A Thermophysical Perspective of the Inter-relationship between Debye Temperature and Electron Density
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INTRODUCTION
THE concept of the Debye temperature (hD) is central to many issues of thermophysics and thermochemistry of materials.[1–4] Though came into existence primarily as a model parameter to explain the low-temperature behavior of specific heat (Cp) of crystalline substances, in terms of the vibrational spectrum of connected atomic springs,[3,5] the impact of hD in modern solid-state sciences goes beyond the realm of heat capacity and is in fact manifold.[1,3–6] Owing to its genetic link with lattice vibration phenomenon, hD is also a fundamental equation of state (EoS) parameter[7] and it can be estimated theoretically either on an ab initio basis[8] or through the method of interatomic potentials.[9,10] In addition, diverse physical quantities, such as cohesive and vacancy formation energies, melting point, phase transition temperatures, thermal expansivity, elastic modulus, indentation hardness, band gap values, Fermi energy, work function, surface energy, acoustic sound velocity, plasmon frequency etc., are all relatable to hD in one way or the other.[1–3,6,11–14] These
SUBRAMANIAN RAJU is with the Physical Metallurgy Division, Materials Characterization Group, Metallurgy & Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, India. Contact e-mail: [email protected], [email protected]. Manuscript submitted June 01, 2020; accepted October 17, 2020.
METALLURGICAL AND MATERIALS TRANSACTIONS A
phenomenological linkages can also be exploited to obtain reasonable estimates of hD as a function of several variables such as crystal structure, composition, temperature, pressure and magnetization etc.[1,2,6,11,12] Despite such ubiquitous status, it is somewhat surprising to know that there are only few research works, which attempt to highlight directly the fundamental link between Debye temperature and another fundamental quantity, namely, the bonding electron density (ne), in a simple and transparent manner.[13,14] The present study is a small attempt in this direction. The relevant theoretical development is outlined briefly below.
II.
THEORETICAL DEVELOPMENT
The Debye temperature hD, as related to the maximum cut-off angular frequency xD = 2pmD of the vibrational frequency spectrum, is defined as follows:[3] hwD ¼ hvD ¼ kB qD :
½1
The symbols h = h/2p and kB stand for the Planck’s and Boltzmann constants, respectively. Traditionally, in the simple harmonic oscillator model of quantized atomic vibrations, the cut-off frequency xD may be expressed in terms of the mass (m) and effective force constant kf of atomic springs involved.[2,3] The force constant kf can be estimated from a simple parametrized
functional form of the effective interatomic potential / (r) or through suitable relations involving physical quantities.[8–10,13–15] In this manner, it is possible to establish clear phenomenological links between hD and various other physical quantities like shear (G) and bulk modulus (BT), average sound velocity (cav), coefficient of volume thermal expansion (ap), melting temperature (T
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