Phase Diagrams for Incommensurate Systems
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PHASE DIAGRAMS FOR INCOMMENSURATE
SYSTEMS
P. Toledano Groupe de Physique th~orique, Facult9 des Sciences d'Amiens 33, rue Saint-Leu, 80039 Amiens Cedex, France
ABSTRACT Phase diagrams, for systems undergoing one or several incommensurate and lock-in transitions, are discussed in the framework of the Landau theory of phase transitions. It is shown that their essential features can be deduced from the explicit forms of the free-energy density ý and the k-dispersion of the coefficient c(k)of the quadratic contribution of the order-parameter components in 4. Two families of phase diagrams are distinguished depending on symmetry considerations.
INTRODUCTION In systems undergoing incommensurate phase transitions, the range of stability of the successive phases (the phase diagram) can be obtained by minimizing the Landau free-energy F, which is written as the sum over the volume V of the crystal of a local density 0 depending on the order-parameter (OP) components n.(x.), and on their derivatives with respect to the space coordinates x. [l,2]J: J
(n, -nr-i ) dv
F
S can
be divided in
3x.n
two parts
3n J where 0, has the form of a classical Landau expansion of the ni, and 2 depends on the n. and on their derivatives. The possible stable states of the crystal correspond to values of the ni, given by the absolute minima of F, which are solutions of the Euler equations
Z J a
a )(2) j
a ix.
J
3.
As these non-linear equations cannot be solved in the general case, approximations have to be used for each particular system. In this respect it has been shown, by introducing trial functions in (2) [3,4], that the existence in 0 of gradient terms of the type :
Mat. Res. Soc. Symp.
Proc. Vol.
21 (1984) QElsevier Science Publishing Co., Inc.
76
T
i
aj
DXk
-
(Lifshitz invariant),
T1
I
ax-k
or
fl
2
Xk
give rise to an incommensurate structure, while the anisotropic part of *1 (anisotropic in the OP space) favours the onset of a lock-in transition towardSa commensurate phase. However, because of intrinsic mathematical difficulties in working out the minima of F, a closer description of systems displaying incommensurate phases, has not been performed. In particular, complex phase diagrams involving successive incommensurate and (or) lock-in transitions (which take place in materials such as SC(NH2 ) 2 , TTF - TCNQ, (N(CH))4 2 MCI4 , with M = Zn, Co, Mn, Cu) could not be described by a Landautype theory. In this paper, we show that the qualitative features of the various types of phase diagrams that may be encountered in incommensurate systems can be obtained by considering : 1) the explicit form of '| in (1) in which high degree invariants should be taken into account when more than one incommensurate and lock-in transitions are found in the system 2) the k-dispersion, in the vicinity of the lock-in vector k , of ct(k) (i.e. the coeffiI), which minima provide, cient of the renormalized quadratic invariant in for each value of the external parameter ýe.g. temperature or pressure) the equilibrium value of th
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