The following points highlight the three main methods for determination of crystal structure of materials. The methods are: 1. Laue Spot Method 2. Rotating Crystal Method 3. Powder Method.
1. Laue Spot Method:
In this method, a single crystal specimen is held stationary in a beam of X-rays of continuous wavelength. The crystal selects out and diffracts the discrete values of λ for which the planes of spacing d and incident angle θ satisfying the Bragg law. A source producing a beam of X-rays over a wide range of wavelengths, preferably from 0.2 to 2 Å is used in conjunction with a pinhole arrangement to produce a well collimated beam. The dimensions of the crystal are usually less than 1 mm.
A flat photographic film is placed to receive either the transmitted diffracted or reflected diffracted beam. The diffraction pattern consists of a series of spots (images of beams) and must show the symmetry of the crystal in the orientation used; thus, if a crystal with four-fold axial symmetry is oriented with the axis parallel to the beam, then the Laue pattern will show the four-fold symmetry. This feature makes the Laue pattern particularly convenient for checking the orientation of the crystals in solid state experiments, or in other words, to orient them suitably.
If the photographic film is at a distance R from the crystal, it will show reflection spots at various distances S from the direct beam. Here,
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S = R tan 2θ
Each spot will be due to all the order of reflection n = 1, 2, 3 … from a single plane. Thus each spot can be considered to be a reflection from planes of spacing d, d/2, d/3 … The Laue photograph gives a series of values of θ for different crystal planes, together with the orientation of each S relative to horizontal and vertical directions on the photograph. The Laue method gives only the symmetry, the axial ratios a/b, c/b and the axial angles α, β and .
This method is, however, not suitable for determining the crystal structure. This is because out of a continuous range of wavelength several wavelengths are reflected in different orders from a single plane, so that different orders of reflection may overlap on a single spot.
This makes difficult the measurement of the reflection intensity of individual spots and hence of the missing reflections. The later would have been an important step in the determination of the crystal structure. The shape of the unit cell can, however, is established from the symmetry of the pattern.
2. Rotating Crystal Method:
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In this method, a single crystal is rotated about a fixed axis in a beam of monochromatic X-rays. The rotation brings different atomic planes into position for Bragg reflection. An X-ray beam made nearly monochromatic by a filter or by reflection from an earlier crystal is used to irradiate a single crystal specimen mounted on a rotating spindle.
It is customary to rotate the crystal about a direction that is normal to the incident beam, and the crystal is oriented so that one of its crystallographic axes is parallel to the rotation axis. The dimensions of the crystal are usually less than 1 mm. A photographic film is mounted in a cylindrical holder concentric with the rotating spindle.
The incident beam is diffracted from a given crystal plane whenever in course of rotation the value of θ satisfies the Bragg equation. The diffracted beams from all planes parallel to the vertical rotation axis will be in the horizontal plane and those from planes having other orientations will be in layers above and below the horizontal plane.
However, there will be no reflected beams from a plane which always contains the incident beam during the whole rotation and from ones whose spacing is so small that λ/2d > 1. The reflected spots on the film form parallel lines. This is easily understood with reference to the Fig. 2.67.
To explain the general nature of the diffraction, consider a crystal mounted so that c-axis is parallel to the axis of rotation, then diffraction cannot occur from the planes of atoms parallel to this axis unless [See Fig. 2.68 (a)]
c cos ɸn = nλ
where, n is an integer. The diffracted beam will, therefore, be along the surfaces of a family of cones whose vertices are at the crystal, and whose semi-vertical angles are given by the above equation.
Of course, the diffracted beams will only occur along those specific directions lying on the cones for which the correct phase relationship also holds for planes parallel to the other two co-ordinates axes. When the film is flattened out after development, these diffraction images will lie on a series of lines called layer lines as shown in Fig. 2.67.
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All the images on the zero layer line come from planes parallel to the axis of rotation, i.e. planes with l = 0 in this example, and the other layer lines arise from planes with I = +1, ±2 … etc. Diffraction images from planes with the same values of h and k but different values of l, all lie on one of a series of curves known as row lines which are transverse to the layer lines; and in the particular case when the a and b axes are perpendicular to c, they intersect the zero layer line at right angles.
If Sn is the separation of these layer lines and R is the radius of the camera, then as seen from Fig. 2.68(b).
Sn =R tan (90 – ɸn) = R cos ɸn
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From last two equations, we have,
or
By subsequently orienting the crystal with the a and b axes parallel to the axis of rotation, the other unit cell parameters may be determined. Therefore, the rotating crystal photographs can be used directly to determine the unit cell dimensions of a single crystal.
3. Powder Method:
In this method a finely powdered specimen is placed in a monochromatic beam, often Ka radiation of X-rays. Just by chance, some of its microcrystals will be oriented at correct diffraction angle for a particular set of planes and a diffraction beam will result.
The incident monochromatic radiation strikes the finely powdered specimen or fine grained polycrystalline specimen contained in a capillary tube. A photographic film is wrapped around the inside of a cylindrical chamber concentric with the sample. The rays are diffracted from individual microcrystals which happen to be oriented with planes making Bragg angle θ with the beam; the various diffracted rays lying, of course, along the generators of cones are concentric with the incident beam.
To understand the point clearly, consider the same set of planes (hkl) in each microcrystal of the powder. Since the microcrystals are oriented in all possible directions, these planes have all possible orientations and the rays diffracted by this set of planes (hkl) in the powder pass through various points forming, clearly, cone that is concentric about the incident X-ray beam.
The half-opening angle of the cone is 2θ, where θ is the Bragg angle. Different (hkl) planes produce different similar cones. Now, since the film is wrapped around the inside of a cylindrical chamber concentric with the sample, a certain portion of these diffracted cones will be intercepted and a series of arcs is produced on the film. A typical diffraction pattern is shown in Fig. 2.70 (a).
Let us see how the diffraction patterns of this method are used to determine the crystal structure. With reference to the Fig. 2.70 (a) and (b) suppose S is the distance on the film between the diffraction arcs corresponding to a particular plane and 4θ is the full-opening angle of the corresponding cone, then we have-
S = 4θ R, (θ in radians)
Where, R is the specimen-to-film distance, usually the radius of the camera housing the film. For easy conversion of the distance S measured in mm to Bragg angle in degrees, the camera radius is often chosen to be 57.3 mm as 1 rad = 57.3°. A list of θ values can thus be prepared directly from the measured values of S. Since the wavelength is known, substitution of θ and λ in 2d sin θ = nλ gives a list of spacing of d.
Use of the geometrical relations between the crystallographic axes, the Miller indices, and dhkl can now be made to assign the appropriate indices to each reflection and to determine the unit cell dimensions. We shall illustrate the procedure for the cubic system. For this system the interplanar spacing is-
It is rather much convenient to use the graphical form of this relation; this is shown in Fig. 2.71. Since the possible values, that the indices h, k, I can have are the same for all cubic crystals, this graph can be used to index all the cubic crystals. Now, to use this graph, the ordinates are drawn corresponding to the measured values of d and the intersection of these ordinates with the lines of graph is sought along the same horizontal line as is explained in the figure.
In this, line can be expected to pass through the intersection of the ordinate corresponding to the largest value of d with (100), (110), or (111). Once this match is obtained, intersection of the horizontal line with the vertical axis of the graph marks the value of ‘a’ of the crystal examined. Thus, the unit cell dimensions and the indices of the reflecting plane are determined at the same time. Similar graphical methods have also been developed for indexing powder photographs of crystals belonging to other systems.
The exposure in a powder camera must be sufficiently long to give reflected lines of good intensity. The exposure time is usually a few hours. After the film is exposed and developed, it is indexed to determine the crystal structure. It is easily seen that the first arc on either side of the exit point corresponds to the smallest angle of reflection. The pair of arcs beyond this pair have larger Bragg angles and are from planes of smallest spacings, recall d = λ/ (2 sin θ).
In the powder method, the intensity of the reflected beam can also be recorded in a diffractometer which uses a counter in place of the film to measure intensities. The counter moves along the periphery of the cylinder and records the reflected intensities against 2θ. Peaks in the diffractometer recording (Fig. 2.72) correspond to positions where the Bragg condition is satisfied by some crystallographic planes.
