Scattering of Debye Phonons by Substitutional Defects in Solids

Abstract

We consider a three-dimensional Bravais crystal with volume V containing N atoms such that p lattice sites are occupied by substitutional defects each of mass M’, while the remaining (N-p) lattice sites are occupied by identical host atoms of mass M. The introduction of impurities leads to simultaneous changes in mass and force constants between the host atoms and defect atoms. It is assumed that the defects are distributed randomly and their concentration (p/N) is small. The changes in the force constant between the impurity and the host atoms may be assumed to be significant only to nearest neighbors. The Hamiltonian of such a system (defect crystal) in the harmonic approximation can be written as $$ H = \mathop{\Sigma }\limits_{{n\alpha }} \frac{{p_{\alpha }^{2}(n)}}{{2M}} + \frac{1}{2}\mathop{\Sigma }\limits_{{n\alpha }} {{\mathop{\Sigma }\limits_{n} }_{{'\beta }}}{{\Phi }_{{\alpha \beta }}}(n,n'){{u}_{\alpha }}(n){{u}_{\beta }}(n') + \mathop{\Sigma }\limits_{{i\alpha }} (\tfrac{1}{{2M'}} - \tfrac{1}{{2M}})p_{\alpha }^{2}(i) + \tfrac{1}{2}\mathop{\Sigma }\limits_{{n\alpha }} {{\mathop{\Sigma }\limits_{n} }_{{'\beta }}}\Delta {{\Phi }_{{\alpha \beta }}}(n,n'){{u}_{\alpha }}(n){{u}_{\beta }}(n') $$ (1) where n denotes the position of an atomic site and i the position of the impurity; uα (n) and pα (n) are the α-Cartesian components of the displacement and momentum vectors of the nth atom, M and M′ are the masses of the normal and impurity atoms, Фαβ(n,n′) andФαβ′(n,n′) are the harmonic force constants for the pure and impure crystal respectively and ∆Фαβ(n,n′)= Ф αβ′(n,n′) − Фαβ(n,n′).KeywordsLattice SiteForce ConstantMass ChangeInterference TermConstant ChangeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.