THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS OF ELECTRON MICROSCOPY AND ELECTRON DIFFRACTION

(1)

HAL Id: jpa-00216528

https://hal.archives-ouvertes.fr/jpa-00216528

Submitted on 1 Jan 1976

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE STUDY OF PHASE TRANSITIONS AND THE

RESULTING DOMAIN STRUCTURES BY MEANS

OF ELECTRON MICROSCOPY AND ELECTRON

DIFFRACTION

S. Amelinckx

To cite this version:

S. Amelinckx. THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN

(2)

JOURNAL DE PHYSIQUE Colloque C4, suppliment au no 10, Tome 37, Octobre 1976, page C4-83

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN

STRUCTURES

BY MEANS OF ELECTRON MICROSCOPY

AND ELECTRON DIFFRACTION

S. AMELINCKX

S. C. K.-C. E. N., Mol and RUCA, Antwerpen, Belgium

RC.sumC.. - L'utilite de la microscopie Blectronique et de la diffraction electronique dans l'ktude des transitions de phase et des structures en domaines qui en resultent, est demontree et illustrke a l'aide de quatre ktudes representatives.

L'avantage de cette technique est surtout bade sur la possibilitk d'examiner de petites regions, ne contenant qu'un seul domaine et d'effectuer des expkriences de contraste permettant de determiner tous les paramztres importants des interfaces qui constituent la microstructure rksultant d'une transition de phase. I1 est egalement dQ au fait que les diagrammes de diffraction Blectroniques constituent des images pratiquement indeformes du reseau rkiproque.

Des structures modulees, mEme incommensurables, peuvent Etre revel6es facilement ; ceci est illustre a l'aide d'une etude des transformations intra- et interpolytypiques dans le TaS2.

I1 est demontrk que la transition a S /l dans le In2Ses est accompagnee de I'apparition d'une structure modulke intermediaire, fragment& en domaines.

On demontre que les macles du type Dauphine jouent un role preponderant dans la transformation a

+

B du quartz.

Les ions de cuivre intercallks dans le TaSz ou le NbSz subissent une transition du type ordre- dksordre. Au del& de la tempkature de transition, un ordre a courte distance subsiste; on demontre qu'il y a une forte correlation entre les ensembles d'ions dans des couches adjacentes.

Abstract. - The usefulness of electron microscopy and electron diffraction in the study of phase transitions and the resulting domain structures is demonstrated and illustrated by means of four case studies.

The advantages of this technique are based on the possibility of examining very small mono- domain regions and of performing contrast experiments which allow to determine all relevant parameters of the interfaces of the microstructure resulting from phase transitions ; as well as on the fact that the electron diffraction patterns give a practically undistorted image of all features in reciprocal space. Incommensurate modulated structures are readily revealed ; this is illustrated by means of a study of intra- and interpolytypic phase transitions in TaS2. In indium sesquiselenide the a-B phase transition is shown to be accompanied by the formation of an intermediate long period modulated structure, which is fragmented into domains.

Dauphine twins are shown to play a dominant role in the a S B phase transition in quartz. The intercalated copper ions in TaS2 and NbS2 are shown to undergo an order-disorder transi- tion. Above the ordering temperature short range order subsists. A strong correlation between clusters in neighbouring layers is found.

1. Introduction. - Electron microscopy and electron diffraction have recently become important tools in elucidating microstructural aspects of phase transformations.

Phase transformations are often accompanied by changes in symmetry of the crystal ; in such cases usually domain structures consisting of translation and orientation variants result. The combination of electron microscopy, selected area diffraction and lattice resolution allowd to analyse in detail the resulting microstructure. The relevant parameters such as the displacement vectors of the translation interfaces and the symmetry relations between the orientation variants can be determined in this way. It is in particular possible to obtain diffraction patterns from single domains.

Using electron diffraction the formation of superlattices and of incommensurate modulated structures

can relatively easily be studied, since even very weak reflections can be detected in the presence of very strong ones. Also the interpretation is simple since electron diffraction provides a nearly undistorted planar section of reciprocal space.

The analysis of the distribution of diffuse scattering in reciprocal space is therefore somewhat easier in electron diffraction than for other diffraction methods. Summarizing one can state quite generally that the use of electron diffraction has definite advantages over other diffraction methods when it comes to study geometrical features. We shall now illustrate the appli- cation of such techniques to four typical problems : a) Inter- and intrapolytypic phase transitions in transition metal dichalcogenides.

b) Phase transitions in indium sesquiselenide.

c) The a-p phase transformation in quartz.

(3)

C4-84 S. AMELINCKX

d)

The order- disorder transformation in copper 2.2 PHYSICAL PROPERTIES OF 1T 14, 51.

-

A num- intercalated niobium disulphide. ber of physical properties have been studied extensi- vely and a number of transitions have been found ; 2. Phase transformations in tantalum disulphide.

-

these are summarized in table I, for the polytype 1T We shall limit our discussion to TaS, which exhibits which is of most interest in this respect.

the greatest variety of phenomena ; information on The electrical resistivity changes abruptly at 350 K other transition metal dichalcogenides can be found in and at 190 K. The latter transition is accompanied by a ref. [l, 2, 31. large hysteresis covering a temperature range of about

2.1 CRYSTALLOGRAPHY. - The 'crystals of TaS, are grown by vapour transport under reduced pressure, using iodine as a transporting agent. On moderately rapid cooling, the high temperature modification, IT, is obtained which has the octahedral Cd(OH), structure, represented by the stacking symbol

aybayb

...

(IT)

However this phase is metastable and on heating to above 720 OC it may finally be transformed by a shear transformation into the 2H form which is stable at room temperature and which has the stacking

apacpc..

.

(2H)

At intermediate temperatures other polytypes such as 4H, and 6R can be formed as a result of the propaga- tion of partial dislocations along the c-planes between and within the TaS, sandwiches.

The 4H and 6R polytypes are represented respectively by means of the stacking symbols

apacpacpcapc..

.

(4Hd and

apabyaby bcabcaca~c..

.

(6R) Note that 6R and 4H, contain octahedral as well as trigonal prismatic layers ; we shall call them mixed polytypes.

100 K, on increasing the temperature it occurs at 240 K whereas with decreasing temperature it takes place at 145 K. The transition at 350 K shows little hysteresis and corresponds to a metalsemiconductor transition, whereas the 190 K transition corresponds to, a change in the band gap of the semiconducting phase. The magnetic susceptibility changes at 350K, the material remains paramagnetic however. The same hysteresis, as found for the electrical resistivity, occurs in the magnetic susceptibility.

2.3 ELECTRON DIFFFRACTION PATTERNS OF THE 1T

OCTAHEDRAL POLYTYPE. - The basal sections of the different diffraction patterns obtained for the 1T polytype are shown in figure 1. The correlation of these patterns with the different phases revealed by the physical properties is shown in table I. We have proposed to call the four phases a, p, y and 6 in the order of decreasing temperature.

We shall now discuss briefly the different patterns that are produced with decreasing temperature as a result of intrapolytypic transitions.

Close to the transformation temperature one observes the pattern of figure la. Apart from the main spots which are due to the basic structure, one notes incommensurate satellites around the main spots as well as curves of diffuse scattering passing through these satellite spots. The diffuse scattering can be attributed to a Kohn anomaly [6] and informa-

Phase transitions in IT-TaS,

1 / 1 1 . I / / / / / / / /

,//<c<;

temperature i

-

100 K > (315 K) 350 K shear transition 1 Ta 1 T y 1 TB 1 T= / l / / / / / / / / / / / phase 4Hb,6R, 2H type of I V I I1 I11 diffraction l I _pattern superlattice large triangles small diffuse triangles scattering dominant feature n ~ / / , 7 electrical

(4)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-85 slightly with decreasing temperature but it is at most of the order of 10 (Fig l c and 2a).

At 350 K the pattern suddenly changes reversibly into that shown in figure Id, i. e. the David star suddenly changes its orientation, producing now a larger triangular arrangement of spots in the centers of the triangles of main spots (Fig. 2b). This pattern is characteristic of the lTy phase. As well the pattern of figure Ic as that of figure Id present spacing and orientation anomalies ; they are clearly incommen- FIG. 1. - Diffraction patterns of different phases of IT-TaS2. a) surate.

Regions of diffuse scattering around each basic spot. b) Octahe- _{Since the hi& temperature phase IT, has hexagonal} dral complex of satellites around each basic spot. The satellites

are at level i 3 c. (lTa-phase). c) Weak satellites due to incom- symmetry, whereas lTB and ITy have lower symmetry, mensurate modulated structures. The small triangles of weak the Iwo phases lTp and can Occur with equal points are not exactly parallel with the main directions probability in at least two different orientations. (1Tpphase). d) Satellites due to incommensurate modulated struc- _{Domains of two variants have been observed in the} tures (ITy-phase). e) Commensurate superstructure (1Ts-phase). _{1~~ phase}_;_{a diffraction pattern across a boundary} between two such domains is shown in figure 3b ; it consists in the superposition of two patterns of the tion concerning the geometry of the Fermi surface can

type one for each variant. be obtained from the diffuse scattering.

These satellites are not situated in the basal section ; from tilting experiments [l] one can conclude that they are situated in planes at f 113 c* forming an octahedral complex of points in reciprocal space.

On lowering the temperature the diffuse scattering decreases in intensity and small triangular arrangements of spots in the centers of the triangles of main spots (Fig. lc) become more prominent.

These spots are situated in the basal section ; together with the spots due to the spikes of the octahe-

dral cOmplex they form a David star of spots _RG._3._-_{a) Superposition of patterns due to the 1Tg and 1Ty} centered On each main 'pot (Fig. 2a). This pattern is _phases_;_{the pattern was taken across the interface between}_B

typical of The IT, phase. and y-phase. b) Diffraction patterns across the boundary between At high temperature the main directions of the David the two orientation variants of the ITy phase. c) Diffraction star pattern enclose a very small angle with the main Patfern due to the 4Hb P O ~ Y ~ Y P ~ . The same pattern would result by superposition of the diffraction patterns due the two directions of the basic pattern ; this angle increases _{orientation variants of the superstructure 1Ts.}

I

li TYPEII(I$)

On passing through the phase transition lTB -+ ITr

!i

_{the same domain of IT, always transforms into the}

same variant of the IT, structure, proving that already in the lTp phase the hexagonal symmetry is lost and at least two variants occur. This is consistent with the small orientation anomaly observed in ITB.

Figure 3a shows the superposition of the two types of patterns. By establishing a temperature gradient

!

T Y P E I ( l T y )

I

_{which have already been transformed into the ITB}across the specimen it is possible to observe domains

phase adjacent to domains which are still in the T, phase. The transformation front can be revealed by

- - making a dark field image in a spot due to the IT,

phase (Fig. 4). A pattern taken across the transformation front will produce the superposed pattern of figure 3a.

On cooling the specimens below room temperature

I

_{the pattern of figure Id gradually changes in orienta-} FIG. 2. - Schematic representation of the David star of satel- tion it

(5)

LINCKX

FIG. 4. - Dark field images of transformation front between the /l and y phases ; the diffraction pattern of figure 3a was

taken across such an interface.

crystal now has acquired a commensurate superstructure. The base vectors A: and A; of the reciprocal lattice of the superstructure IT, and those of the IT, phase and

z2

are related by the relations :

Also the superstructure can clearly occur in two variants inherited from the ITp and the IT, structures. The superlattice appears and disappears around 190 K with the same hysteresis as mentioned above for the electrical properties.

2.4 ELECTRON DIFFRACTION PATTERNS OF MIXED

POLYTYPES : 4Hb AND 6R. - On heating specimens in the temperature range around 570 K an interpolytypic shear transformation starts as can be deduced from the movement of partial dislocations along the c-plane. In this way partially transformed specimens containing as well octahedral as trigonal prismatic layers are formed. Such specimens produce complicated diffraction patterns due to superposition. Interesting observations can nevertheless be made in such specimens when studying the temperature behaviour of the diffraction pattern.

The succession of events on cooling from the transformation temperature to room temperature is shown in figure 5. All these changes happen in a temperature interval of some 20 degrees, they are presumably

FIG. 5. - Evolution with decreasing temperature of the pattern of satellites around a basic spot in a partly transformed specimen of TaS2 containing octahedral and trigonal prismatic layers.

a) Octahedral complex (l), b) a satellite (2) develops on one side of the spots (1) of the octahedral complex, c) satellites (2) and ( 3 ) develop symmetrically on each side of the spots of the octahedral complex, d) the spots (1) decrease in intensity, e) the pattern only contains the spots ( 2 ) and (3). This pattern is the

same as that of figure 3c.

related to the discontinuity at 315 K in 4Hb as revealed by the magnetic measurements 171.

(i) An octahedral complex of incommensurate spots 1 is formed around each basic reflection. The radius of the circle passing through these spots is 0.271 a, where a, represents the distance between basic spots (Fig. 5a).

(ii) On one side of the spots 1 an additional set of spots 2 is formed (Fig. 5b). In a given area of the speci- men the additional spots are invariably formed all on the same side of the spots of the octahedral complex. (iii) On further cooling the symmetrically situated spots 3 are formed ; first they have a smaller intensity, then gradually the intensities of 2 and 3 become equal (Fig. 5c).

(iv) The spots of the octahedral complex move inwards with respect to the circle passing through the twelve spots 2 and 3. This results from a decrease in the projected size of the octahedral~complex, it becomes 0.260 a, (Fig. 5 4 .

(v) Finally the spots 1 disappear completely leaving only spots 2 and 3 with equal intensity (Fig. 5e and also 3c).

This sequence of events is completely reversible with temperature.

In ref. [g] a detailed discussion of this behaviour is presented ; we shall only summarize the conclusions here. The behaviour can be understood if one notes a number of facts and makes a number of assumptions. (i) The crystals are partly transformed, i. e. they are probably disordered and contain presumably superposed lamella having structures resembling the 4Hb and the 6R structures.

(ii) The majority of octahedra have maintained the orielitation they had in the 1T polytype ; in 4Hb half of them have been transformed into their mirror image. (iii) The polytype 6R, which has all its octahedra in the initial orientation, is assumed to form as an intermediate step in the formation of 4Hb which has an equal number of octahedra in both orientations ayb and bya.

(iv) The superlattice is formed in 4Hb at room temperature, whereas it is formed below rocm temperature in IT ; it is assumed to form in 6R at a temperature slightly above that for 4Hb.

The sequence of events can now be explained as follows. The octahedral complex I is common to all polytypes containing octahedral layers. On cooling first the octahedra in the same configuration as those in IT, i. e. layers in the

6R

configuration, produce deformation modulated layers giving rise to spots 2. This can be deduced from the superposed diffraction patterns of specimens containing still some untrans- formed IT.

(6)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-87 in a 4H, configuration become deformed giving rise to

spots 2 and 3.

The pattern formed by the spots 2 and 3 is the superposition of the two superlattice patterns of the type formed in IT related by the same symmetry operation

FIG. 6. -The monoclinic structure of NbTez (according to Brown 191). a) Projection on the layer plane, b) view of the

structure along the b-axis.

as the octahedral layers in 4Hb. A pure 4Hb specimen also produces such a pattern.

2.5 THE TELLURIDES OF NIOBIUM AND TANTALUM.

-

2.5.1 Crystallography and defect structure.

-

At room temperature niobium and tantalum ditelluride have a monoclinic deformation modulated cadmium iodide structure. This structure, as determined by Brown [9] is represented in figure 6. Such crystals are invariably fragmented into a domain structure, which was analysed in detail in [IQ] (Fig. 7): Although it has not been possible to transform such specimens back into the high temperature modification in the specimen holder of the microscope, one can safely assume that the fragmentation is due to the fact that at the

FIG. 7.

-

Domain structure in NbTez observed at room temperature. The arrows indicate the projections of the c-axis. The

interfaces are coherent twins.

(7)

C4-88 S. AMELINCKX temperature where the crystals are grown, they have

hexagonal symmetry and that they transform on cooling into the monoclinic form. Since the sixfold symmetry axis is lost on transformation, six orientation variants are possible, which give rise to three essentially different interfaces (Fig. 8).

oreo over

since the projected size of the unit cell increases by a factor of three there are three translation variants giving rise to antiphase boundaries which can form triple points (Fig. 9).

FIG. 9.

-

Antiphase boundaries in NbTez.

2.5.2 Interpolytypic transformations. - On pulse heating monoclinic crystals of NbTe2 by means of the electron beam it is possible to cause an interpolytypic shear transformation which produces crystals which exhibit slightly above room temperature the incommen- surate diffraction pattern of figure 10a. On further cooling this incommensurate pattern is transformed into the commensurate superlattice pattern of figure lob. This superlattice is different from that observed in TaS,. The relations between the base vectors A: and A: of the reciprocal superlattice and the base vectors A; and A, of the reciprocal basic lattice are given by

Like in TaS, the doubly positioned superlattice pattern is also observed in NbTe, (Fig. 10c). Also the same symmetry relation as in TaS, exists between the two patterns, i. e. a mirror with respect to the (10i0) plane. Also in this case the doubly positioned superlattice pattern is formed below a slightly higher temperature than the single superlattice pattern. The structure giving rise to this pattern is probably the 4H,.

2.5.3 Lattice images.

-

The circumstances for obtaining images of the deformation waves are parti- cularly favourable in NbTe, and TaTe,. In these crystals the superlattices, the commensurate as well as

FIG. 10. -Diffraction patterns of polytypes in NbTez. a ) Incommensurate modulated structure, b) commensurate superstructure, c) doubly positioned superstructure ('%&-type).

the incommensurate ones, can be obtained at room temperature due to the large hysteresis of the transformation. In this case heating or cooling of the specimen can be avoided and hence a source of creep of the specimen and of its holder is eliminated. Moreover the wave vectors of the deformation waves are in the basal plane of the reciprocal lattice.

Lattice images were produced for incommensurate as well as for the commensurate phases in NbTe, (Fig. 11). From the fact that well defined images could be obtained one must conclude that the deformation waves, which give rise to satellite reflections and to the superlattice are static in nature at room temperature.

(8)

THE STUDY O F PHASE TRANSITIONS AND THE RE ISULTING DOMAIN STRUCTURES BY MEANS C4-89

FIG. 11. - Lattice images of charge density waves in NbTez. a) Incommensurate (cf. Fig. IOU), b) commensurate (cf. Fig. lob),

c) doubly positioned superstructure (cf. Fig. 10c).

superlattice, one obtains the lattice image of figure 1 lc. This image exhibits a striking moirC pattern of which the mesh size is consistent with the orientation diffe- rence between the two superposed superlattices [ll].

3. Interpretation of incommensurate spot patterns and diffuse scattkring.

-

The interpretation of the incommensurate diffraction spots in terms of lattice deformation waves is straightforward [12]. The diffraction pattern of a periodically deformed crystal consists of the spots due to the basic, undeformed structure and of a set of satellites around each basic spot. The vectors connecting these basic spots with their satellites are the wave vectors of harmonics of the deformation waves.

The larger the number of satellites, the more harmo-

nics the deformation wave contains. The intensities of the satellites are proportional to the squares of the amplitudes of the harmonics of the deformation waves as well as to the square of the lenght of the basic diffraction vector with which the satellites is associat- ed [12].

A simple way of understanding why a crystal structure may become periodically distorted was given by Peierls [13]. We take for simplicity a linear crystal with a period a. The well known E versus k relationship is illustrated in figure 12a. Discontinuities occur at the

FIG. 12.

-

Relation between Fermi surfaces and incommensurate diffraction patterns. a) E vs. k relationship for a linear crystal. b) Discontinuity in E vs. k curve at k = kg resulting

from the creation of a periodic distortion. c) Relation between g,-vectors and Fermi surface. d) and e) Fermi surfaces for two

layer structures TaS2 and VSe2 (after ref. [14]).

Brillouin zone boundaries (i. e. at

+

nla

;

+

2 z / a ; etc.). Assuming that the crystal is a metal, the energy levels are filled up to a certain level, the Fermi-level

E,, corresponding with the wavevector k,. Should one now introduce a new zone boundary at around k = k,, the E versus k curve would acquire discontinuities as shown in figure 12b. The energy of the occupied levels in the vicinity of, but below

kF,

(9)

( 3 - 9 0 S. AMEI

would not affect the electronic energy. The net decrease in electronic energy will allow to overcompensate a small increase in energy of elastic origin, due to the deformation with period n / k , required to introduce the additional, possibly incommensurate, periodicity. The total energy of the deformation modulated structure may thus be lowered. The resulting band structure has now become that of an insulator (or a semiconductor). Although in three dimensions the situation is cer- tainly more complicated, the essential features remain and the wave vectors q, describing the distortion waves, are simply related to the Fermi surface of the undistorted crystal. In our one-dimensional model the period of the distortion is such that q = 2 k,. This relation is still valid in three dimensions where it gives the wave vector g, of waves for which the Kohn anomaly occurs [ 6 ] . Such vectors q = q, span the Fermi surface, i. e. connect points on the Fermi surface along which the tangent planes are parallel. Clearly, if we can measure q, as a function of direction, we may map out the Fermi surface. Some complications are possible because it will not always be clear whether g, in a given direction corresponds to either a minimum or a maximum in the diameter of the Fermi surface (Fig. 12c). Also visible diffuse scattering is expected to occur only for q vectors connectingflat parts of the Fermi surface. Therefore only a rudimentary image is obtained in most cases. Such measurements have been made in TaSe,, TaS, and VSe2 [2, 31 and compared with computed results [14]. The agreement is satisfac- tory (Fig. 12d, e).

4. The a $

fl

transition in SiO, (quartz) and in

AIPO, [15]. - 4.1 INTRODUCTION.

-

The U ~t

P tran-

sition in quartz, the prototype of a displacive transformation, is undoubtedly the most studied phase transition. A large number of techniques have been applied to this problem and it may seem surprising that electron microscopy has only recently contributed significantly in elucidating some details of the transformation mechanism. The electron microscopic examination of quartz has strongly been hampered by the fact that at room temperature it becomes rapidly damaged by the electron beam. However, at tempera- tures close to the a p

p

transition it is damaged a t a much smaller rate and examination becomes possible. Specimens are prepared by sawing, followed by grinding, and finished by ion beam thinning ; they are examined in a heating holder which allows to heat the specimen to above the transition temperature (573 O C ) . Figure 13 shows the domain structure in a crystal cut at 450 to the c-axis. The domains are Dauphin6 twins and we shall see that they play an important role in the phase transition in quartz.

4.2 CRYSTALLOGRAPHY AND DOMAIN STRUCTURE OF

QUARTZ. - The room temperature form, a-quartz, belongs to the non-centro symmetrical point group 32 whereas the high temperature from

P,

belongs to the

FIG. 1 3 . - Dauphin6 twins in a-quartz. Dark field image taken in the 3031 spot.

point group 62. On going through the phase transition with decreasing temperature the symmetry is lowered due to the loss of a symmetry operation ; a rotation over 1809 about the main symmetry axis. As a result the low temperature form can be formed in two variants a, and a,, which can be derived one from the other by the symmetry operation 'lost in the phase transition, i. e. by a rotation over 1800 about the c-axis. Such domains are said to be in the Dauphin6 twin relationship.

4 . 3 DIFFRACTION EFFECTS DUE TO DAUPHINB TWINS. - The following diffraction experiments 'allow to identify the domains shown in figure 13 as Dauphin6 twins.

(i) Contrast is absent in the BF image for all reflections in the c-zone, also in multiple beam situations. This is due to the fact that the simultaneously excited reflections, in the two crystal parts, have extinction distances of equal magnitude ; only the phases of the Fourier coefficients are different for certain reflections. (ii) Multiple beam dark field images using reflections in the c-zone produce contrast which is to be attributed to the violation of Friedel's law in non- centro symmetrical crystals in multiple beam situations [16].

(iii) Reflections of a general type such as ( 3031 }, { $071 ) and ( l0i2 ) for which the extinction distances are different for the simultaneously operating reflections in the two variants a , and a, produce contrast as well in the BF as in DF images. All of the images were made under these conditions, making DF images using the (3031) reflections.

(iv) The interfaces are imaged by U-type fringes. These experiments are sufficient to conclude that the observed domains are Dauphin6 twins. This is also consistent with the fact that only two variants are observed, i. e. triple points are absent.

(10)

THE STUDY O F PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-91

related to that of quartz, the silicon atoms in the chains Figure 15 refers to a quartz specimen. A small along the c-axis are replaced alternatively by alumi- temperature gradient is maintained across the speci- nium and phosphor, leading to a doubling of the c- men, causing part I to be in the P-phase, i. e. above parameter of the unit cell. At 580 0C aluminium and part I1 below the transition temperature, i. e. in phosphate undergoes a similar phase tra~-rsformation

as quartz. The low temperature phase is fragmented into Dauphin6 type domains (Fig. 14) which have been identified in the same way as in quartz.

FIG. 15. -Variation of Dauphine domain size in a small temperature gradient. Part I is below the transition temperature (a-phase) and part I1 is above the transition temperature

(B-phase).

the a-phase. It is clearly visible that the a-part is bro- ken up into domains which gradually decrease in size so as to become unresolved in the P-phase. Since moreover the walls vibrate constantIy the volume which continuously flips from a, into a, increases, with decreasing domain size. This strongly suggests that

FIG. 14. - Dauphine twins in AIP04. Only the extreme _{the P-phase is in fact the time average}_of_the_a,_and_a, positions of the vibrating interfaces are recorded.

structures.

4.6 THE a P

P

TRANSITION AND THE DIFFUSE SCAT- Due to the doubling of the c-parameter antiphase TERING. - The diffraction pattern at temperatures boundaries can be formed as a result of the ordering of close to the transition temperature exhibits diffuse A1 and P. These boundaries remain in place above the streaks of the type shown in figure 16 which refers transition temperature proving that the transformation to a section perpendicular to the c-axis in quartz ; the is not accompanied by atomic disorder, which is same effect is observed in aluminium phosphate consistent with its displacive nature. however.

4.5 GEOMETRICALFEATURES OF THE PHASE TRANSITION.

-

As the temperature of the transition is approached the Dauphin6 twin interfaces start vibrating violently. Sometimes they vibrate between two well defined extreme positions, under these conditions it is possible to photograph the area swept by the interface which then shows up in an intermediate shade (Fig. 14). Its structure in this area changes periodically from a, into a,. It has also been possible to make a motion picture of this oscillating movement (l).

The mobility of the twin is especiaIly large in AlPO,. Close to the transition temperature a large number of small Dauphin6 twins is nucleated in quartz as well as in AlPO, and sometimes a dense network of more or less triangular domains is formed. Such networks contain extra rows of domains leading to dislocation like defects, called domain dislocations.

(1) This movie was shown at the conference.

FIG. 16.

-

Diffuse scattering in the section of reciprocal space perpendicular to the c-axis. Note the strong scattering along the

(11)

'24-92 S. AMELINCKX

The diffuse scattering is not related to the domain structure since it is also present in the diffraction pattern of a single domain. In ref. [l51 it is shown that the geometry of the diffuse scattering can be explained qualitatively on the basis of the same type of motion which drives the phase transition and whichconsists in a vibration of SiO, tetrahedra about their twofold axis. This motion is shown schematically in figure 17a. The transformation of a, into a, only requires a cooperative reorientation of coupled strings of SiO, tetrahedra in the manner represented in figure 17b.

FIG. 17. - Schematic representation of the vibration of Si04 tetrahedra about their twofold axis. The two a-orientations and the 8-configuration are illustrated. b) Cooperative vibration of

SiOz tetrahedra in quartz.

5. Phase transformations in indium sesquiselenide.

-

5.1 STRUCTURE AND STRUCTURAL DEFECTS. - The literature data concerning the structure of In,Se, are not consistent ; for a review of the structural data we refer to ref. 1171. We shall mainly be concerned with the phase transformations in the vicinity of room temperature ; it is of interest to describe the structures involved. According to Osamura [l81 the room temperature form has the structure

where the capital letters represent selenium and the greek letters indium. Above 200 OC the /?-form with structure

(1

AyBaC ( BaCPA ( C/?AyB

I(

A

is stable. Both forms consist of the same close packed planes ; the a-parameter is the same for both :

a = 4.05

A

; the c-parameter is slightly different : c = 28.77 for a and c = 29.41

A

for

P.

According to Semiletov [l91 the room temperature form consists of a hexagonal close paclced arrangement of Se-atoms with the indium ions in the tetrahedral interstices according to the scheme

The superlattice with a = 16

A,

c = 19.24 L% is due to the fact that 1/16 of the indium ions lie in the octahedral interstices within the same layer that contains the indium in tetrahedral interstices. The indium ions in octahedral interstices form an hexagonal arrangement with a fourfold repeat distance.

In the high temperature form, i. e. above 200 OC the selenium ions still form six layer hexagonal arrangement with the indium ions all in tetrahedral interstices of the same type. The lattice parameters are a = 7.1 1, c = 19.30

A

[20].

The cl -,

P transition is accompanied by a large change

in the electrical conductivity. Its high temperature form has a large resistivity whereas the low temperature form has small resistivity. Also the Seebeck coefficient changes suddenly (from 150-200 pV/deg to 600-700 pV/deg just above 200 OC). The transition exhibits a large hysteresis as deduced from the thermal dilatation ; the high temperature form can even be kept at room temperature.

The material used in this investigation was obtained by direct synthesis of the elements in an evacuated quartz tube. The crystals were grown by vapour transport at 600 OC in iodine as a transport agent. The specimens were prepared by cleavage with adhesive tape.

5.2 PHASE TRANSITIONS. - 5.2.1 LOW temperature. - On cooling below room temperature the a-modification undergoes a phase transition, leading to a phase of which the structure has not yet been determined. Since no extensive movement of dislocations along the basal plane is noted during this transformation, the stacking of the selenium layers presumably remains unchanged, the transformation being intrapolytypic. The basal section of the diffraction pattern of the a-modification is simple hexagonal ; at-125 OC this pattern transforms into the one shown in figure 18a ; from which it is clear that a new structure has been formed. From the image (Fig. 19) one can easily deduce that the crystal has been fragmented into domains by twinning on the (1070) planes as referred to the hexagonal lattice of the a phase.

(12)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-93 Ignoring lattice deformations the relation between the base vectors a,, a, and a, of the hexagonal unit mesh in the basal plane of the a-phase and the base vectors a and b of the rectangular unit mesh are given by

a = a, - a,

b = - 2 a , .

The resulting structure is clearly a superstructure of the a-phase. From the splitting of basic spots one deduces that a slight deformation of the Se-lattice occurs.

5.2.2 The intermediate temperature transition.

-

When heating the as-'grown crystals, exhibiting in the basal section a hexagonal pattern with a = 4.05

A,

to above 200OC diffuse streaks are formed, along the (a) main direction and through the Bragg spots (~i~.-20a).

FIG. 1'8. - a ) Diffraction pattern of the low temperature phase of In2Se3. The pattern consists of the superposition of three patterns with orientation differences of 120". b) Diffraction pattern from a single domain, (a) is the superposition of three

patterns of this type.

FIG. 20. -Diffraction patterns of In2Se3 around the cc ZJ3 transition. a) Pattern of diffuse scattering consisting of non- radial streaks through the Bragg spots (above 200 "C). b) Satel- lites are developed on each side of the basic spots around 200 "C.

c) The distance between basic spots is divided into eight equal intervals. This pattern is due to a domain structure consisting of three orientation variants. d) Single domain diffraction

pattern.

(13)

C4-94 S . AMELINCKX

initially be irrational it ultimately becomes rational. The distance between basic spots is then subdivided into either eight or nine intervals-(Fig. 20c).

Linear arrays of closely spaced spots are formed along the three main directions only, suggesting that the crystal has a domain structure. Electron microscopy reveals the domain structure as well as the long spacing in each of the variants of which the structures differ by 1200 in orientation (Fig. 21). Selected area

FIG. 21. - Domain structure showing the lattice images in the

three variants (2 and 4 belong to the same variant).

diffraction of a single domain produces the patterns of figure 20d. Sometimes the long spacing is not well defined ; this is revealed by broadening of the superlattice reflections in the diffraction pattern and by a variation of the fringe spacing in the lattice image (Fig. 22a). In figure 22a the variation is pseudoperio- dic, whereas in figure 22b. bands of two different

spacings occur. The underlying hexagonal lattice of the new structure is apparently somewhat deformed since the spots due to the basic structure are split in the outer parts of the diffraction pattern.

On further cooling to room temperature, below about 60 OC, some crystals transform slowly back to the initial form ; all superlattice spots disappear and the simple hexagonal pattern is reproduced. A similar hysteresis is observed for a number of physical properties as mentioned above. Other crystals of the same batch transform directly into the low temperature form describe above.

These crystal can be made to go through this cycle again and again. The transformation is thus reversible to some extent.

During the heating cycle movement of dislocations along the basal plane is noted above 200 OC suggesting that the transformation may require reshuffling of the layers, i. e. the transformation may be of the shear type. However the movement of dislocations might also be due to the stresses generated by the phase transformation. The reversible character also suggests that the transformation should not be of the shear type although it does not exclude it.

5.3 THE NATURE OF THE a

+

P

TRANSITION AROUND 200OC.

-

The interpretation of the complex behaviour during the a

P

transition is made difficult by the absence of firmly established structures for the a and forms. Two alternative models have been advanc- ed in [l71 :

(i) the intermediate form (called

p')

is a deformation modulated structure,

(ii) the superstructure

p'

is due to a particular arrangement of indium atoms and vacancies in the interstices of the Se sublattice.

According to the first model the diffusestreaks formed above 200 OC could be due to the scattering by trans- verse acoustic phonons with wave vectors perpendicular to the close packed chains of atoms, and with a polarization vector roughly parallel to these chains, i. e. parallel to the layer plane. Such phonons are known to produce planes of diffuse intensity in reciprocal space which are perpendicular to the close packed chains of

atoms.- he

intersections of these planes with the Ewald sphere then produce the observed non-radial streaks [21]. The modulated structure would then be a frozen-in wave of this type ; the corresponding mode having become soft.

According to the second model the superstructure results' from the ordering of a fraction. of the indium atoms which have occupied octahedral sites. Accor- ding to [l91 1/16 of the indium atoms. would occupy such sites at room temperature and would give rise to a superlattice in the basal plane with four times the

( b ) _{a-spacing. We have never observed such a superlat-}

(14)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-95

at room temperature in which one long spacing had developed, equal to 4, 5 or 6 times the basic spacing. Since the structure was presumably determined from a multi domain crystal [l91 the author would of course conclude to the presence of a hexagonal lattice with a parameter of four times the basic parameter. A more detailed interpretation requires a complete redeter- mination of the structures of the phases involved, since previous determinations have ignored the presence of a domain structure. Such work is underway (Likforman : private communication).

6. Order-disorder transformation and short range order in copper intercalated TaS, and NbS,.

-

6.1 INTRODUCTION. - Copper (and also silver) can be intercalated in several metal dichalcogenides such as TaS, and NbS, [22]. The copper ions occupy the tetrahedral interstices between the two sulfur layers bound by Van der Waals' forces [23] ; they behave in many respects like a separate two dimensional phase, which resembles either a liquid or a solid depending on the temperature. We shall describe the phase transition in a typical compound of this type Cu,,,NbS,. A detailed account is t o be published shortly [24].

The structure of the host crystal in intercalated NbS, is different from that of NbS, ,it can be represented by the stacking symbol

where the capital letters represent sulphur ; the small latin letters niobium and the greek letters copper. 6.2 OBSERVATIONS. - 6.2.1 DifSraction patterns.

-

At room temperature and above the diffraction pattern exhibits next to the Bragg spots due to the host structure, continuous streaks of diffuse scattering along lines which in the c')-section are midway between the rows spots and parallel with the direction of the basic vectors (Fig. 23).

This diffuse scattering can also be described as being located along hexagonal prisms parallel to the c-direction and centered on the basic spots. Around the central spots and around the basic spots of the second hexagon, i. e. of the type 1130, the intensity of the diffuse scattering is relatively much smaller or disappears.

On cooling below

-

30 OC the streaks disappear and supplementary spots appear midway between the basic spots in the c-section @-phase) (Fig. 24). It

?IG. 24. - Diffraction pattern of the pphase. a) Three variants present, b) a single variant.

turns out that the pattern is in fact the superposition of three diffraction patterns with a rectangular mesh but with orientation differences of 1200 (or 600) simulating hexagonal symmetry. The pattern of figure 24a is thus due to a domain fragmented crystal containing three orientation variants. The c-section of the diffraction pattern due to a single variant is shown in figure 24b. From observations of the higher levels of reciprocal space it follows that the unit mesh in the c-section is determined by the relations :

where a,, a, and a, are the base vectors in the c-plane of the hexagonal lattice. In the c-direction the unit cell size is 13.1 1

A

it contains two NbS, layers. It is clear from figure 24b that in the c-section, i. e. for l = 0, the rows with k = odd are systematically absent.

On further cooling to about-80 OC the c-section of the diffraction pattern becomes as shown in figure 25 (a-phase). The spots which were systematically absent in ,the P-phase now appear and increase in intensity with decreasing temperature (Fig. 25b). The pattern of figure 25a is again due to a domain fragmented structure, consisting of three variants differing 1200 (or 600) in orientation.

The observation suggests that the crystal exists in

FIG. 23.

-

Diffuse scattering in copper intercalated NbS, different phases which w e and 7 with (y-phase). Note the absence of diffuse scattering around spots of increasing temperature and which are ~haracterized

(15)

C4-96 S. AMELINCKX

G. 25. - Diffraction pattern of the c-section of ,the a-phase.

a ) Three variants present : b) a single variant.

Phase transitions in Cu intercalated Ta/NbS, Type of

transition Displacive order-disorder

-

- Phase CC I3 Y Diffraction pattern Main feature

Interspot distance Superlattice Diffuse equal to one fourth spots halfway lines of the distance between

between basic spots basic spots

6.2.2 Microstuucture.

-

In the y-phase the crystal presents no resolvable microstructure. On cooling to within the P-phase a microstructure consisting of three orientation variants is formed. This can be established by making dark field images in spots belonging to each of the three subpatterns (Fig. 26).

FIG. 26. - Dark field image of the microstr~~cture in the B-phase. Note the orientation variants imaged in a different shade. Within a given orientation variant several translation variants

occur.

Within the same orientation variant one observes a further fragmentation by interfaces separating domains which exhibit the same shade. -These translation interfaces as well as the interfaces between orientation variants migrate when heating the crystal to close to the 2 y transition. Moreover the boundaries widen and strips of disordered material are formed on either side of the boundaries, when heating into the y-phase. In this process a configuration such as figure 27a can be transformed into a configuration such as figure 27b proving that at least four orientation variants are present.

FIG.

such

27.

-

Translation interfaces in the p-phase. A configuration as (a) can be transformed into one such as (b) proving that

at least four translation variants are present.

Within the same orientation variant one sometimes also observes regions which have a different shade. This may be due to overlap of domains which do not extend completely through the foil thickness or to other reasons to be discussed below.

On further cooling into the K-phase no further fragmentation into domains is observed.

6.3 THE STRUCTURE OF THE

B-PHASE.

-

A structural

model of the P-phase which is in agreement with all the crystallographic data is shown in figure 28. From the electron diffraction data we conclude that the orthorhombic unit cell is related to the unit cell of the host structure in the manner shown in figure 28a. In view of the composition one has to assume that the layers of tetrahedral interstices represented by a and

P

in figure 28b are only filled for 114. The simplest filling pattern which is compatible with the required compo- sition is represented in figure 29. The average distance between copper ions is as large as possible, minimiz- ing the Coulomb energy. The copper ions form a puckered double layer, occupying two different levels, as presented in figure 29.

(16)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-97

possible structures. With respect to Fig. 29 they can be represented by means of the stacking symbols

P1 a 1 a 3 P3 (1)

0 Cu B1 a1 a2 P2 (2)

P1 a4

P,.

(3)

Only sequence (1) leads to the observed systematic a3

i a] +<2 extinctions. For the general structure amplitude F,,,, of the

arrangement (l) one finds

~ ~ ~ ~ = j ~ , , ( e - ' ~ i " { l

+

exp[~ai(i--$)]

)

+

k h

+

e+ 2nizz

(

exp [2 ni

(?

-

?)]

+

eni(k+l)

) )

where z has the meaning indicated in figure 28b. B

...

IAb~pcrBaBapIAbA

...

~t is clear that for I =

o

this reduces to

F,,,, = fCu(l

+

e [l

+

BXp

(

-

y)

]

.

0

For the arrangement (1) all reflections for which l = 0 and k = odd should systematically be absent. This is

FIG. 28.

-

Structural model for the 8-phase. a) View along the c-axis. The small circles represent copper whereas the larger ones represent sulphur in two levels. The niobium positions coincide partly with the Cu-positions ; they have not been represented. One has to imagine the Nb-ions in the centers of the trigonal prisms of S-ions. The double arrows indicate possible displacements of the Cu-ions ,in the a-phase ; the single arrows indicate the simultaneous deformation of the sulphur tetrahedra. b) Cross-sections along XY. The tetrahedral interstices in the

layers a and B are only filled for 3.

FIG. 29. - Schematic representation of the different stacking positions in the two sublattices of tetrahedral interstices a and 8.

The open figures belong to the a-sublattice whereas the full figures refer to the 8-sublattice.

in fact observed (Fig. 24b).

For l # 0 on the other hand the reflections lc = odd shodd be present ; this can be deduced to be the case from the first level of the reciprocal lattice : it is further confirmed by non-basal sections.

The filling sequences (2) and (3) do not lead to the observed superlattice reflections along oko since the only reflections which should be present are those for which k is a multiple of 4.

We shall see below that also the microstructure is consistent with structure (1).

6 . 4 STRUCTURE OF THE LOW TEMPERATURE PHASE :

a-PHASE. - The additional reflections characteristic of the a-phase are very weak and somewhat diffuse just below the transition. The lower the temperature the more intense the spots become and the sharper they are ; they always remain very weak however.

These observations suggest that the additional reflections result from a small deformation of the P-structure, which is nevertheless large enough to destroy the systematic extinctions and such that within a given domain of the P-phase it can only take place in a single way. This is required because no additional domain fragmentation or boundary movement is observed in the a-phase.

(17)

C4-98 S. AMELINCKX

However, since the Cu-ions are in the centre of a tetrahedron of sulfur ions such a displacement would cause a deformation of this tetrahedron ; in particular the projected distance between the S-ions along the b-direction would alternatively increase and decrease as represented in figure 28a where the displacements of the S-ions are indicated by small arrows.

As the temperature is lowered, the effect of the Coulomb interaction becomes relatively larger as compared to the effect of thermal agitation, which tends to average the structure of the tetrahedra. This model is therefore consistent with the temperature behaviour of the intensity of the additional spots and the continuous nature of the transition.

It is interesting to note that according to the model the additional spots of the a-phase are due to a modu- lation of the host structure, caused by the presence of the intercalated ions. Since the deformation is uniquely determined by the arrangement of copper ions no additional domain fragmentation takes place.

6.5 DOMAIN STRUCTURES. - 6.5.1 Orientation variants. - The point group of the host crystal is 6/m 2/m 2/m which is of order 24. On ordering of the copper ions the orthorhombic P-structure is formed with point group symmetry 2/m 2/m 2/m of order 8. There- fore three orientation variants are to be expected, the variant generating group being 3 [25].

These different orientation variants have in fact been observed as described in

5

6.2.2.

6.5.2 Translation variants.

-

The projected size of the orthorhombic unit cell of the P-phase is four times larger than that of the lattice of tetrahedral sites (i. e. that of the host crystal) leading to four essentially different translation variants, indicated by 1,2, 3 and 4 in figure 29. The displacement vectors of the interfaces are R,,

R,

and R, (Fig. 30) they therefore be termed antiphase boundaries.

Some domains are further subdivided into regions of different shade, the sharp boundary between such regions being clearly of a different nature than the usual boundaries between orientation variants. It is assumed that such shade differences are due to stacking faults in the copper arrangement.

The fact that distinct translation domains can be observed suggests a high degree of correlation between successive copper double layers, i. e. long range order, also in the c-direction.

6.6 THE SHORT RANGE ORDER STATE. - The patterns of diffuse scattering shown in figure 23 can be explained by assuming that above the order-disorder transition temperature of the copper arrangement, the crystal contains a predominance of seven point planar clusters of two kinds [24]. The first kind consists of a copper ion surrounded by six nearest neighbour vacancies, whereas in the second kind the centre is a vacancy, with

FIG. 30.

-

Models for antiphase boundaries in the B-phase. The four translation variants give rise to three essentially different anti-phase boundaries with displacement vectors RI, R2

and R3.

configurations ; either .the copper ions are in a para or in a meta configuration.

The formation of such clusters is due to the fact that the Cu-ions remain on the lattices of tetrahedral interstices and to the Coulomb interaction between the copper ions, which tend to maintain a maximum average distance compatible with the composition so as to minimize the energy.

The extinctions in the pattern of diffuse intensity can be attributed to the correlation between clusters in a and

p

layers.

(18)

THE STUDY OF PHASE TRANSITIONS AND THE RESULTING DOMAIN STRUCTURES BY MEANS C4-99 described as a binary system having the properties of a

two-dimensional liquid undergoing a solid-liquid transition.

This transition is quasi-independent of the host crystal, except that the host crystal determines the sites which can be occupied either by a vacancy or by a copper atom. The Coulomb repulsion between Cu-ions causes a certain degree of order to be maintained even in the liquid phase, (y-phase) since copper-ions tend to avoid occupying neighbouring sites, minimiz- ing in this way the Coulomb interaction energy. Melting of the solid is preferentially nucleated along

APB's, and to a smaller extent also along boundaries

between orientation variants. These boundaries then give rise to narrow strips of disorder on each side of their original location ; this leads to widening of the image, which moreover becomes different in nature. The two phases are now present simultaneously.

The observed diffuse scattering is due to the liquid phase. Since the interaction time of fast electrons with the moving copper ions is very small, the electrons see a static configuration which is comparable to the transition state in an ordering binary system. The analysis of the diffuse scattering making use of a me- thod developed for the transition state is thus justified.

The liquids in the two layers of the same double layer are apparently highly correlated as a result of Coulomb interaction, leading to the weakening of diffuse scattering around the (1120) spots, i. e. to systematic extinction in the diffuse scattering.

As the temperature increases the correlation within a double layer will presumably become smaller and smaller, until finally the Cu-ions start behaving more like a two dimensional gas.

On the other hand one may wonder whether th correlation between the arrangements of double layers on either side of a NbSz sandwich breaks perhaps down in a separately discernable manner. This is likely to occur, if it does, in a narrow temperature range around the order-disorder transition.

It would manifest itself by the occurrence of nume- rous stacking faults (i. e. filling mistakes) within the domains. In such a situation one would observe within one orientation variant domains related by a pure translation, but nevertheless differing in shade for other than trivial reasons (such as surface steps). Such situations were in fact observed, as described in 6.2.2.

Below - 300C a superstructure develops as a result of the solidijication of intercalated copper ions In this superstructure, the P-phase, the same correlation exists between the ordered arrangements in the two layers of tetrahedral interstices within the same sandwich. Further cooling below - 80 OC leads to a continuous transformation into the a-phase, in which the host lattice becomes gradually modulated with the period of the copper intercalate.

The two superstructures show the same three orientation variants and the same four translation variants, in accord with the fact that a one to one correspon- dence exists between the a and P-phases.

Acknowledgments. - I would like to thank my collaborators Prof. Dr. J. van Landuyt, Dr. R. de Rid- der, Dr. G. van Tendeloo and Mr. D. van Dyck for the use of photographs and of results prior to pubIi- cation.

References

[l] VAN LANDUYT, J., VAN TENDELOO, G., AMELINCKX, S.,

Phys. Stat. Sol (a) 26 (1974) 359.

[2] WILSON, J. A., DISALVO, F. J. and MAHAJAN, S., Phys.

Rev. Lett. 32 (1974) 882.

[3] WILLIAMS, P. M., PARRY, G. S. and SCRUBY, C. B., Phil.

Mag 29 (1974) 695.

[4] DISALVO, F. J., MAINES, R. C. and WASZCZAK, J. V., Solid

State Commun. 14 (1974) 497.

[5] THOMPSON, A. H., GAMBLE, F. R. and REVELLI, J. F., Solid

State Commun. 9 (1971) 981. '

[6] KOHN, W., Phys. Rev. Lett. 9 (1959) 393.

[7] DISALVO, F. J., MAINES, R. C. and WASZCZAK, J., Solid State Commun. 14 (1974) 497.

[8] VAN LANDUYT, J., VAN TENDELOO, G. and AMELINSKX, S.,

Phys. Stat. Sol. (a) (in the press).

[9] BROWN, B. E., Acta Cvystallogv. 20 (1966) 264.

1101 VAN LANDUYT, J., REMAULT, G. and AMELINCKX, S., Phys.

Stat. Sol. 41 (1970) 271.

1111 VAN LANDUYT, J., VAN TENDELOO, G. and AMELINCKX, S.,

Phys. Stat. Sol. (a) 29 (1975) K1 1.

[l21 VAN LANDUYT, J., VAN TENDELOO, G. and AMELINCKX, S.,

Phys. Stat. Sol. (a) 26 (1974) K9.

[l31 PEIERLS, J., Quantum T h e ~ o y of Solids (Oxford Claredon Press) 1955.

[l41 SCRUBY, C., WILLIAMS, P. M., PARRY, V. S., PARKIN- SON, V. M., TATLOCK, G., Developments in Electron

Microscopy and Analysis, Ed. J. Venables (Academic Press, London) 1976, p. 377 ; also MATHESB, L. F.,

Phys. Rev. B 8 (1973) 3719.

[IS] VAN TENDELOO, G., VAN LANDUYT. J. AMELINCKX, S.,

Phys. Stat. Sol. (a) 33 (1976) 732.

1161 SERNEELES, R., SNYKERS, M., DELAVIGNETTE, P., GEVERS, R.

and AMELINCKX, S., Phys. Stat. Sol. (b) 58 (1973) 277. [l71 VAN LANDUYT, J., VAN TENDELOQ, G. and AMELINCKX, S.,

Phys. Stat. Sol. (a) 30 (1975) 299.

[l81 OSAMURA, K., MURAKAMI, Y. and TOMIE, U., J. Phys. Soc. Japan 21 (1966) 1848.

[l91 SEMILETOV, S. A., Soviet Phys. Crystallogr. 6 (1961) 158. [20] SEMILETOV, S. A., ibid 5 (1961) 673.

[21] HONJO, G., KODERA, S., ISOBE, E., TOMAKA, M., TAGAKI, M.

and KITAMURA, N., J. Phys. Soc. Japan 17 (1962) 1199. [22] BOSWELL, F. W., PRODAN, A. and CORBETT, J. M., Phys.

Stat. Sol. (to be published)

[23] KOERTS, K., Acta Crystallogr. 16 (1963) 432.

[24] DE RIDDER, R., VAN TENDELOO, G., VAN LANDUYT, J., AMELINCKX, S., Phys. Stat. Sol. (to be published). [25] VAN TENDELOO, G. and AMELINCKX, S., Acta Crystallogr.