Stationary States in Inhomogeneous Ordered Binary Alloys: Long-Period Superlattices
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STATIONARY STATES IN INHOMOGENEOUS ORDERED BINARY ALLOYS : LONG-PERIOD SUPERLATTICES
M. AVIGNON and B.K. CHAKRAVERTY Laboratoire d'Etudes des Propri6tgs Electroniques des Solides, C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France
ABSTRACT We examine the nature of all possible stationary states in an inhomogeneous ordered binary alloy. For this purpose, we use a development of the free energy in terms of the gradient of the non linear Euler equations is determined from the nature of its singular points. This method allows us to study these solutions for arbitrary expressions of the free energy of the homogeneous system as well as of the gradient coefficients. In general, periodic solutions which can be identified with long period superlattices are found. In specific cases, analytic solutions can be obtained. Fourier components are calculated and compared with experimental values determined for CuAu II.
The existence of ordered structures in binary alloys is generally fairly well understood with short-range interactions between atoms [D,2]. These interactions are usually taken as pairwise although many body interactions might be of importance to account for stability of certain structures and properties of phase diagram [3]. However, more complicated structures the socalled 'long-period superlattices' exist in a number of alloys [4], the first example being discovered by Johansson and Linde E5]. These structures are stable between the 'normal' ordered (for example in CuAu-CuAu I, LIo type) and the disordered structures. They can be described by a one-dimensional variation of order parameter S(x) along one of the crystallographic axis [6]. The period M of this modulation is characteristic of the alloy and generally nonintegral [4,6,7]. We want to show that such order-parameter modulations appear as stationary states in a binary alloy with inhomogeneous order. The stability of these long-period superlattices is clearly electronic in origin [,8], this is consistent with the long range character of pairwise interactions in these systems E9]. The energy of the conduction electrons is lowered when the Fermi surface touches new energy gap introduced by the additional periodicity created by the function S(x). For arbitrary values of the period M, the structure is an 'almost periodic' one and it is difficult to predict the band structure a priori. Attempt has been made to account for non integral values of M in the one-electron energy spectrum [IO]. We consider essentially an inhomogeneous system with respect to the order parameter S and we describe the free energy by the Landau and Lifshitz form in terms of spatial variations of this parameter F = fV [f(S) + K (1S) +.. dV (I) A more general expression should include simultaneously variations of concentration and order ff(c,S) + K, (VC)
+ K2 (7S)
+ K3 ( VC) (VS)
+ ...
} HV
(2)
where f(c,S) simply represents the free energy of an homogeneous system with concentration c and order parameter S. The gradient coefficients K 1 , K2 , K3 can he function of c and S. The statio
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