On the symmetry properties of the incommensurate phase in thiourea

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On the symmetry properties of the incommensurate phase in thiourea

J.M. Perez-Mato, G. Madariaga

To cite this version:

J.M. Perez-Mato, G. Madariaga. On the symmetry properties of the incommensurate phase in thiourea. Journal de Physique, 1987, 48 (11), pp.2019-2021. �10.1051/jphys:0198700480110201900�.

�jpa-00210645�

2019

On the symmetry properties ^{of the} incommensurate phase in thiourea

J. M. Perez-Mato and G. Madariaga

Departamento ^de Fisica, Facultad de Ciencias, Universidad del Pais Vasco, Apdo 644, Bilbao, Spain

(Reçu ^le 23 janvier 1987, révisé le 2 juin 1987, accept6 ^{le 24} juillet 1987)

Résumé.

L’analyse ^{de la} symétrie ^{de la} phase modulée incommensurable de la thiourée, proposé ^{dans le J.}

Physique ⁴⁶ (1985) 2187-2195,

révisée. On corrige certaines des conclusions qui surestimaient l’importance

des modes secondaires pour les propriétés ^de symétrie de la modulation.

Abstract.

The symmetry analysis of the incommensurate modulated phase of thiourea reported ⁱⁿ [J.

Physique ⁴⁶ (1985) 2187-2195] is revised. Some of its conclusions overstating the relevance of secondary ^modes

for describing ^the symmetry properties of the modulation

corrected.

J. Physique ⁴⁸ (1987) ^2019-2021

1987,

Classification

Physics ^Abstracts

64.90

61.50K

64.70K

1. Introduction.

A symmetry analysis ^{of the} incommensurate (IC)

modulated structure of thiourea has been recently reported ⁱⁿ [1]. ^{In that work, the} symmetry adapted

modes involved in the structural distortion

sponding ^{to the IC} phase ^{of thiourea}

^determined

and the possible space groups of the commensurate lock-in phases

discussed. In their symmetry considerations, the authors have judged ^of special

relevance the effect of secondary ^modes

^faint

variables [2]. Among others, ^the following important points

^{claimed :}

i) The order parameter ^{in the}

of thiourea

«

contains

than

I.R. (irreducible represen-

tation) ^of Go (the space group of the parent paraelec-

tric phase).

ii) The relation between the phase ^{of the} primary

mode amplitude and those of higher ^harmonics (secondary modes) ^{in the IC} phase depends

^the

actual values of

coefficients in the Landau free energy expansion. ^{The relation} is then in general

rather arbitrary.

iii) ^As

consequence of ii), the presence of

secondary ^{modes in the IC} phase ^of thiourea may break the superspace group resulting ^{from the} primary ^mode.

iv) ^{For the} commensurate lock-in phases, ^umk- lapp ^terms involving couplings between the primary

mode and higher ^harmonics originate ^{that the mini-}

ma

in the free energy associated ^to these phases ^do

not correspond ^to simple

symmetric

values of the

phase ^{of the} primary ^mode amplitude, ^{but in} general

to rather arbitrary values. Hence the commensurate distorted phases ^have the minimum allowed space group symmetry.

In the present communication, these four conclu- sions from [1]

^refuted.

2. Primary ^and secondary modes in thiourea.

Point i) apparently ^{situates the}

of thiourea in conflict with

of the basic hypothesis ^{of the}

Landau theory ^of phase transitions [3]. ^{This conflict}

is however only ^{due to}

unrestricted

in [1] ^of

the term

order parameter

^to designate ^{the whole}

structural distortion. This latter is indeed made up

not only by the distortion corresponding. ^{to the} primary ^mode (the actual order parameter) ^trans- forming ^{in fact} according ^to

^{I. R. of} Go, ^{but also} by contributions from other

secondary » ^modes

associated to other I.R. of Go.

The presence in

distorted structure of secondary

spontaneous ^variables belonging ^{to I.R. different}

from that of the order parameter ^is

well known fact in the frame of Landau theory [2, ^4, 5]. ^{One of the} simplest

^{is the} polarization ⁱⁿ

improper

ferroelectric [6]. ^The only difference in the

of IC structures is that the number of these secondary

modes is in principle unlimited. These types ^of

variables

essential to describe the whole structur-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110201900

2020

al distortion. They ^{should not be} considered, ^how-

. ever,

part of the order parameter, ^{this term} being

reserved to design ^the degrees of freedom of the system having

^critical susceptibility ^{at the} phase

transition (the primary mode(s)). ^{In addition, the}

symmetry properties of the order parameter

pletely ^{describe the} symmetry breaking ^{at the} phase

transition without having ^to consider the effect of the always existing secondary ^modes [5]. ^The

dary ^modes

^indeed

symmetric

^{than the}

order parameter ^{in the}

^that they

compat- ible with the space group resulting ^{from the} primary

mode distortion and therefore their presence in the whole distortion does not further reduce the symmet- ry of the crystal [2, 7]. As discussed in [7] ^and

contrary ^to point iii) ^{above, this is} also valid in the

case

of IC phases if superspace symmetry ^is

sidered.

The argument ⁱⁿ [1] leading ^to the conclusions listed above

ii) ^and iii)

^{the need to}

minimize the free energy ^not only ^with respet ^{to the} phases ^{of the} primary ^and secondary modes, ^{but also}

with respect ^{to their} amplitudes (moduli). ^{In the} following ^{the correct} mathematical manipulations

are

briefly ^indicated.

The coupling ^terms of the three modes considered in [1] (Qqs, T4, Q2.q,,,,, Q3 qs, T4) ^to be included in the free energy

(see p. 2192 in [1]) :

where Qqs, T4 ^{is the} complex amplitude of the order parameter (primary mode) ^{with modulation}

wavevector qs and small I.R. T4, while Q2qs,1"1 ^and Q3 qs’ 1"4

^those corresponding ^to secondary ^modes (second ^{and third} harmonic).

Using ^the following ^{notation :}

to designate ^the amplitudes ^and phases ^{of the} modes, the free energy expansion

^{be written}

A straightforward minimization procedure ^{of the} preceding ^{function with} respect ^to all the variables p i, P 2, P3’ CPt, CP2, CP3, ^{leads to the} equations :

Combining (3) ^and (4), ^{it is} easy ^to

that for

Pi :A 0

while 0

is arbitrary ^and corresponds ^{to the}

«

phason

degree of freedom.

Therefore, ^it

be concluded that, contrary ^to the statements in [1] (point ii) above), ^the phase

relations between the primary ^and secondary ^modes

are

rather simple ^and independent of the actual values of the coefficients in the Landau potential.

Moreover, this relation is just ^the adequate ^{to make}

the secondary ^modes compatible with the superspace symmetry ^{of the} primary ^mode distortion,

^it

be easily ^checked using ^the procedure described in

[7]

^the arguments in section 5 of [1]. ^Hence,

contrary ^to point iii), the superspace group of the IC

phase in thiourea is completely determined by ^the symmetry properties (I.R.) ^{of the} primary ^mode (order parameter), ^the secondary ^modes having

relevance in this matter. This is in accordance with what

already ^{stated in} [7] ^for

general

Finally, the conclusion in [1] listed above

iv) ^is

also closely ^{related to the} preceding

^{It is} argued ⁱⁿ [1] that for the 6=1/9- lock-in phase

additional umklapp ^term of the form

must also be considered in the free energy expansion,

besides the usual umklapp ^term

as

both

in fact of the

order. It is further

argued ^{that the addition of this term} originates ^that

the minimum of the free energy

^at

unknown general value of the primary ^mode phase ( P 1)’ This conclusion (demonstrated ^{below to be} erroneous)

special problems ^when

pared ^with experiment, ^{since in the} particular

^of

the I. R. _{T 4} the space group allowed by ^{the order} parameter ^{is the}

^for

arbitrary order par- ameter phase

^{for the} special values 01 = - ’TT’ /2.

However, for other I.R. with analogous ^free

2021

gies, the conclusion in [1] ^would imply

lower space group symmetry ^{for the} commensurate phase ^than

that allowed by ^the

symmetric

^directions ø1 = 0, 7T/2, ... (see Table Va in [1]), ^{where the}

free energy ^minima

situated, ^if only ^the primary

mode coordinates

considered in the Landau

expansion.

According ^{to this} argument,

^have

again

that the consideration of secondary ^{modes in the}

Landau analysis ^would

ⁱⁿ general

^drastic change ^{in the} symmetry restrictions for the distorted

phase ^with respect ^to those obtained by only

sidering ^{the order} parameter ^{terms. The} reasoning

in [1] leading ^to this conclusion is however incorrect,

as a

minimization procedure ^{similar to the}

discussed above for the IC phase

show. In this case, the free energy (1) ^{has to be} completed ^with

the terms (6) ^and (7). ^The resulting additional terms in the derivatives give place ^{to the} following ^relations

where A, B

real numbers depending ⁱⁿ general

on

the values of P2 and P3. Similarly :

If

substitute (8) ⁱⁿ (6) (or ^{in the} equations corresponding ^to the derivatives of the free _energy with respect ^to p 1 and cp 1), ^{it is obvious that the term} (6) only introduces

renormalization of the coeffi- cient V 9 of the usual umklapp ^term (7) ^{and other} isotropic ^{terms, like} V12 qs 4 - qs, ’r4’ in ^{the free}

energy restricted ^to the order parameter coordinates.

The

happens ^when expression (9) ^is

sidered. Hence if, for the free energy restricted to the order parameter ^terms, the directions in the order parameter space along which the minima take

place ^{do not} depend

the actual values of the

expansion coefficients, ^{this situation is not altered}

whdn secondary ^modes

introduced in the free energy.

3. Conclusions.

Summarizing,

conclude that the influence of

secondary ^modes

^the symmetry properties ^of

thiourea in both the IC and commensurate mod- ulated phases

overestimated in [1]. ^In fact,

of the main utilities of the order parameter concept

in Landau type transitions is that the symmetry

properties

^be analysed ^{in terms of} it, ^without having ^to consider the secondary ^modes [5]. ^{This is}

also true for IC phases ^{when the} concept of supers- pace group symmetry ^is employed,

^it

already

shown in [7].

Secondary ^{modes can} however favor the relevance in the order parameter free energy expansion ^of higher ^order terms, which may finally ^stabilize phases (of ^lower symmetry) ^different from those favored by free energy expansions ^{truncated at}

lower order. Recently ^Aizu [8] ^{has also} pointed ^out

this possibility ^{in the}

^{of IC} phases. ^{But in the}

case

of thiourea, ^the stability ^{of IC and}

rate lock-in phases ^{of the} symmetry described in [1]

would require ^to consider in the free energy expan- sion terms up ^to the 18th degree in the order parameter amplitude, instead of the nineth order

expansion ^{discussed in} [1] and in this comment.

However, ^such

high ^order expansion

hardly ^be expected ^to be relevant for describing ^and explaining

the modulated phases ^{in thiourea.}

References

[1] SIMONSON, T., DENOYER, ^{F. and} CURRAT, R., J.

Physique ⁴⁶ (1985) ^2187.

[2] KOPSKY, V., Ferroelectrics 24 (1980) ^3.

[3] LANDAU, L. D. and LIFSHITZ, E. M., Statistical Physics (Pergamon ^{Press, New} York) ^1969.

[4] ^{AIZU, K., J.} Phys. ^Soc. Jpn ³⁷ (1974) ^885.

[5] TOLEDANO, ^{J. C. and} TOLEDANO, P., Phys. ^{Rev. B 21} (1980) ^1139.

[6] TOLEDANO, ^{P. and} TOLEDANO, J. C., Phys. ^{Rev. B 14} (1976) ^3097.

[7] PEREZ-MATO, ^J. M., MADARIAGA, ^{G. and} TELLO, M. J., Phys. Rev. B 30 (1984) ^1534.

[8] AIZU, K., ^J. Phys. ^Soc. Jpn ⁵⁴ (1985) ^4213.