Debye-Waller factors of alkali halides

Abstract

Using very-high-intensity (ϳ70 Ci͒ 183 Ta Mössbauer sources, we have measured the Debye-Waller factors ͑DWF's͒ of sodium chloride, potassium chloride, and potassium bromide single crystals for several of the (h00) and (nnn) Bragg reflections. We have used an approach which properly accounts for thermal expansion over the temperature range of our experiment, from 90 K to 900 K, about 100 K below the melting point of our crystals. We have found that a procedure used to analyze data by earlier workers leads to incorrect parameters in the Debye-Waller factor exponent, and our procedure does not require empirical parameters to account for the effects of thermal expansion. Additionally, we find three items of significance. Contrary to earlier results, we observe that the cations and the anions have identical DWF's in NaCl and also in KBr. We observe terms in the expansion of the DWF exponential which are quartic in the scattering wave vector Q ជ in NaCl and KCl, with some evidence for a Q 4 term in KBr. The size of the Q 4 contribution is reported and varies with the direction of momentum transfer. We also observe that the Debye temperature and the coefficient of the anharmonic Q 2 term also vary with the direction of momentum transfer. We believe our data are the definitive evidence for a nonspherical thermal cloud in a cubic crystal; the ions have a larger amplitude of oscillation in the ͓h00͔ direction than in the ͓nnn͔ direction, contrary to the commonly held view of crystallographers that the most general form of the mean-square thermal motion is of an ellipsoid shape.

Quartic anisotropy of the recoilless fractions in NaCl

“…4, and for various Bragg reflections. The latter results have been derived from the Rayleigh scattering of the Mössbauer radiation ͑RSMR͒ measurements performed upon single crystals at various temperatures and for various Bragg reflections having wave-vector transfer either along the main crystal axis or along the cell diagonal.…”

“…The numerical results are summarized in Table I The numerical values of the parameters are listed in Table II. Additionally, the ratio b 40 (4) /͗R (4) ͘ was calculated versus temperature for sodium and chlorine. The functions ͗R (2) ͘/a 2 , ͗R (4) ͘/a 4 , b 40 (4) /a 4 , and b 40 (4) /͗R (4) ͘ are plotted versus temperature in Fig.…”

“…It seems that neither extinction nor the static disorder have any meaning for the samples investigated. 4 Isotropic contributions to the nonlocal scattering amplitudes have been accounted for during the original data evaluation. 4 However, the anisotropic contributions due to the solid-state effects cannot be accounted for reliably leading to the mixing between genuine recoilless fractions and the ionic scattering amplitudes ͑distributions of scattering electrons͒.…”

“…4 Isotropic contributions to the nonlocal scattering amplitudes have been accounted for during the original data evaluation. 4 However, the anisotropic contributions due to the solid-state effects cannot be accounted for reliably leading to the mixing between genuine recoilless fractions and the ionic scattering amplitudes ͑distributions of scattering electrons͒. This effect becomes stronger at low temperatures where the ''thermal'' motion is small, while the electronic distribution remains practically unaffected for the sodium chloride.…”

Higher-order anharmonic contributions to the Debye-Waller factor

“…This unit for hu 2 i is chosen because the previous QH and ¶ 2 contributions (Shukla 1994) and MC values (Shukla et al 1996 ) were presented in these units. 1, to obtain the ® nal contributions , we have evaluated contributions of all the 14 diagrams at the equilibrium volume corresponding to rˆr 0 and the highest volume r 6ˆ1 :0464r 0 to compare hu 2 i with the MC results.…”