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STACKING FAULTS, TWINS, AND THE
STRUCTURAL STABILITY OF VAN DER WAALS SOLIDS
J. Venables, C. English, K. Niebel, G. Tatlock
To cite this version:
J. Venables, C. English, K. Niebel, G. Tatlock. STACKING FAULTS, TWINS, AND THE STRUC-
TURAL STABILITY OF VAN DER WAALS SOLIDS. Journal de Physique Colloques, 1974, 35 (C7),
pp.C7-113-C7-119. �10.1051/jphyscol:1974711�. �jpa-00215867�
JOURNAL DE PHYSIQUE Colloque C7, supplkment au no 12, Tome 35, Ddcembre 1974, page C7-113
STACKING FAULTS, TWINS, AND THE STRUCTURAL STABILITY OF VAN DER WAALS SOLIDS
J.
A. VENABLES,C.
A. ENGLISH (*),K. F.
NIEBEL (**) andG. J.
TATLOCKSchool of Mathematical and Physical Sciences
University of Sussex, Falmer, Brighton B N 1
9 QH,
Sussex, EnglandResume. - Des dislocations et des fautes d'empilement ont pu &tre identifiees par microscopie Blectronique en transmission, B basse tempkrature, dans des gaz rares solides, ainsi que dans a-02
et a-N2. La relative stabilite des diffkrentes structures cristallines de ces solides a Ctk Cgalement calcul6e.
Dans le xenon solide des fautes d'empilement ont Cte observkes ; leur energie a ete estimee partir des largeurs de dislocations et des nceuds Btendus ; la diffirence relative d'Bnergie entre les phases h. c. et c. f. c. du xenon a pu &re aussi 6valuBe, et est en bon accord avec la theorie au moins pour l'ordre de grandeur.
La structure de b-02 est rhomboBdrique et peut &re representee par un empilement ABC de mol6cules 0 2 parallhles. Macles et fautes d'empilement sont frequentes sur le plan basal (OOOl), du fait de la g b l e diffkrence d'knergie entre les empilements ABC ou ABAB des molCcules. Des macles sur (1014) se rencontrent aussi frkquemment, et leur Bnergie est estimee.
On a montrC que la phase de basse tempkrature de a-NZ posskde la symktrie cubique Pa3, c'est-a- dire que la structure est c. f. c. a ceci prks qu'en chacun des quatre nceuds de la maille c. f. c. les axes mol&culaires pointent dans une direction
<
11 1>
differente. Des macles et non des fautes d'empi- lement, se forment facilement, qui sont probablement du 2" type d'aprh la diffraction et la micro- scopie 6lectronique.Les calculs indiquent qu'il est possible de comprendre qualitativement les structures de BOz et de a-Nz I'aide des forces de dispersion et quadrupolaires, et I'extension de cette methode aux configu- rations des dCfauts est en cours.
Abstract. - In the course of a programme of investigating van der Waal's solids by low tempe- rature transmission electron microscopy, dislocations, stacking faults and twins have been observed in the rare gas solids, a-6-02 and a-Nz. Calculations of the relative stability of different crystal struc- tures of these solids have also been performed.
Stacking faults have been observed in solid xenon and the width of dislocations and extended nodes used to estimate both the stacking fault energy and the relative energy difference between the h. c. p.
and f. c. c. forms of xenon. A theory of this energy difference has been given which is in order of magnitude agreement with the observations.
The structure of j3-02 is rhombohedral, and can be visualised as an ABC packing of parallel 0 2
molecules. Twins and stacking faults on the basal (0001) plane are frequently observed, which arise because of the low energy difference between ABC and ABAB packing of the molecules. Twins on (1074) are also observed frequently and their energy is estimated.
The low temperature phase a-Nz has been shown to have the Pa3 cubic structure, in which the structure is f. c. c., except that at each of the four lattice points, the molecular axis points in a different
<
111>
direction. Twins, but not stacking faults, are readily formed, and electron diffraction and microscopy experiments indicate that these are probably type I1 twins.Calculations have indicated that the structures of D - 0 2 and a-NZ may be understood qualitatively in terms of dispersion plus quadrupole forces, and extension of these calculations to understand defect configurations is underway.
1.
Introduction. - A t the University of Sussex, molecular solids are interesting because they are bound we have been using electron microscopy and diiTraction together largely by van der Waals', or dispersion, t o investigate the structures and microstructures of the interactions which havea
relatively well-known solids formed a t low temperature by condensing simple form. Thus the attraction of studying these solids gases, such as the rare gases,0,
andN,.
These simple lies in our ability t o perform calculations of expe--
-rimentally measurable quantities. It
is
because o f (*) Present Address : A. E. R. E., Harwell, Didcot, Berkshire,England. this interplay between theory and experiment that
(**) Present Address : I. B. M. Laboratories, Biiblingen, the subject has advanced so rapidly i n the last
ten
W. Germany. years.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974711
C7-114 J. A. VENABLES, C . A. ENGLISH, K. F. NIEBEL AND G. J. TATLOCK
Electron diffraction and microscopy enables us pri- marily to study the crystal structure of - and defects in
-these solids. Although the crystal structure has usually been determined definitively by X-ray diffrac- tion, this is not always the case, as our work on a-N, shows (see 5 4). In particular, it is difficult to obtain single crystals of low temperature phases which are large enough for single-crystal X-ray work, and selected area electron diffraction and related techniques can be used to overcome this problem.
The study of defects, particularly stacking faults and twin boundaries, is interesting because it enables us to explore molecular packing configurations which are not present in the perfect crystal, but which usually have only a small energy above that of the perfect crystal. Thus even a rough measure of the energy of a stacking fault or a twin boundary enables some fine details of the intermolecular interaction to be deter- mined. This point is illustrated by the problem of the crystal structure of the rare gas solids. Despite the fact that interatomic potentials derived from gas phase data and theoretical calculation, coupled with a suitable version of lattice dynamics, can predict the thermal and elastic properties of these solids with approaching one percent accuracy, these same potentials predict the wrong crystal structure (h. c. p. instead of f. c. c.) by about 1 part in lo4.
A description of the intermolecular interaction which predicted the observed crystal structure, and gave the order of magnitude of fault energies, would, therefore, improve our understanding of intermolecular forces in solids at and below the 1 % level of accuracy. This is the motivation for the experimental and theoretical work we have done so far, which is described in this paper. In section 2, we describe our attempts to measure the stacking fault energy of xenon, and a new theory for the stability of the f. c. c. structure of the rare gas solids.
Section 3 describes observations of twins in solid
0,and calculations on the phase diagram, of, and twin boundaries in, this solid. In section 4, the work done on solid N, is discussed. The structure of a-N, has been determined and twins have been observed and charac- terised. Much more work of this type remains to be done, but the general direction that the work is going is by now quite clear.
2.
Stacking faults and structural stability of the rare gas solids. -We have examined solid xenon in order to measure its stacking fault energy, and to relate this energy to the relative stability of the f. c. c. and h. c. p.
structures. The experimental techniques used have been described previously. A liquid helium stage for a Hitachi HU 11 B 100 kV transmission electron micro- scope was used (Venables
etal., [15]) and the xenon was condensed from the vapour onto an electron transparent amorphous carbon film. This film was the bottom window of the specimen holder, which was in the form of an enclosed cell, with a top window, usually made of formvar or parylene-N. This design was a
slight modification of that described by English and Venables [3].
When xenon is condensed between 40-60 K, it forms highly faulted polycrystals. These films were then warmed up to 95-100 K for about 10 min by an elec- trical heater attached to the specimen holder. During the heating the faults annealed out to a large extent and the grain size increased. The pressure cell is neces- sary to prevent the samples subliming in the microscope vacuum
-the vapour pressure of xenon at 100 K is
rr.0.55 torr.
An example of an annealed xenon film is shown in figure 1. In this bright field micrograph we see (220)
FIG. 1. - Bright-field micrograph of annealed solid xenon, showing (2%) and (220) bend contours, and dislocation crossing
them at D. Additional dislocations are seen at C .
and (220) bend contours in a foil orientainted near
(111). As it crosses the contours, a dislocation D is
visible. In addition there are dislocations C, which have
weak images far from the contours. The images at D
show that the dislocation is quite narrow, certainly less
than 20 nm in the case shown. This finding has been
confirmed by a limited amount of dark field work, and
an example is shown in figure 2, which is taken with the
(220) spot indicated. Far away from the (220) contour
it is clear that the splitting of a dislocation (such as D)
into Shockley partials has a separation of < 4 nm. In
the centre of the contour there are some three-fold
nodes N, although the images are not particularly
symmetrical. The approximate size of these nodes
(2,in the analysis of Jassang et al., [S]) is around
10 nm.
STACKING FAULTS, TWINS, AND THE STRUCTUR .AL STABILITY OF VAN DER WAALS SOLIDS C7-115
FIG. 2. - Dark-field (2%) micrograph of xenon showing dislo- cations D and extended nodes N.
The interest in these stacking faults can be explained with reference to figure
3.If we assume that the atoms in each (I 11) plane stacked in positions A, B or C, are only sensitive to the angular arrangement of their
partial
dislocations
c repel
-
FIG. 3.
-
Schematic diagram of a stacking fault in the f. c. c.lattice.
nearest neighbours, then each layer feels either an f. c. c. (f) or a n h. c. p. (h) environment. Within this simple model a stacking fault is equivalent to 2 planes of h. c. p. phase and the contribution of these angular dependent interactions to the energy difference per atom between the two structures, BEf -,, can be related to the stacking fault energy, y, as
where a, is the lattice parameter. The factors in brackets arise because ao/ J3 is the spacing of the { 11 1 ) planes and 4/a: is the number of atoms per unit volume. Thus a measure of the stacking fault energy also measures that part of the energy difference between the h. c. p.
and f. c. c. structures which is due to angular depen-
dent, short-range interactions. In fact, this is likely to be the major part of this energy difference, since many calculations have shown that long range 2-, 3- and many-body interactions scarcely distinguish the two structures and even stabilize h. c. p. by amounts varying from nothing to 1 part in lo4 (see Niebel and Venables [lla] for a review).
The approximate value of
ycan be worked out using elasticity theory (Jsssang
et al.[a]) and the elastic constants of xenon (Gornall and Stoicheff [5]), extra- polating these values using the calculations of Klein and Hoover
[9])to the temperature of interest.
Assuming the unsplit dislocations in figure 2 to be in near-screw orientation a genuine lower limit of
is obtained, which corresponds to
using eq. (1) and a value of Ef
=2.7 x 10-l3 erglatom for xenon. Similarly, a node of size Z0
=10 nm gives a value of
y 1.
2.0 +
0.6erg/cm2 ,
and
A - 6 + 2 x
The errors quoted do not include the current contro- versy about the relationship between stacking-fault energy and dislocation width, discussed in these proceedings by Cockayne, nor do they include any allowance for node anisotropy considered by Bacon.
Such effects would be likely to increase
Afurther. On the other hand if angular dependent forces which have a longer range are taken into account in relating
yand BEf-,, these are likely to decrease A somewhat.
On the present model two stacking faults have the same energy whereever they are in the crystal whereas one might expect the energy of two adjacent faults to be somewhat lower.
However, these possible sources of error cannot obscure the fact that the value of
A is- 5 x
which is some 50 times greater (and of opposite sign) to
that obtained from the interatomic potentials nor-
mally used to describe the thermodynamic properties
of the heavier rare gas solids. With this discrepancy
in mind, we have recently proposed a theory (Niebel
and Venables; [ l l ] ) which accounts for the stability
of the f. c. c. structure and gives estimates of
A (- 3 x lop3for xenon) which is similar in order of
magnitude to the present experimental results. The
central idea is that the excited electronic states, which
are mixed into the ground stated by the van der Waals
interaction, have energies in the crystal which differ
from their energies in the isolated atom. In particular,
excited states with lower energy are mixed in more
strongly, and there is a relatively large difference in the
lowering of the d-state energies between f. c. c. and
h. c. p. Because there is a centre of symmetry at the
atomic sites in f. c. c., these d-functions overlap much
more strongly with nearest neighbours than they do in
C7-116 J. A. VENABLES, C. A. ENGLISH, K. F. NIEBEL AND G. J. TATLOCK
h. c. p., where-the centre of symmetry is between the atomic sites
;this is the principal reason why f. c. c. is favoured.
It is interesting to note that the interaction proposed
isof the short-range angular dependent type assumed in deriving eq. (1). It will be interesting to see, as measu- rements of quantities such as the single crystal elastic constants become more accurate
(-1 %), whether an interaction of this form is also needed to explain those results satisfactorily.
3. Twinning and structural stability in solid oxygen.
- Solid oxygen exists at low pressure in three forms
: a,which is monoclinic, and is stable below T
=23.8 K
; /I,rhombohedral, stable between 23.8 and 43.8 K
;and
y,which is a complex cubic structure stable between 43.8 and the melting point of 54.8 K. The P structure
consists of hexagonal rafts of molecules with the mole- cular axes perpendicular to the raft
;these rafts are then stacked in ABC packing to form the rhombohedral structure. An alternative way of viewing the structure is to imagine an f. c. c. arrangement of molecular centres. On each site the molecular axis is parallel to the same [I 111 direction
;as a result of the molecular shape this axis becomes elongated into the hexagonal c-axis. The a-structure is very similar, but a slight rectangular distortion within the hexagonal rafts lowers the symmetry to monoclinic.
Calculations have been performed of the structures of the low temperature phases of the diatomic solids N,, O,, F2 and the halogens using a simple model inter- molecular potential which describes the shape of molecule and the relative importance of quadrupole- quadrupole interactions and dispersion forces (English and Venables [3]). In the case of O,, whose quadru- pole moment is small, the preferred structure is either the observed
j3structure R3m or ~ x m 2 , a hexagonal structure which results from stacking the rafts in an ABAB sequence rather than ABCABC. Once again, the calculations prefer the hexagonal two-layer repeat by A - as against the observed three layer repeat, which again has a centre of symmetry at the molecular positions. So it appears that, just as in the rare gas solids, there must be a small angular dependent interaction present which stabilises the observed structure. However, in this case there are no experi- mental or theoretical indications as to how large this interaction is.
The a structure has been shown to be due to the onset of antiferromagnetic ordering of the (spin 1) 0, molecules (English
et al. [4]). This ordering leadsto a slightly greater attraction between neighbours of opposite spin and a repulsion between neighbours of the same spin within a hexagonal raft, which lowers the symmetry to rectangular. The ABC stacking is then slightly distorted, and the resulting structure is mono- clinic.
The y - 0 , structure ( ~ o r d a n
et al.[6] is undoubtedly caused by the onset of hindered molecular rotation, as
is the analogous fl-F2 structure. The facts that cl-F,, stable below 45.5 K, does not transform to another structure at low temperatures and that it can be described rather well by our simple model, strengthen our feeling that the P-a-0, transformation is caused by magnetic ordering. On this view, spinless 0, molecules would have the ,!I-structure down to
0K.
Experimentally, it is easiest to observe /I-0, in the electron microscope, because the vapour pressure is reasonably low, and the temperature is not too low.
Below about 17 K the films quickly become amorphous due to radiation damage, without transforming to a-0,.
The lack of transformation could be due to the inhibit- ing effect of either the thin film or the high magnetic field in the objective lens of the microscope. In any case, irradiation damage and chemical reactions are a pro- blem with solid 0,. In the present work it was essential to use amorphous A120, substrates rather than carbon. Amorphous carbon was quickly transformed to CO in the irradiated area, as in the work of Bostan- joglo and Goertz (1969).
One of the most interesting aspects of the observa- tions on P-O2 was that annealing produced a wide variety of twinned structures. Much of this evidence is reported elsewhere (English [2-31). Evidence has been obtained that twins are produced, not only on the expected (0001) planes, but also on the (1031), (1012) and (1014) planes. Twins on (0001) correspond to the ... ABCBA ... stacking of the hexagonal rafts, and
FIG. 4. - Anneaed , B - 0 2 crystal, showing twins on (0001) at D, and coherent (1014) twin boundary AB and incoherent (1014)
boundary AC.
STACKING FAULTS, TWINS, AND THE STRUCTURAL STABILITY OF VAN DER WAALS SOLIDS C7-117
these boundaries must have a low energy, as in the
corresponding f. c. c. case. However, the (1014) twins are also very common, and often large areas of the foils are twin related by this twinning law. An example is given in figure 4, which shows a (1014) twin boun- dary along AB
;each crystal contains (0001) twins and probably stacking faults at D.
As can be seen, the(0001) planes in the two crystals are almost exactly at right angles to each other. The non-coherent boundary AC lies
03a (i014) plane. This is only at -
20to the
corresponding plane in the twin and so AC should also be a low energy boundary with a small dislocation constant. Such twins with approximately rectangular boundaries were frequently observed. The diffraction pattern corresponding to figure 4 is shown in figure 5, which shows that two types of twins are indeed present.
e Twin
FIG. 5. -(a) Diffraction pattern from the crystal of figure 4.
(b) Indexing of the pattern in figure 5(a).
Since the Lennard-Jones potential used by English and Venables [3] is a reasonable starting point for describing P-02, a start has been made to use the same potential to study the structure and energy of these
twin boundaries. Not surprisingly, the (0001) is cnl- culated to have the lowest energy,
y,-- 0.18 ergs/cm2.
This, however, must be an underestimate, since an angular dependent interaction necessary to stabilize the rhombohedra1 structure has not been included. As we can see, this calculation gives an answer which is an order of magnitude smaller than the measured
yfor stacking faults in xenon, whereas the experimental value may well be similar in the two cases. Calculations of the (10i4) twin boundary energy are more compli- cated, in that it is necessary to relax the molecular positions and, in principle, the orientations of the molecules. A suitable programme has, however, been written recently and preliminary values of the twin boundary energies calculated. For the (1014) boun- dary, this programme gives y, 5 1.05 ergs/cm2. In this case, the calculated energy is likely to be an overestimate due to inadequate relaxation. A scale drawing of the calculated (10i4) boundary configura- tion is given in figure 6, which is shown in the same orientation as figure 5. Note that only the crystallo- graphic axes are mirrored across the twin plane
;the boundary structure itself involves a translation parallel to the boundary and a slight change in the (1014) plane spacing locally, in order that the molecules fit snugly together.
\
'.
Matrix Twin
FIG. 6.
-
Calculated molecular positions in a (1074) twin boundary (in the same orientation as Fig. 5) using a Lennard-Jones atom-atom potential.
A direct experimental measurement of these twin boundary energies is not directly in prospect. It is interesting, however, that the observed boundaries are calculated to have a low energy, and that the calculated ratio of
y,to the surface energy
y, (--los2) is comparable with the same ratio experimentally observed for metals.
4.
Twins in a-N2.- Nitrogen forms two phases a t low pressure. The cubic cr-structure is stable below 35.6 K, while hexagonal P-N, exists between 35.6 K and the melting point of 63.3 K. The exact structure of the cr-phase has been in doubt since the 1930's when Ruhemann (1932) proposed that it was Pa3. In this structure the centres of the N2 molecules lie on an f. c. c.
lattice, but the axes of the 4 molecules per cell each
point in a different < 111 > direction. However,
C7-118 J. A. VENABLES, C. A. ENGLISH, K . F. NIEBEL AND G. J. TATLOCK
Vegard [13] had observed some reflections not allowed by Pa3 symmetry and suggested a similar structure, but which had the centres of the molecules displaced by a small distance, r parallel to the molecular axis
;this lowers the symmetry to P2,3.
Many diffraction experiments have tried to resolve this discrepancy, but preparation of single crystals of the a-phase is difficult, because the large contraction during the
j - atransition shatters the crystals. A detailed X-ray study by Jordan et al. (1964) showed 2 reflections not allowed by Pa3, { 052 } and { 051 }, and the data was analysed to give a P2,3 structure with the displacement
r =0.16 + 0.02 A (La Placa and Hamilton [lo]). However, it must be borne in mind that the data was obtained from one single crystal, out of over twenty attempts. This emphasises the diffi-
shows dislocations, and small dots which are caused by radiation damage below about 20 K. These defects.
have not been analysed yet.
A start has been made to analyse the nature of the.
twins themselves. The diffraction pattern of a (100) crystal with a twin on (111) is shown in figure 8a. Its.
interpretation in terms of the types of twins we think.
are present is shown in figure 8b. As can be seen, a (212) twin pattern is superimposed on the (100) pattern an&
extra twin reflections are visible at the Pa3 forbidden positions 032(324),, 033(303),, 014(Z2), and 012(m),..
Double diffraction is in fact a serious problem i n interpreting these patterns, since not only are the 001 ,.
003 type spots forbidden for either structure, but also.
the 303, and Eo, spots should also be forbidden.
culty of producing good single crystals of the low temperature phases of molecular crystals.
In a recent paper, we have shown that selected area electron diffraction patterns from a-N, are consistent with the Pa3 structure, or more rigorously that if the structure is P2,3, then r is certainly less than 0.05 A
and probably < 0.02 A (Venables and English [16]).
The Pa3 structure is also the structure calculated to be stable for NZ, which has a high quadrupole moment.
So it is extremely likely that the structure is indeed Pa3.
We have shown, however, that the crystals form twins very readily on annealing, and that these twins give rise to Pa3-forbidden diffraction spots. An example is shown in figure 7, where a foil of a-N, in approximately (100) orientation shows twins T, and T, on { 11 1 } planes. Some other examples were given by Venables (1970). In addition to the twins, figure 7
FIG. 8.
-
(a) (100) diffraction pattern of a-Nz containing a twiln FIG. 7. - Twins on { 11 1 } planes in a-Nz. on (111). (b) Indexing of the pattern in figure 8a.STACKING FAULTS, TWINS, AND THE STRUCTURAL STABILITY OF VAN DER WAALS SOLIDS C7-119
This problem can be overcome by combining the
selected area diffraction pattern with an expanded diffraction pattern, which is a low magnification picture of the distribution of diffracted intensity across the selected area aperture. Since the crystal is bent, genuine diffraction spots become bend contours while the spots which arise from double diffraction remain as a weakly excited region around the intersection of two contours.
In this way it is possible to detect the reflections which are forbidden.
In the case of Pa3 a-N,, the twins form on { 11 1 ) planes as in the f. c. c. structure. But whereas the centres of the molecules are defined by the ABCBA stacking sequence, this does not uniquely define the orientation of the molecules. In fact there are two possibilities. Either the twin can be related to the matrix by a rotation of
nabout [ l l l ] (a reflection in (1 11) is equivalent) or by a rotation of
nabout [I121 (a reflection in (112) is equivalent). The only distinc- tion in the diffraction pattern is that if a spot would be, indexed as a hkl for a twin of the first kind (type I twin) it would be khl for the second kind (type 11). Since these two reflections are not equivalent in Pa3, the types of twin are different.
To identify which type of twin is present we focus on reflections of the type Okl, which are forbidden if k is
odd. A (140) pattern turns out to be particularly useful, because this contains an allowed 4i0 spot in the matrix, as well as (for a type I1 twin) allowed 023 and forbidden 320 type spots in the ( 322 ) pattern of the twin. If the twin is of type I the geometry of the pattern is unchang- ed but the positions of the allowed and forbidden spots are interchanged. Similar rules can be worked out for a (410) matrix pattern containing a (1ZO) forbidden spot and the determination is therefore unambiguous. The patterns obtained to date suggest to us that the type I1 twins are the ones actually observed. However, it will be necessary to take more (140) patterns to be sure that this type of twin is always obtained.
It is noticeable that in a-N, simple stacking faults are never observed. It will be interesting t o compute the energies of twin boundaries of both types and stacking faults in the future, and see whether the results tie in with the experimental findings.
Acknowledgments. - It is a pleasure to acknow- ledge Mr. J. S. Notton for his continued help in the laboratory. We thank the Science Research Council (C. A. E. and G. J. T.) and the Studienstiftung des deutschen Volkes and A. E. R. E. Harwell (K. F. N.) for financial support.
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