Morphological Importance of Crystal Faces in Connection with Growth

Morphological Importance of Crystal Faces in Connection with Growth Rates and Crystallographic Structure of Crystal Jolanta Prywer*

CRYSTAL GROWTH & DESIGN 2002 VOL. 2, NO. 4 281-286

Institute of Physics, Technical University of Ło´ dz´ , Wo´ lczan´ ska 219, 93-005 Ło´ dz´ , Poland Received April 5, 2002;

Revised Manuscript Received May 28, 2002

ABSTRACT: Analysis of correlation between growth rates and interfacial structure of a crystal is the key to understanding the growth behavior of crystal faces. The subject of this paper is research on the morphological importance of crystal faces in connection with their growth rates and crystallographic structure of the crystal. Results of this analysis show that the largest crystal face does not necessarily correspond to the slowest growing face. Even the fast-growing faces can bound the crystal. From this, it follows that morphological importance of faces is not inversely proportional to their growth rates. 1. Introduction Crystals take a variety of shapes, depending on internal factors, such as the crystallographic structure of crystal, and external factors, such as the growth environment. Both the internal and external factors influence the growth rates, and, therefore, they modify the crystal morphology. For growing crystals, the growth rates of crystal faces are proportional to the distances from the center of the crystal to the respective hkl crystal faces. In this meaning, the prediction of the growth morphology is equivalent to the prediction of growth rates in different crystallographic orientations. In the past, many attempts have been made to predict crystal morphology. The Bravais-Friedel law1,2 reveals the a priori correlation between the morphological importance (MI) of a crystal face and its interplanar distance dhkl. The MI of a crystal face is commonly understood as its relative size in a given crystal habit.2 The interplanar distance dhkl is the distance that separates physically identical surfaces. According to the Bravais-Friedel law, the observed crystal faces are those with the largest interplanar distances. The larger the interplanar distance, the more important the corresponding crystal face. This law is violated sometimes; therefore, Donnay and Harker3 extended it by considering the screw axis and the glide planes. In this way, the Bravais-Friedel-Donnay-Harker law (BFDH law) was introduced. This law often gives a satisfactory description of crystal morphologies. Later, Hartman and Perdok4 developed a more general theory based on an energetic hypothesis. The Hartman-Perdok (HP) theory introduces the concept of periodic bond chains (PBCs), which plays a key role in this theory. A PBC is an uninterrupted chain of bonds representing strong interactions between growth units in the crystal lattice. Three types of faces are distinguished by the classical HP theory: flat faces (F faces), parallel to at least two nonparallel intersecting PBCs; stepped faces (S faces), parallel to only one PBC; and kinked faces (K faces), not parallel to any PBC. According to this theory, only the F faces are important for the crystal morphology. * E-mail: [email protected].

Later, Burton, Cabrera and Frank described the transition of flat crystal faces to roughened faces at equilibrium as a function of temperature.5 Further, the Hartman-Perdok theory was integrated with the theory of surface roughening,6,7 and the concept of connected net was introduced.8-10 A connected net is a set of growth units connected by bonds constituting a network. Equivalent connected nets are separated by the interplanar distance dhkl, corrected for the space group symmetry (according to the BFDH law). Connected nets can be derived from a so-called crystal graph, which is defined as an infinite set of points corresponding to the centers of the growth units with strong bonds between these points. Connected nets have edge free energies larger than zero in all crystallographic directions parallel to the net. Faces parallel to a connected net grow as flat faces (called F faces in the HP theory) below a specific roughening temperature, and as rough rounded off faces above this temperature. In most cases, the slowest faces of crystals are growing below their roughening temperature as flat faces with well-defined hkl orientation. The growth occurs by a layer mechanism, like spiral growth or two-dimensional nucleation. Fast-growing faces are not parallel to a connected net and have a roughening temperature equal to 0 K, and, therefore, they grow as rough rounded off faces. According to all these theories, the faces limiting the crystals are the slow-growing faces. In other words, the largest, i.e., the most morphologically important crystal faces, are those that grow the most slowly. The aim of this paper is to show, based on analytical description, that the most important faces do not necessarily correspond to the slowest growing faces. Even fast growing faces can bound the crystal. This assertion is in contrast with the conventional crystal growth theories. All our considerations are in connection with the growth rates and crystallographic structure of crystals. 2. Morphological ImportancesTheoretical Analysis The correlation between growth rates of crystal with its size has been the object of experimental and theoretical studies for a long time. Some purely empirical

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Figure 1. Cross-section of a hypothetical crystal; lhkl, size of a hkl face; h1k1l1, h2k2l2, neighboring faces of hkl face; Rhkl, Rh1k1l1, Rh2k2l2, the normal growth rates of hkl, h1k1l1, and h2k2l2 faces, respectively; R and γ, interfacial angles; GB, growth bands; GSB, growth sector boundaries; GS, growth sectors.

models have been proposed (e.g., ref 11). However, earlier, in 1957, the paper of Kozlovskii12 appeared. In this paper, the author derived the dependence of size lhkl of a given hkl face on its growth rate and the rates of the neighboring faces and appropriate interfacial angles. Unfortunately, besides a few papers,13-15 which profit from the results of this paper, it has remained unnoticed. On the basis of geometrical dependences, the size lhkl of a given hkl face, in the case of crystal crosssection, is expressed by the formula:12

lhkl ) Rh2k2l2 sin R + Rh1k1l1 sin γ - Rhkl sin(R + γ) sin R sin γ

t + l 0hkl (1)

where Rhkl, Rh1k1l1, Rh2k2l2 are the normal growth rates of hkl, h1k1l1, and h2k2l2 faces, respectively, R and γ are the appropriate interfacial angles illustrated in Figure 1, l 0hkl is the initial size of the hkl face (not marked in Figure 1), and growth time t corresponds to the change of the size of the considered face from l 0hkl to lhkl. The initial size l 0hkl may be considered as the size of a given face in the seed or after growth of the crystal layer (growth band). The constant growth rates Rhkl, Rh1k1l1, Rh2k2l2 are not required throughout the whole growth process but only during the growth of a layer of crystal, which corresponds to the change of the size from l 0hkl to lhkl. It should be remembered that the above formula was derived based on trigonometric dependences, not assumed a priori. Later, based on this formula, the 13 critical growth rate Rcrit hkl was introduced and derived:

Rcrit hkl )

Rh1k1l1sin γ + Rh2k2l2 sin R sin(R + γ)

(2)

The physical meaning of the critical growth rate Rcrit hkl is explained below. Taking into account eq 2 and properly transforming eq 1, we obtain the size lhkl of hkl face in more suitable form: 0 lhkl ) (cot R + cot γ)(Rcrit hkl - Rhkl)t + l hkl

(3)

The above formula is the basis for analysis of the size of crystal faces and consequently of their MI depending on the interfacial structure of crystals and growth rates of faces. First, let us focus on the quantity Rcrit hkl - Rhkl. The value of the quantity Rcrit hkl - Rhkl determines the existence, or nonexistence, of a given face in crystal morphology. For Rcrit hkl - Rhkl < 0, the hkl face grows with the growth rate Rhkl greater than the critical. This means that the hkl face decreases its initial size l 0hkl or it does not appear if it was absent in the habit. For Rhklcrit Rhkl ) 0 the hkl face grows with the growth rate Rhkl equal to the critical, and the hkl face does not change its initial size l 0hkl (lhkl ) l 0hkl). If it was absent in the habit, it still does not appear. For Rcrit hkl - Rhkl > 0 the hkl face grows with the growth rate Rhkl smaller than the critical. This means that the hkl face increases in size. If it was absent in the habit, it starts to appear. The value of the quantity Rcrit hkl - Rhkl depends both on interfacial structure (the interfacial angles R and γ, cf. eq 2) and on growth rates of hkl, h1k1l1, and h2k2l2 faces, which vary with growth conditions. The dependence of Rcrit hkl - Rhkl on the interfacial angles R and γ for different values of growth rates Rhkl, Rh1k1l1, Rh2k2l2 is shown in Figure 2. Additionally, the plane Rcrit hkl Rhkl ) 0 is shown. It should be noticed that our analysis is restricted to the interfacial angles satisfying the condition R + γ < π, because only for such interfacial angles, a given face may increase, preserve, or decrease its initial size. The condition R + γ ) π corresponds to the hkl face situated between two parallel faces. As follows from eq 2, for R + γ ) π, the quantity Rcrit hkl f ∞. This means that in such a case, the hkl face never disappears.15 In the case for R + γ > π (for example, tetrahedral habit), a given hkl face can only increase, and the idea of the critical growth rate loses its sense.15 Figure 2a presents this dependence for Rhkl ) Rh1k1l1 ) Rh2k2l2 ) 5.00 µm/h. It is seen that in this case the quantity Rcrit hkl - Rhkl is always greater than zero. It starts to take negative values for R + γ > π, but we do not analyze such angles because, as it is explained above, for such interfacial angles the idea of the critical growth rate has no physical meaning. From this, it follows that if the hkl face and the neighboring h1k1l1 and h2k2l2 faces grow with the same growth rates, the given hkl face increases its initial size only. In other words, if the hkl face grows equally fast as the neighboring faces, preserving or decreasing its size is not possible.15 (The greater the value of all these growth rates Rhkl, Rh1k1l1, and Rh2k2l2, the faster the given hkl face increases.) Figure 2b presents the quantity Rcrit hkl - Rhkl for Rh1k1l1 ) Rh2k2l2 ) 1.00 µm/h and Rhkl )1.50 µm/h. In other words, the hkl face grows faster than the neighboring h1k1l1 and h2k2l2 faces. It is seen that there exists a range of interfacial angles, for which the quantity Rcrit hkl - Rhkl takes values greater than zero (above the plane Rcrit hkl - Rhkl ) 0). This means that for such a range of interfacial angles the hkl face increases its size. However, for interfacial angles R and γ lying on the curve being the intersection line of the surface given by Rcrit hkl - Rhkl (for Rh1k1l1 ) Rh2k2l2 ) 1.00 µm/h and Rhkl ) 1.50 µm/h) with the plane Rcrit hkl - Rhkl ) 0, the quantity

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Figure 2. The dependence of Rcrit hkl - Rhkl on the interfacial angles R and γ for different, theoretically assumed, values of growth rates Rhkl, Rh1k1l1, Rh2k2l2. (a) Rhkl ) Rh1k1l1 ) Rh2k2l2 ) 5.00 µm/h; (b) Rhkl ) 1.50 µm/h, Rh1k1l1 ) Rh2k2l2 ) 1.00 µm/h; (c) Rhkl ) 1.52 µm/h, Rh1k1l1 ) 1.95 µm/h, Rh2k2l2 ) 0.00 µm/h; (d) Rhkl ) 2.10 µm/h, Rh1k1l1 ) 2.20 µm/h, Rh2k2l2 ) 1.00 µm/h; (e) Rhkl ) Rh1k1l1 ) 1.45 µm/h, Rh2k2l2 ) 1.00 µm/h.

Rcrit hkl - Rhkl is equal to zero. This means that for these interfacial angles the hkl face preserves its size. Furthermore, the quantity Rcrit hkl - Rhkl can take values smaller than zero, which means that for appropriate values of the interfacial angles R and γ, the hkl face decreases its size. It should be emphasized that, in this case, the hkl face, although growing faster than the neighboring faces, may increase in size. Such a phenomenon is possible for faces of particular geometry, i.e., those for which the sum of interfacial angles R and γ is close to π.16

Figure 2c presents the dependence of Rcrit hkl - Rhkl on the interfacial angles R and γ for Rhkl ) 1.52 µm/h, Rh1k1l1 ) 1.95 µm/h, and Rh2k2l2 ) 0.00 µm/h. This means that the hkl face grows faster than one neighboring face and, at the same time, more slowly than the other neighboring face. It is seen that in such a case the hkl face may behave in three different ways: it may decrease (for crit Rcrit hkl - Rhkl < 0), preserve (for Rhkl - Rhkl ) 0, corresponding to the curve being the intersection line of the crit surface Rcrit hkl - Rhkl with the plane Rhkl - Rhkl ) 0), and crit increase (for Rhkl - Rhkl > 0) in size depending on the

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need not be slow growing faces, because even growing faster than the neighboring faces, they increase their sizes. The face between parallel faces is the limiting case. Such a face, even if it grows a few times faster than the neighboring faces, does not decrease its size. From this, it follows that the faces appearing in final crystal morphologies, need not be, as it was thought so far, the slow growing faces. Therefore, the MI is not inversely proportional to the growth rates. The presented analysis is not limited to growth rates constant in time. It is possible to consider any function as the growth rate. In such a case, the size lhkl of the hkl face is given by:

lhkl ) (cot R + cot γ) Figure 3. The dependence of geometrical factor (GF) on the interfacial angles R and γ.

values of interfacial angles R and γ. As it was derived in ref 15, for interfacial angles satisfying one of the conditions 2R + γ < π or 2γ +R < π, the given hkl face may decrease its size growing more slowly than one of the neighboring faces, the h1k1l1 face or h2k2l2 face, respectively. A similar situation may be observed looking at Figure 2d. Here, the growth rates Rhkl, Rh1k1l1 and Rh2k2l2 are equal to 2.10, 2.20, and 1.00 µm/h, respectively. Also in this case the hkl face decreases its size growing more slowly than one of the neighboring faces. Figure 2e is obtained for growth rates Rhkl, Rh1k1l1, and Rh2k2l2 equal to 1.45, 1.45, and 1.00 µm/h, respectively. This means that in this case the hkl face grows equally fast as one neighboring face and faster than the other neighboring face. It is seen that also in this case, the hkl face may decrease, preserve, or increase its size depending on the values of interfacial angles. The above analysis shows that, to exist in crystal habit, the given hkl face need not be the slow growing face. Even fast growing faces may develop their sizes. This is in contrast with the contemporary crystal growth theories. As eq 3 shows, the size lhkl of the hkl face depends also on the geometrical factor expressed by cot R + cot γ. This geometrical factor (GF) is not decisive in the matter of existence or nonexistence of a given face in crystal habit, because this factor does not influence the sign “+” or “-” of the size lhkl of the hkl face (eq 3). However, this factor influences the size lhkl of the hkl face. Figure 3 illustrates the dependence of GF on the interfacial angles R and γ. It is seen that GF takes positive values for the whole considered range of interfacial angles R + γ < π (for R + γ ) π, GF ) 0, then the size lhkl of hkl face is equal to l 0hkl independently on the growth rates). The above analysis demonstrates that the size of a given face and, consequently, its MI depends on the GF and the quantity Rcrit hkl - Rhkl. The bigger the value of the product of GF and the quantity Rcrit hkl - Rhkl, the bigger the size lhkl of the considered hkl face and, therefore, the more important the corresponding crystal face. The product of the quantity Rcrit hkl - Rhkl and GF reaches its highest values for the interfacial angles R + γ close to π. The faces with such interfacial angles

(

∫Rh k l

1 1 1

sin γ +

∫Rh k l

2 2 2

sin R

sin(R + γ)

)

∫Rhkl dt + l 0hkl

-

(4)

It is also possible to estimate the rate of face increase. We are able to calculate such a growth rate based on the formula:

dlhkl ) (cot R + cot γ)(Rcrit hkl - Rhkl) dt

(5)

The bigger the value of the result of a subtraction Rcrit hkl - Rhkl, the bigger the value of dlhkl/dt and the faster the size lhkl of the hkl face increases. Both the constant and varying in time growth rates are important in the field of crystal growth. Constant growth rates are important in the case of crystals grown in nature. For such crystals, it is assumed that each growth band grows with a constant growth rate. Variable growth rates are important in the case of modeling of the growth process. 3. Morphological Importance-Computer Experiment To illustrate the introduced development, let us consider a triclinic potassium bichromate (KBC) crystal (space group symmetry P1 h , unit cell parameters a ) 7.445 Å, b ) 7.376 Å, c ) 13.367 Å, R ) 97.96°, β ) 96.21°, γ ) 90.75°).17 Let us concentrate on Figure 4, which illustrates the cross-sections of this crystal. In the case of Figure 4a, the growth begins from a seed, but in the case of Figure 4b, it begins by spontaneous nucleation. First, let us consider the crystal shown in Figure 4a, whose growth process begins from a seed. It is seen that all faces, which appear in the seed, are present in the final crystal morphology. For simplicity, all these faces grow with constant, but different for every face, growth rates. All these growth rates are assumed theoretically, and they can be estimated as distances between consecutive growth bands. Table 1, column 2 presents these growth rates normalized to the growth rate of the (001h ) face. Two facts are worth noticing. First, that the (001) face (interfacial angles R ) 54.90°, γ ) 67.29°), growing faster than the neighboring (011) and (01h 1) faces, does not decrease, but it increases in size. Second, that the (01 h2 h ) face (interfacial angles R ) 16.54°, γ ) 38.36°),

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Table 1. Theoretically Assumed Relative Growth Rates Rhkl/R(001h ) (Column 2) for which the Cross-Sections Shown in Figure 4 Are Obtained; MI of Faces Determined by Rhkl/R(001h ) (Column 3), the Size of Individual Faces (Column 4) and eq 3 (Column 5); the Quantity Rcrit hkl - Rhkl (Column 6) crystal face hkl

relative growth rate R(hkl)/R(001h )

MI determined by R(hkl)/R(001h )

MI determined by the relative size of crystal face

column 1

column 2

column 3

column 4

(001) (01 h 1) (01 h 0) (01 h1 h) (01 h2 h) (001 h) (011 h) (010) (011)

2.45 2.23 2.15 2.20 2.07 1.00 2.33 1.95 2.00

9 7 5 6 4 1 8 2 3

Crystal Shown in Figure 4a 2 3 6 7 9 1 8 4 5

(001) (01 h 1) (01 h 0) (01 h1 h) (01 h2 h) (001 h) (011 h) (010) (011)

2.45 2.23 2.15 2.20 2.07 1.00 2.33 1.95 2.00

9 7 5 6 4 1 8 2 3

Crystal Shown in Figure 4b 3 2 6 7 absent 1 absent 5 4

Figure 4. Cross-sections of KBC crystals. Plane of observation (1 h 00). (a) The growth begins from seed; (b) the growth begins from a point (this may be considered as spontaneous nucleation). Both these cross-sections are drawn for the same relative growth rates, which are presented in Table 1.

growing more slowly than the (01h 1 h ) facesone of the neighboring facessdecreases its size. From this, it follows that based on the growth rates we cannot

lhkl/l(001h ) calculated based on eq 3/MI determined by eq 3

crit R(hkl) - R(hkl)

column 5

column 6

0.64/2 0.48/3 0.32/6 0.08/7 0.02/9 1.00/1 0.05/8 0.43/4 0.41/5

1.88 1.04 0.36 0.05 -0.05 2.48 0.00 0.51 0.73

0.50/3 0.52/2 0.33/6 0.07/7 -0.06/absent 1.00/1 0.00/absent 0.44/5 0.47/4

1.88 1.04 0.36 0.05 -0.05 2.48 0.00 0.51 0.73

conclude about the size of faces and, consequently, about their MI. According to the improvement proposed in this paper, we are able to predict the behavior of faces and their MI based on the quantity Rcrit hkl - Rhkl. This quantity for the crystal shown in Figure 4a is presented in Table 1, column 6. From the values of this quantity, it is seen that the (01 h2 h ) face decreases its size during the growth process (Rcrit hkl - Rhkl ) - 0.05), and the (011 h ) face does not change its initial size (Rcrit hkl - Rhkl ) 0.00). In other words, this face preserves its seed-size. The values of this quantity for the other faces are greater than zero. This means that all these faces increase in size till the end of the growth process. Also, for the (001) face, the quantity Rcrit hkl - Rhkl is greater than zero (it is equal to 1.88; cf. Table 1, column 6). From this, it follows that its increasing in size, although it is a fast-growing face, is predicted by the mathematical description proposed in this paper. The quantity Rcrit hkl - Rhkl is important and decisive for the matter of existence of a given face in crystal morphology. However, based on this quantity we are not able to conclude about MI. We are able to predict MI based on the expression presented by eq 3. For the KBC cross-section shown in Figure 4a, such an MI of individual faces is presented in Table 1, column 5. Comparing the real MI (i.e., MI according to the size of crystal faces) and MI according to eq 3 (column 4 and 5 in Table 1), we see that these MI are the same. From this, it follows that eq 3 describes well the MI of faces. MI according to the relative growth rates Rhkl/R(001h ) (Table 1, column 3) does not reflect the reality. Therefore, we can put forward a proposal that, perhaps, the seed from which the growth process proceeds influences the MI of faces. To verify this, let us focus on the KBC cross-section shown in Figure 4b. Here, the growth process begins from a point; it may be interpreted as spontaneous nucleation. In this case, the values of the quantity Rcrit hkl - Rhkl are the same as in the previous case (cf. Table 1, column 6 for both cases). Let us focus on Rcrit h2 h) hkl - Rhkl for the (01

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face. Now, the value -0.05 means that this face does not appear in crystal habit (in the previous case, it decreases its size because its initial size was different than zero). Similarly, in the case of the (011 h ) face, this face also does not appear, as the value of the quantity Rcrit hkl - Rhkl is equal to zero. For the other faces, the quantity Rcrit hkl - Rhkl takes positive values and all of them appear and exist in the final habit. MI estimated based on eq 3 is shown in Table 1, column 5. It is seen that this MI is in excellent agreement with those estimated based on the size of individual faces. However, MI is different than in the previous case of the crystal that began the growth process from seed. Besides these two faces, which do not show up in cross-section presented in Figure 4b, but exist in the cross-section shown in Figure 4a, there are other differences in MI. For example, the (001) and (01 h 1) faces exchange places. For the cross-section with seed (Figure 4a), the (001) face is second, and (01h 1) is third, while for the crosssection without seed (Figure 4b) contrariwise. Similarly, the (010) and (011) faces exchange places in this sequence. From this, it follows that the seed and its morphology may influence, to some degree, the final crystal habit, especially in a relatively short growth time. The seed influences, first of all, the faces for which the quantity Rcrit hkl - Rhkl takes negative or zero values. Such faces, depending on the seed morphology and growth time, exist in final crystal habit or not. It should be pointed out that the seed morphology is taken into account in eq 3 as the quantity l0hkl, the initial size of a given hkl face. This quantity represents the initial size of a given face, in seed or after some time of growth. However, we will not analyze in detail the influence of seed morphology on the final crystal habit, as this is not the main subject of this paper. From the above analysis, we are able to draw the conclusion that the (001) face is one of the most important crystal faces even though it grows faster than the neighboring (011) and (01h 1) faces. Such a behavior is determined by the crystallographic structure, i.e., by the interfacial angles R and γ. For another crystallographic structure, for example, for interfacial angles R and γ appropriate for the (01h 2 h ) face of KBC crystal, this face, growing more slowly than one of the neighboring faces, decreases its size. Therefore, we are able to put forward a proposal that the MI of crystal faces is not inversely proportional to their relative growth rates. 4. Conclusions From the above analysis, we are able to draw the conclusion that the relative size of a given crystal face and, consequently, the MI of this face, depend on the quantity Rcrit hkl - Rhkl, crystallographic structure represented by cot R + cot γ, growth time t, and the initial relative size of this face. For constant growth rates, the MI is proportional to the value of the product of the quantity Rcrit hkl - Rhkl and GF ) cot R + cot γ in unit growth time. The bigger the value of this product, the more important the corresponding crystal face. For interfacial angles R + γ close to π, the value of this

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product reaches the highest values. Therefore, in most cases, crystals are bound by faces, which produce such interfacial angles. Such faces need not be the slow growing faces, because even growing faster than the neighboring faces, they increase their sizes. The face situated between parallel faces is the limiting case. Such a face, even if it grows a few times faster than the neighboring faces, does not decrease its size. On the other hand, if a given face produces interfacial angles satisfying the condition 2R + γ < π or 2γ + R < π15 another interesting phenomenon occurs. Namely, a given face growing more slowly than one of the neighboring faces, the h1k1l1 face and the h2k2l2 face, respectively, may decrease its size and, in consequence, disappear. From this, it follows that the faces, which appear in final crystal morphologies, need not be, as it was thought so far, the slow growing faces. Therefore, the MI is not inversely proportional to the relative growth rates, and, based on the relative growth rates, we are not able to infer the MI. All these conclusions are drawn based on derived equations, not empirical ones. All our theoretical predictions are verified by computer experiments performed for the KBC crystal. It is shown that one of the most important faces of this crystal, the (001) face, growing faster than the neighboring faces, is still one of the largest faces. Additionally, it is shown that the (01 h2 h ) face, growing more slowly than one of the neighboring faces, decreases its size. Both these two facts confirm that based on the relative growth rates we cannot conclude the MI of crystal faces. References (1) Bravais, A. E Ä tudes Cristallographiques; Gauthier-Villard: Paris, 1913. (2) Friedel, G. Lec¸ on de Cristallographie, Hermann: Paris, 1911. (3) Donnay, J. D. H.; Harker, D. Am. Mineral. 1937, 22, 446467. (4) Hartman, P.; Perdok, W. G. Acta Crystallogr. 1955, 8, 4952. (5) Burton, W. K.; Cabrera, N.; Frank, F. C. Trans. R. Soc. London Ser. A 1951, 243, 299-358. (6) Bennema, P.; van der Eerden, J. P. In Morphology of Crystals; Sunagawa I., Ed.; Terra Scientific: Tokyo, 1987; Part A, pp 1-5. (7) Bennema, P. In Handbook of Crystal Growth; Hurle, D. T. J., Ed.; Elsevier: Amsterdam, 1993; Vol. 1A, Chapter 7, pp 477-500. (8) Grimbergen, R. F. P.; Meekes, H.; Bennema, P.; Strom, C. S.; Vogels, L. J. P. Acta Crystallogr. A 1998, 54, 491-500. (9) Meekes, H.; Bennema, P.; Grimbergen, R. F. P. Acta Crystallogr. A 1998, 54, 501-510. (10) Grimbergen, R. F. P.; Bennema, P.; Meekes, H. Acta Crystallogr. A 1999, 55, 84-94. (11) Abegg, C. F.; Stevens, J. D.; Larson, M. A. AIChE J. 1968, 14, 118-122. (12) Kozlovskii, M. I. Kristallografiya 1957, 2, 760-769. (13) Szurgot, M.; Prywer, J. Cryst. Res. Technol. 1991, 26, 147153. (14) Szurgot, M. Cryst. Res. Technol. 1991, 26, 555-562. (15) Prywer, J. J. Cryst. Growth 2001, 224, 134-144. (16) Prywer, J. J. Phys. Chem. Solid 2002, 63, 491-499. (17) Brandon, J. K.; Brown, I. D. Can. J. Chem. 1968, 46, 933-941.

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