Recoilless-Fraction Ratios for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">Fe</mml:mi></mml:mrow><mml:mrow><mml:mn>57</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>in Octahedral and Tetrahedral Sites of a Spinel and a Garnet

We have determined the ratio of recoilless fractions, $\frac{f_{B}^{}{}_{}{}^{\ensuremath{'}}}{f_{A}^{}{}_{}{}^{\ensuremath{'}}}$, of ${\mathrm{Fe}}^{57}$ nuclei in octahedral ($B$) and tetrahedral ($A$) sites in ${\mathrm{Fe}}_{3}$${\mathrm{O}}_{4}$ and in yttrium iron garnet (YIG), using the M\"ossbauer effect, by measuring the area ratios as a function of temperature. We find $\frac{f_{B}^{}{}_{}{}^{\ensuremath{'}}}{f_{A}^{}{}_{}{}^{\ensuremath{'}}}=0.99\ifmmode\pm\else\textpm\fi{}0.01$ at 0\ifmmode^\circ\else\textdegree\fi{}K and $\frac{f_{B}^{}{}_{}{}^{\ensuremath{'}}}{f_{A}^{}{}_{}{}^{\ensuremath{'}}}=0.94\ifmmode\pm\else\textpm\fi{}0.02$ at room temperature. Since, in the Debye approximation, a difference in recoillesss fractions implies a difference in the Debye temperatures, we have defined two Debye temperatures for such materials corresponding to Fe ions in $A$ and $B$ sites. These Debye temperatures have been determined independently and are consistent with the ratio of recoilless fractions. The Debye temperatures are, however, considerably lower than those found by specific-heat measurements. Possible explanations for this difference are discussed.

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