-\left({1}/{\sqrt{3}}\right)V_{20}\right]&\cr V_{yy} & =C \left[-V_{22+}-\left({1}/{\sqrt{3}}\right)V_{20}\right]&\cr V_{zz} & =C \left({2}/{\sqrt{3}}\right)V_{20}&\cr V_{xy} & =C V_{22-}&\cr V_{xz} & =C V_{21+}&\cr V_{yz} & =C V_{21-}&(2.2.15.16)}]where [C=\sqrt{{{15}/{4\pi}}}] and the index M combines m and the partity p (e.g. [2+]). Note that the prefactors depend on the normalization used for the spherical harmonics.

The non-spherical components of the potential [v_{LM}] come from the non-spherical charge density [\rho_{LM}]. For the EFG only the [L=2] terms (in the potential) are needed. If the site symmetry does not contain such a non-vanishing term (as for example in a cubic system with [L=4] in the lowest [LM] combination), the corresponding EFG vanishes by definition. According to the Gaunt numbers in (2.2.15.14)[link] only a few non-vanishing terms remain (ignoring f orbitals), such as the pp, dd or sd combinations (for f orbitals, pf and ff would appear), where this shorthand notation denotes the products of the two radial functions [R_{\ell m}(r)R_{\ell^{\prime }m^{\prime}}(r)]. The sd term is often small and thus is not relevant to the interpretation. This decomposition of the density can be used to partition the EFG (illustrated for the [V_{zz}] component), [V_{zz}\approx V_{zz}^{p}+V_{zz}^{d}+\hbox{small contributions},\eqno(2.2.15.17)]where the superscripts p and d are a shorthand notation for the product of two p- or d-like functions.

From our experience we find that the first term in (2.2.15.15)[link] is usually by far the most important and often a radial range up to the first node in the corresponding radial function is all that contributes. In this case the contribution from the other two terms is rather small (a few per cent). For first-row elements, however, which have no node in their 2p functions, this is no longer true and thus the first term amounts only to about 50–70%.

In some cases interpretation is simplified by defining a so-called asymmetry count, illustrated below for the oxygen sites in YBa2Cu3O7 (Schwarz et al., 1990[link]), the unit cell of which is shown in Fig. 2.2.15.1[link].

[Figure 2.2.15.1]

Figure 2.2.15.1 | top | pdf |

Unit cell of the high-temperature superconductor YBa2Cu3O7 with four non-equivalent oxygen sites.

In this case essentially only the O 2p orbitals contribute to the O EFG. Inside the oxygen spheres (all taken with a radius of 0.82 Å) we can determine the partial charges [q_{i}] corresponding to the [p_{x}], [p_{y}] and [p_{z}] orbitals, denoted in short as [p_{x}], [p_{y}] and [p_{z}] charges.

With these definitions we can define the p-like asymmetry count as [\Delta n_{p}={\textstyle{1\over 2}}(p_{x}+p_{y})-p_{z}\eqno(2.2.15.18)]and obtain the proportionality [V_{zz}\propto\left\langle {{1}/{r^{3}}}\right\rangle _{p}\Delta n_{p}, \eqno(2.2.15.19)]where [\left\langle {{1}/{r^{3}}}\right\rangle _{p}] is the expectation value taken with the p orbitals. A similar equation can be defined for the d orbitals. The factor [1/r^{3}] enhances the EFG contribution from the density anisotropies close to the nucleus. Since the radial wavefunctions have an asymptotic behaviour near the origin as [r^{\ell}], the p orbitals are more sensitive than the d orbitals. Therefore even a very small p anisotropy can cause an EFG contribution, provided that the asymmetry count is enhanced by a large expectation value.

Often the anisotropy in the [p_{x}], [p_{y}] and [p_{z}] occupation numbers can be traced back to the electronic structure. Such a physical interpretation is illustrated below for the four non-equivalent oxygen sites in YBa2Cu3O7 (Table 2.2.15.1[link]). Let us focus first on O1, the oxygen atom that forms the linear chain with the Cu1 atoms along the b axis. In this case, the [p_{y}] orbital of O1 points towards Cu1 and forms a covalent bond, leading to bonding and antibonding states, whereas the other two p orbitals have no bonding partner and thus are essentially nonbonding. Part of the corresponding antibonding states lies above the Fermi energy and thus is not occupied, leading to a smaller [p_{z}] charge of 0.91 e, in contrast to the fully occupied nonbonding states with occupation numbers around 1.2 e. (Note that only a fraction of the charge stemming from the oxygen 2p orbitals is found inside the relatively small oxygen sphere.) This anisotropy causes a finite asymmetry count [(2.2.15.18)[link]] that leads – according to (2.2.15.19)[link] – to a corresponding EFG.

Table 2.2.15.1| top | pdf |
Partial O 2p charges (in electrons) and electric field gradient tensor O EFG (in 1021 V m−2) for YBa2Cu3O7

Numbers in bold represent the main deviation from spherical symmetry in the [2p] charges and the related principal component of the EFG tensor.

Atom[p_{x}] [ p_{y}] [ p_{z}] [ V_{aa} ][V_{bb} ][ V_{cc} ]
O1 1.18 0.91 1.25 −6.1 18.3 −12.2
O2 1.01 1.21 1.18 11.8 −7.0 −4.8
O3 1.21 1.00 1.18 −7.0 11.9 −4.9
O4 1.18 1.19 0.99 −4.7 −7.0 11.7

In this simple case, the anisotropy in the charge distribution, given here by the different p occupation numbers, is directly proportional to the EFG, which is given with respect to the crystal axes and is thus labelled [V_{aa}], [V_{bb}] and [V_{cc}] (Table 2.2.15.1[link]). The principal component of the EFG is in the direction where the p occupation number is smallest, i.e. where the density has its highest anisotropy. The other oxygen atoms behave very similarly: O2, O3 and O4 have a near neighbour in the a, b and c direction, respectively, but not in the other two directions. Consequently, the occupation number is lower in the direction in which the bond is formed, whereas it is normal (around 1.2 e) in the other two directions. The principal axis falls in the direction of the low occupation. The higher the anisotropy, the larger the EFG (compare O1 with the other three oxygen sites). Excellent agreement with experiment is found (Schwarz et al., 1990[link]). In a more complicated situation, where p and d contributions to the EFG occur [see (2.2.15.17)[link]], which often have opposite sign, the interpretation can be more difficult [see e.g. the copper sites in YBa2Cu3O7; Schwarz et al. (1990[link])].

The importance of semi-core states has been illustrated for rutile, where the proper treatment of 3p and 4p states is essential to finding good agreement with experiment (Blaha et al., 1992[link]). The orthogonality between [\ell]-like bands belonging to different principal quantum numbers (3p and 4p) is important and can be treated, for example, by means of local orbitals [see (2.2.12.4)[link]].

In many simple cases, the off-diagonal elements of the EFG tensor vanish due to symmetry, but if they don't, diagonalization of the EFG tensor is required, which defines the orientation of the principal axis of the tensor. Note that in this case the orientation is given with respect to the local coordinate axes (see Section 2.2.13[link]) in which the [LM] components are defined.

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