247

J. Appl. Cryst. (1977). 10, 247-251

Investigation of the Ferroelastic Domain Structure in Lead Phosphate

BY M. CI-IAB~ AND F. GILLETa'A

Laboratoire d'Etudes Physiques des MatOriaux (Equipe de Recherche associOe au CNRS n ° 13), UniversitO d'OrHans, 45045 Or[Oans COdex, France

AND J. P. ILDEFONSE

Laboratoire de GOochimie-MinOralogie Institut de Recherches sur les Ressources et MatOriaux MinOraux, UniversitO d'OrHans, 45045 OrlOans COdex, France

(Received 3 November 1976; accepted 4 February 1977)

The ferroelastic domain pattern observed in Pb3(PO4)2 at room temperature is analysed in terms of twinning by pseudosymmetry. The characteristics of the three kinds of domains and the two types (W and W') of walls are determined and agree with the experimental observations. The variation with temperature of the spontaneous strain is deduced from the investigation of the orientation of W' walls.

Introduction Lead phosphate [Pb3(PO4)2-1 undergoes a purely ferroelastic transition at 180.5°C (Keppler, 1970; Brixner, Bierstedt, Jaep & Barkley, 1973), from a high- temperature trigonal phase (phase/~, space group D~) to a monoclinic low-temperature phase (phase ~, space group C6h). This transition is first order (Cao-Xuan An, Hauret & Chapelle, 1975; Toledano, Pateau, Primot, Aubree & Morin, 1975; Gilletta, Chabin & Cao-Xuan An, 1976). According to Aizu's classification (Aizu, 1969), lead phosphate belongs to the 3mF2/m ferro- elastic species. Aizu (1969) has shown that three orien- tation states or domains can be generated in the ferro- elastic phase, because of the disappearance of some symmetry elements of the high-temperature prototypic phase.

The ferroelastic domain pattern has been previously described by Brixner et al. (1973) and Dudnick, Sinyakov, Gene & Vagin (1975); they observed three kinds of domains (or twins) and two different types of walls separating adjacent domains.

On the other hand, Aizu's model, considering that the low-temperature lattice can be described as the prototypic lattice spontaneously distorted, enables Sapriel (1976)* to determine the equations of the walls as functions of the components of the spontaneous strain tensor.

The analysis of the characteristics of the various kinds of domains and walls in lead phosphate can be performed in another way, by considering the pos- sibility of twinning by pseudosymmetry. We have then investigated these characteristics at room temperature, with the aim to account for our results by the laws of formation of pseudosymmetry twins.

* We thank Dr Sapriel for sending his paper prior to its publica- tion.

Experimental results The samplest are thin plates cleaved along the mono- clinic (100) plane. The domain pattern was studied at room temperature with a polarizing microscope pro- vided with a four-axis universal stage (accuracy in the measurement of angles" 1°). Some investigations were performed with a simple polarizing microscope pro- vided with a temperature-stabilized heating stage. The domain pattern is generally complicated; nevertheless (Fig. 1), three kinds of domains [labelled D1,D2,D3 on Fig. l(b)] and two types of walls (W and W') appear.

1. Optical characteristics of domains The orientations of the monoclinic crystallographic

axes T1,7"2 (twofold axis) and T3, relative to the prin- cipal axes of the index ellipsoid, have been determined in each kind of domain and agree with previous results (Torr~s, Aubree & Brandon, 1974) (Fig. 2). The angle 2V=81 +2 °, between the optic axes O1 and 02, con- tains the principal index axis Ng; it follows that ~-Pb3(PO4)z is a positive biaxial crystal. The index axis N,,, lies along the monoclinic twofold axis T2.

2. Associations of domains We have observed two kinds of association between

adjacent domains. (a) Domains separated by a subvertical wall (W wall).

In this association, the wall appears as a line under the microscope, and is almost perpendicular to the cleav- age plane (Fig. 3a). The (100) planes of both domains (D1 and D2) are not coplanar (Fig. 3c) and the angle between these two faces, measured by an optical goniometer, is 178.60___0.1°. The directions of the crystallographic axes, and of the axes of the index

+ We thank Professor Chapelle for providing the crystals, grown in the Laboratoire de Physique Cristalline, Universit+ de Paris XI, Orsay.

JAC 10-3"

248 THE FERROELASTIC DOMAIN STRUCTURE IN LEAD PHOSPHATE

D 3

w t~

" ~ ~

(a) (b)

Fig. 1. (a) Domain pattern of a (100) cleaved plate of Pb3(PO4)2 at room temperature. We notice three kinds of domains, DID2D3, and two types of walls, W (as a line) and W' (as a stripe with inter- ference fringes). (b) Scheme of the different domains and walls of (a). Double arrows: twofold axes of domains. The subscripts 12, 13, 23 on walls W and W' are related to the kinds (D~,D2,D3) of domains that the walls separate.

\ \ / / ~ 2V = 81"± 2*

T ~

o?- Fig. 2. Orientations of axes in a domain. TI,T2,T3: crystallographic

axes; Nm, N~, Np: principal axes of index ellipsoid.

ell ipsoid are reported on a stereographic projection (Fig. 3b) performed on the (100) plane of domain D1. It appears that the wall is also a mirror for s imilar axes (T~ and T 2, for instance). The angle between the two- fold axes (T~ and T 2) of the domains is 58 °.

(b) Domains separated by a tilted wall (W' wall). This tylbe of wall appears as a stripe which contains inter- ference fringes under the polarizing microscope (Fig. 4a). The (100) plane is c o m m o n to both domains (D2 and D3) (Fig. 4c). The usual stereographic projection performed on this (100) plane shows that the similar axes such as T 2 and T23 derive one from the other by a b inary symmetry a round an axis located at the inter- section between the wall and the (100) plane (Fig. 4b). The angle between the twofold axes (T 2 and T23) of domains D2 and D 3 is 118 °.

The or ientat ion of the wall has been determined either, at room temperature, with the help of the four- axis universal stage, or by measur ing the apparent width x of the stripe (Fig. 4c): the thickness, e, of the sample being known, the value of the angle 0 between the wall and the normal to the cleavage plane is readily computed. Fig. 5 shows that tan O=x/e remains the same up to the transit ion, within the experimental accuracy (10-2); we obta in t a n 0 = 0 - 3 1 5 or 0 = 1 7 °. The variations, A e, of sample thickness e, due to thermal expansion have been neglected [(Ae)/e, cal- culated from di la tometr ic results (Toledano et al., 1975), is est imated at 4 x 10-3, from room temperature to the transi t ion].

3. Relative positions of walls Inspect ion of Fig. 1 shows that two kinds of domains

(D2 and D 3 for instance) can be separated either by the first type (I4/23) or the second type (W23) of walls described above; the angle between the traces of these two walls on the cleavage plane is 92.5 +0.5 ° .

Note also that the angle between the wall 14123 and

(I00~ ~

trace of W12

( N~I x N~ /

(b)

D 1

(a) (c)

(100)~

I vy2 02

178.6"

Fig. 3. (a) A magnified part of Fig. 1 : domains D1 and D2 separated by the wall W12 almost perpendicular to (100). (b) Stereographic projection of crystallographic and index-ellipsoid axes of the two adjacent domains D~ and D2, (subscripts 1 and 2). The projection is performed on the (100) plane of domain D~. T~ and T] are located in the projection plane whereas T22 and T ] lie along a plane which is slightly tilted relative to the projection plane. N] and 2 1 N~, Nm and N 2, N~, and N 2, point upwards. The wall W12 is the mirror-symmetry element for the two domains D~ and D2. (c) Side-view scheme of sample exhibiting the gutter aspect of (100) adjacent faces of D~ and D2.

M. CHABIN, F. GILLETTA AND J. P. ILDEFONSE 249

D2\

1301 ~ : \D3

3 ce of w~s

i N ; / /~N; Nm; T:~ D2

000)'/

I i

°O/W' I 23 , ~ D3 ',e I

000)3/

(a) (b) (c) Fig. 4. (a) Magnified part of Fig. 1" domains D2 and D3 separated by a wall W2s tilted at 17 ° relative to the perpendicular to (100) plane.

(b) Stereographic projection of crystallographic and index-ellipsoid axes of domains D2 and D3 (subscripts 2 and 3). The projection is performed on the common (100) plane. T 2, T 2, T~, T 3 are located in the common projection plane. N2,N 2 point upwards whereas N~,N~ point downwards. One domain derives from the other by a binary rotation around the axis labelled I . . . . . . I. (c) Side-view scheme of this type of association.

_17!

I

A sam#e 1 o sam#e 2

i i I

0 0

I I I -150 -100 - 50

Fig. 5. Tan 0 versus (T- T~).

~ o 2 °

I (T-Tc~ "K 0 lo

\

i '

T2 (a)

q • \ l x

x~ a

t

O------O - - o -O. - - -

/ / /

Vl T31. d ' ' . i i I I

P

T2

,/[011] ~' . / ~ - .

~ ~ t r a c e of 0 1 3 )

-~---e----4-----e [010]

\, t,ace o~ (ir~) x x

• %41 • • [OlT]

(b)

Fig. 6. (a) Pseudo trigonal lattice Lpt defined by elementary vectors Vl,V2,Va and built on the monoclinic lattice L,,(Tt,T2,T3). (b) (100) monoclinic-lattice plane. 1-011] and rol 1-] are pseudo binary axes and (113) and (1i3) pseudomirrors of Lpt.

walls W 1 3 and 14112 (belonging to the associations D1D3 or DID2 respectively) is found to be 30 °.

D i s c u s s i o n

We want to analyse our results by considering the possibility of twinning by pseudosymmetry.

1. Possibility of twinning by pseudosymmetry Pseudosymmetry twins can appear in the monoclinic

phase if the symmetry of the monoclinic lattice is very 'near ' a higher-symmetry lattice (Friedel, 1964).

We can build on the monoclinic lattice Lm (basis vectors T1,T2,T3) a superlattice Lpt , with elementary translations: Vl = T a + T 1 , V2=T1 -T2, V3 = T t +T2 (Fig. 6a).

Table 1 compares the parameters defining: the lat- tice, Lpt, at room temperature, and a high-temperature trigonal lattice Lt (obtained by doubling the basis vectors t 1, t2, t 3 of the primitive lattice of the prototypic phase).

Inspection of Table 1 shows that Lpt is a slightly distorted form of Lt. Lpt will be called pseudotrigonal. The lattices Lm and Lpt possess exact symmetry ele- ments which are the monoclinic twofold axis [010] and the mirror (010); moreover, some directions or reticular planes of Lm are almost symmetry elements of Lpt; they will be called pseudosymmetry elements: [011] and [011-] monoclinic directions which are pseudobinary axes of Lpt; (11~) and (1T~) monoclinic planes, which are pseudomirrors of Lpt (Fig. 6b). These pseudo- symmetry elements obviously transform into exact symmetry elements at the transition when Lpt trans- forms into Lt.

To sum up, monoclinic (113) or (1i~) planes and

250 THE FERROELASTIC DOMAIN STRUCTURE IN LEAD PHOSPHATE

Table 1. Parameters of the superlattice Lpt at room temperature and the trigonal lattice L, of the prototypic phase

L p t a V 1 = 14"96 A (Vt,V3)=(Vt,V2)=43"22 ~ Monoclinic [301] 20~C V, = V 3 = 14"94 A (V2, V3)=44"78"- direction at 46' from

L, b 200~C 2t~ = 2 t z = 2 t 3 = 14"96 A (t~,t2)=43"4: threefold axis of L,

(a) Computed from Brixner's (1973) results. (b) Keppler's (1970) results.

[011] or [01]] directions are permissible twin planes or twin axes respectively.

2. Twinning by pseudomirrors (113) or (l-f3)

One twin (or domain) derives from the other by a mirror symmetry. The pseudo mirror plane [(113) or (1]-3)] is common to the two adjacent twins, and is the wall itself.

This kind of twinning accounts for W walls (explain- ing, for instance, the stereographic projection).

The computed value of the angle between the (100) plane and the wall [(113) for instance], is 89-3Y, lead- ing to an angle of 178"66 ~ between the (100) faces of adjacent domains. This result agrees with observation and explains the gutter aspect of the (100) faces (Fig. 3c).

The angle ~0 between [010] and the trace of the wall in the plane (100), (i.e. the [031] direction) is given by tan ~p = 7"3/3 T 2, (Fig. 6b) or q~ = 28-92 ° at room temper- ature.

It follows that, taking into account the gutter dis- position of (100) faces of adjacent domains, the angle between their twofold axes is 57"85 ° , in agreement with the experimental observation 58 ° .

This angle varies with temperature, because of the changes (Toledano et al., 1975) in 7"_, and Ta. We have obtained, for instance, 58"40 ~ at 100~C, 59-55: just below the transition, with a jump to 60" at the transi- tion when the crystal goes into the prototypic phase.

3. Twinning by pseudo binary axis [011] or [011] These two directions give similar results" we shall

work with the [011] direction. One domain derives from the other by a twofold

rotation around the pseudo binary axis [011]" they possess the same (100) plane.

N ...L to 0oo)

(100!3,. / ~ " [011],,.,

Fig. 7. Construction of the rhombic section.

This kind of twinning accounts for W' walls, (see the stereographic projection). The angle between the two- fold axes of adjacent domains is twice the angle tp be- tween the [010] and [011] directions (Fig. 6b). With tan 0 = T3/T 2, we obtain, at room temperature, 2~0 = 117"81 ~" (experiment gives 118~). A discontinuity in 2~ occurs also at the transition: 2~ = 120 ~ in the proto- typic phase, and 119.58 ~" at just below the transition.

The wall, defined as the "rhombic section' by Friedel (1964) is a plane which contains the pseudo binary axis [011] and the intersection D between two planes (Fig. 7): the monoclinic lattice plane almost perpendicular to 1011] [in our case ~1131] and the plane ~ exactly normal to [011].

The indices of the plane ?~ are easily calculated in the monoclinic lattice, leading to ( -0-86357,1,2.74914). It follows that the indices of the direction D are [ - 0"25086,0"15843,- 0" 13643]. The wall, defined by [011] and D, is tilted by an angle 0, relative to the normal N to the monoclinic plane (100). The indices of N being [1,0,0.31412], we obtain 0= 17.28 ° in good agreement with the experimental value, 17.

On the other hand, Sapriel (1976) has obtained the equations of the walls as functions of the two compo- nents a and c of Aizu's spontaneous strain tensor (Aizu, 1970) expressed in orthogonal coordinates of the proto- typic phase, and show that the angle 0 is given by

¢ tan 0 - (1) a"

These components, a and c, are related (Aizu, 1970) to the components eu of the strain tensor of the ferro- elastic phase expressed in the usual axes system com- mon to the two phases: x3 threefold axis, x2 twofold axis common to the two phases, x l perpendicular to X 2 and X3;

'£22 --/311 6/= C=/313 .

2

We have previously shown (Gilletta et al., 1976) that the spontaneous strains e u which appear in the ferro- elastic phase are all proportional to the square of the order parameter Q of the transition (eu=auQ 2 where a u are assumed to be constant). It follows that a and c are proportional t o Q2, tan 0 = - c / a does not depend on Q; 0 is temperature independent, as experimentally shown.

Furthermore, the spontaneous strain xs, defined (Aizu, 1970) by xs=(2a2+2c2) 1/2, is proportional to Q2 and can only be expressed in terms of a [equation (1)]:

M. CHABIN, F. GILLETTA AND J. P. ILDEFONSE 251

0.04

0 03

0 02

00~

I I l l

0 I I I _175 _150 -I00 - 50 (T-Tc) °K 0 10

Fig. 8. Spontaneous deformation Xs--(2a2 + 2c2) 1/2 versus ( T - T~).

between these traces is equal to 92.16 ° at room tem- perature, in agreement with the observed value 92"5 + 0-5 °. e becomes 90 ° at the transition.

In the same way, the angles between the traces on the (100) plane of walls belonging to different domain associations can be calculated: for instance, returning to Fig. 1, the traces of walls 14/23 (association D2D3) and W12 (association D2DI) correspond on the (100) plane of domain D2, to the directions [011] for 14/23 and [031] for W~2 (Fig. 6b). The angle fl between these directions is equal to 29.97 ° at room temperature, the experimental value being 30 ° .

C o n c l u s i o n

The existence of a pseudo trigonal lattice in lead phos- phate at room temperature has led us to describe the characteristics of ferroelastic domains in terms of twinning by pseudosymmetry. The agreement with the experimental data is good. When the crystal tempera- ture increases, this pseudo trigonal lattice approaches the high-temperature trigonal lattice, and the charac- teristics of domains tend towards the values predicted by considerations of symmetry in the prototypic phase.

xs =a[2(1 + t a n 2 0)]1/2= 1.482a.

Its variation with temperature, T, can be deduced (Fig. 8) from the curve a(T) reported by Toledano et al. (1975) and exhibits a temperature dependence charac- teristic of a first-order transition.

4. Relative positions o f walls

The walls between two domains (i.e. D2,D3; Fig. 1) are either the W' wall containing a pseudo binary axis ([011] for instance) or the W wall almost perpendicular to this axis [pseudomirror (1]3) in the chosen example]. The same considerations hold for the pseudo mirror plane (113) and pseudo binary axis [01i].

The traces of these two types of walls on the (100) plane of one domain are the [031] direction for the W wall, and obviously, [011] for the W' wall. The angle c~

R e f e r e n c e s

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