Rhombohedral-hexagonal transformation

The triiodides of La to Nd have an orthorhombic structure (space group Ccmm), the triiodides of Sm to Lu have a hexagonal/rhombohedral structure (space group A3). High temperature transformations have been reported for the lanthanide triiodides from Ndfo to Dyfo. The orthorhombic NdL probably transforms into the rhombohedral structure (Dworkin and Bredig,... 

The cubic 2inc blende form of boron nitride is usually prepared from the hexagonal or rhombohedral form at high (4—6 GPa (40—60 kbar)) pressures and temperatures (1400—1700°C). The reaction is accelerated by lithium or alkaline-earth nitrides or amides, which are the best catalysts, and form intermediate Hquid compounds with BN, which are molten under synthesis conditions (11,16). Many other substances can aid the transformation. At higher pressures (6—13 GPa) the cubic or wurt2itic forms are obtained without catalysts (17). 

The initial interlayer anion also plays a strong role in determining the pathway of the reaction. Neither the hexagonal nor rhombohedral forms of LiAl - NO3 exhibit staging the alpha vs. time curves cross at o 0.5, strongly suggesting a direct transformation from host to product. This is illustrated in Fig. 13b,c. 

For rhombohedral unit cells, it is best to transform indices to hexagonal indices, and to use the above formula for the spacings of planes. 

Suppose we have rhombohedral indices hR kR lR and wish to transform them to hexagonal indices hH IcH iHlH. The c axis of the hexagonal cell is 00 (through the middle of the rhombohedron). Pass from O to O, first of all directly the phase-difference between waves from O and those from O js In. Now go from O to O by way of rhombohedral axial directions-—for instance, via OD, DK, and KO. Waves from D are hR wavelengths ahead of those from 0, those from K are k.R wavelengths ahead of those from D, and those from O are lR wavelengths ahead of those from K. The total is h R- rk R -l r- Thus lH = hRA-kRA lR ... 

Consider now the reverse transformation, from hexagonal to rhombohedral indices. It is necessary to go from the origin to some point, first by way of rhombohedral axes and fhen by way of hexagonal axes. Such a point is M if we go. there directly, the distance is 36, and thus the phase-difference between waves from O and Mf is 3fcjj. Now go from O to M by way of hexagonal axes—for instance, via B and M. From OtoB the phase-difference is Jcf, from B to M it is —hH (since we are going in the negative aH direction), and from M to M it is lH the total is kH—ha+ln. Thus %kR -- kH—hH+lH. 

The atomic layers from c-BN (ABCABC) have to rearrange into an ABAB stacking sequence of h-BN during the solid state phase conversion. A possible mechanism would be the intermediate formation of the rhombohedral BN phase (r-BN) with ABCABC stacking. The r-BN phase is structurally related to the hexagonal phase, but only differs in the d-values (h-BN d = 6.66 A r-BN d = 10.0 A) of the layers (Fig. 5b). Subsequently the rhombohedral phase is transformed into the hexagonal modification at the reaction temperatures [10]. 

Here we use hexagonal axes description of the R3c space group. The transformation to monoclinic and rhombohedral axes can be found in Ref. [5]. 

Silicon carbide is covalently bonded with a structure similar to that of diamond. There are two basic structures. One is a cubic form, /i-SiC which transforms irreversibly at about 2000 °C to one of a large number of hexagonal polytypes, and the other is a rhombohedral form also with many polytypes. Both the hexagonal and rhombohedral forms are commonly referred to as a-SiC. 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

The pentavalent oxide, Pa205, is obtained when the hydroxide is heated in air or oxygen above 650°C. It is a stoichiometric phase and is useful in analytical procedures. Recent investigations (118, 128, 129) have shown that several crystal modifications can be prepared, depending on the temperature to which the pentoxide is heated, but the existence of the orthorhombic phase reported by Sellers et al. (125) and also by Kirby (99) has not been confirmed. The properties of the various crystal modifications are listed in Table XII. It has been suggested that the low temperature cubic modification may, in fact, be the tetragonal phase T3 in a poorly crystalline state (118). Above 1000°Cthe T3 phase transforms to the hexagonal phase and a rhombohedral phase (R2) has been identified above 1240°C. 

Crystals in the rhombohedral (trigonal) system can be referred to either a rhombohedral or a hexagonal lattice. Appendix 4 gives the relation between these two lattices and the transformation equations which allow the Miller indices of a plane (see Sec. 2-6) to be expressed in terms of either set of axes. 

When the pattern of a rhombohedral substance has been so indexed, i.e., with reference to hexagonal axes, and the true nature of the lattice determined, we usually want to know the indices hkl) of the reflecting planes when referred to rhombohedral axes. The transformation equations are... 

Further information on the rhombohedral-hexagonal relationship and on unit cell transformations in general may be obtained from the International Tables for X-Ray Crystallography [G.l l], Vol. 1, pp. 15-21. 

---