Faraday Discuss., 1993, 95, 3-10 Explanation for the Occurrence of {hklm}Faces on Modulated Crystals P. Bennema,* M. Kremers, H. Meekes, K. Balzuweit and M. A. Verheijen RIM Laboratory of Solid State Chemistry, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands The concept of a morphological theory extended to include a superspace approach is presented. The aim of this theory is to be able to treat the case of incommensurate modulated crystals and consequently to understand the existence of satellite faces. The case of a one-dimensional modulated crystal is discussed. The science of crystallography started with the discovery that crystallographic faces fulfil the law of rational This law implies that each crystal face is characterised by a set of integral indices (hkl),the so-called Miller indices, which determine a reciprocal vector k perpendicular to the face.k = ha* + kb* + lc* (1) where a*, b*, c* are the basis vectors of the reciprocal lattice. The set of symmetry- equivalent faces to which (hkl)belongs is called a crystal form and is denoted by {hkl).The relative morphological importance (MI) of crystallographic forms {hkl) on growth or equilibrium forms (habits) of crystals can be derived from the law of Bra~ais,~ Friede14 and Donnay and Harker5 (BFDH), which can be formulated as follows: Ik,I 'Ik2l +MI1 <MI2 (2) that is, the smaller the norm Jkl,the higher the MI. The corresponding interplanar distance dhklis equal to the inverse of k. Allowed values of k obey the space-group selection rules.The BFDH law has been quite successful in predicting the morphology of numerous crystals."* Recently, the law of rational indices and to a certain extent also the BFDH law has been generalised to describe crystal faces that occur on incommensurate displacively modulated crystals with an average p-K,S04 structure such as Rb2ZnBr4,9 ((CH3)4N)2ZnC1410>1 and the mineral calaverite (AuTe2).12-15 For these incommensurately modulated crystals, the vector k perpendicular to a crystal face does not belong to a three-dimensional reciprocal lattice, but to a so-called Fourier module, which consists of all integral linear combinations of the three basic vectors a*, b*, c* and the modulation wave vectors ql,..,qd.Thus the rank of this module is equal to 3 + d and its dimension is three.Accordingly, the corresponding crystal face can be defined by 3 + d integral indices (hklml...md)such that: k = ha* + kb* + lc* + m,q, + ... + mdqd (3) The examples mentioned above are one-dimensionally modulated, so that in these cases d = 1. The modulation wavevector is indicated by q, defined as: q = aa* + pb* + yc* (4) where, at least one of the coefficients a,p, y is an irrational number. Note that allowed values of k again obey the selection rules imposed by the symmetry of the crystals. In the case of 3 (hklm) Faces on Modulated Crystals incommensurate modulated crystals the symmetry is described by a 3 + d dimensional superspace group.I6 This can be seen as an extension of the BFDH law.The morphological importance of a crystal face, however, seems to be determined not only by the interplanar distance because of the new types of faces that occur on modulated crystals which are called satellite faces and are characterised by having some non-zero index m. Indeed, one then expects that the orientation and magnitude of the modulation wave also represents relevant parameters for the MI of those faces. In any case, the morphology reflects the modulation properties and it has been possible to determine, from the orientation of satellite faces, the modulation wavevectors q with high precisi~n.~-~~J~.~~The morphologically determined values of q are consistent with values determined from X-ray19,20 and neutron diffraction experiments.21 The selection rules following from the relevant four-dimensional superspace group, leaving the incommensur- ate crystals given above invariant, could be shown to be compatible with the available morphological data.22 In some cases, experimental information on the MI is even sufficient to fix the superspace group itself.One can therefore claim that at the present level of knowledge the superspace group approach is supported not only by X-ray structure determination but also by crystal morphology. The occurrence of (hkl)faces on classical crystals fulfilling the symmetry of one of the 230 three-dimensional spacegroups can be explained with the integrated roughening and Hartman-Perdok theorie~.~~-'~In order to predict the morphology of these crystals the following procedure is carried out.(1) For the crystal structure under consideration the overall bond energies between (first-nearest) neighbour growth units (atoms, molecules, complexes) is determined. (2) The growth units are then reduced to centres of gravity or charge and the crystal graph is determined. This crystal graph consists of the centres of gravity or charge and the bonds between these centres. (3) The crystal graph is partitioned in slices with an overall thickness dhkl,the interplanar distance. It is checked whether within these slices the so-called connected nets can be identified. By definition a connected net consists of a net with an average thickness dhkl,where all points of the crystal graph are connected to each other by bonds.Connected nets show in principle a roughening transition. This implies that an absolute roughening temperature TRcan be identified. By definition a connected net consists of a net with an average thickness dhkl,where all points of the crystal graph are connected to each other by bonds. Connected nets show in principle a roughening transition. This implies that an absolute roughening temperature TRcan be identified. If a crystal face is growing below this roughening temperature it will grow with a layer mechanism (spiral growth or a 2D nucleation mechanism) with an orientation (hkl).If a certain orientation (hkl)is growing at a temperature above the roughening temperature, this face will grow as a roughened, rounded face with no fixed orientation (hkl).(4) It is possible to define a hierarchy of faces in the sense that faces having, in principle, the highest dhkl,which means the highest energy content per growth unit, will have the lowest growth rates and hence will have the highest MI. Using such hierarchies growth forms with a high heuristic value can be constructed. It is interesting to note that connected nets correspond to cusps in a raspberry Wulff plot according to Herring,23 26 It has also to be observed that the faces that dominate the 8rowth or equilibrium forms are those having the highest dhkl,the highest energy content (Eir),the lowest surface energy or attachment energy and the deepest cusps in the Wulff- Herring plot.So far only a mathematical recipe for describing the (hklm)orientation of satellite faces that occur on modulated crystals is known. No physical explanation for the occurrence of satellite faces is developed so far. Dam2' found from in situ observations that a satellite face that occurs on a modulated ((CH3)4N)2ZnC14 crystal growing from an aqueous solution P.Bennema et al. showed rotating spiral-like patterns. This suggests that this satellite face behaves as a pseudo-classical face. It is the aim of this paper to show, starting from simple physical principles and using the higher-dimensional crystallography of de Wolff, Janner and Janssen, 16,28 that the occurrence of (hklm)faces on modulated crystals can be explained.It will be shown that in a sense the integrated Hartman-Perdok roughening transition theory can be generalised to include the case of modulated crystals. Modulated One-dimensional Crystal Embedded in a 2D Superspace Fig. 1 shows the simplest possible modulated one-dimensional crystal embedded in 2D superspace. The 2D translational invariant superstructure is spanned on the basis of a, and d, according to the superspace- description. The Fourier module of the 2D space corresponding to a ‘standing-wave’ crystal consisting of transversal waves can be recognised. The d, axis is chosen to be perpendicular to the real a axis. The points of the modulated crystal are generated by the intersections of the real one-dimensional space (the horizontal axis) with the transversal ‘world lines’.The translational symmetry of the 2D superspace corresponds to translations na combined with a phase shift nq*aalong the d, axis. The 2D vectors a, and d, are characterised by their 1D components in the following way: a, = (a, -ad) (5) ds = (07 d) (6) where d is a vector in the d, direction. The reciprocal 2D vectors are then defined as: a: = (a*,0) d,* = (4,d*) Fig. 1 Two-dimensional superspace representation of a modulated one-dimensional crystal. The hatched area corresponds to the bonds cut by the faces (01). The symbols are referred to in the text. (hklm)Faces on Modulated Crystals Thus, qrepresents the projection of d$ on the real a axis. Pis chosen such that the following conditions are fulfilled: a*a*= I &d*= 1 (9)a-d = 0 a*9d = 0 Consequently: a,-a: = 1 d;d: = 1 as-d,*= 0 a$*d,= 0 Eqn.(10) represents the usual relations between the direct and reciprocal lattice. It follows from the conventions used that Iql = b*l = laial (1 1) We assume that for a real crystal a is fixed. If we choose the angle between the real axis and a, to be p = 45", then the phase along the d, axis is measured with the same unit of length as that of the real axis a. The only free parameter left is then the amplitude A of the world lines. Thus, the two periodicities present in the crystal, namely the average lattice parameter (la/) and the modulation wavelength (l/lql) are represented by a and ds, respectively. Concept of Chemical Bond in a Modulated Crystal: Selective Cuts Returning to Fig.1, which represents a modulated real one-dimensional crystal embedded in a two-dimensional superspace, the points of the crystal are given as: X-n, X-,+l,...,XO, XI, x2,...x, ,... with x, = na + A sin(q*na+ t) (13) where t represents a phase shift of the modulation with respect to the origin. The distance a(n) between the adjacent points x,+ and x, is given by a(n) = a + A sin[q-(n + l)a + t] -A sin(q*na+ t) (14) Using eqn. (1 1) we find a(n) = n + A sin[a(n + 1) + t]-A sin(an + t) (15) a(n) is a measure of the bond strength @[a(n)] between the neighbouring points x, + and x,. We implicitly restrict the bonds to the direction of the real space, i.e. in Fig.1 only the horizontal bonds are relevant. In order to obtain a survey of all possible bonds the information of the 2D space will be used. If the phase factor t is varied (01t12~) by travelling along the d, axis the whole continuum of horizontal bond distances is engaged within the period d, of one elementary cell of superspace. Travelling an infinite distance along the real space of the modulated crystal, all distances, a(n), of the continuum of distances 'stored' in the elementary cell of superspace will be met. Nevertheless, the order in the continuum of distances which is realised along the real axis is, though completely determined, mathematically complicated. The points of the modulated crystal are completely determined by a, d, and A. P.Bennema et al.In order to illustrate the identification of distances, equidistant netlines characterised by the Miller indices (Ol), (li)and (10) are drawn in Fig. 1. Since the 2D space is translationally invariant, concepts of classical mathematical crystallography like net lines can be used. We generalise the notion of mesh area hfhk[, for which it holds that the volume v of the unit cell is equal to hfhk&&, in the case of our 2D superspace. Then hfhm stands for a length and V is the area of the unit cell in superspace. Looking at Fig. 1, one can see that a grid of equidistant parallel (01)lines cuts the real modulated 1Dcrystal in equidistant pieces. These observations lead to the following results, which yield the essential ingredient of this paper. The grid (01) cuts only a fraction of the whole continuum of distances (bonds). This is because, as can be seen from Fig.1, only those distances which are cut by the mesh area of (01) occur in the cuts of the grid with the modulated crystal. Thus, it can immediately be seen which fraction of the continuum of distances stored in superspace will be cut by the grid (hatched areas). Changing the phase t will implicitly change this fraction. The total energy involved for a grid (hm)is therefore a function of the phase t. We will refer to those grids (hm)as faces (hm). In order to calculate bond energies the interaction potential between adjacent points @[a(n)]of the modulated crystal is used. With this potential the specific energy Eh,,,(t), i.e.the sum of the energies of all bonds cut in a mesh area divided by the area, can be determined for each face (h,m)as a function of the reference phase t.In this way a minimum specific energy E:,~ is found for a certain phase The modulated one-dimensional crystal is cut in pieces, having on the average a lowest cut (surface) energy and a highest energy within the pieces. Here a kind of generalisation of the Hartman-Perdok theory occurs for a modulated ID crystal embedded in a 2D superspace. This generalisation implies that the modulated 1 D crystal is cut in pieces having on the average a lowest Eattand a highest Eslice. We note that looking at other grids determined by equidistant lines (hm) like for example the (17) grid in Fig.1,it can be seen that the mesh area becomes larger than that of (01). The relevance of the phase t and the mesh area becomes clear in this figure. The findings of this section concerning the selectivity of cuts for a face (hm)can be generalised to a modulated 2D crystal embedded in 3D space and a modulated 3D crystal embedded in 4D space. In the latter case it can be checked whether connected nets are present within a thickness dhkrrn.l4 wulff Plot 2D Plot for Non-modulated Crystals Consider a one-dimensional unmodulated and translationally invariant crystal. This crystal can be obtained from Fig. 1 by making the amplitude A zero. The continuum of bonds of Fig. 1 now becomes a set of equal bonds. In order to calculate specific energies expressed by bond energies divided by some unit of length, we proceed by using the construction as suggested by Herring.26 Thus we write Y($hrn) = cos 4hm (16) where y is the surface energy unit of length and $ is the angle between the normal of the face (Am)and the horizontal line giving the orientation of the face in reference to the continuum of horizontal bonds (h,m are considered as continuous variables) and D is a proportionality constant.Making a polar plot of y yields a plot consisting of two circles. Note that for the face (10) 4 is zero, corresponding to the maximum value of 7. The face (hm)with h = -ma corresponds to the only very steep cusp. The reason is that for the latter face no bonds are cut. The only function of the extra dimension (dsaxis) is to store data (bonds) with their proper phase in 2D space.Using the principles of a Wulff plot construction it can be seen that all orientations (hm) cut the real axis in the same point. This means that all orientations (hm)have the same (hklm)Faces on Modulated Crystals contribution to the equilibrium form, which is a piece of a line with a surface energy zero corresponding to the orientation (10) and an edge energy @(a). As could be expected, it does not make any sense to embed the one-dimensional truly translationally invariant crystal in a 2D space, because no new faces (hm)show up. This situation becomes radically different if a modulation is switched on. Principles of 2D Wulff Plot for a Modulated Crystal In order to proceed, we have to calculate the specific energies for the different faces of a modulated crystal.Instead of labelling the different bonds with n, that is in real space, we use the phase t for this purpose. Implicitly we use that all relevant bonds are engaged, through in a different order, in the correct number of occurrence along the d,axis. Therefore we write a(t)= a + A sin[a(n + 1) + t] -A sin(an + t) (17) The integration runs over a complete mesh area Mhmof the face (hm)in formula Now the reference phase has to be chosen in such a way that the energy becomes minimal. In the exceptional case of (hO) the energy has been minimised by varying the position of the face along the real axis. Results of the Calculation In order to perform the calculations mentioned in the previous section, we chose the example of a 1 + 1 dimensional superspace, describing a modulated crystal with q = 0.3a* and an amplitude of ]AI = 0.21al.For the interaction we used a Coulomb-like potential.In Fig. 2 the results are summarised in a Wulff plot. The spread in energy as a function of the reference phase is indicated by the radial lines. The minimum values correspond to the cusps of a Wulff plot. For these minima, the habit-determining lines are drawn. The actual value of the amplitude determines the size of the spreading interval. When the amplitude approaches zero, this interval shrinks to the value on the two-circle unmodulated case. For Fig. 2 Wulff plot for the crystal in Fig. 1, Faces with 3 as maximal index have been indicated.Note the steep cusp along the d, axis and the spread in energies for the different faces. P. Bennema et al. 9 our choice of the modulation wave vector the face (01) corresponds, apart from the face (am,a),to the deepest cusp. The order of the minimal energies obviously depends on the modulation wavevector. Conclusion In this paper we have demonstrated that the attachment energy of a face on a modulated one-dimensional crystal does not involve the whole spectrum of bonds present in such a structure. Depending on the face (hm)and on the phase t of the modulation only a well determined subset of this spectrum turns out to be relevant. The phase t acts as a minimising factor for this energy. These results are used to make a generalised Wulff-Herring construction in order to obtain a prediction of the equilibrium form of the crystal.In a forthcoming paper the effect of the modulation wave vector on the specific energy will be treated. Furthermore, it will be shown how our two-dimensional model which in principle explains the occurrence of ‘faces’ (hm) can be generalised to three and four dimensions in order to explain the occurrence of faces (hklm).Analogously, we then assume that a four-dimensional grid of hyperplanes (hklm) is partitioning a three-dimensional crystal graph in slices with thickness dhkbn. Assume that a 3D crystal graph is cut in slices with a thickness djklmand that the cuts have the lowest surface energy. It is then reasonable to assume that if the corresponding nets are connected by the bonds of the 3D crystal graph, the corresponding faces (hklm) will grow as flat faces, if the actual growth temperature is below the roughening transition temperature. Since, as a rule the value of djklmof a large number of faces is quite high, many (hklm) faces will be connected and will have relatively high roughening temperatures.This explains to some extent the very large variety of {hklm)forms that occur on calaverite crystals. The relatively high amplitude of the modulation in calaverite probably enhances this effect. In case the amplitude of the modulation is small, cusps in a 2D (and 4D) Wulff plot will be less steep. Consequently, the chance for the occurrence of satellite faces is smaller.This may explain the much lower number of (hklm) faces that occur in the structures having an average potassium sulfate structure, growing from aqueous solution. References 1 J. G. Burke, Origins of the Science of Crystals, University of California Press, Berkeley, 1966. 2 G. J. Schneer, Crystal Forms and Structures, Dowden, Hutchinson and Ross, 1977. 3 A. Bravais, J. Ecol. Polytech., 1850, 19, 1. 4 G. Friedel, Bull. Soc. Fr. Miner. Le Couscle Crystallographie, Herman, Paris, 1911. 5 J. D. H. Donnay and D. Harker, Am. Mineral., 1937,22,446. 6 F. C. Philips, An Introduction to Crystallography, Longman, London, reprinted 4th edn., 1978. 7 P. Hartman, Crystal Growth: An Introduction, ed. P. Hartman, North Holland, Amsterdam, 1973, pp.367402. 8 P. Hartman, Morphology of Crystals, Part A, ed. I. Sunagawa, Terra Scientific Publishing and D. Reidel, Dordrecht, 1987, pp. 271-319. 9 A. Janner, T. Rasing, P. Bennema and W. H. van der Linden, Phys. Rev. Lett., 1980,45, 1700. 10 B. Dam and A. Janner, 2. Krist., 1986, 165, 274. 11 B. Dam and A. Janner, Acta Crystallogr., Sect. B, 1985, 42, 69. 12 B. Dam, A. Janner and J. D. H. Donnay, Phys. Rev. Lett., 1985,551, 123. 13 B. Dam and A. Janner, Acta Crystallogr., 1989, 45, 115. 14 L. J. P. Vogels, K. Balzuweit, H. Meekes and P. Bennema, J. Crystal Growth, 1992, 116, 397. 15 K. Balzuweit, A. Hovestadt, H. Meekes and J. L. de Boer, J. Crystal Growth, submitted. 16 T. Janssen and A. Janner, Adv. Phys., 1987,36, 519. 17 L.J. P. Vogels, M. A. Verheijen, H. Meekes and P. Bennema, J. Crystal Growth, 1992, 121, 697. 18 B. Dam and P. Bennema, Acta Crystallogr., Sect. B, 1987, 43, 64. 19 E. Colla, P. Muralt, H. Arend, R. Perret, G. Godefroy and C.Dumas, Solid State Commun., 1984,52,1033. 20 W. Schutte and J. L. de Boer, Acta Crystallogr., Sect. B., 1988, 44, 486. (hklm)Faces on Modulated Crystals 21 K. Gezi and M. Izumi, J. Phys. SOC. Jpn., 1978,45, 1777. 22 P. Bennema, K. Balzuweit, B. Dam, H. Meekes, M. A. Verheijen and L. J. P. Vogels, J. Phys. D, 1991,24, 186. 23 P. Bennema and J. P. van der Eerden, Morphology of Crystals, Part A, ed. I. Sunagawa, Terra Scientific Publishing, Tokyo and D. Reidel, Dordrecht, 1987, p. 1. 24 P. Bennema, Sir Charles Frank, An Eightieth Birthday Tribute, ed. R. G. Chambers, J. E. Enderby, A. Keller, A. R. Lang and J. W. Steeds, Adam Hilger, Bristol, 1991, pp. 47-78. 25 P. Bennema, Handbook ofCrystal Growth, ed. D. J. T. Hurle, Elsevier, Amsterdam, 1993, vol. I, ch. 7. 26 C. Herring, Phys. Rev., 1951, 82, 87. 27 B. Dam, Phys. Rev. Lett., 1985, 55, 2806. 28 P. M. de Wolff, T. Janssen and A. Janner, Acta Crystallogr., Sect. A, 1981,37, 625. Paper 3/00139C; Received 8th January, 1993