195

J. Appl. Cryst. (1992). 25, 195--198

Phase Identification of Single Crystals Using a Self-consistent Indexing Scheme Based on the Laue Method

By JINSHENG PAN

Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People ~" Republic of China

(Received 11 February 1991; accepted 2 October 1991)

Abstract

The application of the Laue method is extended to phase identification of single crystals. This extension is made possible by a new method of indexing Laue patterns, which is based on computerized rotation of the reciprocal axes of the crystal and a new criterion for judging true solutions. The new method and cri- terion have been tested successfully for crystals with various structures, including triclinic. An example is given for the phase identification of molybdenum oxide which has six structural modifications.

Introduction

So far the conventional Laue method has been used mainly for the determination of the orientation of single crystals. It is seldom used for phase identifica- tion except for special cases where the incident X-ray beam can be made parallel to (100) (Amoros, Buerger & Canut, 1975). Even for orientation determinations, problems exist with the indexing methods. Current methods are based on matching the measured angles between the prominent (low-Miller-indexed) Laue spots and zones in the pattern with the calculated values between low-index planes and directions. Such 'angle-matching methods' have two serious draw- backs. Firstly, the matching procedure is usually com- plicated and requires a computer with a large memory capacity and high speed (see, for example, Huang, Christensen & Block, 1971: Christensen, Huang & Block, 1971; Anazia, Lee, Jerner, Christensen & Peavey, 1975; Cornelius, 1981). Secondly, the results may be unreliable. The procedure sometimes does not give a unique solution, depending on whether one has correctly preselected enough prominent spots and zones with low Miller indices from the pattern (Ploc, 1978).

To accelerate the analysis of Laue patterns and, in particular, to obtain reliable results, we earlier presented a new indexing procedure called the rotation method (Pan, 1982) and have since developed an associated criterion. The new method and criterion have been successfully used for the orientation

0021-8898/92/020195-04503.00

determination of a number of single crystals. Recently, we found that the new technique can be extended to the phase identification of unknown crystals. In this paper, the rotation method is briefly described and the new criterion for judging the correctness of the indexing scheme is discussed in detail. An example is given to show the application of the procedure to phase identification.

The rotation method In essence, the determination of the orientation of a single crystal requires the position of the crystal axes (the axes of the unit cell) within the specimen to be found and this can be achieved with the rotation method.t The outline of the method is as follows. First, the reciprocal-lattice constants a*, b*, c*, ~*,/~* and 7* are calculated from the estimated lattice constants a, b, c, ~, 13 and y of the assumed phase of the crystal. The actual positions of the reciprocal axes a*, b* and c* within the specimen are of course not known, but initially these axes may be set at an arbitrarily chosen position called the initial position, then successively rotated by computer until they reach the actual position. To simplify the rotation transformation matrix of the reciprocal coordinates of the gnomonic projections of the Laue spots, an orthogonal coordinate system, X YZ, is fixed to, and rotated with, the reciprocal system a*h*c*. For direct reading of the initial coordinates of the Laue spots, the initial position of the set of orthogonal axes is selected such that the Z axis is perpendicular to the plane of the Laue photograph while the Y axis is vertical as shown in Fig. 1. Fig. 1 also shows the orientation relationship of the two sets of axes, a*, h*, c* and X, Y, Z, which remains unchanged during rotation.

+ The details of this method have been deposited with the British Library Document Supply Centre as Supplementary Publication No. SUP 54674 (9 pp.). Copies may be obtained through The Technical Editor, International Union of Crystallography, 5 Abbey Square, Chester CHI 2HU, England.

~-~)~ 1992 International Union of Crystallography

196 PHASE IDENTIFICATION OF SINGLE CRYSTALS

With respect to the orthogonal axcs at their initial position, the Cartesian coordinates x, y, z of any Laue spot and X o, Yo, Zo of the gnomonic projection corresponding to that spot can be determined directly from the Laue pattern. The initial reciprocal coordinates, h o, k o, l o, of the gnomonic projection of the Laue spot, i.e. the coordinates referred to axes a*, b* and c* in the initial position, can be calculated from the relative orientation of the two sets of axes. When the axes have been rotated to their final (actual) position, the coordinates h o, k o and lo become h', k' and / ' . The apparent Miller indices h, k, l are then determined from h', k', r with the condit ion that they should be the smallest integers such that the following relation is satisfied within an allowed error E:

h:k: l~_h' :k ' :r . (1)

Here E is the calculated angle between (h'kT) and (hkl) for the assumed lattice constants.

The earlier work proposed a criterion to judge whether the axes had been rotated to their actual position within the crystal. The criterion was that at the actual position the selected control points have low Miller indices with min imum average error, i.e. ( ~ E)/n is minimal, where the summat ion is taken over the n control points. Here the so-called control points refer to the prominent crystal zones or the prominent Laue spots, or both, in the pattern which are presumed to have low Miller indices. By prominent zone we mean a zone which passes at least four Laue spots and by prominent spot we mean a spot which lies at the intersection of two or more prominent zones but does not necessarily have high intensity. Since it was found later that the indexing

, _ + X - r a y ~

(hkl)

~ r y s t a l

f i l m

Fig. I. The arbitrarily chosen initial position of the reciprocal-lattice axes a*, b* and c* and the orthogonal axes X, Y, Z. (P Laue spot, P' the gnomonic projection of P.)

scheme with min imum average error was not necessarily the true scheme, a new criterion was needed.

The new criterion

The new criterion for determining a true indexing scheme can be derived from the following arguments:

(1) The orientation of the crystal should be determined from the arrangement of all Laue spots rather than from several control points only. Therefore, a proper criterion should involve informa- tion from all Laue spots.

(2) The Miller indices and the errors associated with the Laue spots as defined previously are interrelated. Therefore, the new parameter for judging a true solution should combine Miller indices with error.

(3) There is no line of demarcat ion between ' h igh ' and ' low' Miller indices. It can be shown that the larger the axial ratio of the crystal, the higher the Miller indices of the Laue spots which can occur in the pattern. Therefore, the apparent Miller indices, (hkl), should be replaced by the weighted indices (hw k., lw), which are defined as

hw = (L,./a)h, kw = (Lm/b)k, lw = (Lm/c)l, (2)

where a, h and c are the lattice constants and L,. is the smallest of a, b and c.

Combin ing the arguments ment ioned above, a parameter, P, can be introduced which is defined as the mean product of the weighted index, M, and the error, E.

P = ( i = ~ I M i E i ) / N (3)

where N is the total number of Laue spots to be indexed and

M i = ([hwi [ + [kwi ] + Ilwi[)/3 (4)

The parameter P can be used as an indexing criterion such that

P = P~ (5)

for a true solution. This means that, for a true solution, the mean product of the weighted index and the error should be minimal.

| t must be emphasized that the true solution refers to a solution which gives not only the true Miller indices and crystal orientation but also the true lattice constants which form the basis for phase identification from the Laue pattern. The new criterion imposes no restriction on the number of control points, a l though three to four control points are usually desirable for reducing the comput ing time.

J I N S H E N G P A N 1 9 7

Table 1. Results from the analysis of a Laue pattern of an MOO:, single co'stal usin9 the rotation method

F o r m u l a S t ruc ture Lat t ice cons t an t s (A) P Surface

MOO2.8 T e t r a g o n a l a = 45.99 c = 3.937 2.61 (001) M o 4 0 ~ 1 O r t h o r h o m b i c a = 24.4 b = 5.45 c = 6.723 2.18 (100) Mo1~O47 O r t h o r h o m b i c a = 21.615 b = 19.632 c = 3.9515 2.18 (0011 M o O 3 O r t h o r h o m b i c a = 3.962 b = 13.858 c = 3.697 0.95 (010) M o 4 0 1 1 M o n o c l i n i c a = 24.54 b = 5.439 c = 6.701 2.29 (100)

/3 = 94.28 M o 9 0 2 6 Triclinic a = 8.145 b = 11.89 c = 19.66 2.23 (001t

= 95.47 [~ = 90.39 " = 109.97 M o O 3 H e x a g o n a l a = 10.531 c = 14.876 3.2 (10]-0t

To test the new criterion, both electron-diffraction and double-exposure methods were used. In the latter case, two Laue photographs were taken for the crystal in the same orientation, one with white radiation for searching out all possible indexing schemes, the other with monochromatic radiation for checking which scheme is in accordance with Bragg's equation, 2d sin 0 = n).. The crystals tested included Ta (b.c.c.), A1 (f.c.c.), Pt (f.c.c.), ZrC (f.c.c.), Sc (hexagonal), quartz (hexagonal), La2_:,SrxCuO4 (tetragonal), YBa2Cu3OT_x (orthorhombic), La2CuO,, (ortho- rhombic), sapphire (rhombohedral) and ZrO 2 (mono- clinic). Both transmission and back-reflection Laue patterns have been analyzed. All results yielded the minimum mean-product criterion.

Example A thin single crystal of molybdenum oxide is used as an example. The film was formed by heating pure molybdenum to 1173 K in oxygen. The chemical formula of the resultant film is MoOx with x varying from 2.75 to 3, corresponding to different structural modifications (see Table 1). The transmission Laue pattern is shown in Fig. 2(a). From the pattern only four prominent zones (n = 4) can be discerned and these are selected as control points as marked in Fig. 2(b), where the spots 1 and 2 constitute the first zone, 3 and 4 the second, 5 and 6 the third and 7 and 8 the fourth. Using the rotation method in connection with the minimum mean-product criterion [equations (2)-(5)], the pattern was indexed in turn for each possible structural modification with a small IBM personal computer. The input data are the estimated lattice constants and the Cartesian coordinates of 25 selected Laue spots (N = 25) referred to the orthogonal axes in the initial position (Fig. 1). The results are shown in Table 1 which clearly shows that the oxide is M o O 3 (minimum P value) with an orthorhombic structure and the surface of the film is (010). The electron diffraction pattern of this thin film is shown in Fig. 3 which is consistent with the results mentioned above.

(a)

~b) Fig. 2. (al The t r ansmiss ion Laue pa t te rn of the MOO.,. single crystal.

ib) The same pa t te rn with mark ings showing the con t ro l points and all the Laue spots to be indexed. The con t ro l po in ts here have four zones : the first zone passes t h r o u g h spots 1, 9, 10. 2 and 11: the second t h r o u g h 3, 15 and 4: the third t h r o u g h 5, 12. 13 and 6; the four th t h r o u g h 7, 23, 17 and 8.

198 PHASE I D E N T I F I C A T I O N O F S I N G L E CRYSTALS

Concluding remarks

A quantitative criterion has been presented for use with the rotation method for eliminating ambiguity from the indexing of Laue patterns. The new criterion has also made it practicable to identify the phase of a single crystal with several possible structural modifications by a small personal computer. This is not only due to the effectiveness of the new criterion, but also to the fact that the amount of calculation in

,J

l O • •

• I • • • D •

• • ' • •

o

the rotation method is less than that in the angle-matching method by many orders of magnitude. In fact, our computer code based on the rotation method is written in Basic and runs on a small IBM PC-compatible 286 computer (640K, 12 MHz, with- out 80287). The typical computing time required ranges from a few seconds to several minutes. In contrast, the codes based on angle-matching methods are mostly written in Fortran and require computers with a large memory and high speed [IBM 360, CDC Cyber 175 or CDC 6600 computers; see Anazia et al. (1975) and Ploc (1978)]. The computing times required are approximately the same.

The author is grateful to Mr J. Z. Zhang for supplying Laue and electron-diffraction photographs of the molybdenum oxide single-crystal thin film.

Fig. 3. The electron diffraction pattern of the crystal in the same orientation as in Fig. 2. (The camera constant is L2 = 13.72.)

R e f e r e n c e s

AMOROS. J. L., BUERGER. M. J. & CANUT, M. L. (1975). The Laue Method, ch. 7. New York: Academic Press.

ANAZIA. C., LEE. C., JERNER. R. C., CHRISTENSEN, J. H. & PEAVEY. J. H. (1975). Metall. Trans. A6, 1751 1753.

CHRISTENSEN, J. H., HUANG. W. H. & BLOCK, R. J. (19711. Metall. Trans. 2, 2295 2296.

CORNELIUS, C. A. (1981). Acta Cryst. A37, 430-436. HUANG. W. H., CHRISTENSEN, J. H. & BLOCK. R. J. (19711. Metall.

Trans. 2, 1367-1370. PAN. J. S. {1982). Acta Metall. Sin. 18, 703 712. (In Chinese.l PLOt, R. A. (1978). J. AppI. Crvsr. 11, 713 715.