Single Crystal Elastic Moduli of Disordered Cubic Alloys
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expressed in terms of appropriate derivatives of a pairwise interatomic potential. In a material possessing metallic bonding, the internal energy per atom can be approximated by the sum of two terms: 1) the sum of pairwise interaction energies between a limited number of neighbors, Up, and 2) a many body term, Uv, that depends on the total volume of the crystal and accounts for the interaction of individual ions with the electron "gas" [15,16]. To determine Up, choose any atom as the origin of a coordinate system with directions as the associated coordinate axes. Expanding Up in terms of atomic displacement about the origin [20], +O(ni)( 3)) ( U.(ii) = +S u +qt)u~ +-JXPVtn~ujuj+(d Upj=-{U 0 +:P 2 n=l 2 n= where s is the number of atoms interacting with the atom at the origin and the primes indicate the degree of partial differentiation of the pairwise interatomic potential, p, with respect to components of the position vector connecting the origin to each neighboring atom, evaluated at the equilibrium spacing between the atoms. Since interatomic forces are generally of short range,
we consider pairwise interactions over only the first and second neighbor atoms. The term involving first derivatives corresponds to the total force exerted by neighboring atoms on the atom at the origin. To avoid imposition of the Cauchy condition on the elastic constants of the crystal it is customary to choose Uv such that the first order term in its Taylor series expansion exactly cancels this force. Higher order terms are subsumed into appropriate higher derivatives of (P. (n)= fij(n) is the force exerted The second derivatives form a force constant matrix such that (Ni" on the atom at the origin in the xi direction when an atom at the n lattice point experiences a unit displacement in the xj direction. Independence of the order of differentiation requires that the force constant matrix be symmetric. Neglecting terms O(1u13) results in the harmonic approximation for the total potential energy of the crystal at absolute zero. The quasi-harmonic approximation, in which elements of fij n) are regarded as temperature-dependent material parameters, is a similar form that describes the potential energy at temperatures above zero K. In the present treatment, we assume that the magnitude of the pair potential for an atom and its first and second neighbors depends only on the position vectors connecting the center of the atom with those of the corresponding neighbors. This central potential gives rise to non-central interatomic forces characterized by two axisymmetric force constants (ASFC) for each neighbor shell: ox, the coefficient for stretching bonds between nth neighbors, and P3,that for bending such bonds. In terms of appropriate derivatives of the pair potential:
S
-' '-r1 r"
n --
rr _ '(2)
where rn represents the distance between an atom at the origin and an atom in the shell. Derivatives appearing in equation (1) can be expressed in terms of the ASFC:
nth
neighbor
where f£ = xi /r. Applying to this model conditions that in
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