The Bragg Method of Crystal Analysis (The Bragg Equation)

 In 1913, William Bragg and his son Lawrence Bragg worked out a mathematical relation to determine inter atomic distances from X-ray diffraction patterns. They showed:- 

1. The X-rays diffracted from atoms in crystal planes obey the laws of reflection.

2. The two rays reflected by successive planes will be in phase if the extra distance travelled by the second ray is an integral number of wavelengths.

The Braggs treated the crystal as a reflection grating rather than a diffraction grating for monochromatic X-rays. When X-rays are incident on a crystal face they penetrate into the crystal and strike the atoms in different planes. From each of these planes, X-rays are reflected. If the reflected waves, from successive layers, are in phase; constructive interference will occur and a diffraction spot will be detected from these planes. On the other hand, if the diffracted waves from successive layers are out of phase, destructive interference of the two waves will occur and no diffraction will be detected from these planes.

In order to understand the theory of this method, consider the parallel, equidistant planes AA, BB, CC, etc which constitute atomic planes of a crystal, with inter planer distance d as shown in Fig.4.9. If a beam of X-rays of wave length strikes these planes at angle 9, some of the rays are reflected from the top-most plane AA', while the rest penetrates the crystal and suffers reflection from the second, third plane etc (i.e., from the succe planes). For constructive interference, it is necessary that the waves reflected from sucenssive planes be in phase. This however would be possible only if the path difference of the two waves is an integral multiple of wavelength

The wave P'Q'R' reflected from the plane BB travelled a greater distance than the wave PQR reflected from the plane AA', since the plane BB' is below the plane AA'. However, the parts of the reflected wave QR and MR must be in phase to get constructive interference. This means that extra distance travelled by the wave P'Q'R' must be equal to an integral multiple of the wavelength of the incident X-rays. The extra distance travelled by the wave P'Q'R' can be obtained by drawing perpendiculars QL and QM on to the wave P'Q'R' from the point Q. From the figure, it is clear that PQ = P' * L and QR = M * R' The extra distance travelled by the wave P'Q'R' is then simply L * Q' + QM

For constructive interference this extra distance, ie., the path difference must be an integral multiple of the wave length, në Mathematically it follows:

L * Q' + QM = n*lambda

(1) where a is an integer value 1,2,3, and is known as the order of reflection

reflection. If follows from triangle LQQ that (L * Q')/(Q * Q') =sin theta^

or Similarly, for the triangle MQQ', we can (3) L * Q' = Q * Q' * sin theta = d * sin theta Q' * M = d * sin 0

Substituting Eqs. (2) into Eq.(1), we get d in*theta + d * sin theta = n * sin n*lambda = 2 * d * sin theta

Equation (4) is known as Bragg's equation which relates the wavelength, order of reflection a, interplanar distance d and the angle of maximum reflection 0. Thus, knowing the wavelength of X-rays and the angle of incidence 0, the interplanar distance d of a crystal can be calculated.

 

The Bragg equation states that for a given wavelength, the constructive interference would be possible only at definite 0 values and these & values depend upon the interplanas spacing d. The interplanar spacing d is independent upon the unit cell dimensions (a,b,c) and the Miller indices of the faces or planes (hkl). This relationship is expressed in different equations for the different crystal systems. For a cubic system, the interplanar distance d is related to the unit dimensiona and the Miller indices hal by the relation.