R A P I D A L I G N M E N T O F C R Y S T A L S I N X - R A Y C A M E R A S

Br S. RAMASWAMr Am~ S. RAMAS~SHA~, F.A.Sc. (From the Delmrtmem of Physics, Indian Institute of Teelmology, Madras-36, India)

Rcccivcd May 25, 1964

1. INTRODUCTION

THE afigrmaent of erystals on the Weissenberg Goniometer in the quiekest possible time would always be an advantage. Ir is espeeially so in low tem- perature erystallography in laboratories with an inadequate supply of liquid air. Therefore, although the question of alignment has been repeatedly studied, it was re-examined with a view to ¡ if possible, routine proee- dures, that will eonsiderably eeonomjse time. The paper presents some ~ew results.

2. THE EXaSTING METHODS

When the alignment ]s perfeet, i .e., a erystallographie axis coincides exaetly with the a~xis of rotation, the zero layer spots ate reeorded on the equator in a eylindrieal camera. The spots lŸ on a straight line on the spread out film. II" the erystallographie axis is slightly inelined to the axis of rotation the spots do not lŸ on the equator. If the displaeements of the reeorded spots from the equator ate measured the errors in alignment can be ealeulated. When the goniometer head is set sueh that the short ate is horizontal with its seale faeing upwards, the long ate is vertical and its seale faees the observer. This is defined as the "parallel-perpendieulax setting ". For ah error ir in the vertical plane alone, the displaeement dl from the equator of a reflee- tion whose Bragg angle is 0 is given by

dl = R sin 20 sin iv (1)

where R is the radius of the eylindrical camera. For an error iH in the hori- zontal plane alone, the displaeement of this refleetion from the equator is

d~ = R (1 -- cos 2�91 sin iH. (2)

Ir both errors axe present, the resultant displaeement is

D ----- d~ + d~ -= R [sin 2�91 sin ir + (1 -- cos 2�91 sin iH]. (3)

These equations were first given by Hendershot (1937).

~,4 115

116 S. RAMASWAMY AND S. RAMASESHAN

In the parallel-perpendicular setting the errors in the arcs will also be ir and iH in the vertical and horizontal arcs respectively.

and

From thc above equations we get

at 0 = q- 45 ~ Dx = R [sin la q- sin iv]

at 0 = -- 45 ~ D2 = R [sin iH -- sin iv].

Thus by making measurements of the displacements D1 and D~ at 0 ~ 45 ~ and - - 4 5 ~ respectively, the errors can be easily computed. Since Bragg reflections may not exist at these specifie angles, unfiltered radiation is used. Ii" two unequal exposures with the crysta! rotated exactly through 180 ~ are given on the same film, the distances at O = 4- 45 ~ between the same reflection in the two traces give 2D1 and 2D2 from w•ich the errors are calculated [Fig. 1 (a), (b), and (c)]. This procedure obviates the difficulty of locating the trace of the ideal central line [Weisz and Cole (1948)].

In a modification of this method Davis (1950) has taken the double exposure with goniometer head turned through 45 ~ in the anticlockwise diree- tioa from the parallel-perpendicular setª [Figs. 1 (A), (B) and (C)]. This we shall call the "45 ~ set t ing"

In this setting let the errors in the vertical and horizontal planes be ir' and in'- The separation is now given by

2D = 2R [sin iH' (I -- COS 20) q- sin 20 sin iv']

ir ' and in' can be related to ir and iH, the actual errors in the two arcs by

[ s i n i v ' ] [ c o s 4 5 - s i n 4 5 ] [ siniv ] .

sin iH' [ s in 45 COS 45 sin in

Substituting this we get for the separation

2D ----- v '2R [sin 20 (sin ir -- sin in) + (1 -- cos 20) (sin ir + sin in)l-

Ar 0 = + 45 ~ this reduces to 21)1 = 2v '2R sin ir

and at 0 = -- 45 ~ this becomes 2D~ = 2v '2R sin in.

Thus the errors may directly be calculated from the measurement of the separations 2D at 0 = 4- 45 ~ thereby eliminating any furt•er additions ertd subtractions as in the previous method [Fig. 1 (C)].

Rapid .41ignment of Crystals in X-my Cameras 117

In most o f the cases the spots are not obtained at these exact angles 0 ----- -4- 45 o. Garayeoehea and Dresdner (1961) suggested that in sueh cases the Hendershot formula can be recast as

sin ir sin in - 2D + 2D ~ 1" (4) 2R sin 20 2R (1 -- cos 20)

Treating sŸ ir and sin in as variables along the eoordinate axes straight-lines may be drawn, one for eaeh dŸ spot. The intercepts along the eo-ordi- nate axes can be ealculated for eaeh spot from its 2D and 0 values. The different straight lines a.re expeeted to interseet at the same point and the oo-ordinates of the point of interseetion of these straight Unes give the errors

~rge/ i ~,,,, . 6 r

I I

t

+3d I ! w

Q, , i

i

I I

I | /' I ,

t

t I t

-gd

I I

I I I r

V ! ii

i i I I i i i r i r

| n

i i

i i i

A

~ L [ F T I R I � 9 1

i i ! ! ! !

| I , .

, , I : i i i i i

tf "l i ~ u

, , ;,, I I |

" /ii I I t

I I

" ] t,d IP-- i i

k: I'l, b B e

C FIo. 1. (a), (b) and (e) represent typical misalJgned pictures for the parallel-perpendicular

setting of the arcs. 1 (A), (B) and (C) represent the corresponding cases in the 45 ~ setting of a~cs. (a) and (A): Traces ob/ained when there is misalignment in the horizontal aro only

(ir = 0). It may be noted that- the curve in 1 (a) is always a (1 - c o s 2 0) curve and in I ( A ) 2 D x = 0 a t 0 = + 4 5 ~ .

(b) and (B): Traces obtained when the misaligarnent is in the vertical aro (in ~ 0). 1 (b) is always a sine r and in 1 (B) 2 Dt = 0 at 0 = -- 45 o.

(c)and ((2): Traces obtained when the misalignment is in both the aros. The sign eonvention is a/so iIlustrated in 1 (CO.

of the two arcs. Davis (1961) has suggested that this modification o f Garaycoehea et al. may be applied to his method also, provided the inter- eepts are marked along the two lines sin ir = sin in and sin ir = -- sin in taking propr sigas.

118 S. RA~ASWAMY AND S. RAMAS~SHAN

3. THE NEW METHOD

For the parallel-perpendieular setting, treating 2D and sin20 as variable eo-ordinates, the Hendershot equation is reeast as

X a (a ~ + b ~) + Y~ -- 2bXY + 2abX -- 2aY ----- 0. (5)

This is an equation to an ellipse passing through the origin. Here X ---- sin 20; Y ---- 2D; a ---- 2R sin in and b ----- 2R sin ir. In the case of a erystal which is misaligned in both the aros the plot of 2D against sin 20 is ah ellipse having its axes inelined to the axes of eo-ordinates. When in = 0, the graph is a straight line inclined to the eo-ordinate axes a n d a s in inereases the straight line opens out into ah ellipse, and when iv = 0 the ellipse has its axes parallel to the axes of co-ordinates. In every case the eurve passes through the origin and if one has three or four points (preferably on either side of the central spot) a good ellipse can be drawn. Reading the values of the displacements eorresponding to sin 20 ---- + 1 from the graph one gets 2D~ = a + b and 2D2 = a -- b from whieh the errors in the aros can be calculated to a fair degree of aecuracy.

Ir is found that this method is applieable also in the 45 ~ setting used by Davis. The plot o f 2D against sin 20 again givcs an ellipse

X 2 (A ~ + B ~) + Y~ -- 2AY -- 2BYX + 2ABX = 0

passing through the origin. But here A = v ' 2 R (sin ir + sin in) and B = v ' 2 R (sin iv -- sin/ra). The advantage in working with the 45 ~ setting of the ares is that the calculation of the errors of the ares is much simplified. For, ir one measures the values of 2D at 0 = • 45 ~ one gets

at 0 = + 45 ~ 2D1 = 2 v ' 2 R sin ir

and at 0 = -- 45 ~ 2D2 = 2x/2R sin ira-

It can also be seen that when iH = 0 (in' v e 0) the displaeement is zero at 0 = - 45 ~ and when ir = 0 ( i r ' v e 0 ) the displaeement is zero at 0 = + 45 ~ and the ellipse touehes the axes at these points. When both the errors are equal the values of 2D at 0 = + 45 ~ and -- 45 ~ are also equal.

4. ROUTINE EXPERIMENTAL PROCEDURE

The goniometer head may be in the parallel-perpendicular setting or the 45 ~ setting. A 15 ~ oseillation picture is taken giving an exposure of 30 minutes. The erystal is rotated exaetly through 180 ~ from the lirst mean position a n d a seeond exposure is given for 10 minutes for the same

Rapid Alignment of Crystals in X-ray Cameras 119

oscillation range. As soon as the film is removed from the camera a small (tot is made at the right hand top comer of the film. I f the diameter of the camera is 57.3 mm. then the distance in mm. of any reflection from the centra l ' spot gives the Bragg angle 0 in degrees. A transparent scale graduated in mm. with zero at the centre may conveniently be used for the measurement of the Bragg angles. A low power microscope with a graticule attached to the eye-piece may be usr to measure the distance 2D directly. The following sign convention is adopted. 0 is + ve when the spot is in the upper half and -- ve when it is in the lower hall. 2D is positive if the darker spot is on the right half and -- ve if it is on the l ~ t half. [2D1 is + ve and 2D2 is -- ve in Fig. 1 C].

The graph is drawn with 2 D and sin 20 values for three or four spots on either side of the centre and the errors are determine& Having calculated the errors in the setting it is convenient to make the necessary corrections

. . . . : " ~ " ~ I ~ S,W 18,

2O

1

Fio. 2 l~o. 3

Fios. 2 and 3 refer to case 1 of the tabulation given above. Fig. 2 gives the ellipse method in the parallel perpendicular setting and Fig. 3 in the 45 ~ setting of the arcs. 2 D values ate taken from the graph at sin 2 0 = + 1 and -- I.

in the arcs with the goniometer in the paxallel-perpendicular setting. Ir the calculated error ir along the long arc is positive then for the correction, the tip of the crystal should more down; Ir the error ir is negative, the tip of the crystal should move up. If the calculated error i~ along the short arc is positive the tip of the crystal should move away from ttte beato; if the error in is negative, the tip should be moved towards the beato. When spots ate not obtained near and above the value of 0 == d: 45 ~ it is quite difficult to determine the angular errors accurately by the earlier wo m ethods, The straight li he method suggested by G a m y c ~ et al.,

120 S. RA.MA$WAMY AND S. RAMASESHAN

and the ellipse me thod are useful even i f there a re on ly three or

our spots on e i ther side o f the central spot .

The wet fi lm m a y i tself be m o u n t e d on the t r ansparen t scale and measure-

ments made . T h e g raph is also d rawn wi thou t any fur ther calculat ions. The

t ime t aken to de t e rmine the errors by this me thod is found to be muela less than by the o the r me thods .

The errors calculated by the four me thods in five ac tual and different

tases in this l a b o r a t o r y are tabula ted be low:

TABLE I

No.

Ellipse method Straight line method

Parallel- Parallel- perpendi- 45 ~ perpendi- 45 ~ Experi-

cular setting eular setting mental setting setting

Case 1 . . ir 3 ~ 38' 3 ~ 30' 3 ~ 42' 3 ~ 44' 3 ~ 40' i , lO 21' 10 8' lO 30' lO 52' lO 20'

Case 2 . . i,, 2 ~ 34' 2 ~ 30" 2 ~ 39' 2 ~ 36 2 ~ 30' i . - - 37' - - 38' - - 48' - - 18' - - 40'

Case 3 . . ir lO 42' 10 48' lO 41' lO 43' lO 40' i . 48 46 51 51 50

Case 4 . . ir - - lO 54' - - lO 51' lO 12' - - lO 53' - - lO 50' i , 12' 25' 30' 13' 10'

Case 5 . . ir - - 15' - - 25 ' - - 24' - - 34' - - 20' i . - - 4 ~ 15' - - 4 ~ 36' - - 4 ~ 12' -- 4 ~ 36' -- 4 ~ 40'

5. SUMMARY

The misa l ignment in the setting o f a crys ta! in the Weissenberg gon iomete r

is de te rmined f r o m the m e a s u r e m e n t o f the d i sp laeement o f the spots f r o m the central l i n e a t 0 = -4- 45 ~ with the aros set in the para l le l -perpendicular

posi t ion (Hendersho t , 1937) or in the 45 ~ posi t ion (Davis~ 1950). When refloe-

t ions are no t p resen t a t 0 = • 45 ~ a g raphica l s t ra ight li;,e m e t h o d has been

8uggested by G a r a y c o c h e a et al. (1961).

Rapid ,4lignment of Crystals in X-ray Cameras 121

Another graphical method called the "EUipse method" suggested in the paper does not involve too muela calculation after the measurement and henee is easy in application, aecurate in the determination of errors and economical in time.

6. ACKNOWLEDGEMENT

The aut,hors wish to record their grateful thanks to Dr. S. Swaminathan and Mr. Anil Kumar Singh of the Physics Department, Indian Institute of Technology, Madras, for the many useful diseussions they had with them. This work was supported by the Council of Scientific and Industrial Research.

7. REFERENCES

1. Hendershot, O. P.

2. Weisz and Cole

3. Davis, P. T.

4. Isbel Garaycochca e t al .

5. Davis, P. T.

. . R e y . o f S c L h ~ t . , 1937, 8, 436.

. . J . S c L I n s t . , 1948, 25, 213.

. . l b i d . , 1950, 27, 330.

. . A c t a C r y s t . , 1961, 14, 206.

. . l b i d . , 1961, 14, 1295.