TY - JOUR

T1 - Analysis of the thermal expansion of anisotropic solids: Application to zinc

AU - Munn, Robert

AU - Barron, T.H.K

PY - 1967

Y1 - 1967

N2 - For anisotropic solids, separate Grüneisen functions γλ can be defined for each strain coordinate ηλ: γλ = (∂s/∂ηλ)η′, T/C′ . Each independent γλ can then be analysed in the same way as γ is analysed for a cubic solid (Barron et al. 1964). In particular, parameters γλ(η) =- (l/n)(∂ In < ωη>/∂ηλ)η′ are obtained giving the strain derivatives of the moments of the vibrational frequency distribution G(ω); these can be used to estimate the strain dependence of various crystal properties.
The method is used to analyse the data of McCammon and White (1965) for the thermal expansion of zinc. Plots of α⊥/T 3 (T 3 and α――/T 3 against T indicate that the apparently anomalous behaviour of α⊥ at low temperatures is due to a combination of electronic and vibrational effects. Analysis of γ⊥ and γ―― gives for the electronic components γλe = 0·6±0·2, γ――e = 4·5±0·5; the density of states at the Fermi level is thus strongly dependent on axial ratio.
Heat capacities are taken from Eichenauer and Schulze (1959) and Martin (1966a); the C t-C t correction is recalculated. The <ωη>, θ(η) and the low-temperature expansion for G(ω) are then derived from the vibrational heat capacity C η l, and the γλ(n) from the γλl, for-3⩽⩽4.
γλ and γλ(n) are weighted averages of the γλ(r) for individual modes. Apart from the specifically electronic effects, the principal features of the thermal expansion of zinc are accounted for if most of the high-frequency modes have γλ(r) considerably greater than γ――(r), most of the modes in the low-frequency peak have γλ(r) considerably less than γ――(r), and for the lowest frequency modes the average of γ――(r) is somewhat greater than that of γ――(r). These requirements are in general agreement with what is known about the lattice vibrations from neutron diffraction measurements.

AB - For anisotropic solids, separate Grüneisen functions γλ can be defined for each strain coordinate ηλ: γλ = (∂s/∂ηλ)η′, T/C′ . Each independent γλ can then be analysed in the same way as γ is analysed for a cubic solid (Barron et al. 1964). In particular, parameters γλ(η) =- (l/n)(∂ In < ωη>/∂ηλ)η′ are obtained giving the strain derivatives of the moments of the vibrational frequency distribution G(ω); these can be used to estimate the strain dependence of various crystal properties.
The method is used to analyse the data of McCammon and White (1965) for the thermal expansion of zinc. Plots of α⊥/T 3 (T 3 and α――/T 3 against T indicate that the apparently anomalous behaviour of α⊥ at low temperatures is due to a combination of electronic and vibrational effects. Analysis of γ⊥ and γ―― gives for the electronic components γλe = 0·6±0·2, γ――e = 4·5±0·5; the density of states at the Fermi level is thus strongly dependent on axial ratio.
Heat capacities are taken from Eichenauer and Schulze (1959) and Martin (1966a); the C t-C t correction is recalculated. The <ωη>, θ(η) and the low-temperature expansion for G(ω) are then derived from the vibrational heat capacity C η l, and the γλ(n) from the γλl, for-3⩽⩽4.
γλ and γλ(n) are weighted averages of the γλ(r) for individual modes. Apart from the specifically electronic effects, the principal features of the thermal expansion of zinc are accounted for if most of the high-frequency modes have γλ(r) considerably greater than γ――(r), most of the modes in the low-frequency peak have γλ(r) considerably less than γ――(r), and for the lowest frequency modes the average of γ――(r) is somewhat greater than that of γ――(r). These requirements are in general agreement with what is known about the lattice vibrations from neutron diffraction measurements.

U2 - 10.1080/14786436708230352

DO - 10.1080/14786436708230352

M3 - Article

SN - 1478-6435

SP - 85

EP - 103

JO - Philosophical Magazine

JF - Philosophical Magazine

ER -