Faraday Discuss., 1993,95,97-107 Hartman-Perdok Theory: Influence of Crystal Structure and Crystalline Interface on Crystal Growth Cornelis F. Woensdregt Institute of Earth Sciences, Utrecht University, P.O.Box 80.021, 3508 TA Utrecht, The Nether lands Hartman-Perdok theory enables classification of crystal faces as F, S or K faces. Only F faces should be present on growth forms as they grow slowly according to a layer mechanism. Such a classification can be quantified for ionic crystals by the calculation of attachment energies in electrostatic point- charge models. The attachment energy of a crystal face (hkl)el,is the energy released per mole, when a new elementary growth layer, called a slice, with a thickness of dhklcrystallizes on an existing crystal face.Theoretical growth forms can be constructed by making use of the supposition that ek'is directly proportional to the growth rate. Often these growth forms are very similar to the morphologies of crystals grown in nature or in laboratory experiments (zircon (ZrSiO,), alkali feldspars ((K,Na)AlSi,O,), natural silicate garnets [A:' B;+(Si0,)3]}. The structure of the crystalline interface is very important for the crystal growth processes. The derivation of F faces provides the atomic topology of the crystalline interface as well. Sometimes there is more than one possible surface of a slice with a thickness of dhk/, e.g.zircon (01 I), garnets (1 10) and (1 12), which may lead to growth via slices of thickness 3dhk/,causing a significant increase of the growth rate.Ordering of the ions which are situated on the slice boundaries may reduce the growth rate (alkali feldspars and YBa,Cu,O, ,). Experimental crystal growth of PbClz shows that adsorption of ~ OH- and H,O+ ions on special sites of the crystalline interfaces may cause habit modifications. This effect can be explained for PbClz in terms of adsorption of Pb(0H)Cl on {21 l}. The influence of the solvent on the growth can also be explained in terms of impurity adsorption on the crystalline interface. The growth of flattened PbCl, crystals from HC1 due to the reduction of the growth rates of (010) and { 121) is such an example. Hartman-Perdok Theory Crystalline materials are often bounded by planar surfaces called crystal faces. In the early days of crystallography these crystal faces were already connected with the internal crystal structure.In 1784Hauyl concluded from the cleavage planes of calcite that the crystals are composed of molecules intigrantes after what Steno2 in 1669 had already formulated as the law on the constancy of interfacial angles. Crystal faces are crystalline interfaces during the process of crystal growth. Sometimes habit changes occur due to alterations of the external growth conditions such as supersaturation and/or impurity concentrations. The surface topology of the crystalline interfaces during the growth determines the surface diffusion, incorporation of the growth units into kinks or steps, adsorption etc. Hartrnan and Perd0k~3~ described the relation between the internal crystal structure and the external crystal morphology.They classified crystal faces either as F faces, which contain at least two periodic bonds (PBCs) in a slice of thickness dhkl,as S faces parallel to only one PBC, or as K faces not parallel to any PBC at all. The F faces are the slowest to grow according to a layer mechanism and are therefore the only important ones for the crystal morphology. 97 Hurtmun-Perdok Theory The PBC is an uninterrupted chain of strong bonds between the crystallizing units, such as ions, molecules atoms or clusters thereof, formed during crystallization. Assuming that strong bonds are confined to the first coordination sphere, a limited number of PBCs can be distinguished, so the identification of PBCs and subsequent classification of crystal faces as F, S and K faces can be of great value in understanding the crystal morphology (see for a review hart mar^^.^).The slice with an F character defines, however, not only the elementary growth layer in case of layer growth, but also the topology of the crystalline interface during the crystal-growth processes on the scale of atoms, ions or molecules. Hence the influence of the crystal structure on the surface configuration of the crystalline interface can be derived with atomic precision by the Hartman-Perdok method. Electrostatic Point-charge Calculations Crystals grown in nature and in the laboratory are mostly not the equilibrium forms, which satisfy the thermodynamic conditions of the minimum specific surface free energy formulated by the Gibbs-Curie law.They are growth forms of which the habit is controlled by kinetic processes and prevailing growth conditions (impurity concentrations, degree of supersaturation etc.). Only under special conditions, such as a closed system in which Ostwald ripening can take place, will equilibrium crystals be formed. The attachment energy, which is assumed to be directly proportional, at least for the slowly growing F faces, to their growth rates' can be computed in an electrostatic point- charge model. Hence calculations of attachment energies can provide data about the growth rates of individual crystal forms {hkl).The attachment energy (E.,)is defined as the energy per mole released, when a new slice of thickness dhk[crystallizes on an already existing crystal face (hkl).dhklis equal to the period with which the same surface configuration is repeated, so it is subjected to the same extinction conditions as X-ray reflections. In order to calculate the electrostatic attachment energies of cubic sphalerite, Hartman* introduced the formulae given by Madelung9 which describe the potential induced in a point P by an infinite row of equally spaced point charges. Further details about the computation of the attachment energies in an electrostatic point-charge model have been published elsewhere, lo Theoretical Growth Forms In the following section examples of theoretical growth forms will be given.They have all been obtained by a three-dimensional plot of the attachment energies according to the principles of the Wulff p1ot.l' In this plot the distance from the central point to {hkois taken to be directly proportional to the absolute value of the attachment energy ,$'I. The innermost closed surface is the theoretical growth form. Zircon (ZrSiO,) In the tetragonal zircon (ZrSiO,) crystal structure the silicon-oxygen tetrahedra are bonded to six different zirconium ions. On the other hand each zirconium ion is surrounded by six silicate tetrahedra. When only Zr and silicate ions are taken into account as crystallizing units, four different PBCs can be described,I2 viz. [OOl], (loo), (H)and (+,+,1 +).F faces are (100)and {01l}.Calculation of attachment energies has been performed in electrostatic point-charge models: I, Zr4+Si4+Oi-; model 11, Zr4+Si2+0i.5-;and model 111, Zr4+SioO:-. The theoretical growth form is short prismatic following { 100) and is terminated by the dipyramid (0111. The lower the oxygen charge, the more elongated is the crystal parallel to the c axis (Fig. 1). Detailed study of crystalline interfaces reveals that sometimes two different surfaces can be distinguished, which differ from each other in height by one submultiple of the slice C. F. Woensdregt Model I Model II Model IllOOl0010 1oi Fig. 1 Theoretical growth forms of zircon (ZrSiO,). Growth via slices d,,a: (a)-(c);growth via slices 420: (4-0. thickness.l3.l4 The difference between these slices is caused by their distinct surface configurations.Examples are the slices dollof barite15 and dollof ADP.16 If more than one slice configuration can be traced, the growth may also take place by slices with a thickness of a submultiple of dhkrprovided that they have also an F character. These slices have a higher negative attachment energy, which results in much higher growth rates than would be expected if the growth proceeded by means of slices with normal thickness dhkl. The slice do,1, which can be defined either as slice dtl bounded by Zr ions or as slice d:l bounded by oxygen ions of the silicate tetrahedra (Fig. 2), is such an example. The slice d:, is not only undulated in the plane of projection, but also in the third dimension.This has been indicated by the shaded areas of Fig. 2, which contain two boundaries. One with silicon ion Si(l), linked to oxygen ions O(1) and O(2,OiO) that are all situated at height 0, and another with Si(4) bonded to O(13) and 0 14,OiO) all at height 0.5. The slice configuration with the lowest attachment energy is 6oil, e.g. model I: -1479 kJ mol-l. The attachment energy of d:, is much higher ( -2 165 kJ mol- l). As these slice boundaries differ in height a half slice with thickness d022,and these half slices dt12and are F faces, the growth of (01 1) may also take place by slices of thickness In that case the growth rates of (01 1) increase and the growth models are even more prismatic. Alkali Feldspars [(K,Na)AISi,08] The reduction of effective charges changes the absolute values of the attachment energies, and sometimes also the relative order of magnitude, as has been shown for zircon.In the same manner ordering of A1 during the growth could have an effect on the theoretical growth form of potassium feldspar (KAlSi,O,). The alkali feldspar [(K,Na)AlSi,O,] crystal structure consists of a corner-sharing three- dimensional network of SiO, and AlO, tetrahedra. The electrostatic imbalance resulting from the replacement of Si4+ by A13+ will be restored by the incorporation of monovalent cations, such as K+and Na+, which occupy the large cavities in the structure. In the triclinic 100 Hartman-Perdok Theory Zr= 0 . SI = 0 0.1 nrn Fig. 2 Projection of the zircon crystal structure along the a axis ordered structure there are four non-equivalent tetrahedral sites, i.e.T1(0), Tl(m), T2(0) and T2(m). In this structure all the A1 is located on T1(0), while Tl(m), T2(0) and T2(m) are occupied by Si. In the monoclinic high sanidine A1 is randomly distributed over all T1 and T2 sites. In the monoclinic ordered orthoclase A1 is divided in equal amounts over the Tl(0) and Tl(m) sites. The F, faces of alkali feldspar,17.18 which are at least parallel to two PBCs completely built-up from (Si,Al)-0 strong bonds, are { 1lo}, {OOl}, {OlO), (301) and {ill). The F, faces which are parallel to only one of these PBCs and to another having both (Si,Al)-0 and additional weaker (K,Na)-0 bonds, are {130}, {021}, {221), (7121, (100) and {TOl).The growth forms of the models with the formal point charges are strongly dependent on the degree of ordering of silicon and aluminium. When the Si,Al is completely disordered the growth form is prismatic following { 1 lo), with additional (001) and {OlO), while (021) is almost invisible [Fig. 3(a)].The partial ordering of Si,A1 yields the presence of ('ZOl) and { 1If)as additional almost invisible crystal forms on the growth form of ordered orthoclase [Fig. 3(b)].The fairly triclinic growth form of low microcline [Fig. 3(c)] is thick prismatic following { 110) and shows the influence of the completely ordered Si,Al on the crystal morphology. Garnets In the cubic silicate garnet structure of A:+B:'(SiO,), the tetrahedrally coordinated Si ion shares each of its oxygens with the adjacent B06 octahedra. Each B06 octahedron shares six of its twelve edges with adjacent AO, dodecahedra. Natural garnets can be classified into two distinct groups according to their chemical composition.The first group has the chemical composition Ca,B,(SiO,),, where B stands for trivalent cations, such as A13+, Fe3+ and Cr3+. The chemical formula of the second group is A,Al,(SiO,),, where A stands for divalent cations, such as Fe2+, Mn2+and Mg2+. Synthetic garnets have the general chemical formula A:+B:+(B3+O:-)3. As trivalent A ions, Y,Gd and other rare-earth-metal elements, and as trivalent B ions, Fe, Gd, Al and Ga may C. F. Woensdregt Fig. 3 Theoretical growth forms of potassium feldspar polymorphs.Disordered surface configu- ration: (a) high sanidine, (b)orthoclase (c) low microcline. Ordered surface configuration: (d) low microcline. occur. Yttrium ion garnet (YIG), gadolinium gallium garnet (GGG) and yttrium aluminium garnet (YAG) are examples of synthetic garnets. The F forms of all garnets are { 1121, { 1lo}, { 123}, {OOl), { 120) and {332}.19 Boutz and WoensdregtZ0 analysed the surface structure of the slices of Mg,Al,(SiO,), (pyrope). For the crystal form { 112) there are also two different slice configurations as is shown in Fig. 4. The slice d, contains only complete silicate tetrahedra and its boundaries are occupied by alternating ions of A1 and Mg. It differs from the slice configuration of dlI2*in height by half the slice thickness of dIl2.The slice d1122cuts through silicate tetrahedra and its boundaries are occupied with alternating Si and A1 boundary cations. The slice dzz0can also be traced in two different ways (see also Fig. 4).The slice d220,is bounded by Si and Mg ions, while that of d2202is bounded by A1 ions only. The difference in height between the former and the latter is exactly half the thickness of slice d,,,. '/ Fig. 4 [1TO)projection of pyrope I02 Hartman-Perdok Theory The attachment energies of the F faces have been calculated in different electrostatic point-charge models in which the point charges of the ions forming Si-0 tetrahedral bonds vary from the completely ionic model with formal charges, [Si4+Oi-], to the covalent model with reduced charges, [SiOO;].In all models the formal point charges of A1 and Mg have been used. The theoretical growth (see also Fig. 5) form based on the formal charges shows only { 112). However, on the more covalent-like models, with lower effective charges of oxygen, qo,(loo}and { 1lo} appear, while at the same time, (21 1) disappears slowly. If the effective charge of qo = -lei, the growth form is merely bounded by (I lo}. Crystalline Interfaces Influence of Two Different Coexisting Surface Structures The possibility that two different surface topologies could sometimes be present on one and the same crystalline interface must not be discarded. Especially, when the specific surface -2.0 -1.75 -1.5 -1 .o Fig.5 Theoretical growth forms of pyrope C. F. Woensdregt energies of the two surfaces are very similar, they are both present separated by steps which have a height equal to a submultiple of the slice thickness dhkl.If these slices with reduced thickness also have an F character, the layer growth proceeds at higher growth rates than under normal circumstances. Examples of these crystalline interfaces are (01 1) of ADP,I6 zircon,12 barite,15 aragonite2' and (112) and (1 10) of garnet.20 Ordering Boundary Ions Sometimes ions are located exactly on the boundary of two adjacent F slices. Such an F face can be defined as one with statistically disordered boundary ions. In this situation the n boundary ions are randomly distributed over 2n sites, which means that the occupancy of all the sites is 0.5.Another possibility is that only half of the boundary sites are occupied by ions. In the latter situation an ordered configuration of ions and vacancies is energetically the most stable situation. Such an ordered surface can be present only if the growth is proceeding slowly. Alkali Feldspars The oxygens of the alkali feldspars on the boundaries of the slices dm,, d2,,,,and d20jcould be ordered. The same ordering can also take place for the potassium ions on the surface of d020.Particularly when Si,Al is fully ordered, the effect is very strong for (001)and (010) [Fig. 3(d)]. They are morphologically much more important than on the correspond- ing model with disordered surface configuration [Fig.3(c)]. YBa2Cu,0, -(YBCO) YBa2Cu30,-(YBCO) has for x = 1, the following F forms:22 (OOl), (01 l},(0131,{ 112) and ( 1 14). The theoretical growth form of this tetragonal phase is tabular following {OOl), with (01 1) as lateral forms. There are two different slices for (OlO), i.e. dolo,and dolob[Fig. 6(a)]. Their surface energies are exactly identical owing to the presence of a symmetry centre at the origin. The slice with thickness of d020is, however, an S face as the [loo] PBCs are not connected parallel to [UOW] within this slice. When x = 0, YBCO becomes orthorhombic. The slice d020[Fig. 6(a)],in which the [loo] PBCs are connected parallel to [301] and [30l]in the case of Yf3a2Cu,0,, determines the Fcharacter of (010).This slice has in its centre Y ions and half of the copper oxide layer (Cu3+Cu:'Oi-) on either side.The copper and oxygen ions in the outermost layer show presumably an ordered arrangement of which the (1 x 2) structure is shown in Fig. 6(a).Calculations22 of the attachment energy indicate that (010) with an ordered boundary structure could be present on the theoretical growth form. Exactly on the slice boundaries of (001) are situated Cu+ ions (x = 1) or Cu3+ and 02-ions (x = 0). The most stable configuration is that of a quadratic lattice with sides 2a, of which the Cuf ions occupy the nodes and centre of the lattice [see Fig. 6(b)].When ordering of the Cu+ ions can take place, the growth rate of (001) is reduced by a factor of 0.74 in the case of YBa,Cu,O,.For YBa2Cu,0, the reduction is even higher (0.53). Adsorption Zircon Caruba et al.23proved experimentally that in the crystal structure of zircon oxygens are partly replaced by OH due to the adsorption of protons on the crystal surfaces when they grow from hydrothermal solutions. There are four equivalent 0-0 distances of 0.2840 nm in the zircon crystal structure, which are short enough to comply with the conditions for a hydrogen bond O-H-*-O.These relatively short bonds between oxygens are drawn in Fig. Hartman-Perdok Theory I I 09 0 0,,@0 occupancy : 112 0 occupancy : 1 w '(4 I 00 I 0[OOlT + 0 0 0 + 0 + cu *Y 0 Ba 00 additional oxygen Fig. 6 (a) [loo] projection of YBa2Cu,0,-,, (6) ordering of (001).Left: Disordered surface; right: ordered surface. 2 for the oxygen O(3).They could provide the additional strong bonds parallel to (110) in order to establish the F character of { 110)and to reduce its attachment energy. In addition they could change the character of (001).Hydrogen bonds are, however, much weaker than ionic bonds. So the F character of { 1 10)and of (001) will not be so pronounced as in the case of the genuine F forms (100) and (011). According to solubility studies of quartz in H,O-CO, and H,O-Ar systems24there are indications that Si(OH),.2H20 is the dominant aqueous silica species in the supercritical regions of these systems. In this complex four hydroxy groups are tetrahedrally coordinated to the silicon atom with two water molecules attached by hydrogen bonding.Their internal Si-OH and O-H***O bonds coincide with the bonding scheme of zircon. Once incorporated in a kink site, these uncharged solvated silica species could act as impurities and reduce the growth rate of the slices dl and dm1.They are neutrally charged and poison the special kink positions on (001) and { 1 lo), which have already been described as possible hydroxy-group adsorption sites. C.F. Woensdregt I05 Instead of silica complexes water molecules could be adsorbed on the surface. In that case the hydrogen bonds of the water molecule react with the crystalline interface in a similar manner to those of the silica complex. Cotunnite (PbC1J The F forms of cotunnite (PbClJ are25 (110}, (0201, (Oll), (120}, (2001, (1 1l}, (121}, (201) and (21 1).This analysis is also valid for Pb(OH)Cl, Pb(0H)Br and Pb(0H)I which have similar crystal structures. For the closely related SbSI structure the same F faces as for cotunnite are present with exception of (1201, which is an S face. The theoretical growth form of PbC1, is short prismatic following { 1 lo} and has (01 1) as terminal form. The effect of the relative supersaturation, 0,in combination with the presence of impurities has been investigated26 for PbCl,, which crystallizes from an aqueous solution to which an impurity in the form of KC1, NH,C1 or CdC1, has been added. The experimental growth habit is isometric with dominant (21 1) as terminal form for low 0and low levels of impurity concentrations.Higher (T and higher impurity concentrations lead to less OH- ions and more C1- ions. The effect is a more acicular habit with dominant (01 1) terminal forms instead of (21 1). The presence of (21 1) can be explained by the adsorption of OH- ions in the form of a two-dimensional adsorption layer of Pb(0H)Cl. The slice dj, is characterized by sheets of Pb and C1 atoms in its centre and two clearly protruding C1 ions. Other slices do not show such a structure. The supposition is that these C1 ions situated at the slice surface have been replaced by OH-ions. The face (2 1 1) belongs to the zone [1 1 I], ofwhich the period is 1.268 1 nm. The same zone has in case of Pb(0H)Cl an identity period of 1.2665 nm, which is the smallest misfit (0.13%) of all possible lattice rows.In fact, deposits of Pb(0H)Cl have been found besides the crystals of PbC12. Hence the habit with (21 1) as terminal form can be explained by the adsorption of one of the solvent components (OH-ions) or lead hydroxyl complexes such as Pb(OH)+ or Pb2(0H)3+, which also may be present in these solutions. Crystallization from HCI-containing solutions shows26 that the habit depends both on (T and the concentration of HCl. The effect of the presence of HCI is twofold, both the growth rate of { 121) and (010) decrease. When the supersaturation increases, the habit becomes elongated with dominating (010) and (1001. The very pronounced decrease of the { 121) and {OlO) growth rate could be explained by the adsorption of H30+instead of Pb2+ ions, which have similar ionic radii.The Pb ion marked by an arrow in Fig. 7 is situated both in slice d,z, and As its adsorption energy is the highest of all possible adsorption sites, adsorption of H,O+ is very likely to occur and would cause the observed habit modification. Fig. 7 [lOf] projection of PbCl, showing the most suitable site for H,O+ adsorption 0,Pb; 0,C1 Hartman-Perdok Theory Discussion and Conclusions Hartman-Perdok theory provides theoretical growth models which can be compared with crystal morphologies observed in nature or obtained during crystal-growth experiments in laboratory. In the present paper the theoretical growth models derived for zircon, alkali feldspars and garnets illustrate that the relative morphological importance of the crystal faces can be derived not only qualitatively but also quantitatively.In the quantitative electrostatic point-charge model computations the surface of a slice is assumed to be ideally crystalline. It represents the flat part (terrace) of an elementary growth layer. In fact such surfaces are obtained by splitting the ideal bulk crystalline structure. The influence of surface reconstruction or relaxation is not considered. This process of crystal growth cannot be understood properly without a thorough knowledge of the crystalline structure of the interface. Hartman-Perdok theory describes the surface configuration of the crystalline interface with atomic precision.Ordering of boundary ions will always reduce the attachment energy and will sometimes change the relative morphological importance as well. Examples of ordering have been given for the ordered polymorph of potassium feldspar and YBa2Cu307. In the latter case ordering of the boundary ions is absolutely necessary, if not (010) would not be an F face. The probability of the coexistence of two different surface topologies on the same face (hkl) has been describedt3,14 as a Boltzmann distribution of the specific surface energies. Even, if this relation is not completely true, the following observations are still valid. If the difference between the two specific surface energies is small, each of the two surfaces will occupy about one half of the total crystalline surface.If the difference increases, the probability decreases, but will also depend on the crystallization rate. Only when the crystallization proceeds slowly will the crystal tend to have energetically the most favourable crystalline surfaces. If such a submultiple of the slice dhklhas F character the growth rate will increase substantially. In the case of zircon, which always grows in a very impure environment, the negative surface occupied by silicate ions will directly be covered by impurity cations. Therefore the difference between the two specific surface energies will become smaller and thus the probability that two surfaces exist together will increase. During crystal growth the crystalline interface is in direct contact with the solution or melt from which the crystals grow.Factors such as impurities and supersaturation must be the cause of the diversity of habits and crystal forms observed in nature and laboratory experiments. Adsorption of impurities including the solvent can produce additional strong bonds and change the crystalline interface. Although the classic Hartman-Perdok theory takes into account only the strong bonds and the corresponding PBCs derived from the bulk crystalline structure, the effects of adsorption of impurities and solvents can be explained in many cases (e.g. zircon and PbCI,) by taking into consideration their interactions with the slice boundaries that are known with atomic precision. The introduction of scanning tunnelling microscopy (STM) and atomic force microscopy (AFM) enables imaging of the real structure of the crystalline interface to be achieved.On the (001) surface of silicon, which is following the principles of the Hartman- Perdok theory, a K face, surface reconstruction has been observed.27 This results in F character of the crystalline interface. In future, observations made by these and other methods must be taken into account in order to adjust and refine the Hartman-Perdok theory and the computation of attachment energies. The author is very much indebted to Professor P. Hartman for his stimulating interest for many years and his very helpful comments on the manuscript. C. F. Woensdregt 107 References 1 R. J. Haiiy, Essai d’une thtorie sur la structure des crystaux appliqute a plusieurs genres de substances crystallistes, Paris, 1784;read in B.G. Escher, Algemene Mineralogie en Kristallografie, J. Noorduyn & Zn, Gorinchem, 1954. 2 N. Steno, De solido intra solidum naturaliter contento dissertationis prodromus, Florentiae, MDCLXIX; read in B. G. Escher, Algemene Mineralogie en Kristallografie, J. Noorduyn & Zn, Gorinchem, 1954. 3 P. Hartman and W. G. Perdok, Acta Crystallogr., 1955, 8, 49. 4 P. Hartman and W. G. Perdok, Acta Crystallogr., 1955, 8, 525. 5 P. Hartman, in Crystal Growth: An Introduction, ed. P. Hartman, North-Holland, Amsterdam, 1973, ch. 14, pp. 367-402. 6 P. Hartman, in Morphology of Crystals, ed. I. Sunagawa, Terra Scientific Publications, Reidel, 1988, part A, ch.4, pp. 269-3 19. 7 P. Hartman and P. Benema, J. Crystal Growth, 1980, 49, 145. 8 P. Hartman, Acta Crystallogr., 1956, 9, 569. 9 E. Madelung, Phys. Z, 1918, 19, 524. 10 C. F. Woensdregt, Phys. Chem. Mineral., 1992, 19, 52. 11 G. Wulff, Z. Krist. Mineral., 1901, 34, 499. 12 C. F. Woensdregt, Phys. Chem. Mineral., 1992, 19, 59. 13 P. Hartman and W. M. M. Heijnen, J. Crystal Growth, 1983, 63, 261. 14 W. M. M. Heijnen, Geol. Ultraiect., 1986, 42, 11 (Ph.D. Thesis, ch. 2). I5 P. Hartman and C. S. Strom, J. Crystal Growth, 1989, 97, 502. 16 M. Aguil6 and C. F. Woensdregt, J. Crystal Growth, 1987, 83, 549. 17 C. F. Woensdregt, Z. Kristallogr., 1982, 161, 15. 18 C. F. Woensdregt, Z. Kristallogr., 1992, 201, 1. 19 P. Bennema, E. A. Giess and J. E. Weidenborner, J. Crystal Growth, 1983, 62, 41. 20 M. Boutz and C. F. Woensdregt, J. Crystal Growth, submitted. 21 W. M. M. Heijnen, N. Jahrb. Mineral. Abh., 1986, 154, 223. 22 B. N. Sun, P. Hartman, C. F. Woensdregt and H. Schmid, J. Crystal Growth, 1990, 100, 605. 23 R. Caruba, A. Baumer, M. Ganteaume and P. Iaconni, Am. Mineral., 1985,70, 1224. 24 J. V. Walther and Ph. M. Orville, Am. Mineral., 1983, 68, 731. 25 C. F. Woensdregt and P. Hartman, J. Crystal Growth, 1988, 87, 56 1. 26 M. van Panhuys-Sigler, P. Hartman and C. F. Woensdregt, J. Crystal Growth, 1988,87, 554. 27 P. E. Wierenga, J. A. Kubby and J. E. Griffith, Phys. Rev. Lett., 1987, 59, 2169. Paper 2/06649A; Received 14th December, 1992