Enhancement of the Lattice Heat Capacity Due to Low Frequency Resonance Modes in Dilute Al-Ag Alloys

Abstract

The lattice properties of a solid are changed appreciably-due to the presence of impurities which differ from the host lattice in mass or in the nearest neighbor force constant.⁽¹⁾ The introduction of isolated heavy impurities into a relatively light host lattice causes a change in the phonon spectrum, which is characterized by the existence of low-frequency resonance modes localized at the impurity site. If we neglect the change in the force constant, the resonance frequency associated with a heavy impurity is given by using

$$ {\omega _0} = \frac{{{\omega _D}}}{{{{\left( {3\frac{{M - M}}{M}} \right)}^{\frac{1}{2}}}}} $$

((1))

a Debye model. Here, ω_D is the Debye frequency, M and M are the mass of the impurity and of the atoms of the host lattice, respectively. The presence of low-frequency resonance modes enhances the low temperature heat capacity of the lattice. This enhancement has been calculated recently by Kagan and Iosilevskii (2) and by Lehman and De Wames.(3) The theory of Kagan and Iosilevskii leads to the following expression for the enhancement ΔC_L in the lattice heat capacity caused by low-frequency resonance modes:

$$ \Delta {C_L} = \frac{{1.5}}{2}\eta {k_B}{\left( {\frac{\Theta }{T}} \right)^2}\int\limits_0^1 {dx\frac{{{x^{5/2}}\left( {\frac{{3{x_0}}}{x} - 1} \right)}}{{{{\left( {x - {x_0}} \right)}^2}\;\frac{{2.25{\pi ^2}}}{9}{x^3}}}\frac{{\exp \left( {\frac{\Theta }{T}\surd x} \right)}}{{{{\left\{ {\exp \left( {\frac{\Theta }{T}\surd x} \right) - 1} \right\}}^2}}}} $$

((2))

Based on work performed under the auspices of, the U. S. Atomic Energy Commission.