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Structure of the modulation in thiourea. I. symmetry properties
T. Simonson, F. Dénoyer, R. Currat
To cite this version:
T. Simonson, F. Dénoyer, R. Currat. Structure of the modulation in thiourea. I. symmetry prop- erties. Journal de Physique, 1985, 46 (12), pp.2187-2195. �10.1051/jphys:0198500460120218700�.
�jpa-00210167�
2187
Structure of the modulation in thiourea. I. Symmetry properties
T. Simonson, F. Dénoyer
Laboratoire de Physique des Solides, Bât. 510, Université de Paris-Sud, 91405 Orsay Cedex, France and R. Currat
Institut Laue Langevin, BP 156 X, 38042 Grenoble Cedex, France
(Reçu le 22 avril 1985, accepté sous forme définitive le 31 juillet 1985)
Résumé. 2014 Ceci est la première partie d’une étude structurale de la modulation dans la thiourée. La modulation est décrite par un mode normal Qq de symétrie 03C44, accompagné de modes harmoniques Q2q, Q3q... Les dépla-
cements correspondants sont déterminés en vue de l’affinement de structure. Dans les phases commensurables,
ceux-ci donnent accès au groupe d’espace ; dans les phases incommensurables, à la supersymétrie. L’approche
de la symétrie grâce au supergroupe d’espace, est comparée à l’approche par la théorie de Landau. La discussion de l’énergie libre montre qu’une phase accrochée ferroélastique peut être stable sous contrainte. Elle fournit aussi de l’information sur les phases des modes Qq, Q2q, ..., importante pour l’affinement structural.
Abstract. 2014 This is the first part of a structural study of the modulation in thiourea. The modulation is described by a normal mode Qq of 03C44 symmetry along with harmonic modes Q2q, Q3q, ... Corresponding displacements are
derived in view of structure refinement. In commensurate phases, these give access to the space group; in incom- mensurate phases, to the supersymmetry. The superspace approach to the symmetry of the modulation is compared
to the Landau theory approach. Discussion of the free energy shows that a ferroelastic lock-in phase may be stable under stress and gives information as to the phases of the component modes Qq, Q2q, ..., important for structure
refinement.
J. Physique 46 (1985) 2187-2195 DTCEMBRE 1985,
Classification
Physics Abstracts
64.70 - 77.80 - 61.50E ’
1. Introduction.
Thiourea is a molecular crystal which is ferroelectric below T~ = 169 K and paraelectric above To =
202 K [1]. The thiourea molecule SC(NH2)2 has C2v symmetry and an electric moment along the
SC axis. Figure 1 shows the unit cells of the para- electric (P) phase and the ferroelectric (F) phase, projected onto the (a, c) plane, from the structure
determination of deuterated thiourea SC(ND2)2 by
Elcombe and Taylor [2]. In deuterated thiourea, T~ and To are shifted to 190 K and 218 K, respecti- vely. The unit cell is orthorhombic with four mole- cules per cell, labelled (1) to (4) in figure 1. In the P phase, the crystal has Pnma symmetry. Molecules (1)
and (4) are exchanged by inversion, as are (2) and (3).
The equivalent positions of the unit cell are listed in table I. In the F phase, molecules (1) and (4), (2) and (3)
tilt in opposite directions around the b axis, so that
inversion symmetry is lost, and there is an electric moment along the a axis. The space group is P2lma.
In between the P and F phases are a number of modu- lated phases, commensurate and incommensurate, whose stability varies with pressure and external electric field. In a diffraction experiment, for instance,
at ordinary pressure, satellite reflections appear conti-
nuously at To, with the incommensurate wave vector q = b(To) b*; 1/7. Then b(T) diminishes to
a value of 0.115 at Ti = 170 K. At Ti, b(T) locks
into the commensurate value 1/9, until T~, where 6(T) drops discontinuously to zero. Figure 2 recalls
the phase diagrams for pressure, external electric field and temperature [3].
Landau theory describes incommensurate transi-
tions and phases with the help of an order parameter, whose symmetry is described by the irreducible
representations (I. R.) of the space group of the parent, paraelectric phase, Go. Equivalently, the I.R.’s of the space group G,, of the modulation wave vector can
be used. In the case of thiourea, dynamical studies
show that the order parameter is probably largely displacive [4]. This means that the modulation is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460120218700
2188
Fig. 1. - The unit cell of thiourea projected onto the (a, c) plane in the paraelectric (above) and ferroelectric (below) phases. The light molecules are in the y = 0 plane, the dark
ones in the y = b/2 plane (Elcombe and Taylor, 1968).
Table I. - Equivalent positions of space group Pnma
(cell origin as in international tables of crystallography).
(1) and (1’) are exchanged by the (ay, b j2) mirror.
Fig. 2. - Above : (E-T) phase diagram obtained from bire-
fringence anomalies for increasing and decreasing tempera-
tures. The solid and open squares are related to the 6 =1 /9 -
incommensurate transitions for increasing and decreasing temperature (Barretto et al., 1983). The tear drop shaped
area is the d = 1/8 phase. Below : (P, T) phase diagram
from neutron diffraction measurements. Hatched area
correspond to commensurate phase (Denoyer et al., 1982).
made up of a number of normal modes of vibration :
we assume that there is a primary mode, whose
wave vector is the satellite wave vector qs, and a series of harmonics with wave vectors 2 q~, 3 qs, ...
In that case, there is also a contribution from the star of these wave vectors : - qs, - 2 q~, - 3 qs...
It is interesting to note that the order parameter contains more than one LR. of Go. We know from
the extinction rules for Bragg and satellite reflection observed by X-ray and neutron diffraction which are
the relevant I. R.’s.
Since qs = (0, 6, 0), Gqa = Pn21 a; the four I.R.’s of Gq are listed in table II. The relationship between
the extinctions and the I. R.’s of Gq has been expressed
simply by Janner and Janssen [5]. It turns out that,
in Kovalev’s notation [6], the odd-order modes have T4 symmetry and the even-order modes i 1
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Table II. - Irreducible representations of Pn2la at the point q = bb*.
g = (R ; t) exp(iqt)
symmetry. All of this imposes severe restrictions on
the form of the atomic displacements associated
with the modulation; that is, on the polarization
vectors of the various modes which contribute to the displacements : these polarization vectors are simply basis vectors of the appropriate I.R.’s of Go.
We shall see that though the unit cell contains four molecules with 6 complexes, external degrees of free-
dom each, symmetry reduces these to 6 real degrees
of freedom for the entire cell. We shall see that the
expression of the polarization vectors and the matrices of the I. R.’s of Go convey in turn 1) the space groups of all possible commensurate phases; 2) the free
energy terms relevant to these phases; 3) the super- space symmetry of the incommensurate phases.
In other words, the symmetry of the modulation allows us to predict the symmetry of any commen- surate phases and to compare the order of magnitude
of their energies.
The three points above make up the framework of this paper. A preliminary section describes the construction of the I.R.’s and basis vectors. The
simple expression of the modulation contained in these vectors will serve as a basis for structural ana-
lysis of one of the modulated phases, which is to be
described in a forthcoming paper.
2. Construction of the LR.’s and basis vectors.
We assume that the molecules are rigid units. This is based on various facts. Elcombe and Taylor showed
that the molecules are only slightly deformed between the P and F phases; they also appear to vibrate
rigidly around their average positions. What is more, there is a large gap between the internal vibration
frequencies and the energy of those dispersion bran-
ches which are unstable near the incommensurate transition.
If e(q7:) is the polarization vector of a normal mode
of wave vector q and branch index 7:, then is made up of four molecular polarization vectors
= 1 to 4. Each of these is a complex six
vector.(Tj, The displacement of atom k of mole-
cule j in cell 1 can be written :
+ complex conjugate term
(rGj is the average position of the centre gravity of
molecule j, rkj the average position of atom (kj)).
We are interested in the effect of the symmetry operations of Go on U(lj) = exp(iql) ; these
are the basis functions apt to represent Go. We already
know what Gq does to them, thanks to the table of I.R.’s. The next step is to derive the form of e(q1:)
from this knowledge. If g = (R ; t) is an element of Go which changes molecule (lj) into molecule (LJ),
then the effect of g on the displacement field is to change U by (gU) (LJ) = R(U(/j)), as shown in figure 3. If g is part of Gq and U has i4 symmetry,
then this means that 1:4(g). e(q1:4; J) = R(e(q1:4; j)).
We recall that, by definition, R(Ti’ Rj) = (R(Tj),
det (R). R (Rj)).
This is one way to find, after Moudden et al. [7],
the well-known 1:4 polarization vectors, listed in table III. They describe the displacements in a given
cell with the help of only one complex 6 vector,
(Tj, R1).
Elements of Go which change q into - q introduce
time-inversion symmetry, which simplifies e(q1:) fur-
ther. For instance the mirror plane (Uy; 0) changes e(q1:) into another normal mode e’(- q1:), of wave
vector - q, of i symmetry, with the same energy Since the conjugate mode
also has the energy W_q = wql the two are degenerate.
Since the I.R.’s of Gq are one-dimensional, they are linearly dependent : 0) e(q1:) = exp(- 2 i~p). e(q1:)*.
It is convenient to replace e(q1:) by exp(Üp) e(q1:).
Then :
Thus can be expressed in terms of 6 real
constants. Since qs = (0, 6, 0), it turns out that the transverse part of e(qs) (Ta, T c and Rb) is real, whereas
the non-transverse part ( Tb, Ra and is purely imaginary. The imaginary part vanishes at the zone centre q = 0.
Fig. 3. - Definition of the effect of space-group element
( ~x, 0) on the displacement field : atom 1 of cell I is changed
into atom 1 of cell L, then the point element 6x is applied
to the displacement U~ 1 : [(a., 0) = (lx(Ul1)’
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Table III. - Polarization vectors e(qi) of the unit cell.
Products of(0y; 0) with Gq lead to all of Go. In the
basis (e(qT) e(- qT) exp(- iqo), the matrix
and the matrixes of Gq are
given by the I.R. i of G.. For r = zi, ..., i4 we obtain,
after K. Parlinski and K. Michel [8], the four I.R.’s
T(q,,r) of Go, listed in table IV.
For any given harmonic which contributes to the
modulation, we simply adapt q and r and read off the effect of Go on the displacements.
The actual displacements are a combination of the successive harmonic modes, each with the complex
amplitude Qqt = Displacements
associated with the first harmonic, for example are
where q, (P, Ta, ..., Rc are all real.
This kind of expression is an adequate basis for
structural refinement of the modulation. Meanwhile,
the matrixes of table IV give access to the space groups of any and all commensurate phases.
3. Symmetry of commensurate phases.
If q is irrational, there is no way to extract from Go
a subgroup which leaves the modulation invariant.
If q = (mln) b*, with m and n integers, the modulated structure has a space group, which depends on
6 = mln and on ’to To find it, we start by multiplying
the unit cell by n along b, then reduce as many as
possible of the matrixes T(q,,r) to a real diagonal
form by a suitable change of bases. Then we read off the elements of Go which leave each basis vector invariant. Each basis vector
is a possible modulation : experimental symmetry observations and/or structure calculations determine which is actually present in the crystal.
Three cases appear, according to the parities of m
and n, which correspond to 6 = 1/9, 2/9 and 1/8 res- pectively. The groups induced by T(q, T) are listed in
table V, along with the basis vectors. Usually there
are two important basis vectors for each T, which correspond to the two lines of T(q, T), and which
differ by a phase factor. The free energy will sometimes
2191
Table IV. - Irreducible representations of Go = Pnma at the point q = bb* (T is a lattice transition, a = exp(i 2 nq. T), ~3 = exp(i 2 nq(T + b/2)) ; the cell origin is on the Uy mirror plane).
Table V. - Symmetry of lock-in phases.
Table Va. - n and m odd (b = 1/9 for example).
Table Vb. - m odd and n even (6 = 1/8).
Table Vc. - m even and n odd (6 = 2/9).
distinguish between the two. For example the repre- sentation T(8 = 1/3, T~) can induce the space groups Pnma and Pn2la. The two symmetries have the same
extinction rules. The first case corresponds to the
basis vector e(qs i4) exp(iqs I)), the second to e(qs ~4) exp(iqs 1 + in/6). Notice that qJ(q’t’) is only
defined modulo q. b. (This is why we treat n/6 and n/2 as the same. Cf. Table V).
A general vector of the T(6 = 1/3,T~) representation
is a linear combination of the two just mentioned. It induces the intersection of the two symmetries Pnma
and Pn21 a, in other words Pn21a.
Comparison of experiment with table V confirms the choice of the order parameter (Q(q’t’4)’ Q(- q’t’4)’
Q(2 Q(- 2 q~i), ...), since the space groups we find agree with known extinction rules throughout
the phase diagram, as well as with the ferroelectric symmetry of the 6 = 1 /8 phase.
We have just seen that the 6 = 1 /9 (or 1/3, or 1/7) phase can have Pnma or Pn21 a symmetry, according
to whether the phases qJ(q’t’) are zero or not For the crystal to have Pnma symmetry, all the phases
must be zero (modulo 6n). If not, the crystal is piezo-
electric. It is not ferroelectric, since the displacements
2192
show that the electric moment of the supercell is proportionate to
+ { similar terms for higher harmonics },
which is zero.
The (5=1/8 phase appears experimentally through coupling of the modulation to an external electric field along a, and leads to a dielectric anomaly even
at zero external field. It is thus assumed to be ferro-
electric, which is in agreement with the symmetry induced by the order parameter. There are three pos- sible symmetries induced by T(6 = 1/8, 4). In the
first case, all the phases 9(qT) = 0 and the space group is P21 ma. In the second case, all the phases n/2 + 6n/2 and the space group is P 1121 /a.
In the third and general case, the phases are random
and the space group is the intersection of P2,ma and P 1121 /a ; that is the monoclinic group
Phases such as 6 = 2/17 are expected to have the
same set of possible symmetries as 6 = 1/8.
Table V also makes apparent a curiosity of incom-
mensurate phases. One can always consider an
incommensurate phase to be a long-period commen-
surate superstructure, whose period changes discon- tinuously when 6(T) = varies. In fact, this
discontinuous variation of translational symmetry is the essence of incommensurability. In the same way,
we see that the point symmetry of the superstructure changes discontinuously with T, along with the pari-
ties of m and n.
4. Minimization of the free energy.
The free energy is a polynomial function of the
amplitudes 6 of the normal modes which make up the modulation. It will give us information as to the possibility of inducing a given commensurate
phase, as well as information about its symmetry, through the phases
Strictly speaking, the amplitudes QqT transform
according to the representation ’T(q, T) obtained by transposing T(q, 7:). A monomial Vp Qql Tl Qq2T2 ... *
contributes to the free energy F if it is invariant under Go.
The variation of qs( T) and the domain of stability
of commensurate phases result from competition
between two kinds of terms in F : normal terms, where ql + q2 + ... + qp = 0, and Umklapp terms,
where q, + q2 + ... + qp is a reciprocal lattice vector
other than zero. In the incommensurate phase
reflects the dispersion of the coefficients of the low- order invariants such as A(q) This in
turn reflects competition between more or less long
range forces which couple the atomic planes normal
to qs. When nears a simple commensurate
value, Umklapp terms can lock q. into that value.
The 6 = 1/9 phase is typical of one class of phases (b = 1/3, 1/7, 1/9). It is induced by ninth-order
Umklapp terms, such as
or
Thus it is difficult to induce, and requires a very
large order parameter. The phase of the modulation is important in order to discriminate between the
centrosymmetric group Pnma ((p(q. i4) = 0) and the non-centrosymmetric group Pn21 a (9(q. ~4) ~ 0). The
first terms in the free energy which include the phases
are the normal terms
and
It is easy to see that these terms have a peculiar symmetry : they only include the relative phases
a = 2 qJ(qs T 4) - qJ(2 qs t 1) and = 3 w(qs z4) - 9(3 qs T4)’ In order to find the absolute values of the
phases, we must go as far as the first Umklapp terms.
This is natural, since the absolute phase 9(q. rj
is undetermined, except in the commensurate phases.
Unfortunately, we know nothing about the various
Umklapp coefficients, V9 and V9 for example, so
that no exact value of t4) can be derived Nor is
a simple approximation possible : to neglect the V9 term, for example, would mean neglecting
which we have no reason to do. The only simple
conclusion is that (p(q,, t4) is not zero, so that the
6 = 1/3, 1 /7 and 1 /9 phases have Pn21 a symmetry.
In none of the commensurate phases do we know anything more about T(qT). Only in the special
case where there is only one primary mode is there
a simple minimum at = 0.
We will need the relative phases a and fl in order
to discuss the supersymmetry of the incommensurate
phases in the next paragraph.
In thiourea, Q3q is larger than Q2,, [15], and the
three terms mentioned above must be minimized
simultaneously.
If V4, V6 and V6’ are either all positive or all nega- tive, a and fl are either zero or n. This is also true if
one of the terms is very small. However, if not, a careful look at the three terms will show that the
optimal values of a and fl can be anywhere. (The
three terms may be written :
If A = B = C, for instance, the absolute minimum is a = - ~3 = 2 n/3 ! What is more, in the right
