The orientation of interfaces between a prototype phase and its ferroelastic derivatives : theoretical and experimental study

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The orientation of interfaces between a prototype phase and its ferroelastic derivatives : theoretical and

experimental study

C. Boulesteix, B. Yangui, M. Ben Salem, C. Manolikas, S. Amelinckx

To cite this version:

C. Boulesteix, B. Yangui, M. Ben Salem, C. Manolikas, S. Amelinckx. The orientation of interfaces

between a prototype phase and its ferroelastic derivatives : theoretical and experimental study. Journal

de Physique, 1986, 47 (3), pp.461-471. �10.1051/jphys:01986004703046100�. �jpa-00210226�

The orientation of interfaces between a prototype phase

and its ferroelastic derivatives : theoretical and experimental study

C. Boulesteix (1), ^B. Yangui (2), ^{M. Ben Salem} (2), ^{C. Manolikas} (3) and S. Amelinckx (4) (1) Faculté des Sciences de Saint-Jérôme, ^rue Poincaré, 13397 Marseille Cedex 13, France

(2) Département ^de Physique, Campus Belvédère, Tunis, ^Tunisie

(3) Department ^of Physics, University ^of Thessaloniki, Thessaloniki, ^Greece

(4) Universiteit Antwerpen (RUCA), Groenenborgerlaan ^{171, B-2020} Antwerpen, ^and SCK/CEN, ^B-2400 Mol, Belgium

(Reçu ^{le 20} juin 1985, accepté le 23 octobre 1985)

Résumé.

Une méthode est proposée pour déterminer les interfaces ^non déformées entre une phase prototype para-élastique ^et les variantes d’orientation ferro-élastiques qui ^en résultent. Les prédictions ^{de cette théorie sont} comparées ^à quelques exemples d’observations obtenus _par microscopie électronique. ^{L’accord entre les} prédictions

et les observations est raisonnable.

Abstract.

A method is proposed ^{to determine the} strainfree interfaces between a paraelastic prototype phase ^and

its ferroelastic derivatives. The predictions ^{of the} theory ^{are tested} by ^{means of electron} microscope observations

on a number of examples. Reasonable agreement is found between the predictions and the observations.

Classification

Physics ^Abstracts

64.70K

1. Introduction.

The permissible interfaces between the prototype high temperature y-phase and the low temperature ^ferro- elastic p-phase ^{of lead} orthovanadate have been derived in reference [1] on ^the assumption ^that along

such interfaces the two structures (or ^their lattices) ^fit perfectly, i.e. that the interfaces are strain free. It was

found that in this particular ^{case this} assumption ^leads

to a correct prediction. ^{It is the} purpose of this _paper to

generalize ^the reasoning ^{and to} apply ^{it to the com-} plete ^{set of} possible ferroelastic transitions. In this

sense we extend the work of Sapriel [2] ^{who worked}

out a procedure ^{to find all} permissible ^interfaces

between different orientation states derived from a common prototype. ^We shall determine the orienta- tion of the permissible interfaces between a prototype phase and its ferroelastic derivatives based on the

same criterion : the interfaces must be strainfree.

Apart from lead orthovanadate only observations

on the rare earth oxides Nd2o3 ^and Sm203 [3] ^and

on BiV04 [7] ^are available; ^{we shall} compare these with the theoretical predictions.

2. Theoretical considerations.

The procedure ^{we use in} deriving ^the permissible

interfaces between a ferroelastic phase ^{and its} proto-

type phase is similar to that used by Sapriel [2] ⁱⁿ deriving ^the possible interfaces between domains of different orientation states, ⁱⁿ ferroelastic crystals.

The criterion he used was that the permissible ^inter-

faces should be strainfree, i.e. that along ^{all directions}

in the interface the strains should be the ^same for the two orientation states. Sapriel could show that two

types ^{of walls occur.}

The W-walls are entirely determined by symmetry;

they ^are parallel ^to symmetry planes ^{of the} prototypic phase ^{which are no} longer symmetry planes ^{in the}

ferroelastic phase.

The W’-walls are parallel ^{to the twofold} axis, ^which disappears ^on transformation; they ^{are not} entirely

determined by symmetry ^{since one} angular degree

of freedom remains. This angle ^is determined by ^the

condition that the interface should be strainfree.

We shall apply ^{the same criterion to the} present problem limiting ^{ourselves to cases} where there is a

discontinuous change in lattice parameters ^{at the} transition temperature (i.e. first order transition).

Since we are interested in the contact plane ^between

the prototype and the ferroelastic phase ^{we have to}

assume that both phases ^are infinitesimally ^{close to the}

transition temperature Tc (which ^is strictly ^defined only ⁱⁿ the unstrained conditions) ^the prototype being ^at T c ^{+ 8 and the} ferroelastic phase ^at T c - ^E.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703046100

It turns out that such ^a situation can be realized in

practice in the electron microscope since small temperature gradients ^{are often} present. ^{Also under} certain conditions the high temperature phase may be stabilized in some part ^{of the} specimen ^{even below} T c [1].

For the theoretical treatment we have to inquire

which spontaneous ^{strain tensor} is relevant in this context Since we are interested in the situation at T c

the effect of thermal expansion ^{has to} be eliminated and only contributions to lattice parameter changes

as a result of lattice distortions have to be retained (1).

This problem ^{has been treated} by ^Aizu [5] ^{and he has}

defined a spontaneous ^{strain tensor which is meant}

to do this.

According ^{to Aizu’s} formulation the spontaneous strain tensor is calculated with reference to a proto- type ^{which is} considered to be the ^« _{average »} of all

possible orientation states. This procedure ^eliminates

the effect of thermal dilatations which ^are only, ^if they ^occur, secondary effects, ^{in the case of ferro-}

elastic phase transitions (2).

In our concrete case one could conceivably ^mea-

sure the lattice parameters ^{of the} prototype ^and of the ferroelastic phase, ^{both at the} coexistence temperature, ^{i.e. at the} transition temperature ^and hence find the usual strain tensor at this temperature.

In practice this will be difficult to do and in actual fact measurements will always be made above and below the transition temperature ^{so that if a small} change ⁱⁿ

volume occurs it is never very well known. For theo- retical considerations it is therefore preferable ^to

define the strain tensor in the manner suggested by

Aizu.

We shall now generalize ^the reasoning, applied ⁱⁿ

the case of lead orthovanadate, and consider all

possible ferroelastic transitions. The problem ^{can be}

reduced to that of finding the surfaces along ^which

(’) ^{We consider} only ^{the case} of « pure » ferroelastic phase transitions, ^i.e. occurring ^{without a} change ^{of volume of the} sample ^at TC. ^{If the} strain would cause a non-negligible change ^{in volume at the} phase transition, ^{it should have to be}

taken into account using ^{the most} general ^{form of the strain}

tensor : see for example Nye ^{on «} Physical Properties ^of Crystals ^». The method could then be extended to different kinds of transformations such as oxidation phenomena ^for example.

(2) Aizu has defined ^a spontaneous ^{strain tensor} XiSi) ^of state Si which becomes ^zero in the prototype phase ^{at all} temperatures ^{and which is different for all} different orienta- tions states of the ferroelastic phase. ^It is related to the usual definition of the strain tensor X(Si) ^of state Si by ^{the relation}

where q ^is the number of different orientation states Sj. ^The

strain tensor Xs clearly depends ^{on the} symmetry ^of the prototype phase ^{as well as on} that of the ferroelastic phase.

the lengths ^{are conserved on} transforming ^{from the} prototypic into the ferroelastic phase, ^{i.e. the surfaces} along which the strain is zero. The geometrical ^locus

of the directions along ^which lengths ^are conserved is

given by ^{R2 = R R = constant for any vector R}

situated in the interface. This condition can be _express-

ed, taking ^{as a common} origin 0, ^a point ^{of the inter-}

face for all R vectors, ^as

- --

N-/

for _any dR ; ^{this means} geometrically ^{that the} change

dR of R due to the spontaneous ^{strain must be} per-

pendicular ^{to R or be zero.}

Introducing ^the symmetrical ^{second rank}

strain tensor Sij ^we have for the components ^of

dR(dxi, dx2, dx3)

where xl, X2, X3 ^are the components of R. The condi- tion (1) ^{thus becomes}

or finally

This homogeneous quadratic equation represents

a conical surface with its apex in the origin ^{taken at the}

interface.

A simple geometrical interpretation ^{can be} given

to equation (2). ^A sphere ⁱⁿ the unstrained body

becomes a quadratic surface after having undergone

a homogeneous ^strain represented by ^a second rank tensor. The volumes of these two bodies ^are the same

since ^{we assume} that volume is conserved in the deformation. These concentric surfaces intersect in

general along ^a quartic ^curve. Along ^lines leading

from the common centre to points ^{on this curve there}

is no length change and hence the directions of these lines are strainfree. The conical surface with its apex in the ^common centre and with this curve as a generat-

ing ^{curve is thus} equivalent ^{with the conical surface}

considered above.

The apex of the ^cone is situated in the interface but this interface must be independent ^{of the} origin ^cho-

sen so that the cone possessing ^a singular point, ^must

be rejected ^{as a} physically unacceptable ^solution [2].

In the general ^case there will only ^be physically acceptable ^{solutions for} planar interfaces if the homo- geneous equation (2) degenerates ^{into the} product

of two linear factors. The condition for this to occur

is that the determinant of the strain tensor L1 = I Sii ^I

should disappear. ^{Since the} symmetrical part ^{of the}

strain tensor, which is the one of interest here, ^is

represented by ^a traceless matrix (since ^{volume is}

conserved in the transformation) ^{the two} permissible planes ^are mutually perpendicular.

If the conical surface is not degenerated ^{the solution}

is not physically acceptable, ^{but in a} thin foil the solution can still be approximately valid because the intersections of the cone with the plane ^{of the} foil,

assumed to pass through ^the apex of the cone, ^are lines independent ^{of the} origin chosen. However such intersections are not necessarily present

The point group symmetry ^{of the} spontaneous strain tensor is completely determined by ^the point

group of the ferroelastic crystal. ^Since every ^tensor of rank two has inversion symmetry, ^{we must} only

consider centro-symmetric point groups, i.e. the 11 Laue point groups when surveying ^all possible ^tran-

sitions. The point group of the ferroelastic phase Lf

is further a subgroup of that of the prototype phase Lp.

The ferroelastic phase ^{can in} general be formed in different orientational states related by symmetry operations of Lp ^{which were lost} during ^the transition.

The number q ^{of such} orientation states is given by ^the

ratio of the order of the point group of the prototype phase and the order of the point group of the ferro- elastic phase [4]. ^The permissible interfaces between the prototype phase ^{and the} different orientation

states derived from this prototype ^{are related} by ^the

same set of symmetry operations by ^{which the different}

orientation states are related _among themselves since such operations reproduce ^the prototype. ^{With each} orientation state a set of permissible interfaces is associated.

The spontaneous ^{strain tensors which we shall}

use have been derived for all types ^of ferroelastic transitions by ^Aizu [5]. The formalism is then as

follows. For each type ^of transition the spontaneous strain tensor is written as well as the corresponding equation (2); the latter is decomposed ^{in linear}

factors when this is possible (i.e. ^{if L1} = 0). Only ^{in the}

latter cases are planar strain free interfaces present;

if L1 =1= 0 approximate solutions could exist (i) ^{the case}

of thin layers, ^{if d 0} making ^{the cone} rather similar to two perpendicular planes. ^{These are indicated as}

well. Sometimes we shall also indicate solutions which

may ^occur for special values of the strain components, i.e. for specific ^materials only.

3. Systematic survey of all possible ^cases.

We can classify ^{the different cases either} according ^to

the point group of the prototype phase ^{or to that of the}

ferroelastic phase.

Adopting ^{the first} point ^{of view we have to consider}

the eleven Laue point groups which have subgroups

of possible ferroelastic phases; this leads to the

following numbers of prototypes : ¹⁰ cubic, ^{9 hexa-} gonal ^and trigonal, ⁸ tetragonal, ² orthorhombic and 1 monoclinic, ^{i.e. 30 cases} in total. The second point

of view leads to a more concise survey; this is the ^rea-

son why ^{we have} adopted this in table I. The different transitions are classified according ^to decreasing symmetry of the ferroelastic phase.

Since the strain tensor depends ^on the reference system ^{it is} important ^to specify this in each ^case.

We shall systematically ^choose rectangular ^{axis pa-}

rallel with the crystallographic ^{axis of the} prototype whenever such axis exist We shall in fact systematically apply ^the convention adopted in references [5] ^and [6] ; ^{in cases where we} deviate from this ^we shall indicate this explicitly.

.

We shall use the following ^{short comments when} applicable :

(1) ^{There are exact solutions in the} general ^{case :}

noted ES.

(2) ^{There are no exact} solutions in the general ^{case :}

noted NS.

(3) ^{There are exact} solutions in the special ^case indicated, ^{i.e. for} special ^values of certain strain components, noted : ^SS.

(4) ^{There are} approximate ^{solutions if certain}

components of ^the strain tensor are sufficiently ^small (e always represents ^{a small} quantity ^as compared

to the other components). Noted : AS.

It turns out that exact solutions (ES) corresponding

to planar interfaces are only found in the following

cases :

(3) ^{The two} solutions of the quadratic equations ^{are : also} note the two solutions as y ^{= ax} and y ^{= -} (1/a) x

with a ⁼ ( - s ⁺ p2 ⁺ s2) ^{p. The two solutions represent}

mutually perpendicular planes.

In the notation of Aizu [5] ^{the indices} p and s denote

principal ^{and side} respectively. These notations refer to the two possible positions of the diad axis of the ferroelastic phase ^with respect ^{to the} symmetry ^{axis of} the prototype phase. If the diad axis is along ^the principal axis this is noted (p); ^{if it is} along ^{a diad}

axis of the prototype phase, ^{which is} perpendicular

to the principal axis, this is noted (s).

In all these cases the planar interface between the prototype phase ^{and the} ferroelastic phase ^also qualifies ^{as a} permissible interface between two different orientation states of the ferroelastic phase.

There is one more transition for which a general

solution exists, but in this case the transformation interface is not a permissible interface of the ferro- elastic phase :

hexagonal (1) ^-+ orthorhombic (p)

It should be noted that in a number of transitions the permissible interfaces coincide with lattice planes

of the prototypic phase, ^{in other cases} the orientations

depend ^{on the} components ^{of the} spontaneous ^strain

tensor at the transition temperature.

In those cases where no planar solutions exist the interfaces are strained and possibly curved. Curved transformation fronts are often observed.

4. Observations.

A few concrete materials for which observations have been made will now be discussed.

4.1 LEAD ORTHOVANADATE [1].

^{Well defined} planar

interfaces have been observed in lead orthovana- date (Fig. 1). ^The phase transition is of the species :

3m F 2/m. Adopting ^an orthohexagonal ^reference sys- tem for the prototype ^the equation (2) is p( y2 - X2) ⁺

2 txz ⁼ 0 and there are no planar solutions. The observations ^are made on foils obtained by cleavage

and hence parallel ^{with the} plane ^{z = constant.}

Putting ^{z = 0 which} corresponds ^{to the} (001) plane

of the hexagonal ^{structure in} equation (2) ^{one obtains} p( y2 - x2) ⁼ 0, ^i.e. y ⁼ ± x, i.e. the traces of the

cone of zero strain with the plane ^{z = 0} passing through ^its apex ^are the bisectors of the orthogonal

axis. This result is to a good approximation ^{in accor-}

dance with the observations as can be judged ^{from the}

models of figure 2(a) ^and (b) ^and figure 3. The ortho-

gonal ^{axis are} indicated in the two variants separated respectively by ^{a W and a W’ wall. In each case it is}

found that the strain free interfaces are parallel ^{to the} traces y ⁼ + ^x. Furthermore one of the traces along

a W wall encloses roughly ^an angle of 900 with one of the traces along the W’-wall. The good correspon- dance between observations and predictions suggests that the « thin foil approximation » ^is justified empirically.

Fig. 1 a, b, c, d, e, f

Successive stages ^{of the} phase ^transi-

tion of Pb3(V04)2 ^{from the} monoclinic f3 ^{to the} hexagonal y

phase ^with increasing temperature. Notice that the interfaces between high ^{and low} symmetry structures are always paral-

lel to the same directions giving ^{the interface a} zig-zag shape.

The y-phase ^{is here} nucleating along ^a ferroelastic domain wall of the fl-phase (in ^this particular ^case W-walls). ^Ferro-

elastic domain walls between orientation states S1 ^and S2

are numbered 1, 2, ^{3 in} (a).

Fig. ^2.

Models for the interfaces between two orientation states P, ^and P2 of the ferroelastic phase ^{and the} y-phase.

(a) ^{The two} fl-variants ^{form a} W-wall; (b) ^{The two} fl-

variants from a W’-wall. The strainfree interfaces enclose in each variant an angle of - 45° with the ortho-hexagonal

axis. The continuity of the lattice across the interfaces is

emphasized. The situation (a) ^{can be} compared ^with figure ¹ (after ^Ref. [1]).

4.2 RARE EARTH SESQUIOXIDES [3].

^{The transfor-}

mations from the hexagonal ^to the monoclinic room

temperature ^{form in} Nd2o3 ^and Sm203 belong ^to

the same species ^{and hence the same} theoretical

predictions ^{are valid.} They ^are rather well verified in the case of Sm203 ^where doubly ^{twinned mono-}

clinic crystals usually appear inside the hexagonal

Fig. ^3.

Interfaces between the fl-variants separated by ^a

W’-wall and the y-phase. ^This image ^{can be} compared ^with

the model of figure 2, ^b.

matrix (Fig. 4). Agreement is found with the theoretical orientation within 5°.

The agreement is somewhat less good ⁱⁿ Nd2o3 (Fig. 5). The interfaces between the monoclinic phase

and the hexagonal prototype ^enclose angles ^{of about} 30°, ^{60° or 90° which} correspond ^{to the} angles ^between

the different permissible strain free interfaces. How- ever, rather large(- 10°) differences can occur between the theoretical angles ^{and those} observed, ^{as is shown}

in figure ^5.

In the latter case high resolution images ^were

obtained of the interfaces (Fig. 6); the deviation from the theoretical orientation is found to be less than 3°

in these images.

From the absence of inclination extinction contours in the vicinity of the interfaces ^{we can} conclude that the latter are strainfree. It was shown previously ^{that small}

nuclei of monoclinic structure can occur which strain the hexagonal matrix; ^{these cannot} grow however (7).

Taking ^{into account} the fact that a small rotation of angle Ax/x ^{must occur between} prototype ^and ferroelastic structures to superimpose ^{the strainfree}

surfaces making ^the choice of reference directions somewhat uncertain, ^the agreement is thus found to be

satisfactory ⁱⁿ lead orthovanadate and in Sm203 ^but

somewhat less good ⁱⁿ Nd203. ^It should be noted that in the latter case the monoclinic phase resulted from the quenching ^of hexagonal specimens. ^{It is thus} quite possible that the interfaces did not adopt ^their equi-

librium orientations, but where frozen in at some

strained orientation. This was apparently ^{less the case}

in the high resolution specimens where results in better agreement with the observations were obtained.

4.3 BISMUTH VANADATE [9].

B’V04 provides ^an example ^{of the} species 4/m ^F 2/m (case 15) ^{for which}

d ⁼ 0 so that a general ^{solution occurs.} The twofold axis of the monoclinic phase coincides with the fourfold axis of the tetragonal prototype phase.

Fig. ^4.

Domains in Sm203. (a) ^Observed configuration ^of

domain walls in monoclinic SM201 (B-phase), ^{within a}

matrix of the hexagonal A-phase. (b) Model for the configu-

ration of domain walls in (a). The monoclinic part ^{is cross-} hatched ; it contains two orientation states of the B-phase.

The interfaces enclose an angle ^of approximately ^{450 with the} ortho-hexagonal axis of the A-phase, ^as predicted by ^the theory.

Fig. ^5.

Domains in Nd203. - (a) ^Observed configura-

tion of domains walls in monoclinic Nd203 (B-phase)

formed after quenching ^{from the} hexagonal phase (A-phase).

(b) Model for the configuration observed in (a). ^{The mono-}

clinic part ^is cross-hatched; it contains two orientational states. Interfaces are along ^the (310) plane of A and close to

(11. ) plane ^{of A} (dotted line); ^The agreement with the theo- retical prediction ^is only approximate; presumably ^{due to}

the quenching.

Fig. ^6.

High resolution image of the interfaces between monoclinic and hexagonal Nd2ol- ^{In this case the agree-}

ment with the theoretical predictions is better than in figure ^4.

Table I.

The 30 possible species of ferroelastic transitions among the 11 Laue crystal classes obtained by adding

the inversion to prototypic and ferroelastic point groups ^are labelled according ^to reference [2]. ^{For each} type of

transition the table gives

(i) ^the different ferroelastic ^species corresponding ^{to this transition}

(ii) ^the spontaneous ^{strain tensor and when} necessary ^some indication concerning the relative orientation of

the axes of high ^{and low} symmetry ^structures

(iii) the value of ^A

(iv) ^the equation of ^{the cone} and, ^when degenerated ^{into two} orthogonal planes, ^the equations of ^these planes

(v) the indication as to whether or not singular ^and (or) approximate ^{solutions exist.}

Table I (continued.

Table I (continued).

Table II.

Lattice parameters of BiV04.

Tetragonal prototype ^{at 566 K}

(’) SLEIGHT, ^A. W., CHEN, ^H. Y., FERRETTI, ^{H. and Cox} D. E., Mat. Res. Bull. 14 (1979) ^1571.

(b) BIERLEIN, J. D. and SLEIGHT, ^A. W., Solid State Com,

is thus

The orientation of the possible interfaces between the tetragonal and the monoclinic phase ^{are then} given by

with

The angle 0 ^{of the} interface normal with the tetragonal

a-axis is then given by 0 ⁼ arctg ^a.

Introducing the numerical values one obtains an

angle 0 ^{of about 32°,} which is in fair agreement ^with the observations. Figure 7 shows the initial phase ^of

the transformation; dagger shaped ferroelastic do- mains are nucleated at the edge ^{of the} specimen ^and propagate ^{into the} prototype phase.

At the tip ^{of the} daggers the orientation is of ^course not well defined since the interfaces are somewhat curved there. Two quasi-mutually perpendicular ^sets

of interfaces are invariably ^observed, corresponding

with the two values of ^a. Such a configuration ^mini-

mizes the strain energy associated with the deforma- tion pattern.

According ^{to the} theory ^the interfaces between

Fig. ^7.

Daggershaped ferroelastic domains in BiV04,

nucleated at the edges ^{of the} specimen. ^The interfaces enclose

an angle of about 320 with the tetragonal ^a-axis.

the prototype and the ferroelastic phase ^are parallel

with those between different ferroelastic orientation states; this is also observed experimentally.

It should be noted that there is evidence in the literature suggesting that this transition should be of second order (a, b, c), implying ^{that the} spontaneous strains all disappear ^at the transition temperature ^and

one may therefore wonder what determines the orientation of the interfaces.

However, the orientation of the interface is usually

determined by the ratio of spontaneous strain compo-

nents. This is for instance clear from the expression :

which determines the orientation of the interfaces in a

large ^{number of cases; it} clearly only depends ^{on the}

ratio s/p. ^{Even in the limit s -+ 0} the ratio of the strain components may remain constant and different from zero, and therefore lead to a well defined ^answer.

This is in particular ^{true in the} temperature range of interest here, i.e. close to Tc’ ^{where the} changes ^{can be}

described by ^{a linear} approximation. ^One expects nevertheless sharper ^and crystallographically ^better

defined interfaces in cases where there is a disconti-

nuous change ⁱⁿ the lattice parameters ^at TC’

5. Alternative approach.

The theoretical problem ^{can be} treated in a somewhat different _way by inquiring under which conditions a

given plane ^is strainfree, ^{i.e. what are} the relations among the strain components which have to be satisfied in order to make ^sure that the bicrystal ^should

be strainfree along ^the plane

This condition is obtained by eliminating say x3 between (2) ^and (3). This condition should then be satisfied whatever be the values of _xi and _x2, i.e. the coefficients of X2, X2 ^and xi x2 should then be sepa-

rately ^{zero. This leads to three} simultaneous quadratic equations which have to be compatible ^{and which}

determine the ratios of the ai ^{when the} compatibility

conditions are satisfied.

Instead of treating ^the general problem, ^{which leads}

to lengthy expressions ^we shall consider ^a particular

but relevant case.

The condition under which

should be a permissible plane ^is given by

obtained by eliminating x2 between (2) ^and (4). ^This

leads to the following relationship between strain

components :

In the general ^{case there is no} solution for a ; the

compatibility condition is obtained by eliminating ^a

between equations (6) ^and (8). ^{One obtains}

The same relation is found for _x2 ^{= -} (1/a) xl.

The value of a can be obtained either from equation (6) ^{or from} equation (8). ^{It is}

or

If the compatibility equation (9) is satisfied these two values are equal ^{of course.}

With reference to table I it is easily verified that the conditions (9) ^and (7) ^{are satisfied in all cases where}

y ^{= ax} and y ^{= -} (1/a) ^{x are solutions and also that}

the value of a is given by (10) ; (11) ^{leads to an unde-}

termined value.

6. Conclusions.

An exhaustive _survey has been given ^{of all} permissible

interfaces between a prototype phase and its ferro- elastic derivatives using ^the criterion that the interfaces should be strainfree. Strainfree interfaces are found to occur only ^{for a limited} number, 9, of transition

species; ⁱⁿ eight ^{of these the} interfaces also qualify ^as permissible walls between different orientation states.

The only experimental result that has been tested

(bismuth vanadate) corresponding ^{to one of these}

cases fits the theoretical predictions.

Approximate (AS) ^and singular ^solutions (SS) ^can

be found in a number of cases where no general ^exact

solution (ES) ^{exists. It is found} experimentally ^{in two}

different cases that in the foils the permissible ^interfaces

have traces along ^the intersection lines of the strainfree

cone with the plane ^{of the foil} passing through ^the

apex of the cone.

In order to establish the general validity ^{of the}

conclusions _many more experimental ^{results are}

required.

References

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[2] SAPRIEL, J., Phys. ^{Rev. B 12} (1975) ^5128.

[3] BOULESTEIX, C., CARO, P., GASGNIER, M., HEURY LA

BLANCHETAIS, ^{Ch. and} SCHIFFMACHER, G., ^Acta Cryst. ^{A 27} (1971) ^552.

BOULESTEIX, C., ^{in Handbook} of ^the Physics ^{and Che-} mistry of Rare Earths (North ^{Holland Publ.} Cy, Amsterdam) 1982, ^vol. 5, ^Ch. 44, p. 321.

[4] AIZU, K., ^J. Phys. ^Soc. Japan ²⁷ (1969) 387 ; Phys. ^Rev.

146 (1966), 423 ; ^J. Phys. ^Soc. Japan ²³ (1967) ^794.

[5] AIZU, K., ^J. Phys. ^Soc. Japan ²⁸ (1970) ^706.

[6] NYE, ^J. F., Physical Properties of Crystals (Clarendon Press, Oxford) ^1960.

[7] ^BEN SALEM, M., DORBEZ, R., YANGUI, ^{B. and} BOULESTEIX, C., ^Philos. Mag. ^{A 50} (1984) ^621.

[8] BOULESTEIX, ^{C. and} YANGUI, B., Phys. ^{Status Solidi} (a)

70 (1982) 597 ;

BOULESTEIX, C., Phys. Status Solidi (a) ⁸⁶ (1984) ^11.

[9] MANOLIKAS, ^{C. and} AMELINCKX, S., Phys. Status Solidi

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