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SPACE GROUP THEORETICAL DETERMINATION
OF DOMAIN STRUCTURES
M. Guymont, D. Gratias, R. Portier, M. Fayard
To cite this version:
JOURNAL DE PHYSIQUE Colloque C7, supplement au n° 12, Tome 38, decembre 1977, page CI-41
SPACE GROUP THEORETICAL DETERMINATION
OF DOMAIN STRUCTURES
M. GUYMONT
Laboratoire de Cristallographie et Physique des Materiaux, Batiment 490, Universite de Paris XI, 91405 Orsay Cedex, France
D. GRATIAS, R. PORTIER and M. FAYARD Laboratoire de Métallurgie structurale des Alliages ordonnés,
Ecole nationale supérieure de Chimie de Paris, 11, rue Pierre-et-Marie-Curie, 75231 Paris Cedex 05, France
Résumé. — Les structures en domaines (domaines d'antiphase et/ou macles) apparaissent
géné-ralement après une transition de phase avec abaissement de symétrie, c'est-à-dire une transition où le groupe spatial H, d'une des phases est un sous-groupe du groupe spatial Gs de l'autre phase. Pour qu'une telle relation existe entre les groupes spatiaux des deux phases, il est nécessaire qu'aucune modification de la métrique ne se produise (ou alors une modification négligeable) pendant la tran-sition.
En utilisant la décomposition du groupe spatial Gs en complexes associés à H„ on peut déterminer tous les types possibles de frontières de domaines. Une frontière quelconque est caractérisée par un complexe entier et non par un opérateur particulier de ce complexe, car tous les opérateurs apparte-nant à un même complexe décrivent le même type de frontière.
On examine brièvement les conditions d'observabilité des frontières par l'étude du contraste en microscopie électronique.
Abstract. — Domain structures — i.e. antiphase domains and/or twins — in homogeneous crystals usually appear after a phase transition leading to a lowering of symmetry, i.e. a transition for which the space group Hs of one phase is a subgroup of the space group Gs of the other phase. Such a
relationship between space groups of both phases can only occur if there is no — or negligeable — modification of metrics during the transition.
Using space group decomposition of Gs into cosets with respect to Hs, all possible types of domain
boundaries can be determined. Any boundary is characterized by one whole coset and not by a particular operator belonging to this coset, because all operators inside one coset describe the same type of boundary.
Conditions for observability of boundaries through contrast study in electron microscopy are briefly discussed.
1. Introduction. — Numerous examples of domain structures are reported each year in connection with their observation by electron microscopy. Diffe-rent types of domains, called variants, are often classed in two kinds : antiphase domains and twins.
In the sequel it will be given a survey of present state domain theory.
2. Domam structures and transitions. — Domain structures generally arise in a crystal which has undergone some transition from a phase whose space group Gs is a supergroup of the space group
Hs of the given crystal. This corresponds to a lowering
of symmetry of the crystal, whose space group was
Gs, during the transition.
Such a relationship Hs <= Gs between space groups
involves the existence of the same relationship between
the corresponding point groups : H c G, and also between the corresponding subgroups of translation : t / c 7". Of course, the latter relationships are under-stood in the large, i.e. we may have for instance
U = T or H = G.
Those transitions are special cases of more general transitions which are metrics-preserving. By this phrase, we mean that in both phases it is always pos-sible to find one common unit cell which has the same size and form and contains the same sorts of atoms in equal quantities. It is to be noticed that we do not impose to the locations of atoms to be nearly the same in both structures. This condition is necessary — althought not sufficient — for having a group-sub-group relationship between space group-sub-groups Hs and Gs.
Of course, actual phase transitions between crystals are seldom exactly metrics-preserving. But we satisfy
C7-48 M. GUYMONT, D: GRATIAS, R. PORTIER AND M. FAYARD
ourselves with the numerous transitions which occur with negligeable modification of metrics.
Two kinds of such transitions are currently encoun- tered and are well known to give rise to domain structures :
i) Order-disorder transitions constitute the first kind. Generally, the supergroup is shown by the high temperature form (at constant pressure). But this assumption is not necessary for our theory. This class of transitions finds numerous examples among alloys, but is also represented in other solids.
ii) Those special paramorphic transitions which are metrics-preserving constitute the second kind. The term paramorphic, after Mallard and Friedel [I], roughly corresponds to Burger's displacive transi- tions [2], but may also include e.g. reversible marten- sitic transitions. So the metrics-preserving paramor- phic transitions would correspond to the so-called
small displacive transitions. Such transitions are
widely represented, e.g. : quartz, VO,, BaTiO,,
.
. .
Of course not all these transitions follow the group- subgroup relationship. These ,structural transitions are in general very rapid and cannot be quenched contrary to ordering transitions which are generally much more sluggish.
3. Group theoretical considerations. - Our start- ing point is therefore the assumption that the space group H, of the structure'showing domains is a sub- group of the space group G,. A variant V in the struc- ture consequently has the spake group H,, i.e. any operator h of H,, and only these, leave V invariant.
Let g, be an operator of G, which does not belong to
H, : it therefore transforms the variant V into another variant Vl = g, V. All operators of G, which make the variant Vgo over into Vl are of the form g, h, with
h taking any value within H,. This set of operators
which transform V into V, is the left coset gl H, with respect to H,.
Between Vand Vl there must exist a boundary. So a left coset of operators corresponds to a domain- boundary
-
It must be recalled that cosets are not groups - To the same type of boundary will corres- pond the same coset. Any operator inside the coset can be chosen to represent the boundary : this opera- tor, like g, in the symbol g, H,, is called coset repre-sentative.
Another coset g,
H,
is obtained, leading to a third variant V,, by taking an operator g, in G, which does not belong to H,, nor to g, H,. So we get a second boundary. And so on.Although space groups. are infinite groups, this process of forming cosets will end up at some time. In fact, because space groups are discrete and periodic, it is always possible to restrict to a finite number of operators within a unit cell. This is what is done in the standard International Tables for X ray Crystallo-
graphy [3]. A smallest common unit cell can always be chosen for both H, and G, because of the H, c G,
assumption. This smallest unit cell frequently turn out to be the Bravais cell for at least the subgroup H,.
As there are always a finite number of cosets, one obtains a finite decomposition of space group G, into cosets with respect to H, :
G, = H,
+
g, H,+
.-.
+
g,-, H,',n being the index of H, int G,, and also the number of
different variants.
Let us recall the properties of cosets :
i) Any two operators belonging to the same coset are distinct.
ii) Any two cosets of the same decomposition have no common operator.
iii) Every operator of G, belongs to one and only one coset.
iv) Left (resp. right) multiplication of all cosets by any operator of G, merely results in a changing of order of cosets in the decomposition.
Some of these considerations, but applied to point groups, have already been made by van Tendeloo and Amelinckx [4].
4. Diierent kinds of boundaries.
-
We shall now apply the previous results about cosets.4.1 NOTATIONS. - A redundant but convenient combina'tion of Hermann-Mauguin and Seitz nota- tions is used for groups and operators : in the symbol (R
1
z), R is the orientation part and, when needed, will be further specified by taking Hermann-Mauguin symbol with some additions ; for instance : 1 will2 n
denote identity,
1
inversion, 3:,,,,
a-
anticlockwise 3rotation around [I 111 triad axis, 4I2,,,, a n anticlock- wise rotation around PO11 tetrad axis, ccllol a glide n reflection with its normal along [110], 4:I,oo1
a
Z
anticlockwise screw rotation around [loo], etc...
z is the translation part and will be further specified by giving the three components along the three unit vectors of the chosen cell. It must be noticed that z represents 'both translation irreducible part and loca- tion of the symmetry element for the considered setting. 4.2 CASES OF COSET DECOMPOSITIONS. - Accord- ing to the different modes of fulfilment of relationship
H , c G,, three cases can be distinguished [5] : i) Case I : H, and G, differ only by their lattices, i.e.
U c T (e.g. AuCu, : Fm3m -, Pm3m). We get a coset decomposition of the type :
SPACE GROUP THEORETICAL DETERMINATION OF DOMAIN STRUCTURES C7-49
There always is only one pure translation operator in each coset. Therefore, only translation - i.e. anti- phase - boundaries may appear, and are conve- niently labelled z l ,
...,
z , - , .ii) Case II : H, is a subgroup of G, but both lattices
are identical U = T (e.g. quartz : P6222 + P3,21). In this case, we shall never get antiphase boundaries.
According to the types of operators within the cosets, two kinds of boundaries can appear :
If the considered coset contains at' least one translation reducible operator then this operator can be taken as representing the boundary, which is a twin - or orientation - boundary.
If the coset contains only mixed operators - i.e. with irreducible translation part - there appear a mixed boundary [5].
A labelling convention has been suggested [5] for naming the boundary, for several operators can equally well be chosen in the same coset : once the boundary has been identified as twin (resp. mixed)
boundary, the twin (resp. mixed) operator of lower order is chosen .for labelling the boundary - of course there may be several operators of the same order.
A number of theorems can be derived to predict the appearance of mixed boundaries [5].
iii) Case III : H , is a subgroup of G, and the lattices
are no more identical (e.g. LiFe,O, : Fm3m -, P4,32). In this case, any boundaries can occur, whether antiphase, twin or mixed. No simple general rule can be given ; anyway all types of boundaries can be exactly predicted in each particular case, once G, and H, are
known, by decomposing G, into cosets with respect to
H, and using International Tables [3].
5 . Two examples. - 5.1 EXAMPLE OF GeSE I1 WITH
A MIXED BOUNDARY. - Let us consider a structure
with space group P4 which has undergone a transition from a structure with space group P 4 c c . Lattices are identical. Let us take the origin as in International Tables. The operators of P4cc can be written :
The operators enclosed in the frame are the operators of P 4 . The cosets decomposition is :
So we get two variants with a mixed boundary which can be equally well labelled c[,ool, c ~ o l o l , cIilo1, or
C[l l o ] (figure 1).
boundary
C
,& I
5 . 2 EXAMPLE OF CASE 111 WITH ALL KINDS OF
'
+0*0+
+'
+0 0 +BouNDARIBs. - Let ~2
be the space group of the
'
0 1
+ @ '+0*0+
+ structure which is considered. It is assumed that this structure has undergone a transition from a structure O + O of space group Ima2. In order that metrics 'be pre- served, the primitive cell of P2 must be pseudotetra- gonal We choose a setting with b axis along the diad axis as is traditional in Crystallography for monoclinic structures. The operators of P2 are :{ p2
;
= { ( 1I
OOO), (2,010,1
000)I
.
Then the decomposition into cosets is written :
FIG. 1 . - Here is shown a structure with one kind of atom in + { ( ~ [ I ~ O I
1
9 0 ) , ( ( ~ [ o o I ]1
300) )general posltlon of space groupe P4. c axis is perpendicular to the
plane of the paper. The symbol
+
indicates that any value can be {1ma2 f = { ~2 )+
{ ( I l L ' - L 2 2 2 h (21,0,0,I
9%
) taken for the c coordinate; 112+
indicates any value plus c/2.C7-50 M. GUYMONT, D. GRATIAS, R. PORTIER AND M. FAYARD
Four variants are then found with three types of - - antiphase boundary
1 1 1 f f i
boundaries : one twin rnI,,,,, one antiphase
-
222
' and one mixed bI,,,, (figure 2).6. Determination by electron microscopy. - Gene- ral criteria for boundaries contrast extinction have been derived in multi-beam electron microscopy [6]. There are two kinds of contrast for boundaries : i) Domain Contrast : When the sets of excited beams
in the two adjacent variants are different, there is no 0-0+ + 0 O+ + O O+ + O O +
twin
extinction condition : one variant is bright while the
o+
O+ boundaryadjacent one is dark. This is the usual way of detecting 0 ~ [ I O O I
+
microtwins.
o+
ii) Boundary Contrast : When both sets of diffracted
o O
+
beams are superimposed (i.e. there is coherency) the boundary
contrast depends on the phase change of the involved btool, + + 0 -Of Fourier coefficients of the scattering potential. In
t + g O
0 O+
this case, extinction is achieved if for all g vectors
of the diffracting set we have :
o+
+ + O ~ f +
\
antiphase boundary
b t *
F ~ G . 2. -This structure with one kind of atom in general position
(hi
I
ti) being an operator ofHS
and ( RI
T) any opera- has a space group P2. b axis is taken perpendicular to the plane tor of the coset describing the boundary. of the paper. There are 4 variants with 3 types of different boundaries.References
[I] FRIEDEL, G., Legons de eristallographie, 2nd ed. (Blanchard, [4] VAN TENDELOO, G. and AMELINCKX, S., Acta Crystallogr. A 30
Paris) 1966. (1974) 431.
[2] BURGER, M. J., chapter 6 in : Phase Transformations in Solids, [5] GUYMONT, M., GRATIAS, D., PORTIER, R. and FAYARD, M.,
ed. by Smoluchowski, Mayer and Weil (Wiley, New Phys. Status Solidi (a) 38 (1976) 629.
York) 1953. [6] GRATIAS, D., GUYMONT, M., PORTIER, R., and FAYARD, M.,
[3] International Tables for X ray Cristallography, vol. I (Kynoch Phil. Mag. 35 (1977) 1199.
