Laue equation

when the path difference between incident and diffracted beams becomes equal to the integral multiple of the wavelength A, then the interference maxima condition will be satisfied, i.e., BC — AD = acosO — acos p = nA, this is one-dimensional Laue equation. When the other two directions are considered, then the corresponding Laue equations are 

In vector form, these three Laue equations can be written as 

When these three conditions are simultaneously satisfied, the entire diffraction phenomenon may be equivalent to a planer reflection and gives rise to Bragg s law. It will be shown in this chapter that the wavelengths for which the Laue conditions are satisfied are not the characteristic radiation but general radiation. 

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A crystal therefore acts as a three-dimensional diffraction grating for these x-rays, and three equations (the Laue equations) must be satisfied if there is to be constructive interference of these monochromatic x-rays. 

Bragg s law is a special case of the Laue equations which define the condition for diffraction (constructive interference) to occur ... 

These are the Laue equations. Note there are solutions only for special values of 6 and X. It will now be shown that these equations are equivalent to the Bragg law. 

The integers h, k, I in the Laue equations may not be identical with the Miller indices because the h, k, I of the Laue equations may contain a common integral factor n which has been eliminated from the Miller indices. Thus, the Bragg law can be written as... 

The condition for reinforcement, when the path difference is an integral number of wavelengths, is shown in Figure 3.9(a). The equations so derived are known as the Laue equations. 

The geometry of diffraction from a lattice, or in other words the relationships between the directions of the incident and diffracted beams, was first given by Laue (see the footnote on page 31) in a form of three simultaneous equations, which are commonly known as Laue equations ... 

Laue equations once again indicate that a periodic lattice produces diffraction maxima at specific angles, which are defined by both the lattice repeat distances (a, b, c) and the wavelength (A,). Laue equations give the most general representation of a three-dimensional diffraction pattern and they may be used in the form of Eq. 2.20 to describe the geometry of diffraction from a single crystal. 

Both the Laue equations and the Bragg law can be derived from Eq. (7). The former are obtained by forming the dot product of each side of the equation and the three crystal-lattice vectors aj, aj, sl successively. For example,... 

Q.28.1 Show that Equation (29.12) is a general form of the Bragg equation as derived from the von Laue equations. 

A.28.1 The solution starts with the recognition that the cosines given in the von Laue equations represent coordinate transforms with respect to the axes... 

Considering these pairs of data, we are solving two equations for each unknown instead of solving three equations for three unknowns simultaneously. Rearranging the von Laue equation, we have... 

GEOMETRY BRAGG S LAW, THE LAUE EQUATIONS THE RECIPROCAL LATTICE AND THE EWALD SPHERE... 

The Laue equations can be recast by using the concept of the reciprocal lattice. New vectors a, b and c can be defined by the following relationships ... 

The Laue equations which characterize the x-ray diffraction in this case are ... 

Laue equations and, hence, exclusively from the periodicity of the crystal structure. 

In the following we will discuss the Laue equations in a somewhat different manner. This will allow us to compare the diffraction by three-dimensional crystals with that by one- and two-dimensional lattices. 

According to the Laue equation a S = h, the projection of S onto a is qual to hfa. The locus of all the vectors S which satisfy this equation is a series/of planes perpendicular to a and separated by 1/a. These planes make up the reciprocal space of a one-dimensional crystal. Their intersections with the Ewald sphere of radius 1/A (Fig. 3.20) define the directions s of the diffracted beams. This results in a series of coaxial cones around the a axis. The angle between the incident wave Sq and a is ocq. The half-opening angles a of the cones are obtained from the Laue equation,... 

The diffraction by a two-dimensional crystal is determined by two Laue equations which must be simultaneously satisfied. The locus of all the vectors S is... 

Fig. 3.20. Diffraction by a one-dimensional crystal satisfying a single Laue equation. When recorded on a flat screen parallel to a, the diffracted beams form a series of hyperbolic lines...

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