13 X-ray diffraction- II

Introduction

In the previous module, diffraction of X-rays by crystal is explained. We have already studied that diffraction can occur only when Bragg’s law 2  sin   = is satisfied. The directions of diffraction of X-ray beam from a crystal are determined and the factors governing the intensity of diffracted beams are discussed. Also, Bragg’s equation applies conditions on λ and θ for any given crystal. With monochromatic radiation, a single crystal will not produce diffracted beams for arbitrary angle of incidence of x-rays onto the crystal. In order to satisfy Bragg’s law, it is necessary to scan in either wavelength or angle to satisfy diffraction conditions. Experimentally it is done by providing a continuous range of values of either λ or θ during experiment. Thus, in this module the various experimental methods of X-ray diffraction are studied.

Diffraction directions

In order to determine the possible directions i.e possible angles 2θ in which a given crystal can diffract a monochromatic beam of x-rays, a general relation is required which can predict the diffraction angle for any set of planes. This relation can be obtained by combining Bragg’s law and the interplanar spacing equation for the particular crystal under study.

For example: Consider a cubic crystal. The interplanar spacing ‘d’ for particular (hkl) planes is related to the Miller indices (h,k,l) and lattice constant ‘a’ via relation

Combining with Bragg’s law (λ = 2d sin θ), we get

This equation predicts all possible Bragg angles at which diffraction can occur from the planes (hkl) for a particular incident wavelength ‘λ’ and a particular cubic crystal of unit size ‘a’. For example, for (110) planes, equation (11) can be written as

For a tetragonal crystal having the axis as ‘a’ and ‘c’, the corresponding general equation is

Similarly, the equations can be obtained for other crystal systems.

Thus it can be inferred that the directions in which a beam of given wavelength is diffracted by a set of lattice planes is determined by the crystal system to which the crystal belongs and its lattice parameters i.e. diffraction directions depend solely on the shape and size of unit cell. Conversely, the above point can be stated as: “ All that can be determined about an unknown crystal by measurements of the directions of diffracted beams are the shape and size of its unit cell”.

Intensities of diffracted beams

We have seen that the directions in which diffraction occurs according to Bragg’s law is governed entirely by the geometry and size of unit cell. These directions are completely independent of the arrangement of atoms associated with each lattice point (basis). However, the constitution of the atomic structure within the unit cell affects the intensity of diffracted beams. The intensity of the diffracted beams primarily depends on the atomic scattering factor and the position of each atom within the unit cell. Since the electrons are the only components of the atom that scatter X-rays significantly, therefore the X-rays scattered from one part of an atom interfere with those scattered from the other parts at all angles of scattering. As the electrons are distributed throughout the atomic volume, the diffracted beams are obtained as a result of combination of scattered waves from the electrons of all atoms in the unit cell.

This process involves two distinct contributions:

1.    Scattering from electrons in the same atom (called as atomic scattering factor or atomic form factor f).

2.    The summation of this scattering from all atoms in the unit cell (called as geometrical structure factor F or S).

The atomic scattering factor is a measure of the efficiency of an atom in scattering X-rays. It is defined as the ratio of the amplitude scattered by actual electron distribution in an atom to that scattered by one electron localized at a point.

Therefore, if the atoms are assumed to be points only then the atomic scattering factor is just equal to the number of electrons present (that is atomic number of neutral atom Z)

The atomic scattering factor of an atom falls off with increasing value of (sin θ/λ). In order to find out the intensity of X-ray beam scattered by one unit cell in a particular direction where there is diffraction maximum, it is necessary to sum the waves that arise from all atoms in the unit cell i.e. mathematically, waves of the same wavelength but with different amplitudes and phases need to be added. The intensity of the scattered beam can be obtained by squaring the resultant amplitude.

Intensity I ∝ |F(hkl)|²

?_? is the atomic form factor which depends on the property of atoms located at lattice point

The structure factor need not be real; the scattered intensity (I) involves |F|2 i.e. F*.F where F* is the complex congugate of F.

If structure factor is zero, there is no intensity in a reflection G permitted by the space lattice. Thus, the structure factor can cancel some reflections allowed by the space lattice and the missing reflections help us in the determination of structure.

Structure factor of bcc lattice

where f is the scattering power or atomic form factor of an atom.

The value of s is zero when the exponential has value -1. The exponential has value -1 whenever its argument is –iπ times an odd integer.

Therefore, S = 0 when (h+k+l) = odd integer (Missing reflection)

S = 2f  when (h+k+l)= even integer (Allowed reflection)

Eg: Na metal has a bee structure. The diffraction spectrum does not contain lines such as (100) (300) (111) (221) but lines such as (200) (110) and (222) will be present.

The cancellation of (100) reflection occurs in bcc lattice because the planes in Na metal are identical in composition. Therefore, alternate planes cancel the contribution i.e. diffracted beam from these middle plane are out of phase by π with respect to first plane as the effective number of body centered atom is equal to effective number of corner atom in a bcc structure.

In case of CsCl, cancellation does not occur. The planes of Cs and Cl ions alternate but the scattering power of Cs is much greater than the scattering power of Cl because Cs+ has 54 electrons and Cl- has only 18 electrons.

Structure factor of fcc lattice:

Experimental X-ray diffraction methods

We have already studied that diffraction can occur only when Bragg’s law 2  sin   = is satisfied. This equation applies conditions on λ and θ for any given crystal. With monochromatic radiation, a single crystal will not produce diffracted beams for arbitrary angle of incidence of x-rays onto the crystal. In order to satisfy Bragg’s law, it is necessary to scan in either wavelength or angle to satisfy diffraction conditions. Experimentally it is done by providing a continuous range of values of either λ or θ during experiment. The ways in which these quantities are varied distinguish three main diffraction methods:

Method	λ	θ
Laue	Variable	Fixed
Rotating crystal	Fixed	Variable (in part)
Powder	Fixed	Variable

Laue method

In 1912, Max Von Laue was the first of all who suggested the use of a crystal as three dimensional grating to study the nature of X-rays. Thus, Laue method was the first method which was ever used for diffraction and it produces Von-Laue’s original experiment. In this method, a beam of white radiation (x-rays of continuous wavelengths) is allowed to fall on a single crystal which is held stationary. The Bragg angle θ is thus fixed for every set of planes in the crystal and each set selects and diffracts that particular wavelength which satisfies Bragg’s law for the particular values of d and θ. Each diffracted beam thus possess a different wavelength.

In this method, a source is used to produce a beam of X-rays over a wide range of wavelength varying from 0.2 Å to 2 Å. A pin hole arrangement is used to produce a well collimated beam. The dimensions of single crystal are taken to be small (< 1mm). The film is flat and placed perpendicular to the incident beam to receive the diffracted beams. Depending on the relative positions of source crystal and film, the Laue method can be carried out in two ways: Transmission Laue method and back-reflection Laue method. In transmission laue method, which is also the original Laue method, the film is placed behind the crystal to record the beams diffracted in the forward direction as shown in figure 1(a). The name “transmission” is derived from the fact that the diffracted beams are partially transmitted through the crystal. In the back-reflection Laue method, the film is placed between the crystal and the x-ray source. The incident beam passes through a hole in the film and the beams diffracted in the backward direction are recorded as shown in figure 1(b).

In both the cases, the diffraction pattern forms a series of spots on the film. Each reflecting plane in the crystal selects from the incident beam a wavelength satisfying Bragg equation. The obtained pattern will give the information about the symmetry of the crystal. If a crystal having four-fold axis of symmetry is oriented with the axis placed parallel to the beam, then the Laue pattern will show the four-fold symmetry.

Figure 4: Laue X-ray camera

This feature makes the Laue pattern particularly useful for checking the orientations of crystals in solid state experiments. However, it suffers from certain disadvantages for crystal structure determination. Because of the wide range of wavelengths, it is possible for several wavelengths to reflect in different orders from a single plane, and different orders of reflection may superimpose on a single spot. This makes the determination of the reflected intensity difficult.

Laue method is suitable for the quick measurement of crystal orientation and symmetry. It also indicates the crystalline imperfections under mechanical and thermal treatment.

Rotating crystal method

In this method, a single crystal is rotated about a fixed axis in a beam of monochromatic X-rays. The variation in the angle θ brings different atomic planes into position for reflection. A simple rotating crystal X-ray camera is shown in figure 5. The film is mounted in a cylindrical holder concentric with a rotating spindle on which the single crystal specimen is mounted. The dimensions of the crystal should be small, generally less than 1 mm. The incident X-ray beam is made nearly monochromatic by a filter or by reflection from an earlier crystal. The beam is diffracted from a given crystal plane whenever in the course of rotation, the value of θ satisfies the Bragg equation.

Figure 5: A rotating crystal camera

Beams from all planes parallel to the vertical rotation axis will lie in the horizontal plane. Planes with other orientations will reflect in layers above and below the horizontal plane. In this method, not every plane can reflect, nor could one whose spacing is so small that λ/2d > 1. The reflected spots form parallel lines. Since, the wavelength is known, the spacing d may be calculated using Bragg equation.

Reflections from planes in different orientations may overlap on the film and to make the identification of the spots less ambiguous, two modifications of the apparatus are employed (a) the oscillation method and (b)    the Weissenberg method. In the oscillation method, the number of planes which can reflect is restricted by oscillating the crystal through a small angle. In the Weissenberg method, while the crystal rotates, the film is translated parallel to the axis of rotation, the two motions being synchronized. These two modifications are usually used in the complete crystal structure analysis.

This method is best suited for the structure determination when the sample is a single crystal. It si also useful for determining the size of unit cell.

Powder method

The method was devised by Debye and Scherrer in 1916 for the determination of the structure of the small-grained crystallites or finely powdered poly-crystalline materials. An important use of this method is in the study of phase diagrams of alloy systems. In this method, a finely powdered specimen or a fine grained polycrystalline specimen contained in a thin walled capillary tube is held on a movable mount at the centre of the cylindrical camera where incident monochromatic radiations strike. The distribution of crystallite orientation will be nearly continuous. The powder method is convenient because single crystals are not required. Diffracted rays go out from individual crystallites which happen to be oriented with planes making an incident angle θ with the beam satisfying the Bragg equation. Diffracted rays leave the specimen along the generators of cones concentric with the original beam. The generators make an angle 2θ with the direction of original beam where θ is the Bragg angle. The cones intercept the film in a series of concentric rings. To ensure that all possible sets of planes face the incident X-rays, the specimen is rotated slowly during exposure. Experimental arrangement of the powder method is shown in figure 6.

Figure 6: X-ray powder diffraction camera

From the Bragg equation, differentiating with respect to θ

For monochromatic X-radiations (i.e. constant λ)

Thus, the variation in θ for small change in d is very large as θ approaches 0°. The reflected lines occur at 2θ from the incident beam, so that the maximum sensitivity is in the backward direction. This method can be applied to any type of material having crystalline arrangement, and does not require single crystals.