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On the symmetry properties of the incommensurate phase of thiourea
T. Simonson, F. Ddnoyer, R. Currat
To cite this version:
T. Simonson, F. Ddnoyer, R. Currat. On the symmetry properties of the incommensurate phase of thiourea. Journal de Physique, 1987, 48 (11), pp.2023-2026. �10.1051/jphys:0198700480110202300�.
�jpa-00210646�
On the symmetry properties of the incommensurate phase of thiourea
T. Simonson (*), F. Ddnoyer (**) and R. Currat (***)
(*) Laboratoire pour l’utilisation du Rayonnement Electromagndtique, Bât. 209C, Université de Paris-Sud, 91405 Orsay, France
(**) Laboratoire de Physique des Solides, Bât. 510, Université de Paris-Sud, 91405 Orsay, France (***) Institut Lauë Langevin, BP 156X, 38042 Grenoble Cedex, France
(Requ le 2 juillet 1987, accept6 le 24 juillet 1987)
Résumé.
-L’énergie libre de la phase incommensurable de la thiourée est minimisée qualitativement. Deux
classes de solutions sont mises en évidence, avec des propriétés de symétrie différentes. Elles correspondent à
deux phases incommensurables avec des supergroupes d’espace différents ; le système passe de l’une a l’autre par une transition de phase entre phases incommensurables, sous le contrôle de certains termes anharmoniques
du développement de Landau de l’énergie libre, qui mettent en jeu les phases et les amplitudes des harmoniques un, deux et trois de la modulation.
Abstract.
-The free energy of the incommensurate phase of thiourea is minimized qualitatively. Two classes of solutions with different symmetry properties are found and discussed. These correspond to incommensurate
phases with two different superspace groups ; the system goes from one to the other by a phase transition
between incommensurate phases, under the control of certain anharmonic terms of the Landau free energy
development, which involve the phases and amplitudes of harmonics one, two and three of the modulation.
Classification
Physics Abstracts
64.90
-61.50K - 64.70K
Introduction.
The article of Perez-Mato and Madariaga, « On the symmetry properties of the incommensurate phase
of thiourea » [1] reexamines conclusions drawn in the symmetry analysis of the incommensurate phase
of thiourea reported in J. Physique 46 (1985) 2187-
2195 [2]. The crucial point which Perez-Mato and
Madariaga wish to « correct » is the minimization of the free energy F with respect to the phases of
harmonics one, two and three of the modulation.
The four physical conclusions which they contest are
all direct consequences of the way this minimization is done. This is made quite clear in their paper. Their conclusions are, however, an oversimplification.
1. The two classes of possible minima of the free energy.
There are five independent variables in F,
These are the amplitudes of harmonics one, two and three of the modulation and the phase differences between these harmonics.
If, as is usually the case in incommensurate materials,
then, to a good approximation, the minimum of F
necessarily verifies the relations
This is the main class of solutions of the problem.
If, however, P3 is not negligible compared to
p 1, then the solution generally has no reason to obey (,%), and a second, less symmetrical, class of solu- tions appears, which we predicted in the above-
mentioned paper. This class of solutions has very
interesting physical properties, which makes it worthwhile to discuss their existence in some detail.
Let us examine the minimization of F more closely.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110202300
2024
2. Structure of the free energy F.
F can be written as
In this expression, V 4’ V6 and V6 are real. This implies an assumption about the real or imaginary nature of
the anharmonic coefficients appearing in the free energy terms
(ql, q2, ..., qn are multiples of the wave-vector q of the modulation ; A a Dirac function.) Iizumi et al. have shown [3] that when inversion symmetry is present in the space group of the parent phase, the coefficient
V (ql, q2,’..’ q n) is either purely real if n is even, or purely imaginary if n is odd. Thus V (q, q, - 2 q) and V(q, 2 q, - 3 q) are imaginary, and V(q, q, q, - 3 q) is real. However, if we replace (01, 02, ’03) by (4)1’ 02 + 7r/2, 43), V(q, q, - 2 q) and V(q, 2 q, - 3 q) are multiplied by
- i and + i respectively, and our problem is transformed back into one with real coefficients.
Differentiating F then leads to :
3. Minimization of the free energy.
The usual minimization procedure starts by deriving
the phase relationships (R) from (S4) and (S5) ; (S2) and (S3) are then used to express p2 as a function of Pf and P3 as a function of p31. Finally, (Sl) is written as a polynomial of the sole variable p1. As pointed out by Perez-Mato and Madariaga,
this is an approximate procedure which depends on 1>pl, as well as on P2>P3-
Indeed, we can use
to replace (S4) by
whence
and the rest follows.
If, on the other hand, we no longer accept (2) and
solve the system completely rigorously, it is clear that the relations (R,) are sufficient, but not neces-
sary to ensure (S4) and (S5). Therefore, although (R) characterizes one possible solution, other solu- tions also become relevant, with slightly non-zero
values of 2 P 1 - P 2 and 3 P 1 - 0 3. This second class
of solutions can be either stable or metastable. Its
’stability or metastability depends on the actual numerical coefficients of the free energy develop-
ment.
4. Existence of a non-trivial minimum of the free energy.
A complete proof of this can be given with very little calculation.
i) First minimize F using (2) as a first approxi- mation ; we obtain
from (S4) and (S5), then
from (Sl), (S2), (S3). (The exponent (1) means
« first approximation ».)
ii) Now minimize F to a second approximation
with.re.spect to 2 ol - 02 and 3 l/J 1 - l/J 3. To do this,
fix PI’ p 2 and P3 at the values pII), p 2(’), p 3(l). Then
we have to minimize the function
with respect to the phases. This we do in three steps.
- First, notice that the function
has the non-trivial minimum {cos a
=-1/2 A, f3 = - a}, for À >- 0.5, and the
trivial minimum {( a, f3 )
=(0, 0 ) (mod 7r ) I ,
for A 0.5.
-
Second, by a continuity argument, it follows that
also has a non-trivial minimum {( a, (3 ) =F (0, 0)} in at least some open neighbourhood
of (1, 1, + oo) in (v, a, A) space (as well as in
open neighbourhoods of (1, + oo, 1) and (+00, 1, 1)).
- Third, let V4 P 1’1 P 2’1’, V 6 P f1)3 pj1) and
V6 p l(l) p 211) p 311) play the roles of v, JL, and A ; 2 CP1 - CP2 and 3 CP1 - CP3 the roles of a and (3.
iii) The proof is complete :
We can conclude that F has a non-trivial minimum
(that is, not verifying (A )) if
or
or
Moreover, by construction this is not an exhaustive list of conditions which permit a non-trivial
minimum, by any means, merely three sufficient
conditions.
5. Physical consequences : a phase transition between two incommensurate phases.
The general solution we are discussing is physically interesting for two reasons.
First of all, the non-trivial class of solutions is
probably relevant to thiourea, since it is found
experimentally [4] that in the commensurate
1/9 phase, P3/Pl = 0.1, prohibiting use of (2).
Second, and most important, a deviation, however small, of 2 ol - 02 and 3 01 - 03 from zero has a
dramatic effect on the symmetry of the modulated
phase. As was shown in reference 2, the superspace group of thiourea is PP’ma if these phase differences
are zero and P p 11 if they are not. This slight
deviation of 2 ol - 02 and 3 ol - 03 from zero depends on the values of the zone-centre coefficients
V4, V6 and V6. In certain areas of (V4, V6,
V6) space, the deviation is strictly zero, in others it is
slightly non-zero, and the super-symmetry of the system is lowered. Thus our free energy develop-
ment, to the ninth-order in p 1, already contains a
mechanism for phase transitions between superspace groups. It is a general mechanism, not specific to
thiourea. This is a remarkable confirmation of the
physical significance of superspace groups, the value of which had mostly been shown as a formal tool for structure refinement.’ We are in the process of
deriving the phase diagram of thiourea in (V4, V6,V6) space, corresponding to this transition mechanism. In our calculation, the other coefficients of the free energy (aim, a 2’ a 3’ f3 1 , 02, 163, y 1) will be fixed at the values which K. Parlinski and K. Michel derived from their phenomenological
model of thiourea [5], while V4, V6, and V6 are
varied.
Analogous reasoning showed (see Ref. [2]) that
the phases of harmonics two and three lower the symmetry of the commensurate phases of thiourea.
For example, the 5=1/8 phase has apparent or- thorhombic symmetry, but the monoclinic space group Plla, due to the non-zero phases of harmonics
two and three.
Conclusion.
In conclusion, as was shown in reference [2], the
relation between the phase of the primary mode and
the phases of higher harmonics is arbitrary in general, even though the simple relations (R) :
f2 ol - 02 = 3 01 - 03 = 0 (mod 17" )} always rep-
resent :
-
a good first approximation to the stable solu- tion when 1 > Pl >> P2 > P3 ;
-
