Electronic Theory of Order-Disorder Transformations in Binary Alloys
- PDF / 210,185 Bytes
- 6 Pages / 418.68 x 637.2 pts Page_size
- 51 Downloads / 177 Views
ELECTRONIC THEORY OF ORDER-DISORDER TRANSFORMATIONS IN BINARY ALLOYS
F.L. CASTILLO-ALVARADO,* J.L. MORAN-LOPEZ*ý AND J.M. SANCHEZ *Departamento de Fisica Aplicada de la Escuela Superior de Fisica y Matem~ticas, %Departamento de Fisica, Centro de Investigaci6n y de Estudios Avanzados del IPN. Apdo. Postal 14-740, 07000 MHxico D.F. :Henry Krumb School of Mines, Columbia University, New York, N.Y. 10027.
ABSTRACT A microscopic theory of order-disorder phase transitions in binary alloys is oresented. The internal energy of the system is calculated within the tight-binding Hamiltonian and long- and short-range order effects are included by solving the equations of motion for the Green's functions in the Bethe lattice approximation. Results are presented for the electronic local density of states and for various values of the long- and short-range order parameters.
INTRODUCTION Order-disorder transformations in binary alloys have been studied extensively by means of phenomenological models [1-3]. In those studies, the internal energy is calculated by assuming effective interactions between a given number of atoms (two, three, etc.). Then, after the entropy is calculated to some approximation (Bragg-Williams, Bethe, tetrahedron, etc.) the equilibrium configuration is obtained by minimizing the free energy with respect to a given set of parameters. Here, we present a microscopic theory that allow us to study orderdisorder transformations and eventually to calculate phase diagrams from the electronic parameters of the constituents. The method consists in solving the equations of motion in the Bethe lattice approximation [4-61. This model consists of substituting the infinite periodic lattice for an infinite system of connected atoms, with the same coordination number Z as the lattice under consideration, but without closed rings of bonds. This lattice has the property that the one particle Green's functions at a given site can be expressed in terms of the Green's functions at the preceding site in the lattice. In this way, it is possible to write analytical expressions for the Green's functions in terms of transfer functions. This method has been used previously to study long- and short-range order separately [4-6]. Here we study both effects at the same time.
THEORY We consider a binary alloy AxBlx with N lattice sites and we assume that these lattice sites can be subdivided into two equivalent sublattices a and 6, such that there are N/2 lattice sites of type a and N/2 of type 6. Each a site has Z sites of type 6 as its nearest neighbors and viceversa. To describe the state of order we consider four pair probabilities PIJ (I,J = A,B) with I pB
1
(1)
I,J and 1 (
6 +06
SPAA
AB
o6 BA
Mat. Res. Soc. Symp. Proc. Vol. 21 (1984) aElsevier science Publishing Co.,
Inc.
338
We define a long- and a short-range order parameter by
a
a6
PAA
PAB)
-
D
ao
a6
a6
(+Uý aý aý(3) -(AA +PBA)
a
a = PA -PA and
a6
o6
PAB + PBA 2x(l-x) respectively. The range of values of a and
(4)
1 depend on x and are given by
x
Data Loading...
