304 S H O R T C O M M U N I C A T I O N S

formly intense focal spot was assumed.* The effective slit- length weighting function was computed from equation (2). The results are shown in Fig. 2. In addition, the values A~ W[ (u)/A~ rf are shown for each of the Soller slits. For comparison, the weighting function calculated for the same collimation system without the Soller slits is also shown. It should be pointed out that the decrease in the width of WTtf(u) is gained at the expense of a decrease in the detected power by the ratio A[rf/A~=O'199.

Although the theory presented here is rigorous within the approximations of the weighting fttnction theory, the question remains as to how well equation (2) can be made to represent the true weighting function. The most serious limitation is the assumption of a uniformly intense focal spot. It is well known that Io varies along the length of the focal spot. In the author's experience, such variations often do not affect the normalized weighting function seriously. However, they may have a drastic effect on the normaliza- tion constant. In the case of a collimation system without Soller slits the determination of A~ is part of the problem of absolute intensity calibration but is not necessary for relative intensity measurements. However, in the presence of Soller slits, accurate values of .47 must be determined either from theory or preferably from experiment even for relative measurements. The author knows of no Soller slit collimation system for which the complete experimental effective slit-length weighting function can be determined

* If necessary, the effects of focal spot non-uniformity may be readily taken into account in the calculation (Hendricks & Schmidt, 1967).

although with most current systems an experiment can be devised to measure the individual A~' 's.

The major benefit of Soller slits is a decrease in the width of the effective slit-length weighting function. However, it is concluded that this benefit is more than offset by the disadvantages of a loss in detected power and an increased uncertainty in the weighting function. These disadvantages must be reflected in a decreased precision of the desmeared intensity.

I am indebted to Drs J. van Es and J. Kao for providing drawings of their collimation systems and for their discus- sion, and to Drs L. B. Shaffer, C. J. Sparks, and H. L. Yakel for their comments on the manuscript.

References BUCHANAN, M. G. & HENDRICKS, R. W. (1971). J. AppL

Cryst. 4, 176. HENDRaCKS, R. W. (1972a). J. Appl. Cryst. In the press. HENDRICKS, R. W. (1972b). Program WEIGHT: A Fortran

IV Program for the Evaluation of Weighting Functions Used in Small-Angle X-ray Scattering, USAEC Report ORNL-TM-1950 (Rev. 2), Oak Ridge National Labor- atory, Oak Ridge, Tennessee.

HENDRICKS, R. W. & SCHMIDT, P. W. (1967). Acta Phys. Austr. 26, 97.

HENDRICKS, R. W. & SCHMIDT, P. W. (1972). Submitted to Acta Phys. Austr.

SOLLER, W. (1924). Phys. Rev. 24, 158. STOECKER, W. C. & STARBUCK, J. W. (1965). Rev. Sci.

Instrum. 36, 1593.

J. Appl. Cryst. (1972). 5, 304

Consideration of the optimum precession angle for upper-level precession photographs. By J. H. Cgoss* and R. H. FENN, Physics Department, Portsmouth Polytechnic, Park Road, Portsmouth, England and A. J. GRAHAM, De- partment of Mathematics & Physics, Queensland Institute of Technology (Darling Downs), Darling Heights, Toowomba, Queensland 4350, Australia

(Received 11 December 1971; accepted 31 March 1972)

An expression is derived for the precession angle #, such that the circle of reciprocal space projected on to the film touches the edge of the film. This optimizes the exposure time for the number of reflexions collected since the average intensity decreases with increasing p.. Graphs are shown from which # and the screen setting position can be easily determined for the Supper and Stoe precession cameras.

Introduction

In precession photography it is convenient to choose a precession angle, /2, such that the area of the reciprocal lattice projected on to the film makes best use of the film area available. If the circumference of the resulting pro- jection of the net plane touches the edge of the film then we have an optimum exposure time. On the Stoe camera, with its circular film, there is no point in having the dia- meter of the region larger than that of the film. On the Supper camera, which has a square film, the exposure time

* Present address: Department of Science and Electrotech- nology, Guildford County Technical College, Stoke:~Park, • Guildford, Surrey, England.

must be increased substantially to collect a few extra reflexions in the corners of the film. On upper levels the diameter of the region projected will get larger with in- creasing distance from the zero layer if the same value of /~ is used for all levels. Thus a relation is required connecting /2 and (, the axial reciprocal lattice coordinate, such that the diameter of the region collected stays constant.

.. Theory Consider photography of the nth level reciprocal lattice zone, having an axial cylindrical parameter ~.. Using Buer- ger's (1964) notation

. . - . . , ..,. :

~'.= cos # . - cos ~. (1)

S H O R T C O M M U N I C A T I O N S 305

and the maximum radial cylindrical coordinate ~,, is given by

~m = sin/2, + sin ~, .

On the zero layer/20 = ~0 and ¢,n = 2 sin/20. If ~,, is to be the same for all levels, then

2 sin/20 = sin/2, + sin 17,. (2)

Both equations (1) and (2) must be satisfied for the nth layer. The solutions of these two equations are:

sin/2, = s in /20- A, and sin ~, = sin/20 + A, (3) where (n[COS2/20--((n/2)2] 112

An = 2 S in2/2o + ( (n /2) 2 "

70

60

50

40

30

20

10

p.,v.(°) F=50

~ . - lao

0'1 0'2 0"3 0'4 0'5 0'6

(a)

70 la°'v"(°) F=60 o o

50

40

r 0'1 0-2 0'3 0"4 0"5 0"6 b

(b)

Fig. 1. # and ~, obtained from equations (3), plotted against ~, for various values of F(mm). (a) The Supper camera (b) The Stoe camera.

S(mm)

80

70

60

50

40

3O

20

10 r=15 20 25

I t i I i •

20 30 40 50 60 70

v (*)

Fig. 2. A graph of s(=r/tan p), against ~ for r = 15, 20 and 25 mm.

The value of/20 depends upon the crystal-to-film distance, according to the equations ~m = 2 sin/20 and R = F~,,, where R is the radius of the projected area on the film. Table 1 gives the values of/20 calculated for common values of F.

Table 1. Values of#o for common values o f f

F(mm) #0 (Supper)* #0 (St6e)t 50 38 °50' - 60 31 o 30' 30 ° 75 24 ° 32' 23 ° 35'

100 18 ° 16' 17028 , 125 - 13053 ,

* Based~on ~4~--~" square film. t Based on 120 mm diameter film.

Fig. 1 shows the curves of equation (3) for /2, and ~, versus ~, at the appropriate values of/20 for the two camera types.

The position, s, of the annular screen of radius r is given by s=r/tan ~, (Buerger, 1964), and Fig. 2 shows the graph of this function for r = 15, 20 and 25 mm.

With the sets of curves in the figures it is thus an easy matter to select the opt imum/2, together with the required annular screen position.

Reference

BUERGER, M. J. (1964). The Precession Method in X-ray Crystallography. New York" John Wiley.