Diffraction from crystalline solids 78

The discovery that crystalline solids diffract x-rays was a critical in the field of material science as it now provides the basis for the identification of the atomic structure of crystalline materials. The idea that X-rays, having wavelengths similar to that of inter atomic spacings leads to the concept of solid lattices acting as a diffraction grating (with X-ray wavelengths having a magnitude of approximately 1 × 10-10 m or 1 Å) and was initially suggested by von Laue and confirmed by Friedrich and Knipping79. The original idea of X-ray diffraction was based from the concept that crystalline solids being 3-D arrays of atoms could act as a diffraction grating and from the way in which x-rays are diffracted information obtained about the atomic structure.

Figure 17 Schematic representation of Laue diffraction from a lattice in the x direction, where a is the lattice spacing, α0 and αn are the incident and diffracted x-ray beams, the path difference between adjacent beams is AB – CD.

The basic model of x-ray diffraction was based on a single crystal containing only a single type of atom, with each atom acting as a scattering centre at the corresponding lattice points. The crystal is constructed of rows of atoms arranged with spacings of a, b and c in each of the xy and

Chapter 2 Synthetic and theoretical methods

z axis of the crystal respectively. Diffraction is observed when constructive interference occurs between diffracted X-ray beams from adjacent atoms (Figure 17) and the path length of neighbouring x-rays is integer multiples of the wavelengths which results in the following equation:



ABCD



 X_i



cos_ncos₀



n_i (2.1)

Where AB and CD are path lengths in (Figure 17), Xi denotes the atomic spacing along axis i (a,

b or c), αn and αi indicate the angles between the diffracted and incident beam, relative to the crystallographic axis i, ni indicates an integer multiple of wavelength, λ. Since diffraction can

occur with a component in each of the three axis; x, y and z, with the corresponding atomic spacing a, b or c equation (2.1) can be written for the diffraction component in each axis:



_n 



n_x a cos cos ₀  (2.2)



_n 



n_y bcos cos ₀  (2.3)



_n 



n_z c cos cos ₀  (2.4)

Where angles βn, β0, and integer ny correspond to the angles and wave length multiples relative to

the y axis, likewise for γn, γ0 and nz for the z axis. The scheme in Figure 17, only illustrates

diffraction in the plane of the page, in reality, so long as the diffracted angle to the atomic row remains αn, then the diffraction conditions can still be met, this results in diffracted beams

actually forming a cone centred on the atomic row, with the apex of the cone having angle αn.

Since the same result can occur in the other two axes, diffracted beams are only observed where the diffraction cones from each axis intersect with each other resulting in well defined beams. For this analysis, in order to compute the direction of the diffracted beams, all of the angle,

Chapter 2 Synthetic and theoretical methods

spacing and integers from equations (2.2 – 2.4) need to be determined. A simpler model was proposed by W. L. Bragg80, where diffraction was considered to be a reflection of the incident beam from planes of atoms (Figure 18), reducing the problem to two dimensions; although this is not physically correct, it makes geometrical sense, and substantially simplifies the problem.

Figure 18 Schematic representation of Bragg diffraction from planes of atoms, with the path difference between adjacent beams being (AB+BC), θ indicates the incident and diffracted beam angles and dhkl indicates the inter layer

spacing within the crystal with miller indices hkl.

From Braggs’ image of X-ray diffraction, the path difference between the beams scattered from adjacent planes of atoms, separated by the inter plane spacing dhkl is given by (2.5), where hkl

indicates the miller indices of the plane of atoms from which the x-ray beam is diffracting:



ABBC



(d_hklsind_hklsin)2d_hklsin (2.5)

For constructive interference to occur, the path difference has to equate to an integer number of wavelengths, resulting in the following condition for constructive interference:

  2d_hklsin

Chapter 2 Synthetic and theoretical methods

Where, n remains an integer number. Typically, experimentally collected diffraction patterns are plotted with observed intensity against the diffraction angle, 2θ, another common representation, especially when considering multiple data sets is Q, which is the momentum transfer on scattering and is defined by78:

hkl d Q    sin 2 4   (2.7)

For Braggs law, the separation of the atomic planes is the governing factor for constructive interference, rather than the specific atomic coordinates. The equations described above additionally show that Braggs description of x-ray diffraction, unlike von Laue’s, is only in two dimensions, dramatically reducing the number of parameters that require determination.

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