2 3.2 CELL PARAMETERS AND SPACE GROUP DETERMINATION

There are several methods of evaluating the unit cell parameters J

(a,b,c, ÔC , ^ I and Y), ₄

(a) The Oscillation Method (Buerguer,1966;{5l88jWoolfson,1970,pi34jMcKie

and McKieiL974,jj2l+) permits determination of one dimension for the layer ï spacingsî e.g. using x-rays of wavelength, X ,

Bin tan'^ (-i)

A

Where is the index of the layer-line considered. Y is the

separation in mm between corresponding layer-lines and R is the radius of the camera (usually 2R=57«296 mm).

The process of determining the remaining cell parameters from an oscillation photograph is tedious (Mckie and Mckie, 1974, p.226; Glasser, 1977» p.90).

(b) The Weissenber# Method

This is a more convenient method (Woolfson,19TO^i4-i;Jeffery, I97I, p.l88f ), both in terms of assigning indices to the plane generating a particular reflection and for determining cell parameters. There is the additional advantage that it enables one to collect information from one complete layer at one time without the overlap difficulty inherent in a simple oscillation procedure, other layers are screened off.

A typical zero-layer weissenberg photograph obtained by synchronous movement of film and crystal oscillation is shown in fig, 2.3*4a.

Mo .

(a) Zero-layer Weissenberg photograph, (^) Diagram of the b aocis on the zero-layer Weissenberg photograph. Fig 2.3 .4

I

Some cell parameters can be calculated directly from\ero-layer 4 Weissenberg photograph (Buerguer, 1964, p,377ff» Jeffery, 1971» p.264). |

From figs. 2.3.4a,b„ where y ’ is in mm. 4

(2.3'3&)

,1 d^io = kb* = 2 sinO = 2 sin (y'/2R)

Since Y‘* - s' / \fS and usually 2R = ^7.296 mm. r

^ b ~ ^OkO ~ ^ (2.3*3)

where s*/ \/~^ is in degrees , and y'/2R in radians.

The reciprocal angle can be calculated from the relation;

Where D is the separation (in mm) between the two reciprocal axes given the construction that tan = 2.

So far only c,'f^, and / have been determined. To evaluate the

remaining constants one must find ocand p from an upper layer equi- 1

,

inclination photograph about the same axis using the method of "angular

1

The basis of the "angular lag" calculation is the location of the

upper-layer origin with respect to the rotation axis. This is defined j in terms of two shifts &a and

Sb,

^ b

j

1

superposed Weissenberg films of the zero- and first layers. Direct cell j angles then can be calculated from;

~ tan oC = — or -tan 6 - — (2.3*4) j

57

'f . / X

{

Where is evaluated from an oscillation photograph (about c). This procedure suffers from two practical disadvantages. It requires two precisely superposed Weissenberg films of the zero and first layers and the measurements are made on low-index reflections

which may be weak and unobservable.

A second approach has been developed by Hulme (I966). This treatment involves the measurement of angles between general hkl

reflections, which complemented by other constants obtained from oscillaton

and zero-layer Weissenberg photograph permits calculation of the rest of j the cell constants either graphically or analytically using a least-square

computer program. The graphical procedure is illustrated below (Fig. 2.3.5).

This procedure is free from the disadvantages present in the Buerguer *s ? method and is still useful when split-film casettes are employed in low

temperature work. Accuracy is best using reflections with 9 ^ 20° having interangles in the 60-120° range.

Aïiother procedure has been described by Herbert (I978). Here measure­ ments are made on festoons representing either axes of the reciprocal

lattice or lattice lines runing parallel to these axes.

For a crystal rotating about the b axis, ^ is calculated from:

% = 0 0 8 (2.3.5)

c,o

where is the equi-inclination angle of the nth layer, and ^ is the distance between the symmetrically distributed points s^ and s^' in axis as is shown in Fig. 2.3*6.

When a reciprocal-lattice line parallel to the axes is used, e.g. *2» ^0 0 obtained from;

where c* and p* can he obtained from the zero-layer Weissenberg

photograph. Then cell angles can be calculated from | n

- tan ^ ~ , etc. ^c,o

^ is calculated from an oscillation photograph. | This procedure has two main disadvantages. It is sensitive.to errors

when festoon extrapolation has to be performed, especially if the reciprocal spacing are large: and, this method is not applicable when a split-film casette is employed.

An oscillation, a zero Weissenberg and an n-layer Weissenberg photograph thus yield c, | y* and o< , ^ .

The cell Parameter a = ^ sin p sin V* 2.3*?a b = sin V*sin c>c 2.3.7b and )(= cos ^ (cos «.cos P - sin « sin p cos X* ) 2.3.7c so that all parameters have been determined.

Since ? and '5 are proportional to s' (in Weissenberg photograph) and y (in oscillation photograph) respectively, better accuracy may be

obtained by measurement of the highest observable reflection in the axis or highest layer-line distance respectively. However factors such as the divergence of the incident x-ray beam and the progressive reduction in angle between the diffracted beam and the surface of the film as the reflections increase the use of the highest reflections does not

make much difference. In general measurement of layer-line spacing in oscillation photograph or from a Weissenberg photograph yield unit cell dimensions of accuracy about 1% and angles with an error of about 0.5°.

Fig 2.3.5

Angular lag method for calculation of cell parameters; Graphical- evaluation of and , and hence c< and P , for

(Data for SbCl^.l-chloronaphthalene compound,tri­ clinic setting).

the following pair of reflections : _-3

111 - 111 : 91.50" é

lTl 121 : 83.50" _Si"2

Fig 2,3,6 Schematic drawing of an upper-layer equi-inclination Weissenberg photograph about b.

(c) A Method Without Reciprocal Lattice Distortion

An undistorted picture of the reciprocal lattice (RL) may be obtained using two methods (DeJong-Boiimann, 1938; Buerguer, 1944), The common feature of these methods is the mechanical linkage coupling the movement of a flat film casette and the rotation of the arcs

carrying the crystal (the "dial" axis) which is measured on a dial. Such coupling fulfl3-s the requirement for recording an undistorted reciprocal lattice i,e, A principal axis of the crystal is always kept perpendicular to the film, so that the film remains parallel to a set of RL layers during movement.

When a layer is to be photographed, unwanted reflections from other layers of the RL are excluded by positioning of a layer screen having an annular slit such that only reflections from the cone subtended at the crystal by the chosen layer are allowed to reach the film.

The scale of the undistorted picture of the RL depends on the crystal-to-film distance, M, so that typically 1cm = 1 r.l. unit.

The basic feature of the De Jong-Boumann camera is that the relative position of crystal, screen and the rotation axis of the film are unchanged (see fig. 2.3*6a). The upper-layer is brought into the recording position by altering the angle of incidence of the x-ray beam, and the position

I

a

I

fi:

of the film.

In the precession method, the crystal is set initially with a principal axis parallel to the x-ray beam. This axis is then tilted (to be perpendicular to the film) to make an angle /c with the x-ray beam and caused to process about it, so that the axis travels around the beam on the surface of a cone of semi-vertical angle . The layer screen attached to the arcs and the film similarly inclined, follow

the motion of the crystal axis (fig. 2.3,6b).

Upper layers are recorded by changing the position (and sometimes the size) of the layer screen and the film position (Buerguer, 1964^p76).

OF K L

ttOrATlON AXU 9F CKV&TAkLOtRA- AHtC ÀFIS y

Fig 2.3.6a

The de Jong-Boumann camera for zero-layer (After Glasser, 1977 » P ? 8 ) Fig 2.3.6b Arrangement of a Precession Camera

J

I

Evaluation of Lattice Parameters (Buerguer, 1964,p8$ff).

From a photograph obtained rotating the crystal about the c-axis, af , b^ and can be determined by measurement of the row spacing and the angle between them, respectively.

In a similar way from a second photograph obtained rotating about the a-axis, b’** , c*** and c<^ can be found.

The difference between the corresponding dial settings is the angle p . Hie rest of the parameters can be evaluated from the following equations,

cos ^ (COS o^cos sin(X* sin V* cos p ) (2.3*8)

1

cos ^ = (cos X* cos - COS 26'"' )/(sin sin p* ) (2.3*10) a = ^ /(a* sin p sin ) ^ b = ^/(b* sin(Xsin (2.3.11) c = ^/(c* sin c4 sin ) (2.3*12)

(d) Accurate Determination Of Cell Constants Using An X-Ray Diffractometer Best accuracy in parameters evaluation by any of the above

procedures requires appropriate choice of 0 range (or reflections), and is usually improved by subjecting some dozen independent measurements to a Least Square treatment. Such an approach is particularly relevant

'.j

when evaluating cell parameters using a diffractometer (Jeffery, 1971» ■! P.I7O; Manual of Instruction for 2-circle x-ray diffractometer .STADI 2). j

B.3.3 TUB .SPAa& (3Ü%n>

In any object there may be some points which are related to others by rotation of the object about an axis or its reflection in a plane. A particular combination of such symmetry elements, each acting at the

same point is knovm as the point group. In space, an object may be # related to others by additional translational symmetry elements (screw

axes and glide planes), which are not restricted to passing through i J a single point. There are 230 possible combinations of symmetry elements *

i

involving translations (Vol. I of International Tables for x-ray Crystallography,

1952)»

as

Translational symmetry elements are revealed by the ’systematic absences’ caused in the diffraction pattern by destructive interference. Recognition of these absences often leads to complete determination of the space-group which describes the intra spatial relationship of atoms or molecules. In some cases ambiguity may arise depending, for example, upon whether or not there is a centre of symmetry. Thus the systematic absences for Cc and 02/o are the same. Some physical

features (pyro and piezo electricity) however occur only in non-centro- symmetric crystals i.e. in space group Cc. Statistical examination of reflection intensities (Howells, S.R., et.al*, 1950» Lipson and Cochran,1966 p.46) or packing consideration may also help to discriminate between

possible alternatives.