SYMMETRY AND PHASE TRANSFORMATION

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SYMMETRY AND PHASE TRANSFORMATION

R. Portier, D. Gratias

To cite this version:

R. Portier, D. Gratias. SYMMETRY AND PHASE TRANSFORMATION. Journal de Physique

Colloques, 1982, 43 (C4), pp.C4-17-C4-34. �10.1051/jphyscol:1982402�. �jpa-00221946�

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JOURNAL DE PHYSIQUE

CoZZoque C4, suppZe'ment au n o 12, Tome 43, de'cembre 1982 page C4-17

SYMMETRY AND PHASE TRANSFORMATION

R. Portier and D. Gratias

C. E. C. M., 15, m e Georges Urbain,94400 Vitry-swl-Seine and E.N.S. C . P., 21, rue Pierre e t Marie Curie, 75231 Paris, France

(Accepted 9 August 1982)

Abstract.- The symmetry properties are among the simplest deterministic principles which allow us to recognize equivalencies between different crystalline morphologies. The basic idea is essentially that the total symmetry of the resulting morphology reflects the symmetry of the originating process : for example, the different types of variants induced by a phase transition are related to the coset decomposition of the space groups of the involved structures onto their intersection group. A scheme has been propo- sed for the numbering and labelling of the variants nucleated in an anisotro- pic medium. More recently the importance of the Intersection Group has been pointed out as one of the basic ingredients for the study of precipitate morphology.

The symmetry considerations are not only an easy way of reducing the confi- gurational space of a certain physical property to a non redundant subspace, but they also allow for the determination of the "symmetry dictated extrema"

of the considered physical property. This promising way will certainly induce a new development in the understanding of martensite morphologies.

The role of symmetry in the study of phase transitions is well recognized from the determination of the order of the transition as well as the description of the microstructure of the transformed phase. The classical theory of Landau on phase transitions (1) stipulates that for a second order transition, the space group of the product is induced by an irreducible representation of the space group of the parent phase, both phases being group-subgroup related ( 2 ) . Whatever the order ofthe transition, a single crystal of the parent phase transforms to a collection of domains of the product phase called variants that are separated by interfaces. The variants correspond to all possible ways, strictly equivalent, of realizing the transformation. The properties of the transformed phase are closely related to this microstructure.

Martensitic transformations being first order transformations, the parent phase may coexist with the product and two kinds of interfaces have to be considered:

homophase interfaces between two variants of martensite (or parent phase) and heterophase interfaces between martensite and parent phases.

Using symmetry arguments for studying the transformation, the first step consists in determining the number of generated variants and the nature of the interfaces between them. A second step concerns the possible relations between the morphology of the bicrystals of interest (parent phase - one variant of the transformed phase and two variants of the transformed phase) and the interfacial symmetry which has to be determined. Then, a relation between this interfacial symmetry and a term of energy will be given.

In a second part some information about interfaces, like the coincidence site lattice, and several points about the martensitic transformation, like the symmetry induced by displacement waves, will be treated.

There are only a few papers on symmetry and martensite. Cahn in 1977 (3) used symmetry for the phenomenological description of the transformation and discussed

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982402

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C4-18 JOURNAL DE PHYSIQUE

the occurrence of sub and supergroup generation. The present authors, in 1979, used symmetry for finding relations between the parent phase and the product, ignoring the intermediate steps. These two papers will provide the basic material for the present one. Also the present psper is more a sort of "pot pourri" on symmetry and phase transitions and does not pretend to give any solution of the martensitic transformation.

NUMBERING OF THE VARIANTS AND INTERFACE OPERATIONS (4) 1. Numbering of the variants generated by a phase transition

Symmetry arguments distinguish cases in which two physical situations are equivalent. These arguments are only sufficient conditions. Certain physical properties may be accidental.1~ equivalent. Applied to phase transition, the basic idea is essentially that the

total

symmetry of the resulting morphology reflects the symmetry of the original process (Curie's Laws). In the present context, the symmetries involved through two basic groups can be described as :

i) the symmetry group of the parent crystal G which is a space group 0'

ii) the symmetry group of the external medium (g). This latter group reflects the anisotropy of the sollicitation applied to the crystal and leads to the transition when the critical value is reached. This group can be continuous. If the sollicitation is the temperature, its invariance group is the symmetry group of a sphere (Sog); if it is an applied stress field along a n direction, its invariance group is that of a cylinder (Dmh [n ] )

. . .

Obviously, all thermodynamical properties of the transformation are invariant through any symmetry elements belonging simultaneously to both groups : neither through the application of such elements. The intersection group : Ho = G o A (g)

may be considered mathematically as the kernel of the transitior. All its elements act as the identity operator with respect to the thermodynamic properties. In other words, assuming that a certain embryo of the transformed phase has to be formed some- where, any other embryos deduced from the first one through the operations of Ho are strictly equivalent. Therefore, Ho is called the group of isoprobability of nucleation of the transformation product.

Once the embryo has formed, further growth will lead to the new structure of space group GI. The existence of the heterophase interface between the two structures will be studied in terms of symmetry further below.

Due to the growth, it may happen that certain embryos in contact would form a single crystal, whereas other embryos with space groups differing in orientation- translation would form a bicrystal and would be separated by an homophase interface.

In order to number all the different variants of the transformed phase, one has to select in Ho all the subset of elements which do not belong to GI. This is perfor- med by considering the new intersection group N ₀₁

-

_'

N~~ =

H ~ A

G~

which is the set of symmetry elements common to both Ho and GI. Now, the number, n of different possible variants of the G1 product phase is given by the number oF1t)imes NO1 is contained into Ho. Mathematically, nol is the index of the subgroup NO1 into Ho : it is the number of NO1 cosets onto Ho :

n = index (NO1/Ho) 01

The complete group-tree characterizing the transition from G to G is sketched as

shown : 1

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The inverse transformation is described similarly but the sollicitation group (g') may be different from the first one. There is an important distinction between domains and variants : each domain belongs to one kind of variant. All possible variants generated can be numbered but the number of domains will depend on the actual conditions of the transformation. The numbering of variants is for the case of a single crystal of the parent phase Go. For a polycrystal parent phase, the group-tree has to be considered for each grain : two grains being deduced by a symmetry-translation operation, the variants in the two grains will also be deduced by,this operation. Finally, it is assumed that the transformation has no boundary conditions and the parent crystal is a perfect one. If there are some boundary conditions, they must be included in the group of the sollicitation which is then reduced to the common symmetry of both the sollicitation and the boundary conditions.

Moreover, if the parent phase is faulted, obviously at the level of a defect the symmetry will be lowered but defects will essentially play an important role as possible sites for the nucleation of embryos and not for the numbering of variants which is governed by the "location" of G with respect to H ₁

.

2. Space operations and space group notations

The most general isometry in space is written

(air)

where a , is a point operation (identity, inversion, rotation, rotation-inversion)' and

:,

the associated translation (Seitz notation (5)). If two crystals are related by such an operation, that means the homologous points are deduced by :

- (el:)&, = arI + J

EII

- with

cI

belonging to crystal I, fII belonging to crystal 11. If the space group of crystal I is GI, the space group of crystal I1 is then conjugated by (a]:) :

GII = ( a l ~ ) GI ( a 1 ~ 1 - I

That means GI only differs from GI by orientation-translation. The operation caracterises the homophase interface but a crystal is invariant through any operation of its space group : if one operates on crystal I first by GI and then by (a

I r)

one gets the same crystal 11. So, an interface operation is characterized by a coset : the right product of the operation (a]:) initially choosen by the space group of the crystal :

(all) GI (see also appendix 1)

3. Some examples of phase transitions i) Order

-

disorder transition

Au3Cu-transformation from the disordered state (Go = Fm3m (a,b,c;o)) at high temperature to the ordered state (GI = Pm3m (a,b,c;o)) at low temperature is a simple case of group-subgroup transition.

The sollicitation is the temperature and, assuming there is no gradient, its invariance group is S0,QDR

-

3 :

H = G

f\

(g) = Fm3m (a,b,c;o)

0 0

- - -

NO1= Hor\G1 = G = Pm3m (a,b,c;o)

1

- - -

The number of variants is :

"01 = index (Fm3m ($,b,c;o)/Pm3m ($,b,c;o)) = 4 There are 3 different interfaces :

which are the three well known antiphase boundaries, obtained by the decomposition of H into cosets of N

.

At the critical temperature, the symmetry of the parent phasg is broken and t% interface operations between variants are precisely the symmetry operations of the parent phase which were lost.

If, one now considers the reverse transformation by increasing the temperature, the new group of isoprobability of nucleation is :

H~ ⁼G n ( g ) = Pm3m ( ~ , b , c ; o )

1 and

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C4-20 JOURMAL DE PHYSIQUE

N10 = H

/\

G = Pm3m ($,b,c;o)

1 0

these two groups being identical, the number of generated variants is one. So, one starts with a disordered single crystal, and ends with it after the cycle of transformations. This transformation is crystallographically reversible.

ii) Transformations of gadolinium molybdate (G.M.O.)

At high temperature Gd2 (MoO4l3 is paraelectric with a space group Go =

P 4 21m (g,b,c;o). Below 15g°C, it is ferroelectric and ferroelastic with ^: G ₁= P b a 2 (a+b,

b-a,

2;

$ $

⁰⁾

The origin and unit cell vectors of the two space groups in this example are different. By cooling, the groups involved in the tree are : ((g) is the sphere invariance group) :

H = P 4 21m (a,b,$;o) and O

N ₀₁= P b a 2 (a+b,

b-a,

^5;

11-

₄₄⁰⁾

so, four variant.^ are generated :

There are one translation interface and two orientation-interfaces for which the two-fold axes are screw axes.

But if now, a stress is applied along the [loo] direction referred to the orthorhombic cell, the new sollicitation group during cooling is :

(g) = D-h [loo]orth,.

and

Ho ⁼Cmm2 (g+b, 5-g,

-

^C;O)

NO1 = Pba2

(a+!,

^b-3,^{5 )}

Only two variants are generated and the interface is a translation one : Cmm2 = [(1)000)

+

(1)010)] Pba2

That means, due to the applied stress field, one orientation variant disappears.

On the other hand, if the applied stress field is parellel to [I101

variants are obtained again as upon cooling the sample. orth.' four iii) Martensite transformation.

In almost all cases of martensitic transformation, there are no common sub- lattices between the parent and martensitic phases. The number of translation variants may then be infinite. However, in most practical cases, it seems reasonable to neglect the translation group and use only the point symmetry groups; the small mismatches between the lattices being probably accommodated by defects in the boundary (dislocations

...

) . A s an example, the transformation of a B2 structure (space group Pm3m, point group m3m) into a tetratgonal structure (space group P4/mmm,point group 4/mmm) with no superimposed symmetry element excepted that of inversion (NiA1) will give if the sollicitation is the temperature :

NO1 = - 1

The number of orientation variants of martensite, n = index (m3m/i) = 24.

01 If now the reverse transformation is considered : HI = 4/mmm

N~~ =

i

and the number of orientation variants os martensite is only 8.

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If a stress field is applied along the [loo] direction of the parent phase, the group of sollicitation is D mh[lOOl and, assuming the induced martensite has the same symmetry :

So the number of orientation variants of martensite is 8

This last point clearly shows that it is possible to reduce the number of the generated variants of the product by applying a selected global sollicitation which may result in the superposition of several different sollicitations. The ultimate case of this is the single interface transformation.

In any case, the interface operations are the operations of the parent phase which are lost during the transition. These operations constitute the generating operations and for each of them, the coset associated with the interface is the product of the generating operation by the space group of the product, both expressed in the same referential system.

In the case of Ni-A1 martensite ( 7 ) , the recognized [111] transformation twinning does not play a special role, it is induced by the group-tree. For instance - the binary operation 2[01110~ of the B2 phase is lost (H, = m3m; NO1 = 1 then Ho = (432) P I ) . The coset of the interface is 2 011

E . p a .

4/mmm and the mirror m[Oll10 ,belongs to it. When referred to the martensi e 1 is wrltten as m

[llllf.c.t.

SYMMETRY AND MORPHOLOGICAL CONSTRAINTS At this point one has two informations

- the space group Go of the parent phase and G1 of the transformed phase, from which the relative location of the two space groups is also known

-

and the generated variants G1 of the transformed phase (the I G )-orbit) and the interface operations which relate these variants. 1

One can therefore study the constraints on the physical properties of crystalline interfaces through the interfacial symmetry.

1. Interfacial symmetry

Recently, G. Kalonji and J. Cahn ( 8 ) ( 9 ) have pointed out the importance of the interfacial symmetry as a basic ingredient for the study of precipitate morphology. Several methods can be used for the derivation of this symmetry (R.C. Pond and W. Bollmann (10); M. Fayard, R. Portier and D. Gratias (11) and G. Kalonji and J. Cahn ( 8 ) ( 9 ) ) .

1 ) Heterophase interface

There is no isometry which relates both crystals because they are of different structures and only the interfacial symmetry is built up with the elements of symmetry which are common to the two space groups Go and GI. The bicrystal symmetry is :

H =

~ ~ f \

^G1 (Go and G1 are defined to the same reference system) il) Homophase interface

The interface operation is described by the coset (a

I T )

GI. The two space

groups are conjugated :

-

-1 -1 1

G2 = (a/:) G1 (a/:) (with (air)-' = (a / - a - :))

G~ = (a(:) G~ (al:)-l -1

The common elements of symmetry form the subgroup H :

H = Glr\ (a/:) Gl (a/:) -1

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But the set of elements of symmetry which transform simultaneously crystal 1 into crystal 2 and crystal 2 into crystal 1 has also to be considered.

This set E is the common elements of the coset of the interface

((a

1:

) G1) with the inverse coset (a]:)-' = G1 (a[:)-' ) which realises the transformation 2 + 1

E = (al:) Glf\ Gl (a~:)-'

The group of superimposition is the union of H and E :

$ = H U E

It is easy to see that H is an invariant subgroup of order 2 in H f o r ~ s ~s H s 3 :-1 = H

Thus hl H h -1 = H 1 .

So one just has to determine

one

operation of E , say The superimposition group is then written (11) :

~ = H U ~ ~ H =1 H + E H

In fact the elements of H, which exchange the two crystals are better labelled as colored elements (8)(9)(10). In this case, the symmetry group of the bicrystal is a 2 color shubnikov group _:

where the prime designates a colored element 2. The Wulff plot (8)(9)

The basic idea of Kalonji and Cahn (9) is that in an isotropic medium, a growing crystal is constrained to adapt a crystalline form compatible with its point symmetry. Further considering for the growing crystal a crystalline environment, the morphological constraints are changed and must reflect the symmetry of both the crystal and the crystalline environment. Two cases have to be examined : the crystal in an environment with a different structure, which is the case of a phase transition; and the crystal in an environment of the same structure but in a different orientation and location, this is the case of two different variants of the transformed phase in contact with each other.

If the interface were planar with the normal to the plane n, the Wulff plot is the group W, the symmetry operations of which will give the set of interfaces physically equivalent to the first. So, it gives the variation of the interfacial energy as the Interface plane is varied and hence the morphological constraints.

For a heterophase interface, the Wulff plot is reduced to the H.group of common operations between the two crystals. Moreover, we have to take into account only the point symmetry of the involved groups (see Burgers (12)).

For a homophase interface, if the normal to the boundary in crystal 1 is defined, it is easy to see that any colored operation applied on the bicrystal will invert the sense of this normal. To restore it to the right sense, the normal must be inverted again. The. Wulff plot in this case is :

w

= H

U i.

( E ~ H )

where

i

is the inversion operation. Some examples illustrating this theory are given in ref. (8)(9).

It is quite easy to see the distinct effects of the two kinds of operation and hence the morphology. Consider crystal 1 and crystal 2 with a planar interface

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defined by it normal

n.

The mirror m is an operation of the group of superimposition : i)

~ G H

resulting morphology

y'

m m

m is a symmetry element of the resulting morphology m

resulting morphology

The symmetry element of the resulting morphology is the mirror m2 product of m' by the inversion.

The Wulff plot is certainly of major interest in the study of martensitic transformation. Kalonji and Cahn pointed out that if there are asymmetries in the environment of the bicrystal, the Wulff plot is reduced to the intersection of W with the invariance group of the asymmetries. Taking a formal example : for an assembly of variants, every bicrystal has a Wulff plot giving the equivalent orientations of the interface. If now a stress field is applied, every Wulff plot will be reduced to its intersection with the symmetry of the field. The interfaces will be no longer equivalent and a regrouping of the variants would appear without new phase transformation.

3. Symmetry dictated extrema (G. Kalonji and J. Cahn (9)(13))

Another symmetry aspect of interest in the present context concerns the so called symmetry dictated extrema or special points. In numerous cases, states of equilibrium are obtained through variational principle : the physical parameters characterizing the stable state are those which minimize a certain function F (X) (like the free energy). Although it is extremely difficult to give an analytical expression for this function, it is often quite easy to determine the minimal symmetries of the same i.e. the set of all the symmetry elements of the configura- tional space which leave the function invariant. If F (X) is invariant through a G-group, the little group of its gradient at a "point" X is the same as the little group of X : (see appendix 2) which means that the gradient of F at the point X is invariant through all the symmetry elements which leave X invariant. Therefore, if the little group is such that no vector remains invariant through the action of all the symmetry elements, the gradient vanishes. This is the case if X is a so called special point (14)(15)(16). The gradient being zero, the function has a minimum, a maximum or a saddle point. This condition relates symmetry and energy and fixes the symmetry dedicated extrema. Cahn and Kalonji have considered the case of two structures for which it is possible to change the relative orientations.

For each relative orientation the common symmetry (H = Go

/\

GI) is the symmetry group of the bicrystal and it will control the interfacial energy. If it is a symmetry dictated extrema this energy will have an extremum with respect to any change in the relative orientation of both crystals. The list of such groups is given in ref. (13). For instance if H is 4/mmm, it is a symmetry extrema with respect to any modification. If H = 2/m, the condition is only partial and the extrema is with respect to any rotation perpendicular to the axis. If H = 1 or

1

there is

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no symmetry dictated extrema. Cahn and Kalonji gave the example of the-FCC-BCC transition. With the Kurdjumov and Sachs orientation relationships H = 1, there is no symmetry dictated extrema. So, it is not surprising that the exact crystal orientation is never observed. With Nishiyama and Wasserman relations, H = 2/m and with Bain relations, H = 4/mmm. This latter orientation is at a symmetry dictated extremum.

Martensitic transformations have to be examined with the help of this theory not only for the relationships between the parent and the product but also for the accommodation of the martensite variants between them.

SOME PROPERTIES OF INTERFACE

The characterisation of the coset associated with an interface does not give, by itself, any information about the realisation of the interface. Some times, for a planar interface, the plane is parallel to a mirror which belongs to the coset.

But, this is not a rule.

Very often, the observed interfaces in crystals are such that a substructure is unchanged when crossing the interface. It is possible, for a given structure with space group G1 to derive all the interfaces which will leave the chosen substructure invariant. This substructure is invariant under all operations of its space group GS but some of them can also belong to G. So, the interfaces we obtained by the decomposition of G into cosets of the intersection group G (\ GS (17).

The other way for studying such a conservative property starts from a knowledge of the coset to the interface :

("1:)

G. The theory has been developed by Bollman but using only point operations and the translation group of the crystal (18). For two variants related by the coset, the coi'ncidence lattice I is the set of common trans- lations of the two variants :

I = T

r\

iajr) T (ulL)-l = T n u T

where T designates the subgroup of translation of the crystal 1.

The coyncidence site lattice is the set of points tro> of the space which remains invariant after interface operations :

( " 1 ~ )

G

r0

=

r0

This equation is the condition of reducibility of a space operation (see appendix 1). So the coincidence site lattice is built up with the support of all the operations of the coset which are reducible (11). All the atoms of the structure which sit on these supports are invariant through the interface.

FROM THE IDEAL SCHEME TO A MORE REALISTIC ONE

It is well known that often the variants of martensite are grouped to form a self accommodating unit which minimizes the induced strains. That means for a transformation induced by cooling, the initial group of invariance of the sollicitation is SO3 !for temperature) and leads to equivalent embryos. Around each embryo, the local solllcitation symmetry reduced to the intersection of SO3 with the invariance group gs of the strain induced by the embryo. Further formation of embryos in its vicinity will be determined by the intersection of this group with the Go group of crystal. Growth of these selected embryos will then lead to the speci- fic group of martensitic variants. Equal probability of the appearance of the variants is not lost since all the equivalent situations appear with the same probability when the initial embryos are considered.

In this case it is interesting to consider altogether the variants of the self accommodating group. The superimposition group defined for a bicrystal can be gene- ralized. If the group has n variants, n cosets ai Gi (i = 1,

. . .

^{n; al}⁼^I)^will

transform the variant 1 into variants i. We have also to consider the cosets trans- forming the variants between them.

For j + i : " . G -1 -1

l l a j ( j + l : G l a j 1 "i G . ) 1

For i + j it is a.Ga. -1 and the set of elements leaving i and j unchanged simultaneously is : J 1

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For a circular permutation like : (1,2,

...

^i,j,k,

...

ⁿ⁾⁺ ^(1,2,

...,

^j,k,i,

...

ⁿ⁾

the set of elements is :

alGlal-'r\ C Y ~ G ~ Q ~ - ' - ~

. . .

^a._{l l J}G a .-If\ a .G a _{J l k}

r\

a G a

i-'n . . . r\

^{y l a n}-1

Designating by pl(i) the transformed of i, for a permutation labelled 1, the set of elements is :

n

Since n variants define n ! permutations, the superimposition group for the n variants is then :

The group H leaving all the variants unchanged :

n -1

H =

n

^{a . ~}^a. is an invariant subgroup o f 3

.

i = l 1 1 1

Considering a11 the generated variants by a phase transformation in which there is a group-subgroup relation between the two phases, their superimposition group is the parent group itself.

REVERSIBILITY PROPERTIES

A condition of crystallographic reversibility was given earlier (3.i) in which the reverse transformation generated an unique variant of the parent phase. However, with 8 variants of the parent phase (3iii) in the example of a martensitic transformation, no reversibility was found. In fact, the invariance group of the sollicitation was assumed to be the same as for the direct transformation. It is reasonable to think that, due to local strains, this group is reduced in such a way that the number of possible variants decreases. In the case of thermoelastic transformations, which have the property of crystallographic reversibility, the reverse transformation appears to be a reversible movement of the interfaces. Any event appearing at the level of the interface gives rise to an equivalent one, derived from it by an operation of the interfacial symmetry. So, a possible condition for the reversibility is that the symmetry group of the interface must be a subgroup of the symmetry group of the parent phase.

This is obviously the case for an interface between the parent phase and a martensitic variant but when the interface is between two variants, the bicrystal symmetry must be carefully examined.

The case of pseudoelasticity is quite similar. Only the symmetry of the applied stress field also has to be taken into account. Finally, when the shape memory effect exists, the unique way for the reverse transformation is certainly a symmetry dictated effect, taking into account all the constraints induced by the mechanical treatment.

SYMMETRY INDUCED BY DISPLACEMENT WAVES (19)

The main problem associated with the investigation of instabilities of the parent lattice in the vicinity of the temperature of a thermoelastic martensitic tranformation is that a number of p o t e n t i a l l y . c o m p e t e t i t i v e reactions can occur.

For instance, there is extensive evidence that the temperature stiffness (Cll-C12)/2 decreases for B2-NiA1 alloy. A consistent interpretation can be arrived at in terms of displacements associated with <110>, <li0> transverse acoustic phonons.

Mcire generally, the problem however, is how one or several displacement waves, each defined by g- the wave vector and 5

-

the displacement vector, can change the

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crystal symmetry and what total displacements of atoms result from them.

Due to a single displacement wave, any position

c ,

is changed into :

q ' (E) = 5 + sin 2wg.5

The vectorial function

r 1

(f) admits the operator ( YI:) as a symmetry element if

Therefore,

yr

- -

+ ^{Y E}sin 2 n

q.f:

+ _t = yr +

t

+ E sin 2n_q.(~: + _t) Thus,

Y E sin 2wq.f = 5 sin 271 (y -1

q . ~

+ 9.:)

This relation must be valid for every position r and leads to four possibilities

For each possibility, a condition on 2wg.t imposes a possible existence or absence of an irreducible translation t.

-

For instance, if :

for a transverse wave (5 perpendicular to

g)

- E sin 2wg.f: = 5 sin 2 n(g.c

+

gt)

Hence,

2k

+

¹

q.t

- -

=

-

²

The y operation can be a

screw

axis parallel to g or a glide mirror perpendicular to 5 .

So, for a crystal suffering a transverse displacement wave, the only surviving symmetry elements could be :

"Is 1

pure or glide mirror glide mirror

m[s I

I

pure or glide mirror

& I

Screw axis

2 pure or screw axis

1.1

2 pure or screw axis [¶A 51

The surviving symmetry is consistent with the point group of invariance of a transverse wave :

m m m ( m

[a] "[el

"'[~AEI 1

In the case of a longitudinal displacement wave (CJ parallel to 5 ) , the symmetry elements are :

Cmv : pure rotation axis along q

pure or glide mirror par"allel to q_ (but the irreducible translation if any, being perpendicular to 2)

m[:] pure or glide mirror (same remark) 2 (perpendicular to g) : pure or glide axis - 1 inversion center

all these elements are consistent with the point group of invariance of a longitudinal wave :

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Dmh [ g l

When ( 8 , s ) is a displacement wave, all the waves deduced from the first one by the operations of the space group of crystal are equivalent and form the orbit of waves.

Some of these, locked for instance by a defect, can be coupled or superimposed.

In such a case, the total displacement function is written as :

r' (r) = r + Z E . sin 2n(q.r

+\Pi)

+"

- -

^{. - 1}

-

^1-

where the summation is extended to all the ( q . , ~ . ) waves which simultaneously inter- act locally. A phase factor

q i

for each waveka; to be added :

so, is a symmetry operator if IYE. sin 2n(q..r +

qi)

^=:zjsin ^2"

1 1

Y -1

%r

+ q..:

+ f

^j)

-J for every i of the set, there is a j such that : leading to the four possibilities :

YE. = + ^{E .}

-1 - - J

YSi =

+ ^Elj

With such a coupling, the surviving symmetry can increase in comparison with a single wave, because the operations which relate the waves between them have to be considered.

Going back to the B2 structure with transverse displacement waves ( {110},

<li0>) some coupling of the waves can be considered. Two orthogonal waves : (ql = 1 [ilo],

[lie])

and (q2 =

$

^jli~],E~//[T~o]) lead to a resultant structure with P4/mbm for which there are some longitudinal distortions, compression and dilatation, along [loo] and [010]. Two orthogonal waves

1 1

(

, [llo], E ~ / / [lio

1)

^and ^{( ?}^[lio

1,

c2//

[ii~])

will lead to Pmmm or to Cmmm if q2 is out of phase.

The coupling of 3 waves <110> with their q vectors belonging to the same I1111

-

plane will induce at least a three fold axis.

It is then possible to interprete most of the competitive reactions and the diffraction anomalies with a small set of displacement waves.

In the present context, the most important point to consider with respect to premartensitic instability and symmetry, probably comes from the local modification of the symmetry of the parent phase. The remaining symmetry is the intersection of the invariance group of the displacement wave G W with the symmetry group of the structure Go. And for an embryo G I , the equivalent embryos, which will appear around it, will be deduced by applying the operations of symmetry :

which do not below to G1 (assuming the temperature as the only sollicitation).

So, after growing they will form a socalled self accommodating group. It is to be remarked that all the variants generated by the classical group tree with Go and only the temperature as an external sollicitation appear with the same probability. Since all the waves of the orbit exist in the sample, each wave Wi belonging to the W- orbit acts locally in the crystal and its G W i group is the conjugate of GW by an operation of Go. So all the variants will be equally present in different, but equivalent accommodating groups.

Appendix 3 gives an example of this problem.

THE SELF ACCOMMODATING GROUP OF VARIANTS

This typical microstructure has been already examined in terms of symmetry while discussing the lattice instabilities and "a more realistic transformation".

(13)

In many cases of thermoelastic martensites the symmetry of the bicrystal, martensite variant-parent phase, is very low _:1 or

i

(inversion)

.

These conditions are not symmetry dictated extrema and, a priori, such a result is not favourable for the transformation. In fact one must consider a pseudo bicrystal : the parent phase with the self accommodating group of variants for which the symmetry group is now the superimposition group. In such a case, the "bicrystal symmetry" can increase and is able to reach a symmetry dictated extrema. Then, the transformation is more

"reasonable" but it is due to the relationship between the parent and the self accommodating group of variants.

(See Appendix 3 for an example)

Remmk I . 3't ha4 been h e a d y 4 d d Ahat o B e n Ahe ob4enved d e [ e c t 4 in i n y o t a h m e Ahode [on w h i c h a maximum e t o m i c o u b ~ h u c t u n e LA unchanged, when U L O / I . L I ~ ~ ~

.the i n t e n [ a c e . The 4ymmet/cy d i c t c t f e d exhema c o n d b 5 o n .b, in a aenoe, q u d e 4imLLan b e c a u ~ e it p04.~kk~.2k4 a conoenvaf.ion of 4 p & y element4 beiueen -the fruo 4 h u c t m e ~ , and Ahe p e e t a t t h e numben o[ mcLinfcLined elemenid, khe L u n g e d i~ khe chance t o g e f a ~ynun&y d i c t a t e d exakemm.

Remank 2. T h i ~ ne4u1A h a c e n f a i n c y 4orn&g t o do w i t h .the n e v e n ~ i b L e u r y o t d o - p u p h y . 9{ Ahene ^{i d}a 4yunefny d i c t a t e d e x h e m a khe p o u p akee can be wn&en. a4 {oUow4 :

8

(self accommodating group)

10

sym. dict. ext. group

Obviou4clly t h e n m b e n o[ vanian.t4 of.

4

( n I O I i 4 ~ r n d ( I why n o t ! l { o n t h e nevende .t/ran4{onmation.

Remmk 3 . The fruo way 4hape rnemoq e [ [ e c t ha4 obviounLy 4orn&g t o do w i A h

~ p + y : th e nema.ining h i v i n g [once Leado t o an u n i q u e way o[ inan4- [onmetcon.

THE SHUFFLES

In the phenomenological description of the martensitic transformation, the periodic inhomogeneous lattice strain, called shuffle, is described by a static displacement wave. The effect of such a wave on a crystal has already been discussed in terms of symmetry by J. Cahn (3). The problem is the same as with the premartensitic instabilities. The symmetry of the strain due to a single shuffle is mmm for a transverse wave, Dmh for a longitudinal and 2/m when there is an angle between the wave and the displacement vectors. A supergroup formation can appear due to certain magnitudes of the displacement vector. For instance, considering the longitudinal wave :

q = - [Ill] and 2 E = ~ [ l l l ] on B.C.C. structures

-

³

the resulting symmetry is :

P 5ml (a-b, -..

-

^b-c)

,

(a+?+c)/2)

.

[ 1 m 3 m n Dmh,

1

But, one can get a supergroup formation. If X = 0.192 the resulting symmetry is P 6/mmm (a-b, b-c, (?+b+c)/2)

- - - -

This is the B.C.C. + w transformation with two successive (111) planes collapsing into ar? unique plane.

THE ISOSTRUCTURAL TRANSFORMATIONS

A transformation is isostructural when an isometry exists between the atoms of the initial crystal and those of the final one. Let us examine the case of a transformation under a homogeneous straln { g

>.

The deformation of the crystal is such that the resulting symmetry is at least the common symmetry of the crystal and

(14)

of the strain : Ho = G

on

^IgI(3). For a critical value of the strain, uc, a supergroup J of Ho can be reached. On releasing the strain, the relaxation from J will result in several conjugated groups of H and finally to several different orientation-translation of G

0'

The operations of symmetry between the generated variants of G are the operations of J which do not belong to Ho.

For instance : strain [ 100 ] on a tetragonal martensitic crystal referred to the old B.C.C. axis

P4/mmm

-

^J

_'t

⁼^Im3m

h[100] ($+b,a-b,c) when the 3 vectors of the cell are equal

The relation of the strain will result in 3 variants of martensite of related by the three fold axis along [Ill].

This group J with a cubic cell keeps the atoms which sit on 100, 010 and 001 of the tetragonal unit cell, indistinguishable. In fact we do not attach a "mechanistic meaning" to this group which can be called a "transit-group".

This transit group can be used when one considers the succession of

compact-planes derived from the parent B2 structure. These compact planes can adopt 3 positions A, B, C. (ABCABC is F.C.C. ABAB is hexagonal, ABCBCACAB is 9 R

...

)

(111) stacking

AB = first neighbour distance (FCC) old 100

-

B2 (second n.)

AC = second neighbour distance (FCC) odl 001 - B2 (second n.)

(15)

If during the transformation the lengths AB and AC become equal, this unstable struc-

ture (J) can relax into two possible ways :

=

This isostructural transformation (twinning or slip), in fact, is a martensitic transformation (J # B2). Such close packed martensite usually exhibits a number of phases of different stackings and mutual transformations between them are easy.

CONCLUSION

The theory of symmetry is quite difficult to apply to the martensitic transformation, since the basic component consists in taking the intersection of different groups. Such intersections are often reduced to the identity due to small differerces in orientation and translation. This is the reason why space groups are not used and only the point symmetry is taken into account. So, there is a loss of informetion. Also in most cases, the initial crystal is assumed to be perfect and the role of the defects and surfaces is forgotten. Another important point is that very accurate crystallographic informations are needed. Moreover, till now, symmetry has given no information about the actual plane of the interface between the phases (if it is planar ! ) and about its location in the cells of the both crystal. Nevertheless, some results are obtained which tend to prove that symmetry can be efficient for this kind of transformation. The right question about applying symmetry rules to study martensite can be formulated using an analogy : examining a grain boundary between two crystals for which the disorientation is very close to a desorientation leading to grain boundary with high coxncidence between the two crystals, the actual grain is the high coyncidence grain with some periodic defects which correct the small misorientation. So for martensitic transformation, the question may be put if the transformation has to be described with the real orientationships or with closed orientationships forgetting some misorientations corrected in the specimen by some defects ?

REFERENCES

1. LANDAU L. and LIFCHITZ E., Physique Statique Ed. MIR Moscou (1979).

2. FALK F., this conference.

3. CAHN J., Acta Met.

2

(1977) 721.

4. GRATIAS D. and PORTIER R., Proceedings of ICOMAT, Cambridge (USA) (1979) 177.

5. SEITZ F., Ann. Math.

37

(1936) 17.

6. YAMAMOTO N., YAGI K. and HONJO G., Phys. Stat. Sol. (a)

41

(1977) 523.

7. CHAKRAVORTY S. and WAYMAN C.M., Met. Trans. A

2

(1976) 555 and 569.

8. KALONJI G. and CAHN J.W., submitted to Phys. Rev. Letters.

9. KALONJI G., Ph. D., M.I.T. Cambridge USA (1982).

10. POND R.C. and BOLLMANN W., Trans. Roy Soc.

292

(1979) 1395.

11. FAYARD M., PORTIER R. and GRATIAS D., "Symmetries and Broken symmetries in Condensed Matter Physics", Ed. N. Boccara IDET, Paris (1980) 57.

12. M.J. BUERGER, Elementary crystallography, J. Wiley and Sons, N.Y. (1956).

13. CAHN J.W. and KALONJI G., Proc. of the Intl. Conf. on Solid-Solid Phase Trans- formations, Pittsburg (1981) in press, (Pergamon Press).

14. KHACHATURYAN A.G., Phys. Stat. Sol.

60

(1973) 9.

15. DE FONTAINE D., Acta Met.

23

(1973) 9.

16. SANCHEZ J., GRATIAS D. and DE FONTAINE D., Acta Cryst.

A38

(1982) 214.

17. GRATIAS D., PORTIER R., FAYARD M. and GUYMONT M., Acta Cryst.

A35

(1979) 885.

18. BOLLMANN W., Crystals defects and crystalline interfaces, Springer Verlag (1970) 19. PORTIER R., GRATIAS D. and STOBBS W.M., Proceedings of ICOMAT, Cambridge (USA)

(1979) 541.

20. SCHROEDER T.A. and WAYMAN C.M., Acta Met.

25

(1977) 1375.

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APPENDIX 1 ELEMENTARY GROUP THEORY

When one selects all possible operations which leave "something" invariant, one obtains a group. Some important definitions are given in this appendix.

Consider the group G, g is an element of G; m represents a point in space.

-

the orbit of m, G(m) : set.of all the points equivalent to m and deduced from it by the group G :

Glm) ⁺m i f a m ) : I v g 6 G ; gm = mi}

-

the small group of m, Gm = the invariance group of m : Gm + ~ E G { W g C ~ ; g m = m ) ~ :

- when two points belong to the same orbit, their little groups are conjugated : m

6

G(m)

l a g 6 G : m ' = g m then G,, = g Gmg -1 m 1 6 G(m)

- the Kernel of the orbit is the group built with all the elements of symmetry which belong simultaneously to all the little groups of the points of the orbit :

K is an invariant subgroup of G :

g E G : g K g - l = K

- Stratum : all points m, the little groups of which are conjugated by elements of G belong to the same stratum.

let H be the little group of a point m and let [H]be the conjugated class of H into G :

[HI

= Ig H g - l , ~ g

t

G 1 the stratum M

[HI is

M [ ~ ]

= { all points m : G m <

[HI]

Consider a space operator (alr). The operator is called reducible if the associated T can be reduced to zero-after a translation p of the origin of the reference syst&.

-

If the origin is translated by f

,

the (alr) operator is now, referred to the new

-

origin, written as :

( a i r + up-? ) the operator is reducible if

_r +

a:-: = 0 that means : (a[:) P = 0

This is impossible for screw axis or glide mirror. Then, the criterion ofreducti- bility is directly issued from the notation of little group. It means (all) belongs to the little group of p, or that

!

is located on the support of the eTement

of symmetry.

-

Insolated strata define special points. A point p is said to belong to a stratum H if p is located at the intersection of the supports of the symmetry elements of H. If this intersection reduces to a single point, the stratum is then said to be an isolated stratum.

If H = 1,,P is located at the inversion center, this is an isolated stratum.

If H = 2 , ~ is along the binary axis and the stratum is not an isolated one.

The different point groups inducing isolated strata are the following : 7 , 2/m, 222, mmm, 3, 3m, 32, 4, 4/m, 42m, 42, 4/mmm

6 , B/m, 6m2, 622, 6/mmm, 23, m3, 43m, 432, m3m

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JOURNAL DE PHYSIQUE

APPENDIX I1

GENERAL PROPERTIES OF A G- INVARIANT SCALAR FUNCTION Let V (r) be a scalar function of r , invariant through G. That means

- -

: (el ) V

(5)

= V [( al:)

r_ I

⁼^V(r_)

i) relation to the Fourier components of V (r) V

(5)

= IV (E) exp- 2ni

2.5

d r 3

The Fourier transform of V

[(air) - -

r ] is ₃ V (9') = IV [(al~)

- -

r ] exp- 2ni ( q t

- .

^(al:)

r)

d r

= IV [(al~)

-

r

-

] exp- 2ni q q

- - .

^T

exp- 2ni a-I I?' . r d r 3

-

^-1 ³

= IV (r) exp- 2ni q'

.

exp- 2ni (a q'

.

^r)d ^r

Putting q' = aq

V (aq)

-

= IV (r) exp- 2ni a q . ~

- - -

exp- 2ni q.r d r

- -

3 We obtain

If now (air) is an operation of the small group

-

of q

-

: as =

q

and

V

(q)

= V

(q)

exp- 2ni q . ~

- -

V (q) must be zero if 5.:

-

is

not an integer. (This, for instance is the zero-condition for the structure factor of the crystal).

ii) Gradient of a scalar function :

What are the symmetry properties of the vector field V V(r)

- -

?

The gradient in terms of Fourier transform can be written as :

V (E) = IIV

(9

exp. 2ni q.r d q

- -

3

= f2ni

q

V

(q)

exp. 2ni q.r d q

- -

3 At the point r'

-

= (air) r this gradient writes

- -

:

V V (r') = I2wi q' V (q' ) exp. 211i

q f

.f' d3q

- -

Taking

ql.r' = aq.(ar + ^T)= q . ~

+

a?.:

- - ^{- . . -} -

and

y

V (f') = 12ni aq V (aq) exp 2ni a q . ~

- -

exp 2ni q.r d q

- -

3 Then

V V (r') = 12wi aq V (q) exp 2ni q.r d q 3

- - .

- - - -

So, the gradients at two homologous points are related through : V V {(al:) = a I V

(5)

If attention is restricted to those ( a l ~ ) operations which belong to the little

group of r :

-

V

-

V (r)

-

= a V

(r)

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The little group of the gradient at the point

r

is the little group of the point r. -.

Here, if r is a special point, the gradient must be zero since no vector of non zero length may be invariant through all the symmetry elements of the little group.

The important consequence is : At the special points a G invariant scalar function has either a maximum, a minimum or a saddle point.

APPENDIX 111

DESCRIPTION OF THE MARTENSITIC TRANSFORMATION FOR B Cu-Zn ALLOYS

The space group of the parent phase B2 is Pm3m (a,b,c) (Go). The martensitic phase has a 9R structure which is slightly monoclinic : d -=

i

(at,bf,cT). On cooling

I

-

^w ^-.

one has 24 martensite variants.

G = m3m g = SO

3 G = I

1

NO1 = - 1

Such an alloy exhibits some premartensitic instabilities associated with

(F

1 (oli), [olil)

transverse displacement wave for which the invariance group is G

w

= mmm = m [oil] m[~li] m[loo]

So the actual symmetry is locally :

~ = i

mvwi

H = mmm

I _-

N = 1

The number of variantgO&or this wave is locally :

nO1 = index (mmm/i) = 4 The coset decomposition is :

It is observed that the martensite variants are clustered into self accommodating groups of 4 variants which adopt a diamond morphology (20) and have between them the 3 operations found by symmetry considerations.

(110)0 plane

-@

D B (O1l)R twin

(011)0 twin twin (100) twin (011); twin (O1l)B twin plane

I

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C4-34 JOURNAL DE PHYSIQUE

When we consider the interfacial symmetry between G o and G1

,

the group H is reduced to

i

and is not a symmetry dictated extrema. Considering the self accommodating group : the cosets are :

A : l - : m[oll].i C : m

[Oll]

.I

D : m [loo]

.i

The superimposition group S is mmm and now the H group (G

n

^{S )}is mmm which is a symmetry dictated extrema.

Locally, two displacement waves occur :

((IoI), [lei]) and ((IIo), [ilo])

the resulting symmetry is only m

[O1i~

SO the group tree leads to :

The self accommodating group with then,comprise of only two variants forming as a wedge (20).

m[oli]

.i

The plane of the interface is (i55)

.

It is different from the twin plane.

The other observed morphologies can be found in the same manner.

If, the reverse transformation is considered H

= i

1 H

- i

10-

Thus, there is an unique variant of the parent phase and one returns to the initial parent phase. Schroeder and Wayman (20) pointed out that mechanical twinning on (011) and (100) planes appear when the martensitic sample is deformed. These interfaces are those found in thermoelastic martensitic transformation, which shows that ewinning is a stress assisted martensitic transformation (the J group is that of B2 structure). The four variants will collapseinto an unique one by movement of the interfaces.