Anisotropic mean-squared-displacement tensor in cubic almandine garnet: a single crystal ⁵⁷Fe Mössbauer study

Abstract

This paper describes single-crystal measurements on a crystal plate cut from a naturally-occurring almandine-rich single crystal (Alm₆₉Pyr₁₉Spe₈Gro₄) from Wrangell Alaska. The objective was to measure the mean-squared-displacement (msd) tensor precisely using Mössbauer spectroscopy. Parallel quantum-mechanical calculations based on X-ray determined atomic displacement parameters and Mössbauer parameters from polycrystalline measurements indicated that the msd tensor should display significant anisotropy, easily measurable within the precision of the Mössbauer experiment. For each single-crystal orientation the observed Mössbauer spectrum represents a macroscopic quantity that is the average over six symmetry-related (local) dodecahedral sites in which the high-spin Fe²⁺ ions reside. The anisotropy in the measured msd tensor is, nevertheless, unequivocal. Furthermore, the magnitudes of the Mössbauer-determined msd principal values exceed those of the corresponding X-ray-determined quantities by a factor 3.7. The equivalent recoilless fractions are also anisotropic and consequently one observes the Gol’danskii–Karyagin Effect (GKE), as manifested by an asymmetric quadrupole doublet in polycrystalline absorbers. Moreover, the line widths of the two quadrupole lines are markedly different but angle invariant. This is interpreted as implying that, in addition to anisotropy in the msd tensor, differential spin–spin relaxation is present in the $m = \pm 3/2 \leftrightarrow \pm 1/2$ and $m = \pm 1/2 \leftrightarrow \pm 1/2,\, \mp 1/2$ nuclear transitions. While both effects contribute to the quadrupole asymmetries observed in Mössbauer spectra of polycrystalline almandine, the GKE is apparently predominant.

Appendix

Thickness corrections

We outline briefly the equations and procedures for thickness corrections following Housley et al. (1969) and Zimmermann (1983) (see also Golding and Tennant (2008), vol II, p. 810). We define useful quantities in terms of a polarization operator, σ, its eigenvalues σ ₁ and σ ₂ and a “thickness”, p (effectively the cross section at resonance), where

$$ p = (\sigma_{1} + \sigma_{2} )/2 $$

(28)

In the most general case, thickness is polarization dependent and the degree of polarization, a, is defined as

$$ a = (\sigma_{1} - \sigma_{2} )/(\sigma_{1} + \sigma_{2} ), $$

(29)

hence

$$ \sigma_{1} = p(1 + a)\quad {\text{and}}\quad \sigma_{ 2} = p(1 - a) $$

(30)

In the almandine case, since polarization is zero, we have simply σ₁ = σ₂ = p.

Assuming that the gamma-ray source is a single line and unpolarized, the dimensionless absorption areas, S, can be expressed by the Bykov and Hein (1963) equation (Housley et al. 1969; Zimmermann 1983; Golding and Tennant 2008), namely

$$ S(p) = \frac{{\sigma_{1} }}{2}\exp \left( { - \frac{{\sigma_{1} }}{2}} \right)\left\{ {I_{0} \left( {\frac{{\sigma_{1} }}{2}} \right) + I_{1} \left( {\frac{{\sigma_{1} }}{2}} \right)} \right\} + \frac{{\sigma_{2} }}{2}\exp \left( { - \frac{{\sigma_{2} }}{2}} \right)\left\{ {I_{0} \left( {\frac{{\sigma_{2} }}{2}} \right) + I_{1} \left( {\frac{{\sigma_{2} }}{2}} \right)} \right\} $$

(31)

In Eq. 31, I ₀ (x) and I ₁ (x) are modified Bessel functions of the first kind (i.e., with imaginary argument) of order 0 and 1, respectively. An expansion of Eq. 31 utilising Eq. 30 for the zero polarization case and its more useful inversion as detailed by Bull et al. (2009) are as follows:

$$ S = p + \frac{1}{4}p^{2} + \frac{1}{16}p^{3} + \frac{5}{384}p^{4} + \frac{5}{3,072}p^{5} - \frac{7}{30,720}p^{6} - \frac{169}{737,280}p^{7} + \ldots $$

(32)

$$ p = S + \frac{1}{4}S^{2} + \frac{1}{16}S^{3} + \frac{5}{384}S^{4} + \frac{5}{3,072}S^{5} - \frac{7}{30,720}S^{6} - \frac{169}{737,280}S^{7} + \cdots $$

(33)

The inverse Eq. 33, is usually sufficient to obtain a convergent thickness correction. The correction can be checked easily by substitution in Eq. 32 for which a few extra terms, given in Bull et al. (2009), can be useful.