PHYSICAL REVIEW
Order-parameter
16, 5 UMBER
VO LUME
B
symmetries
for the phase transitions
and higher-order ferroics
1
1
of nonmagnetic
JULY 1977
secondary
Pierre Toledano
Groupe de Physique
Faculte des Sciences, 80000-Amiens, France
Theorique,
Jean-Claude
Toledano
Centre National d'Etudes des Telecommunications,
92220-Bagneux, France
(Received 21 january 1977)
A theoretical investigation is presented of the order-parameter symmetries of all possible secondary and
higher-order ferroic transitions, (nonmagnetic). These structural phase transitions, which are neither
ferroelectric nor ferroelastic, involve the onset of a spontaneous macroscopic quantity represented by a polar
tensor of rank higher than 2. The investigation is performed on the basis of the Landau's symmetry criteria
for continuous transitions. Particular attention is given to transitions accompanied by a change of the number
of atoms in the crystal's unit cell (improper ferroic transitions). Accordingly, the irreducible representations
of 147 space groups are examined, corresponding to all the relevant high-symmetry points of their respective
Brillouin zones. Eight tables summarize the results by indicating the irreducible representations which are
compatible with a secondary or higher-order ferroic transition, and specifying the corresponding space-group
change. For each of the predicted transitions, a free-energy expansion is constructed. Twelve different types
of expansions are encountered, relative to one-, two-, three-, four-, and six-dimensional order parameters.
The form of the term representing the coupling between the order-parameter
and the spontaneous
macroscopic quantity is also determined, These results are discussed, and compared to the existing
experimental data relative to the considered type of transitions, such as those in ammonium chloride, iron
sulfide, or quartz~~-case of niobium dioxide is treated in detail, and a new interpretation is given for its
transition's order parameter,
I. INTRODUCTION
Several currently investigated structural phase
transitions, ' ' such as the o. -P transformation in
quartz, do not involve the onset, below the transition's temperature, either of a spontaneous polarization, or of a spontaneous deformation. These
transitions are neither ferroeleetric4 nor ferro-
elastic. '
However, they have in common several important features with the two preceding, extensively
studied' categories of transitions: modification of
the point group of the crystal, presence of twinning
in the low-symmetry phase, and possible removal
of the twins by externally applied forces. ' This
similarity has led Aizu' to define the concept of
a ferroic crystal which covers ferroelectrics and
ferroelastics as well as other types of crystals
such as those mentioned above. In an earlier
work by Indenbom, ' a comparable generalization
was already implied.
A crystal is said to be ferroic when it can exist
in two or more states having equal stabilities in
the absence of external forces, and when it has
crystal structures which only differ in their spatial
orientations. %hen simultaneously present in a
given crystal, these differently oriented structures
constitute ferroic domains. Each state can be considered as slightly modified with respect to a
'
16
"prototypic" structure of higher symmetry, invariant by the set of symmetry operations of all
the differently oriented states. The definition of
a ferroic also specifies the possibility of reversibly switching the crystal from one orientation
state to another by application of external forces.
However, the latter condition can be ignored when
ferroics are considered from a crystallographic
point of view.
Ferroelectrics and ferroelastics clearly comply
with the former definition. In these two types of
crystals, the different orientation states are characterized at macroscopic level by the direction of
a. spontaneous polarization or by the orientation of
a spontaneous strain tensor. In a similar way,
the orientation states of all ferroics ean be characterized by tensorial quantities which take different values in the different states when referred
to a common system of axes. One can distinguish
various types of ferroies on the basis of the rank
of the lowest-rank polar tensor" characterizing
the different orientation states of the crystal.
Thus, polar tensors of rank one (polarization),
two (strain), three (piezoelectric modulus), and
four (elastic modulus), are, respectively, associated with ferroelectrics, ferroelasties, and the
new types of ferroics labelled ferroelastoelectrics
and ferrobielastics.
More complex ferroics are
also conceivable" for which the relevant polar
"
"
386
16
ORDER-PARAMETER
SYMMETRIES FOR THE PHASE. . .
tensor is of rank higher than four. On the other
hand, axial tensors can be considered, leading to
the definition of magnetic ferroic species. For
instance, ferromagnetics are those ferroics which
are characterized by an axial vector.
A classification of ferroics has been based'
on
the form of the difference of free-energy hI" between two orientation states, exyressed as a
the mechanipower series of the electric field
cal stress o„-, and the magnetic field H, A primary ferroic is defined as one for which the
lowest-power term in the
expansion is linear
in E,-, o, &, or H, . Primary ferroics are composed
"
E„
~
.
of ferroelectrics, ferroelastics, and ferromagnetics. When the lowest-power term in AI" is a
bilinear function of the above forces, we have to
deal with a secondary ferroic. Finally higherorder ferroics include all ferroics for which the
lowest-power term in bE is of degree higher than
"
two.
As seen from a symmetry point of view, it is
possible to identify nonmagnetic secondary ferroics as those characterized by polar tensors of
rank three or four. Likewise, nonmagnetic higherorder ferroics correspond to polar tensors of
rank higher than four.
Although the preceding considerations do not
imply that a ferroic crystal undergoes a phase
transition, it is clear that, similarly to ferroelectrics and ferroelastics, most ferroics will
transform at a certain "Curie" temperature into
a phase possessing the symmetry of the prototypic
structure. This phase transition can be studied by
using the Landau's theory of phase transitions"
and, in particular, an order parameter can be
defined for it. In the case of ferroelectrics and
of ferroelastics, and a number of studies" "have
used the Landau's theory in order to account for
the symmetry change observed at the Curie point.
They have also shown that the behavior of the
macroscopic quantities (such as the polarization
or the dielectric susceptibility) on varying the
temperature closely depends on the symmetry
properties of the transition's order parameter.
No systematic investigation has been performed
symmetries and
up to now of the order-parameter
possible symmetry changes relative to secondary
and higher-order ferroic transitions (SHFT). An
additional stimulation to such a work can be drawn
from the recent results" obtained on niobium dioxide, a material which can be classified as a
secondary ferroic" (ferrobielastic). It was pointed
out by Mukamel" that a new situation occurred for
the transition of NbO, with respect to the renormalization-group theory of critical exponents,
owing to the unusualf our-dimensionality
of its order
parameter and to the particular tetragonal aniso-
""
"
tropy of the expansion'
associated with its transi-
tion.
In this paper we present an exhaustive study of
the order-parameter symmetries and spacegroup changes for all possible SHFT. This theoretical analysis has been performed on the basis
of Landau's symmetry criteria for continuous
(second-order) transitions.
It is restricted to
nonmagnetic ferroics. Accordingly we have examined the irreducible representations of 147
crystallographic space groups which are likely to
constitute the invariance group of the prototypic
phase. Consideration of the space-group representations instead of the simpler point-group ones
is known to be necessary if one wants to account
for the transitions which are accompanied by a
modification of the crystal's primitive transla-
"
tions.
The paper is divided into three sections. In
Sec. II we briefly review the method used to apply
the Landau's theory to the SHFT. Extensive use
is made of results obtained in a similar work"
in which we analyzed the order parameters of
ferroelectric nonferroelastic transitions. In Sec.
III the results are summarized in table form for
the different types of ferroics and for the different
crystalline systems. For all the predicted transitions, free-energy expansions are constructed.
Finally, the available experimental data are compared to the theoretical results. Particular attention is given to the controversial case of niobium
dxoxzde.
II. THEORETICAL PROCEDURE
A secondary or higher-order ferroic species is
defined by the point symmecrystallographically
tries of the ferroic and prototypic phases.
Thi. s
set of point groups determines entirely the rank
and particular components of the tensorial quantities which distinguish the different orientation
"
states of the ferroic crystal. Newnham and Cross"
have worked out the 34 secondary or higher-order
ferroic species which can be decomposed into 15
ferroelastoelectric, 5 ferrobielastic, 10 simultaneously f erroelastoelectric and f errobielastic,
and 4 higher-order ferroic species. We have reproduced them in Table I together with the corresponding tensorial components characterizing the
ferroic states. The prototypic phase belongs to
one of 17 crystal classes of the orthorhombic,
tetragonal, trigonal-, hexagonal, and cubic systems. We note that the two point groups defining
each species belong to the same crystalline system, providing that the hexagonal and trigonal systems are considered as a single system.
PIERRE TOLEDANO
AND JEAN-CLAUDE
TOLEDANO
Secondary- and higher-order ferroic species. The set of point groups defining each
species is indicated, as well as the tensoria1 components characterizing the different ferroic
domains. d~~ are components of the piezoelectricity tensor in the Voigt contracted notation
of the indices {Ref. 11). Likewise C;, are components of the elastic stiffness tensor. L~;& is
a polar tensor of rank six (third-order elastic constants). All tensors are referred to the
standard setting of axes (Ref. 11}. For most species the change in the number of symmetry
operations is twofold, and there are two domain orientations characterized by opposite values
of the tensorial components. Some species involve a fourfold decrease of point symmetry and
four types of domains. For these species four sets of tensorial components distinguish the
domains. We have only indicated the set for one domain orientation.
Type of
Ferroic species
ferroic
Fe rr oelas toelec tric
mmm
4/m
4/mmm
4/mmm
3m
6
di4
6/mmm
6
6m2
diii
6/m»z~n
6/mmm
622
23
23
43m
23
dii
d22
d22
di4
d i4
di4
di4'
Liis-Lii2
L i88 —L i55
4/~mm
3m
4/m
6/m
(Ci4' C25)
6/mmm
3
3m
3
42m
4
(di5, d, i,
(di4, Ci8)
3m
6
6m2
622
3
3
6/mmm
32
32
3m
3
32
6/mmm
6/m
43m
m 3m
m3m
23
m3
432
6mm
6mm
ferroics
of Landau's theory to the SHFT
I andau's theory of continuous phase transitions"
shows that the order parameter (OP) of a transition transforms according to a physically irreducible representation (IR) of the space-group G,
of the high-symmetry phase of the crystal. A
physically IR is either a real IR of G, or the direct
sum of two complex-conjugate IR of 6,. In order
Ci8
Ci5
Ci4,. C25
.
i4~
i5~
(dii,. di4,
4mm
4/mmm
A. Application
(di4, dii)
(dii~ d22)
6m2
6/ma'am
Higher-order
(di4~ dw)
'
6mm
m3
432
m3m
m3m
Ferroelastoelec tric
and ferrobielas tic
422
42m
32
(d i4 q d 25 ~ d38)
(di4~ di5q d3iq d38)
6
6
6/m
Ferr obielastic
222
Spontaneous tens orial
components
(dii,
(di4~
(dii'
(dii,.
C,8)
Si~
38 ~
i8
Ci5}
d22, Ci4, C25)
Ci4)
Ci4)
Ci4)
dii~ d22~ i4~
d iis di4~ C i4
i4~
25
(L ii8 L i45)
(
ii3
+ ii2~ L i88
i55)
(L ii3 —L ii2,' L i88 -L i55)
¹nth-rank polar tensor
to determine the OP symmetries of all possible
SHFT taking place between a ferroic phase and
its prototype, it is therefore necessary to examine
the IR of all the space groups associated to the
17 crystal classes mentioned above. There are
147 such space groups.
Each IR of a given space group can be denoted
I'„(0*) and identified by two quantities.
The star
k", represented by a k vector of the first Brillouin
"
SYMMETRIES FOR THE PHASE. . .
ORDER-PARAMETER
ter
and to definite high-symmetry"
points of the
BZ surface. The latter points were listed in TT
for all the crystalline systems but the cubic one.
Table II completes the list and contains the acceptable points of the cubic Brillouin zones. In addition, for each acceptable BZ point, and for each
space group, some of the IR are not compatible
with the (I')' criterion. They can be found by
using the space-group character tables, according to a procedure indicated in TT.
The [I']' criterion is necessarily obeyed by all
the IR whose stars verify the relation
zone (BZ), specifies the translational symmetry
properties of the basis functions of I'„(k*). The n
index specifies the small IR, denoted v„, of the
little group" G$). The dimension of I'„(k*) is the
product of the dimension of v. „and of the number of
arms in the star k*. This dimension is equal to
the number of components of the OP. A given
space group has an infinite number of IR, as an
infinity of k vectors corresponding to distinct
"
stars can be found in the BZ. However, the number of these IR which have actually to be considered in the study of an SHFT is reduced by several
restrictive rules.
In the first place, if we limit the scope of the
present work to continuous (second-order) transitions between strictly periodic crystalline structures, the general theory of Landau and Lifschitz'4
k, +Pc, +%, 40
(1)
for any (k, , %, , k, ) belonging to the considered star
jgg
When the preceding rule is not verified an IR
can nevertheless satisfy the [I']' criterion. However, in this case, one has to work out the matrices of the considered IR in order to determine
the symmetry properties of the basis functions of
I'„(k'). One is then able to check directly condition (b) by finding out if an invariant exists in the
imposes two conditions to an acceptable IR:
square, noted (1']2 of
(a) The antisymmetrized
I'„(k") must not have any IR in common with the
vector representation of G, .
(b) The symmetrized third power of I'„(k*),
noted [I']', must not contain the totally symmetric
IR of Go.
The physical meaning of these criteria, as well
as the detailed procedure of their application have
been recalled recently by the present authors in
the work mentioned above, referred to hereafter
[I']' space.
"
Several additional restrictions have to be considered in the study of an SHFT. As emphasized
a given IR of the space group G, is
by Birman,
compatible with a transition between G, and one
of its subgroups G, if the restriction to G of
I'„(k") contains the totally symmetric IR of G.
When dealing with a SHFT any subgroup G of G,
"
"
as TT.
First, the (1]2 criterion selects a few acceptable
stars in each BZ which correspond to the BZ cen-
TABLE G. Modifications of Bravais lattice occuring in the cubic system and related to
high-symmetry points of the BZ. The considered points correspond to a k vector whose
invariance point group possesses a central point (H, ef. 14 and 17). The labelling of the points
is the one used by Zak (Ref. 24). Column (a): coordinate of the k vector defining each point
and referring to the primitive translations of the reciprocal lattice. The Bravais lattices of
the two phases belong to the same crystalline system.
High
symmetry
Brillouin-
Bravais
lattice
zone
point
P
Ferroic
Bravais
lattice
Number
of alms
in the
(a)
star
P
0 0 0
F
0-,'0
1
2
1
2
1
1
2
2
1
0
2
0 0 0
H
N
Pa
1
P
I
i
1
1
2
2
2
1
1
1
0 0
1 1
4 4
6,
2
4.
0 0 0
1 '1
2 2 0
3 1 1
4 2 4
1
2
Brillouin-zone
1
2
"
1
2
points whose k vector is not equivalent
to
-k.
Unit-cell
expansion
PIERRE TOLEDANO
390
AND
is not acceptable. The following conditions must
be respected:
(c) The change in point symmetry between G,
and G has to coincide with one of the species of
Table I.
(d) As shown in TT, when a transition preserves
the crystalline system, the Bravais lattice of G is
related to that of G, and to the star
unambiguously
of the considered IR. More precisely, the translation group of G is composed of all the translations T which verify the set of equations
\
where k,. is any vector in the star 0*, and Tbelongs
Go These equations correspond to the maximum
loss of translational symmetry compatible with
the star k*. Thus each BZ point is associated
with a change of the Bravais lattice indicated in
Table II for the cubic system and indicated previously in TT for the other systems.
(e) The nonprimitive translation t associated
with each point operation of G has to be equal
either to the corresponding nonprimitive translation t, of G, or to the sum of t, and a primitive
translation of G, not belonging to G, i.e. , not fulfilling Eqs. (2).
In summary, the procedure to determine if an
SHFT is induced by a given IR of G, is to select
the subgroups of G, whose point group, Bravais
lattice, and nonprimitive translations are specified
by conditions (c)-(e), and to check with the help of
the space-group character tables" the compatibility of these subgroups with Birman's condition.
Two rules demonstrated in TT are useful to
systematically discard many IR:
(f) When I'„(k*) is a real and one-dimensional
IR, with k4 0, it induces no change in the point
symmetry of the crystal.
(g) Let I'„(k*) be in IR whose star k* has two
arms with each K equivalent to -%. If the small
no pointIR (7„) is real and one-dimensional,
symmetry change occurring within the crystalline
system of G, is compatible with I'„(k").
We note that the procedure outlined above sets
necessary conditions for the occurrence of an SHFT
and allows us to work out the corresponding
space-group change. It does not warrant that the
determined ferroic phase constitutes the actual
equilibrium state of the crystal. In agreement
with the Landau's theory,
one must check that
this phase corresponds to the absolute minimum
Howof the transition's free-energy expansion.
ever the former procedure restricts the necessity
of constructing a free-energy expansion to only
those cases where the former necessary conditions
to
"
are verified.
"
JEAN-t LAUDE TOLEDANO
B. Proper and improper ferroie transitions
The free-energy expansion E is a function of
the quantities varying rapidly in the vicinity of
the transition, i. e. , the order parameter and the
relevant macroscopical quantities which can
couple to it. I" is an invariant function under the
symmetry operations of the space group G, .
At a SHF T, the lowering of point symmetry
determines the onset of nonzero "spontaneous"
values for certain tensorial components which
vanish by symmetry in the prototypic phase.
These macroscopical quantities are identical to
the ones characterizing the different orientation
states of the ferroic phase, and are therefore
those indicated by Table I for the various species.
They must be included in the free-energy expansion, as their variation is significant near the
transition. Consequently the expansion I' will
contain three kinds of invariant terms.
The first ones are invariants of even powers of
the OP components. In general, for discussing the
symmetry change, one can limit the expansion to
the fourth power. The second ones are invariants
of the relevant tensorial components. Quadratic
terms are sufficient to derive the behavior of these
The third kind of term is comcomponents.
posed of "mixed" invariants of the OP and of the
tensorial components; they represent the coupling
between the OP and the tensorial components.
All the preceding invariant terms can be construcusing the
ted by standard projector techniques,
matrices of the IR associated with the transition.
Similar to the case of ferroelectric and ferroelastic phase transitions, one can distinguish two
types of SHFT on the basis of their OP symme-
"'"
""
"
tries ~5 16
A "proper" SHFT will be characterized
"
y
by an
OP transforming according to the same IR of Go
as the tensorial components relative to the considered ferroic species. Such an IR necessarily
corresponds to the center of the BZ of G, (I' point).
It induces no modification of the translational
symmetry of the crystal. As an SHFT preserves
the crystalline system, the Bravais lattices of
both phases are the same. The free energy E of a
proper SHFT could, in principle, be expanded exclusively as a function of the tensorial quantities
taken as the OP of the transition. In Sec. III we
have preferred to use an OP, physically unidentified, coupled linearly to the various tensorial
components possessing the same symmetry prop-
erties.
An "improper" SHFT has an OP transforming
according to an IR of G, which is distinct from
that of the spontaneous tensorial components.
The onset of nonzero values for the latter quanti-
ORDER-PARAMETER
i6
SYMMETRIES FOR THE PHASE. . .
ties is a secondary effect of the transition, arising
from'the nonlinear coupling of these macroscopic
components to the OP. In particular, all the SHFT
whose OP is associated with a nonzero R vector of the
BZ of G, are expected to be improper. Levanyuk
et a/. have shown in the case of improper ferroelectrics that the mixed invariant in F must deOtherwise no
pend linearly on the polarization.
spontaneous value of this quantity can arise at the
transition. Likewise, at an SHFT, the mixed invariant has to be linear in the relevant tensorial
quantities. Consequently, the power of the OP in
this term has to be greater than one since the
coupling is nonlinear.
The power of the OP contribution to the mixed
invariant has been defined as the faintness index
of the. transition with respect to its ferroic properties. Thus a proper ferroic has a faintness index
equal to one, while an improper ferroic is characterized by a faintness index greater than one.
The value of the faintness index determines the
qualitative behavior of the various macroscopical
quantities near the transition. For instance, at a
ferroelastoelectric transition having a f aintness
index of four, the spontaneous piezoelectric components will vary as the fourth power of the OP.
"
"
III. RESULTS
AND DISCUSSION
In this section we describe the results relative
to the order-parameter symmetries and spacegroup changes for proper and improper SHFT and
compare them to the experimental data.
The OP symmetries for proper SHFT have also
Low-symmetry
space group
Pmmm
P222
P222
P222
P222
Pnnn
Pccm
Pban
Pmma
Pnna
Pmna
Pcca
P 21212
P2212
P2221
P2122
Pb cm
P 21212
P 21212
P 22121
Pnnm
Pmmn
&21212
Pbam
Pccn
Pb cn
21212
P 21221
"
been examined in a recent work by Janovee et al.
who studied the symmetry changes induced by the
point-group representations.
The OP of the proper
SHFT are all one dimensional and they correspond to a lowering of point symmetry of the crystal by a factor of 2. Hence several of the ferroic
species listed in Table I, which involve a greater
decrease of point symmetry, cannot occur during
a proper SHFT (for instance, m3m -23, or
"
-
4/mmm
4).
The results relative to the OP symmetries of
continuous SHFT are summarized in Tables III-X.
For the sake of completeness we have included in
the tables the proper transitions previously treated by Janovec et al.
For these transitions we
have worked out, effectively, the equitranslational
subgroups of each of the considered space groups;
the former authors only indicated the change in
point symmetry.
"
A. Organization
and use
of the tables
Tables III-VI are devoted to ferroelastoelectric
(FEE) phase transitions, Table VII to ferrobielastics (FB), Tables VIII and IX to transitions
which are simultaneously
FEE and FB, and Table
X to higher-order ferroic transitions (HOF). In
each category the tables are ordered according to
the crystalline system common to both phases of
the crystal and to the space group of the high~
symmetry phase.
First, for each space group the IB which are
compatible with the (I'P criterion are specified.
As pointed out in TT, this criterion is necessarily
TABLE III. Ferroelastoelectric transitions in the orthorhombic system. A single transit1on is possible at a BZ boundary point and is not shown in the table: Fddd F222 (72) at
the & point (orthorhombic F lattice). It involves an eightfold expansion of the unit cell, and
corresponds to the (ft} type of free-energy expansion (see Table XI). All other transitions
occur at the BZ center and are indicated in this table. The two phases are equitranslational.
High-symmetry
space group
891
High- symmetry
space group
Pbca
Low-symmetry
space group
P2 12121
Pnma
P 212121
Cmmm
C222
C2221
C2221
'
C222
C222
C222
F 222
F 222
I 222
I 222
Cmc m
Cmca
Cccm
Cmma
Ceca
Fmmm
Fddd
Immm
Ibam
Ib ca
Imma
I 212121
I 212121
PIERRE TOLEDANO
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lw
oo ~ ~
0
o
IW
p
8 K
N
m
h
IW
0O
0
V
cd
h
r
4Q
Cd
N
~4
I
o ~
0 o
cd
A„'g
cd
~ h~ +
cd ~
I&
cd
Q ct
O
cd O
CI
0 0
~ .
g 0
~
O
0
0
cd
0 g
0) M
0
g)
a)
cd
CI
P
O
s
O
cd
~~ 'M
Q
chic
Q
&
0
'g 8
E
ft
co
.M~ 0
~
Q
O
o
~N~
cd
O
cd
4 ~
444
~ ~~ 4
~ 4 ~
~ ~ ~
~
~ 4~~
o
p
0 Hcd
t-
Q)
0
0
Q
CCI
~I
IW
cd
~~ ~4
t-
0) cd
O
I
W
IW W
ORDER-PAB, AMETER SYMMETRIES FOR THE PHASE. . .
16
TAB? E V. Ferroelastoelectric
in Table IV. For the I",
Highsymme try
space-
BZ
(I )2
point
I,
transitions
and Z' points,
Ferroelastoelectrio
space-group
(a)
(b)
r
A
all
L
FI
P3lc
M
all
H
~
r
}
A
all
L
8
r
A
all
M
a
P312(Ts)
3
}
A
all
K
r
g
X
A
f
r
A
H
K
P312
all
Ti ~
T4
B2
space-
(a)
(b)
point
group
all
L
3
PG
K
H
a
3
Ti
T4
r
PGmm
P6(T2)
A
M
8
all
H
P321
1
a
P321
3
8
K
P6cc
1
a
3
8
P6(T2)
P6
2
1
d
a
all
P6
3
8
Tit T2
PG
2
d
P6s
1
a
all
P6s
3
8
P6s
P6(Ts)
Ti
T2
K
r
P62c
L
P321
1
a
P321
3
8
I'
P6scm
A
M
L
R32
0
a
3
e
K
I
P6smc
A
M
PGm2
3
K
I'
s
all
L
P6(T4)
0
0
1
3
H
a
8
PGc2
K
I'
T 1 T4
Ti, T2
K
all
L
H
I
A
M
d
L
H
PGs
3
8
d
1
a
P6(T2)
3
8
0
K
P6s/mcm
I'
A
M
1
3
a
8
8
PG(T2)
2
1
d
PG
P6
3
8
1
a
PGm2(T))t
(P622(T5)t
P 62m(Tp)
3
8
P622(T, ), P62m(T2)
P622, P6c2, P62c
2
d
1
a
P622, P6c2, P62c
3
8
P 6222, P 6c2, P 62m
1
a
P6s22, PGc2, P62m
3
e
P6s22, P62c
2
d
P6s22, P6m2, P62c
1
a
P6222, P6m2, P62c
3
8
a
I
74t T5
all
Tit. T2 P622, P62c
74t T5
all
H
K
Tit
T2
74t T5
P6s/mm c
0
0
a
3
P 62m (Ts)
H
a
2
1
P6
P622(T~), P6m 2(Tip),
all
1
P6(T,)
P6(T2)
T2+ Ts
it T2 PG(T2)
A
M
2
P6
A
M
K
all
P6s
PGs
T4, 75
r
P6/mcc
'
it T2
A
M
8
2
all
I
H
R32
Ti t
(b)
T4t T5
H
1
(a)
TitT2
M
K
H
K
)
~
H
0
t
A
PG/mmm
A
M
all
8
Ferroelastoeleotric
space-group
( I'j 2
r
A
L
P6 (T2)
L
space-group
P62m
A
M
8
~
~
2
I'
P6s/m
~
all
$ I')
point
.
Highsymmetry
Ferroelastoelectric
H
R32
L
H
a
~
R3m
PG/m
1
BZ
~
I
L
R3c
P312(T5)
P312
L
P3c1
space-
~
I
P3m 1
Highsymmetry
group
group
P31m
system. All columns have the same content as
in the trigonal-hexagonal
is indicated for the first space group of each class.
v„
A
M
I
H
K
Ti T2
P6222, P62m
T4t T5
Ti
T4
fulfilled by real one-dimensional IR. Thus, the
tables only indicate the results for multidimensional IR which are identified by the corresponding
BZ point and by their small representation v„,
whose labelling refers to the space-group character tables of Zak et a/. 24 Some of the crystal
classes (for instance, 4/mmm or msm) appear
in several tables. The results relative to the (I'}'
criterion have been reproduced only in the first
table where they occur.
As stated in Sec. II, transitions induced by an
IR not fulfilling the (I'}~ symmetry criterion are
outside the scope of the present work. Consequently, the compatibility of a given IR with an
SHFT has only been investigated for those IR
which fulfill this criterion.
When an SHFT is compatible with I'„(k*), the
space group of the ferroic phase is specified
v„relative
together with the small-representation
to I „(k*). It can happen that an identical spacegroup change is determined by several IR corre-
In that case, the
symbols for the small representations are grouped
together between brackets following the ferroic
space group. It can also happen that SHFT towards
different space groups are induced by the same IR.
This situation is indicated by grouping the symbols
of the various ferroic space groups between
brackets. When I'„(k*) is associated with another
type of symmetry change (ferroelectric, ferroelastic, or equi-orientational), no space group
has been indicated for the low-symmetry phase.
In addition, for each predicted SHFT, the tables
specify the dimension of the OP and the type of
free-energy expansion associated with it. The
detailed forms of these expansions are given in
Table XI. The unit-cell multiplication which accompanies the transition, is determined by the
translation symmetry properties of the OP and
can be found in Table II of the present work for
the cubic system and in Table II of TT for the
other crystalline systems.
sponding to the same BZ point.
PIERRE TOLEDANO
TABLE VI. Ferroelastoelectric
Hrghsymme try
BZ
space-group
point
Pm3
transitions
AND JEAN-CLAUDE
in the cubic system.
Q'3
space-group
F
e try
space- group
symrn.
(a)
(b)
16
All columns have the same content as Table IV.
High-
Ferroelas toelectric
TOLEDANO
BZ
point
Ferr oelas toelec tric
{I'j2
s pace- group
all
X
I23(75, 78), I2(2(76, 77)
F
P23
R
1
all
X
+23
all
P23(&59 TY), P2(3(769 78)
e
W
L
F
all
H
73
1
a
I43m(77, 78)
3
e
I2&3(7 ip)
6
i
a
I2,3
0
1
3
P 43m(7 7)
Pm3m
all
&23
I
L
~
Pn 3n
all
2+
X
R
X
Pm3n
F
X
~
M
F
I2(3
Pn3m
e
all
X
M
P432
X,7, , ~, I23(73),
M
F
X
7, , 7,
.
W
~
I2P (7 2)
E23
P 23(7'2, &3)
3
1
3
e
a
e
I'm3c
L
F
X
7i
q2
7'29 73
)
~7'4, 75
M
F
X
P23
I2)3(v 3, 74)
E23
(729 T3
P2~3(T2 73)
1
0
739 74
I43d (77 7 gp)
I23(76)
P43m
all
Sl
4
I2)3 (74)
73
N
all
I'
I
T2+ 13
71
6
1
e
i
a
P43m(7 ~, 7'8)
P2(3(Tip)
3
+43c
1
3
7f 9 7
73
1.
i
a
e
~
T4
all
P43m(7'8), P43n
(7'6)
4
6
a
e
i
E43m
P2(3(74)
W
e
F
a
X
e
all
39 4
P43m(T f
T4)
+43c
+43c(72, &5) 4
1
P 2~3(74)
f2
6
a
i
6
h
1
a
W
e
F
3
6
&43m
3
3
1
3
i
a
T3978
F
a
749 T5
2
1
P2P(7m)
e
e
0
R
6
~2
73 74
F
I23(72)
P43n
e
7f 972
T3
R
1
4
F
X
P23(7-2)
P43n
72
F
R
~
I
3
T3+ 74
P2(3
all
I
P
M
T3
Ti
F
F
R
I23
I23 (7'3), I 2(3 (7'4)
P4332
a
all
F
X
M
~
I
M
I432
1
e
M
I'4@2
(b)
P23(&5)
R
E432
(a)
~3(7g 9 72)
I23
e
4
1
f2
Im3m
L
F
H
N
a
P
I23(. 4), I2P(73)
Ia3d
F
I43m
39 8
all
19
T3
I43d
73+ T4
P2(3
72
N
P
e
I43m(7, ), I43d (~,)
ef
ORDER-PARAMETER
16
SY
MMKTRIKS FOR THE PHASE.
..
TABLE VII. Ferrobielastic transitions. The only BZ points indicated are those for vrhich an FB transition is found
possible. The results relative to the (I'} criterion can be found in Tables IV and V. Columns (a) and (h) have the
same content as in Table IV.
High-
Highsymmetry
space
gr oup
point
P4/mmm
F
P4/mcc
sym. metry
l
A
z
P4/nbm
P4/n nc
P4/m bm
F
r
r
M
P4/mn c
A
r
M
A
R
P4/nmm
P4/ncc
P42/mme
P42/m c m
P42/nb
c
P4, /nnm
P42/mb
c
P4, /mnm
P42/nm
c
r
r
r
r
r
r
r
M
r
M
R
A
P42/n cm
r
A
I4/mmm
I 4/m c m
I 4(/amd
I 4(/a c d
Ferrobielas tie
space group
BZ
r
r
r
I 4/m
I 4,/m
I4/m(~~,
P 4
~2)
/m (Tf ), P4 2/m (&2)
1
1
a
2
2
b
b
a
a
a
b
b
a
b
b
P31 c
g
P6/m
1
1
1
I 4/m
2
2
P4/m
P4/m (7,),P4/n(7, )
(Tq, &, )
P4/m
P4/m (Tg), P4/n (~6)
I 4/m (7 ), 7'6)
I 4/m (~&, 72}
1
2
2
4
1
1
1
1
1
1
1
P4/n
P4/n
P4, /m
P4,/m
P4, /n
P4, /n
P4,/m
(&6)
2
P42/m (7 g), P42/n (T6)
I 4&/~ p&, ~2)
2
P42/m (v ~),P42/n
P4, /m
P4, /.
I 4, /a (7 „72)
P4, /n
I 4~/a
I 4/m
I 4/m
I 4,/a
P g, ~g}
I 4)/a
1
a.
a
a
a
a
a
a
a
b
a
b
4
1
g
2
b
1
2
1
1
1
1
are
P31 m
P3ml
P 3c1
a
b
a
a
a
a
Ferrobielastic
space group
r
r
M
r
M
r
r
M
r
P3(~2)
P3(v2)
M
P3
P3
P3
P3
P3
P3
1VI
R3c
R3
R3
R3
R3
r
P3(~8)
P3(72)
P3 {74)
P3
P3
P3
K
r
M
K
P 6/mmm
r
M
P6/m cc
K
r
M
X
a
Let us clarify the use of the various tables by
searching the possible SHFT arising from the
high-symmetry phase Em3. Table I shows that
the m3 point group gives rise to a single seeondaryor higher-order ferroic species, i.e. , the FEE
species m3-23, characterized by the onset of the
piezoelectric component + dye Referring to Table
VI, dealing with FEE transitions of the eubie
system, we find that for the Em3 space group,
FEE transitions are only possible at the I' and X
IR at the
points of the BZ. The multidimensional
S' point are discarded because they do not fulfill
the f1 }' criterion, while the IR at the I, point induce symmetry changes which are not FEE. At
the I' point, the (r, ) one-dimensional representation determines the equitranslational symmetry
change Em3-F23.
At the X point, two possible transitions
Bz
point
(b)
P4, /n
P4/n
space
group
(a)
P63/mcm
r
M
P63/mm c
X
r
M
K
P 31m (&4),P3m 1(7'3)
P 31m (~,),P3m 1P-,)
P 31m (7'4), P3m 1(7'5)
P 31c,P3 c1
P 31c,P3 c1
P31c,P3 c1
P31m, P3 c1
P31m, P3 c 1
P31c,P3m1
P31c,P3m1
P 31c,P3m 1
P 31m, P3 c 1
(a)
1
3
1
3
1
3
1
3
1
3
1
3
1
(b)
a
e
a
e
a
e
a
e
a
e
a
e
3
a
e
2
d
1
3
e
2
d
1
a
e
3
a
2
d
1
3
a
e
2
d
3
e
2
d
3
a
e
2
d
1
1
a
predicted: Em3-P23 and &m3 -P2, 3. For both
Table II indicates a fourfold expansion of the unit
cell. The first symmetry change can be induced
(Table VI) by two different IR identified by the
small representations (v, ) and (v, ) of Zak's tables.
Likewise the second transition is induced by (v, )
or (r, ).
B. Predicted
symmetry
changes and experimental
data
F
Newnham and Cross" have listed about 30 substances illustrating the various ferroic species.
These substances ean be divided into two groups.
The smaller group (Table XII) comprises crystals in which the existence of an SHFT is well
established. This is, for instance, the case of
the n Ptransformatio-n in quartz. For these crystals, structural and physical measurements are
"
PIERRE TOLEDANO
396
TABLE VIII. Simultaneous ferroelastoelectric
have the same content as in Table IV.
Highsymmetry
space-
BZ
group
point
P4mm
fr)
r
space-group
(a) (b)
P4(T, )
i
a
BZ
group
point
4
I4cm
i
a
M
P4(T, )
2
b
A
I4(Ti)
2
b
P4cc
P4nc
p42
M
all
R
~
all
r
all
Rj
~
r
X
all
Z
0
point
i
a
P4m2
2
}
r
[P4f, P43i] (T, )
Z
Ti
b
~
all
~
Z
Ti
N
A
~
l
b
b
a
2
b
I4f
2
P4(T, )
i
0
P4
M
all
Z]
T2+ T4
b
r
P42
i
a
T2
r
P42im
M
P42mc
Z
Tf
A
P42bc
&j
r
all
M
Tf
Ti
Z
A
R
~
[P4i, P43](T )
2
I4f/a
2
b
b
i
a
2
2
b
b
(T f)
}
Z
a
A
j
0
P4
J 4, (T, )
[P4i, P43] (Ti)
a
p4
i
a
i
a
i
a
4
~
P4
T3+Tg
T3+74
all
r
X
i
a
N
A
I4c2
r
T, +T,
r
M
Z
i
+ T5
T3+ T4
Tfa T2
I4
all
Nj
P43mc
P4
0
Z
P42c
2
all
M
T2+ T5
T3+ T4
I4m2
a
I4(T, )
a
7,
all
all
b
A
i
73+ T4
~
b
X
2
T2+
a
R
P4
P4(Ti)
all
X
r
(a) (b)
0
r
M $
r
A
X
ll
M
a
space-group
P4
R
P4n2
P42m
i
i
P4b2
[P4i, P43] (Ti)
F e r roelasto electr ic
and Ferrobielastic
Jr)
all
M
[P4i, P43] (T f)
system. All columns
r
z(
N
A
a
2
2
P4c2
~
~
X
X
a
ll
I4f
I6
A
R
i
I4
a
all
~
M
[P4i, P43i] (Ti)
I4, (T, )
P4
A
r
1
p42
P42(Ti)
A
group
M
r
N
A
I4imd
~
r
BZ
(b)
N
P4
z
P43nm
I4
space(a)
all
A
r
Rj
Ferrobielastic
space-group
in the tetragonal
Z
X
P43cm
(rP
I4mm
R
P4bm
and
TOLEDANO
Highsymmetry
F erroela stoelectr ic
space-
X
all
ferrobielastic transitions
and
Highsymmetry
Ferroelastoelectric
and Ferrobielastic
M
Z
AND JEAN-CLAUDE
P4
i
a
P4(Ti)
2
b
I42m
all
r
I4
Z
I4(T, )
A
~
2
X
b
all
N
A
P42fc
r
M
all
Z
+ T4
T3+ T5
i
a
P4(T, )
2
b
I4(T2+ T3, T4+ T&)
2
c
I42d
T2
all
Bj
P4
X
T3y T4
.
4/mmm
Class
a
T2+ T4
T3+ T3
~
all
A
~
generally available for both phases.
The second group (Table XIII) includes crystals
in which no SHFT has been detected up to now.
Also, the ferroic character is still speculative
since no twinning has generally been reported.
Newnham et al. have assigned ferroic properties
to these compounds on the basis of the available
structural data for a single phase. Each of the
listed crystals possesses, besides its actual symmetry, a pseudosymmetry which ean be attributed
to a conjectured prototypic phase. The actual
symmetry Rnd the pseudosymmetry are such as to
allow a classification of the considered crystal in one
of the secondary- or higher-order ferroic species.
In the former procedure, one uses implicitly a
r
z
I4(T„Tq)
~
4
f2
transitions
to the 4 class
No
strQctural bRsls for ferrolelty slmllRl to thRt indicated by Abrahams and Keve" for ferroelectrics
and ferroelasties.
Following these authors, a
crystal is recognized as a possible ferroic if pairs
of atoms exist in its structure unrelated by the
space group G of the crystal, and whose coordin- .
ates (x„y„z,) and (x„y„z,) can be related through
transformations of the form
(x„y„z,) =f(x„y„z,)+la, (x„y„z,),
f
are space-symmetry operations not belonging to G, and 6 are small displacements of
the order of 0.1 A. The cancelling of these displaeements will determine a phase with higher
symmetry whose space group Q, is obtained by
where
SYMMETRIES FOR THE PHASE. . .
ORDER-PARAMETER,
f
combining G and all the above
operations.
At the experimental level, possible ferroic
crystals can often be detected in the available xray data by the fact that an ambiguity remains on
the actual symmetry of the crystal. This ambiguity
can be induced in insufficiently precise measurements by the occurrence of the pseudosymmetry.
It can also be determined by the presence of a
great number of ferroic domains giving rise to
the appearance of the average prototypic symme-
results and the experimental
different types of SHFT,
1. Ferroe1astoelectric transitions
In the orthorhombic system, most FEE transitions are predicted to occur at the BZ center
Table III). One exception is
(proper transitions
Fddd-F222, arising at the II point of the Bg (F
lattice). It corresponds to a four-dimensional OP
and an eightfold expansion of the crystal's unit
cell. In the tetragonal system, transitions are
—
try.
Let us now examine successively the theoretical
TABLE IX. Simultaneous
Highsymmetry
space-
BZ
group
point
P3mi
r
A
(r)
all
Highsymmetry
space-group
1
a
group
point
P64
P6,
A
L
BZ
system,
space-group
r
(a)
(b)
3
e
PSI
all
L
space-
BZ
Ferroelastoelectric
and Ferrobielastic
group
point
space-group
P6mm
r
P6cc
K
I'
P3m 1 (T2), P31m (T4)
P3m 1 (T4), P31m (T2)
PSim(T, )
PSci, PS ic
M
K
P3
P62cm
A
e
PS
r
All columns have the same content as in Table V.
Highsymmetry
Ferroelastoelectr ic
and Ferrobielastic
e
PS
all
space-
in the trigonal-hexagonal
A
PS(T, )
0
r
ferrobielastic transitions
and
F erroelastoelectric
and Ferrobielastic
PS(T, }
L
P31m
ferroelastoelectric
PS
3
r
PSci
o
A
M
all
~
~
L
8
PS ic
K
r
P3
3
K
P6&mc
A
~
M
aQ
e
a
P6i22
all
I
RS(T2)
3
L
e
RS
a
R3
e
P6522
all
e
A
PS(T, )
Af
all
&j
PS(T, )
r
all
~
~
L
SI
all
P3i
e
P32
all
~
~
L
H
P32
all
P6322
3
e
r
4
A
j
all
r
1
P32
P32
P321(T, }
P3i2 1. , PSI 12
1
a
3
a
2
1
all
H
0
K
T j T2
K
r
M
e
K
I'
M
a
P62c
P 3 I 21,p 3i 12
K
r
M
K
6/mmm
PS/21
P3221, P3212
class
P3221, P3212
3
e
P3221
2
d
a
P3221, P3212
P3, 21, P3, 12
2
P3, 21, P3, 12
I'
M
P62m
d
1
3
4
e
P3i21
2
P321, P312
3
e
P321
2
d
d
a
P321, P312
A
L
K
P62
0
PSI21, P3i12
Ti T2
r
P321(T2), P312 (T2)
P3, 21
P6422
K
P65
TI T2
A
e
P3i
A
M
TI~ T2
Mj all
P6222
3
r
ej
P6i
P321(T2), P312(T4)
A
0
Z
X
Ti~ T2
all
a
A
RSc
r
all
A
RS(T, )
Z
Mj
P3
~
I'
Tip T2
P6c2
~
ej
P6m2
P 622
r
K
T
T3y T4
PS
L
R3m
PS
r
M
e
H
K
data relative to the
PSci, PS ic
P31c
PSci, PSim
PSci, PSim
PS ic
P3mi, P Sic
P3mi, PS 1 c
P3im
P312 (T4)
P312(7,)
P321(T4)
P312
P312
P321
P321
p321
P312
P321
P321
P312
No
transitions
towards
the 32 class
(a)
(b)
a
3
e
2
1
d
3
2
1
3
2
1
a
e
d
a
d
a
3
e
2
d
1
a
3
e
2
1
a
e
2
1
3
2
1
a
e
d
a
3
2
d
PIERRE TOLEDANO
AND JEAN-CLAUDE
TOLEDANO
16
- (I4m2, I4c2) provide
possible at several points of the BZ boundary
(Table IV). Most of these improper FEE transitions involve a two-dimensional OP and a twofold
expansion of the crystal's unit cell. Only two
transitions, namely I4, /a-I4, and I4, /amd
a more complex situation.
They occur at the N point of the body-centered
BZ and are associated with a fourdimensional QP
and eightfold expansion of the unit cell.
In the trigonal-hexagonal
system, FEE transi-
TABLE X. Higher-order ferroic transitions. For the elrnmm and m3m classes the results relative to the (I"} criterion can be found in Tables V and VI. In these two classes the only BZ points indicated in the tables are those for
which an HOP transition is found possible. The [r] column specifies the IB which do not fulfill the [r]3 criterion and
lead to discontinuous SHFT. Other columns have the same content as in Table IV.
Highsymmetry
Highsymmetry
space-
BZ
group
point
Pe/mmm
Higher-order
(I'}&
r
P6/m (T4)
P6/m(T))
K
r
Pel
Pe/m
M
I e/m
I e,lm
K
P63/mcm
r
K
Pe, lm
Pe, /m
Pe, /m
Pe, lm
r
»3(T2)
K
P43m
r
[I']
a
e
2
d
1
3
a
e
2
d
space-
BZ
gl oup
point
Pm3m
e
2
d
Pnsn
e
2
d
E43m
X
Pn3m
all
S"
Tg,
6
k
I23(T2, T3),
I2)3(T5)
3
6
e
E23
1
a
3
e
6
i
T, ),
P2~3 (T5)
r
i
Em3c
Tp
I'
all
I23 (Ts), I2(3 (T4)
Im3m
P43n
3'
T4+
T5
Tp + T4
T3+ T5
all
E43c
I43d
r
x
N
P
all
a
E23 (T4+ T5)
2
c
3
e
P2)S(T)+ T4, T3+ T5)
I2, 3(T, , T4),
P2, S(T, )
I2(3
P2(3 (T4+ T5)
{
T4+ T5
~
Tp
1
I23(T, )
E23
P23 (T3, T4),
L
r
0
P23
+ T3
6
i
a
3
e
6
i
1
a
c
2
Pms(T, ), P432(T, )
Pas(T, , T„)
{Pn3, P432
ras(T5)
(a) (b)
a
6
k
e
6
i
a
1
[Ed3, E432] (Tq)
I4S2(T, )
Pm3, P4232
[Em3, E4, 32] (T, )
[Pa3, P4(32, P433 2] (T3, 74)
Ia3(T3, T4), I4)32(T8, Tg)
X
r
X
r
N
2
6
b
i
a
2
6
3
6
b
k
e
Ims(71)
Pns, P4232
[P4)32, P4332] (73, T4)
I4)32 (Tq)
Em3, E432
Pms(T, ), Pns(7, )
P432 (T6), P4232 (Tg)
3
3
Pa3(T5)
6
i
1
a
3
3
e
Em3, E432
Pms(73), Pns(T2)
p432 (Tg), P4232 (78)
T5
Ed3c
X
r
X
P2~3(T5)
{P23(T2,
X
R
r
M
a
3
Higher-order
ferroic space-group
Ims(T2& T3), I432(T6, Tg)
a
3
r
X
M
all
L
(h)
3
Pe, /m
M
P63/mme
(a)
Pe/m(T2)
M
P6/mcc
ferroic space-group
p
s(,)
Eds, E4, 32
I'4(32, &4)» (r3)
Ed3, E4(32
[P4 32 P4 32](T )
Im 3, I432
I
I
Ims (T4), Ias (T, )
I432 (T5) I4 j32 (T6)
Ia3, I4&32
[Pas, P4s2] (T, )
1
i
a
6
6
k
1
a
e
e
6
1
i
i
a
6
i
1
6
a
i
a
6
6
h
h
1
a
2
b
[r]3
ORDER-PARAMETER
SYMMETRIES FOR THE PHASE. . .
tions are possible for the F, E, and M points
respectively, one-, two-, and three-dimensional OP (the unit cell is multiplied by one,
three, and four).
In the cubic system, improper transitions are
found at the M point of the simple cubic BZ, the
X and L points of the face-centered lattice BZ,
and at the X point of the body-centered lattice BZ.
They correspond to three-, four-, and six-dimensional OP and to expansions of the unit cell by
factors of 4 and 8.
In the latter system, a peculiar situation can be
noted for the species m3m -23. As mentioned
above, it involves a decrease of point symmetry
by a factor of 4 which cannot be produced by a
proper FEE transition. Table VI shows that this
species ean arise from improper transitions corresponding to the M point (P lattice) and the X
point (E lattice). In both cases the transition has
a six-dimensional'OP and is accompanied by a
fourfold expansion of the unit cell.
Two FEE species are associated with the m3m
prototypic group: m3m -43m, and m3m -23. The
preceding result shows that FEE transitions do
not necessarily take place towards the larger of
the two subgroups. This result can be compared
to the case of ferroelectrie transitions for which
Ascher" has pointed out that the point-symmetry
change was always towards the maximum polar
subgroup of the paraelectrie phase.
It is also remarkable that among the five species
listed in Table I which involve a fourfold decrease
of the point symmetry the species m3m -23 is
the only one which can arise during a continuous
phase transition.
Phase transitions with FEE symmetry change
have been reported in ammonium chloride (NH, C1),'
iron sulfide (FeS), and cesium copper chloride
with,
(CsCLICIB).
'
NH4Cl undergoes
with the symmetry
"
a FEE transition at -30 'C,
change Pm3m P43m. This
transition is nearly continuous and does not modify
the number of atoms in the unit cell. Thus, it is
a proper FEE transition which is induced, as
shown by Table VI by the one-dimensional IR (7,)
of the m3m point group.
FeS provides an example of an improper FEE
transition. This material transforms at 138 C
from P62c to P6, /mmc. This transition had been
This
initially identified as a ferroelectric one.
assignment is obviously incorrect since the lowsymmetry group 82m, though acentric, is nonpolar
and thus incompatible with ferroelectricity.
X-ray measurements" have shown that a sixfold expansion of the unit cell accompanies the
transition consisting in a doubling of the c parameter and in a threefold expansion in the (001)
"
899
plane. Such a modification of the translational
symmetry is related to the H point of the hexaTable V indicates that for this point,
gonal BZ.
no FEE transition is expected, due to the noncompatibility of the corresponding IR with the (I"j'
criterion. A further investigation shows that,
nevertheless, the observed symmetry change ean
be induced by one of these IR, defined by a twodimensional small representation (the OP is thus
six dimensional). As discussed by several
authors, '4 the nonfulfillment of the (I'P criterion
should lead either to a spatially modulated, structure for the 82m phase or to a first-order transition. This is consistent with experimental data
which have detected a strong discontinuity and a
large transition heat of 450 cal/mole.
Cs CuCl, has been recently reported to experience
the FEE symmetry change P6, 22-P6, /rnmc at
about 150 C. An unusual feature of this transition is a threefold expansion of the c axis associated with the (0, 0, 2m/Sc) point located inside the
first BZ of the high-symmetry phase. As emphasized in See. II, such a point is incompatible
with the (FP criterion. However, similarly to
the preceding material, the transition in CsCuCI,
is observed to be strongly discontinuous with a
transition heat of 650 cal/mole.
A number of other substances have been pointed
out by Newnham et al. as possible FEE (Table
XIII). Some of these crystals provide potential
examples for improper FEE.
Bismuth fluoride BiF, belongs to the same P43m
space group as the ferroic phase of ammonium
chloride.
Its structure has the pseudosymmetry
Em3rn with a four times smaller primitive unit
cell. The restoring of the higher symmetry would
only require displacements of the Bi atoms by
about 0.08 A. A phase transition P43m Em3m is
compatible with the predictions of Table VI. Such
a symmetry change is induced by the (v, ) or (w, )
IR corresponding to the X point of the BZ (E lattice). It is described by a three-dimensional OP.
Other interesting examples are those indicated
for the species m3m -23, namely, high crystobalite" (a high-temperature form of silica SiO, ),
KA10» and KFeO, . As mentioned above this
species is necessarily associated with an improper
transition.
The former three materials have been assigned
the symmetry P2y3 and a pseudosymmetry I'd3m. '
Table VI'indicates the possibility of this symmetry
BZ.
change at the X point of the high-symmetry
The transition would correspond to a six-dimensional OP and a fourfold expansion of the unit cell.
However a recent reexamination" of the structure of high crystoballite has case a doubt about
the validity of the P2, 3 space group. It appears
"
"
"
-
"
PIERRE TOLEDANO
.
I
0
I
Q
g
~
~
M
0
M
~
Cd
Cd
0
Q
Q
CD
A
O
A
~
0
A
A
A
A
O
CQ
A
A
A
+ + +
I
A
V
+ +
A
+
+
A
+ +
O + +
CQ. CQ.
~CQ.
cu
9
'U
g
A
A
A
~
+
V
0
Cd
O
O
O
V
Cd
g
TOLEDANG
I
M
~ 0 8
~
JEAN-CLAUDE
AND
~
+
A
CQ
CQ
CD
ce
A
A
~+
A
A
+ +
CQ.
Cd
hD
~
~
8
0
G
0 V0
M
8
Cd
~
g,
W
Q
Q
M
Q
cd
~
K
O
4 O
0 II
'V
0
g O
cd
—O
AJ)
O
O
O
II
II
II
II
II
O
II
g'
M
cd
W
8
s
0
Q
Q
Q
~
~
M
Q
K
I
U
0
cd
O
5
Q)
M
0
0
Q
W
g}
P Q
p
0
m
M
Q
II
O
0
0
Q
0
0
S
0
—"
0
g)
Q
4
g ~~
Q
M
Q
cd
~
~~
hD
K
Cd
Q
S
g
~
Cd
W
V
0
M"
ed
+
V V
~R
0
+
S
Q
Q
hD
wIN
K
Cd
S
Q
Q
0
0w
I
v
0
~
+
+
~
M
M
wI+
Q
rg
+
+
~ «[m~
CQ
~IV
+
~
~If/
+
+
C4C4
COCe
NW
C4&
+
+
Ice
v-=
+
C4«
«~cv
+
+
CAW
ZL
ZL
wICQ
+
cd
a
~e~
~+
g
cd
V
Q
N
+
CU
Cd
Q
~
K
Q)
Q 8
~
~IN
0
g
A
8$
PQ
'Q
Cd
0
M
Q
0
g
M
Pa
~IN
'~
+
a 04'
M
I
M
Cg
~ +l~~
ce
M
hD
0
+
cd
I
I&
+
Cd
Q
Q
+
M
40 ~™ 0 g
Cd
Q)
+
I
M
M
+
+
«JN
+
AJl
AA
«[a
+
ORDER-PARAMETER
O
N
~
0M
~
W
O
A
O
Q
SYMMETRIES FOR THE PHASE. . .
A
+
+
~+c4
A
~
A
CQ
~ P4
O
+
O
A
A
C4
+
A
Cg
N
cd
g
CQ. Cg
+
A
8
+
+
+
A
A
OO~Cg
A
cd
+
A
+
M
0
C
~ "U
O
o
N
(Q
II
o
Q
8
CD
PO
II
cD
II
II
II
II
II
M W
cd
~
~
0
~
~
g
g
8
~
~
hA
+
ce
O
~
C4
C4
N
0
W
U'
I
M O
~
C4 ca
~0
cd
C4 e~
C4 e~
I
I
I
ce
C4e~
C4e~
I
8
C4 e~
9
I
C4 CD
I
C4
C44g
eeee
+
C4 e~
g'
I
C4
'~
~y
+
C4 4g
CV lib
~ C4C4
+
cd
~ e~
+
~ e~
~
C4 e~
H
:N-„
.
wlee
~irk
+
wiN
+
+
&leg
+
~
H
cd
8
Q
8
cd
M
~
0+
cd
Q
0
PIERRE TOLEDANO
AND
JEAN-CLAUDE
that SiO, has a disordered structure with average
space-group I"d3m, the local structure being tetragonal. Thus, only KA102 and KFeOz remain as
possible illustrations of the anomalous rn3m-23
TOLEDAÃO
'e
Q
~
4
~
Cd
K
8
0
80
species.
K
K K
~
cd
As shown by Table VII, both proper and improper
FB transitions are found possible. For most of
them the OP symmetries are similar to those encountered for FEE transitions. However, a new
situation can be pointed out for the A point of the
simple tetragonal BZ. It concerns the symmetry
changes P4/mnc -I4 jm and P4, /mnm -I4, ja, and
involves a four-dimensional OP and a fourfold expansion of the unit cell.
Lanthanum cobalt oxide LaCoO, undergoes the
FB symmetry change RSc -RS at 375 C. This
crystal has the high-symmetry phase stable below
the transition's temperature, in contrast to the
more usual situation.
Precise x ray measurements" have detected no
modification of the translational symmetry. Thus
this material is an example of proper FB, whose
symmetry change is consistent with the results of
Table VII. Three other speculative FB with proper
transitions are indicated on Table XIII.
Niobium dioxide displays a more complex and
interesting situation. At room temperature, NbO,
"
possesses a distorted rutile structure. Singlecrystal x-ray measurements by Marinder" have
assigned to it the space group I4, /a, with a primitive unit cell eight-times larger than that of rutile,
whose space group is P4, /mnm. On the other
hand, powder x-ray measurements have been performed by Sakata ~t al. , showing that the rutile
structure is restored at high temperature. A
0
K
g
0 0
Ct)
Cd
Q
K~~
Q)
K
~
Cd
cd
Ct)
K
K
Ct)
e e
.S~&
K P-~
~
~
~
g
Q
~
0
bG
~~
4
~
e 0
G4
0
M
K
8
CD
~
0 0
V
Q
s
P
~I
0
CL)
K
Q
cd
0
K
cd
6
~
o~
pE
K
Cd
CO
0
Q
I
O CO
QO QO
O O
Cd
"
phase transition has been detected by these authors
near 800 C through anomalies in the thermal expansion of the material. More recently, neutron
scattering measurements by Shapiro et al. ' have
confirmed the oeeurrenee of a transition at 808 C.
The temperature dependence of the observed reflection intensities could be interpreted satisfactorily by these authors, assuming that the eightfold expansion of the unit cell takes place at this
transition, and is induced by an IR corresponding
', —,') lying inside the first BZ of
to the% vector (», —,
the P4, /mnm group. An unpublished group-theoretical calculation quoted by Mukamel" further
shows that the space-group change P4, /mnm
-I4, /a can be induced by a one-dimensional small
representation associated with the above k vector,
whose star has four arms (the OP is therefore
four dimensional). We have checked the latter
statement and found it correct. However, as
™
0
0
2. Ferrobielastie transitions
~
~.S
bG
~
QO
CD
g
p
Q
~ V 0
0 0
Q V
0
bG
cd
N
0
V
~
0
Q)
Cd
I
(D
bD
g
'~
bo
0
+~
)CD
CD
t
t
~~g
t
t
M
K
CD
CD
00
o o
Q
K
K
0 g
K K
Ct)
cd
5-t
CL)
0V
Q)
'Cf
Q
V
e
P-"l
O
Wo7»
e
~
(X)
cu
(:)
m
ORDER-PARAMETER
SYMMETRIES FOR THE PHASE. . .
403
TABLE XIII. Materials which constitute possible ferroics. The space group of the prototypic phase coincides with
Column Q: maximum atomic displacements involved beof the structure,
the one describing the pseudosymmetry
tween the actual structure and the prototypic one. The other columns have the same meaning as in Table XII.
Ferroic
type
FFE
Material
PNbg02)
CuCr204
C s3A s2Cle
N2
I4/m
I4, /amd
P3m1
Pa3
BiF3
Fd3m
Fm3m
KpOs02 (OH) 4
I4/mmm
KA10)
FB
Speculative
prototypic
space group
A sI3
FesiF,
~
6H, O
R3c
Nature
of the
Ferroic
space group
(A)
I4
P321
P2, 3
P2(3
P4 3m
Na2ThF6
P62m
P321
04
Higher
K3W&Cl&
P6, /mme
P6, /m
0.2
Fm3m
Fm3m
Pa3
Pa3
b
order
K, NaAlF,
N)H6Clp
BZ
point
r
r
1
.
1
1
I'
4
4
X
.
X
r
I
r
0.25
0.0
FEE and
FB
cell
expansion
I
I
0.08
I4/m
R3
R3
P
P
P
P
0.2
0. 15
0. 17
I42d
R3m
transition
Unit-
OP
dimension
0.2
For FeSiF6 6H20 the occurence of two stable states with average symmetry R3m has actually been detected by x-ray
measurements.
The conjectured mechanism of a transition is of the order-disorder type and would not involve atomic displacements.
"
it appears that
already stressed by Mukamel,
the considered IR does not satisfy the (I P criterion.
By contrast to the case of FeS and CsCuC1„
the transition in NbO, is of second thermodynamic
order. ' Besides no spatial modulation of the roomtemperature phase has been detected from x-ray
or neutron scattering measurements. Consequently, this transition complies strictly with the applicability conditions of the Landau theory, and the
noncompliance to the (I'P criterion appears as a
serious drawback of the interpretation of Shapiro
et a/. A reexamination of the existing x-ray work,
in the light of the present theoretical results,
allows us to propose an alternate interpretation
of the experimental data which, moreover, preserves the validity of the Landau symmetry cri-
"
teria.
"
Sakata et a/. have observed that in their powder
x-ray data the superlattice reflections related to
the distortion of the rutile structure disappear
between 850 and 900 C, some 75 'C higher than
the temperature of the anomalies of the material's
thermal expansion (797 'C). The latter temperature
is close to the transition temperature indicated
by Shapiro et a/. ' for the neutron scattering data
(808 'C) and should be identified with it.
On the basis of the x-ray results, we can therefore assume that two successive phase transitions
are present in NbO„one in the range 850-900 C,
and the other near 800 C, the space-group above
900 'C being that of rutile (P4, /mnm). In this mo-
del, the translational symmetry is expected to decrease in two steps with a modification of the
Bravais lattice following one of two schemes:
P-P-I or P-I-I.
By examing the Bravais
lattice changes" associated with the various high-
symmetry
points of the tetragonal
BZ boundary,
it is found that only the first scheme can account
for the observed modification of translational symmetry. For this scheme two possibilities can be
considered:
P-P
(M point of the BZ), then
P —P
(X point), then
P-I
P-I
(8 point)',
(2 point).
Reference to Table IV permits one to rule out
the second sequence, as transitions are forbidden
by the (I P criterion at the X point of the P4, /mnm
space group. Therefore a single possibility remains corresponding to the sequence of transitions:
P4, /mnm P4, /m I4, /a .
The first transition is induced
by a two-dimensional IR (w, ) at the M point of the BZ of P4, /mnm
(Table VII). It involves a twofold expansion of
the unit cell. The translations of the low- and
high-symmetry phases are related by
PIERRE TOLEDANO
404
AND
The second transition is a mere change of translational symmetry induced by a two-dimensional
IR (a real one-dimensional r„and a two-arm star)
corresponding to the B point of the simple tetragonal BZ. The final body-centered tetragonal cell
ls
'.
a, = 2(a', —a,'), a, = 2(aa, + a,'), a, = 2a,
This cell coincides with that of the room-tem-
"
perature phase of NbO, . The preceding model
thus accounts for the space group and primitive
translations of NbO„as well as for the temperature dependence of the x-ray powder spectra. It
also explains the neutron scattering data if a relabelling of the observed reflections is performed
in order to refer these reflections to the BZ of
the intermediate P4, /m phase, and not to that of
the P4, /mnm phase. The reflection assigned to
', —,) has to be relabelled (0, —,', —,')
the wave vector (-,', —,
and thus corresponds to the 8 point of the BZ
which is associated by the present interpretation
Likewise the reflections
with the 800 C transition.
', 0, 0) and (—,', —,', 0) must be relabelled (-,', —,', 0)
at (—,
and (0, I, 0). The proportionality pointed out by
Shapiro et al. between the intensity of the two latter reflections and the square of the intensity of
the former one is obviously consistent with our
interpretation as well, since this proportionality
relies on a geometrical relation between the %
vectors which is not modified by the former relabelling.
A consequence of the conjectured mechanism is
that the 808 C transition possesses a two-dimensional OP instead of the four-dimensional OP assumed by Mukamel.
Accurate single-crystal
x-ray measurements between 800 and 900'C should
decide between the two interpretations.
"
3. Simultaneously
ferroelastoelectric
and ferrobielastic transitions
Tables VIII and IX indicate that the types of OP
symmetries are the same as for FEE transitions.
The n Ptransforma-tion of quartz (SiO, ) is an
example of both a proper FEE and FB transition.
This transformation occurs at 573 C with the
space-group change P3, 21-P6422. Its thermodynamic order is close to second order' (slightly
first order). The primitive translations are the
same for both phases. The symmetry change is
thus consistent with that; predicted for the 1 point
of the BZ of the high-symmetry space group.
AlPO, is isostructural with quartz and undergoes
an identical transition at 579 C.
Two other materials are suggested by Newnham
eg al. as possible FEE and FB. However, one
of them, Ag~HgI„ involves very large displacements of the silver atoms to achieve the prototypic
"
"
JEAN-CLAUDE
"
TOLEDANO
16
"
structure,
and is therefore unlikely to be a ferroic. The pseudosymmetry of the other one,
Na, ThF„permits classification as a proper ferroic. No example for an improper FEE and FB
transition is available.
4. Higher-order ferroic transitions
Two differences can be pointed out between the
results for HOF transitions (Table X) and those
relative to the FEE transitions of the cubic system (Table VI).
In the first place several HOF transitions are
found possible at the X point of the simple cubic
BZ; they correspond to a six-dimensional OP and
an eightfold expansion of the unit cell of the crystal.
On the other hand, the six-dimensional IR at
the M points of the P Brillouin zone and at the X
point of the I' Brillouin zone do not satisfy the
[I'] criterion. They are associated with discontinuous transitions falling outside the scope of
this work. Nevertheless, we have indicated the
corresponding symmetry change (Table X), as
these IR constitute the only exceptions of the cubic
system with respect to the [I']' criterion. This
does not imply that the tables also account for
possible first-order transitions, as these transitions can arise, for instance, when the symmetry
of the OP is described by a reducible representation of the prototypic phase, a circumstance not
considered here.
No phase transition of the HOF type has yet been
experimentally reported. Three compounds can
however be considered as possible examples
(Table XIII).
The structure of K,W, Cl, has the symmetry
P6,/m and a pseudosymmetry P6, /mme. 35 The
prototypic and ferroic unit cells have the same
number of atoms. The symmetry change is consistent with a proper transition indicated in Table
X for the I' point of the high-symmetry group.
N, H, CI, possesses a structure (Pa3) slightly disThe
torted with respect to that of CaF, (+m3m).
modification of the Bravais lattice between the distorted and prototypic structures discloses the
improper character of the corresponding HOF
transition. This symmetry change is in agreement with the indications of Table X at the X point
of the BZ (E lattice). The OP is six dimensional
and the transition is predicted to be first order.
The same symmetry change Em3m -Pa3 is conjectured for elpasolite K,NaAlF, . However, as
shown in detail by Newnham et al. , the mechanism
of a transition in this material would involve an
order-disorder process rather than the displacement of certain atoms: the Em3m space group
"
"
ORDER-PARAMETER
SYMMETRIES FOR THE PHASE. . .
corresponds to the random occupation of unequivalent sites in the cell by the potassium and sodium
atoms, while the ordered pattern is described by
the Pa3 group.
Elspasolite belongs to the cryolite" family in
which several transitions with a ferroelastic symmetry change have been reported by x-ray measurements. Up to 500 C no transition was detected in K,NaAlF, . However, the expected transition, though first order, leaves the crystal in the
cubic system and might have been difficult to detect through powder x-ray experiments.
C. Free-energy expansions associated with the SHFT
The form of the expansion associated with a
transition depends on the form of the set of matrices representing the IR which induces the symIn the ease of the investigated
metry change.
SHFT, twelve different forms of expansions were
encountered. They are listed in Table XI. They
"
correspond to one-, two-, three-, four-, and sixdimensional OP. To each OP dimension are associated one or several types of expansions which
are distinguished by the expression of the various
fourth-power invariants, by the faintness index,
and by the form of the OP contribution to the mixed
invariant, defined in See. II.
It is well known" that several low-symmetry
phases are compatible with a multidimensional
IR. Each one is associated with a set of relative
values for the components of the OP. Table XI
specifies the set corresponding to a secondary- or
higher-order ferroic phase as well as the conditions imposed on the P,- coefficients of the expansion in order to insure that this set of OP values
determines an absolute minimum for the freeenergy expansion. It appears that, for all the
SHFT indicated in Tables III-X, there exists a
range of compatible P, coefficients corresponding
to the stability of the considered ferroie phase.
The inequalities which determine the suitable
range of p, values have been established without
taking into account in the expansion the mixed invariant. The effect of such a term can be to change
the thermodynamic order of the transition from
second to first.
We assumed that the coefficient of the mixed invariant was sufficiently small
to make the influence of this term negligible.
It has been pointed out in Sec. II that the mixed
invariant term depends linearly on the tensorial
components characterizing the considered ferroic
species. Each of these tensorial components .
transforms according to a one-dimensional IR of
the prototypic point group. Consequently, the OP
contribution to the mixed invariant is a polynomial
transforming according to the same IR. This IR
"'"'"
405
is also the one which induces a proper transition
for the considered species. For instance, if we
consider the FEE transition Pm3m -I@3m described by the expansion labelled (e) in Table XI,
the OP contribution to the mixed invariant is
$1ff2ff3 which transforms in the same way as the
piezoelectric modulus d„according to the (r, ) IR
of the m3m point group (Table VI).
The case of the species m3m -23 is more complex since there is no IR of the m3m point group
related to it. The corresponding improper transitions are described by an expansion labelled (i)
in Table XI, for which three OP contributions to
the mixed invariant are indicated:
2
j =1,3
(OJ
2
Kj )P
11 12
13+ 01~2~3&
11 1213
~1~
f23
'
These three terms transform according to the (v, ),
(v, ), and (w, ) one-dimensional IR of the m3m point
group, respectively, related to the species m3m
-43m, m3m-m3, and m3m -432. The three
terms are simultaneously nonzero for the set of
OP values:
I&, I=I&, I= I&, leo
and
c, =o.
The preceding values correspond to a phase
whose point symmetry is the intersection of 43m,
m3, and 432, that is, the 23 point group.
We can note that in most of the other transitions,
the OP can also couple in a similar way to several
IR of the prototypic point group. However, the OP
contribution to all the mixed invariants other than
the one indicated in Table XI vanish for the set of
OP values corresponding to the SHFT.
For instance, the OP of the Pm3m -L 43m transition mentioned above can couple to a two-dimensional IR of the m3m point group by means of the
set of polynomials
(2q,'-71,'-g', , 2q,'-q',
-q,').
It is clear that the corresponding mixed invariant
will vanish for the set of values lail = Inal = In~I
relevant to the transition Pm3m -E43m (Table XI).
Hence, the uniqueness of the m3m -23 transition
among all the investigated SHFT is related to the
fact that three mixed invariants corresponding to
different point-symmetry changes can 5e simultaneously nonzero for a stable state of the crystal.
An expansion similar to (i), which has also been
labelled (i) in Table X, occurs for several discontinuous HOF transitions. It has in common
with (i) the second- and fourth-power terms. In
addition, it possesses a third-power invariant
equal to one of the two terms (q, q, q, + r„,g, f, ) or
(q, q, q, —g, g, g, ). The OP contribution to the mixed
invariant is
PIERRE TOLEDANO
406
AIVD
j=1,3
This expansion has not been reproduced
XI.
in Table
IV. CONCLUSION
The main usefulness of the present work consists
in a tabulation of the results obtained by a systematic examination of the irreducible representations of 147 space groups. The tables indicate
all the space-group changes relative to secondary-
or higher-order
ferroic species which are com-
patible with the Landau theory of continuous phase
Their consultation allows a straightforward determination of the dimension and symmetry properties of the order parameter corresponding to a given space-group change, as well
as the derivation of the associated free-energy
expansion. The tables presented here are complementary to those worked out in TT"; these
two studies exhaust all the ferroic transitions
which keep unchanged the crystalline system,
i. e. , all those which are nonferroelastic.
The present results demonstrate that both
proper and improper SHFT are possible. The
latter transitions are associated with two-, three-,
f our, and six-dimensional order parameter s.
The complexity of these transitions is thus greater
than that of the previously investigated ferroelectric nonferroelastic transitions,
where only two-,
and three-dimensional
order parameters were
transitions.
"
f ound.
A peculiar situation has been pointed out for the
ferroelastoelectric species m3m 23. This
species, as well as four others, involves a decrease of the number of point-symmetry elements
by a factor of 4, and its occurrence is forbidden at
the Brillouin-zone center. However, by contrast
to the other four species, which are also f orbidden
at other BZ points, a continuous transition m3m
-
-
23 has been found to be possible at the BZ
boundary. This prediction is in agreement with
the experimental data, since examples have been
indicated for the species m3m 23, while none is
-
J. P. Bachheimer
(1975).
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J.
JEAN-CLAUDE
TOLEDA50
16
available for the four other mentioned species.
Crystals which constitute proven or speculative
examples for SHFT have been examined with respect to their order-parameter symmetries.
Proper as well as improper transitions were identified in the available materials, in agreement
with the tables.
In the case of two crystals possessing an SHFT
of the improper type, namely FeS and CsCuCl„
it was noted that the irreducible representations
describing the symmetry change do not fulfill the
(I')'
symmetry criterion. However, these materials undergo strongly discontinuous transitions,
and therefore escape the strict application of the
Landau's theory.
In a third crystal, NbO„ the current interpretation of the 800 'C phase transition also implied an
incompatibility with the (I P rule. We have stressed that this crystal poses a serious challenge to
the validity of the Landau's theory as its transition
is continuous and does not give rise to a spatially
modulated structure. By using the tables established here, we have been able to show that an
alternate interpretation, preserving the validity of
the ( 1)2 rule, could be given of the experimental
data for NbO, . It assumes the occurrence of two
successive transitions, one at 800 C and the other
between 800 and 900 'C. As a consequence, the
800 'C transition is predicted to be described by
a two-dimensional order parameter instead of the
four-dimensional one assumed up to now.
FeS and CsCuCl, are two of the three examples in
which an SHFT of the improper type is confirmed
at present. The interpretations of their transitions
denotes the nonfulfillment of the f I'P criterion.
The same situation was pointed out in TT for the
only two available examples of improper ferroelec-
tric nonferroelastic transitions, i. e. , (NH, ),BeF,
and NaH, (Sea, ), . It is puzzling to note that most
actual examples of improper transitions which preserve the crystalline system (i. e. , nonferroelastic)
seem to contradict the jl')' symmetry criterion,
while most improper transitions which involve a
modification of the crystalline system (ferroelas-
tic) respect this rule.
""
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~
The possibility of a polarization reversal does not exist
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such as lithium niobate or barium sodium niobate
which are, nevertheless, classified among ferroelec-
J.
trics.
ORDER-PARAMETER
16
SYMMETRIES FOR THE PHASE. . .
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