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Bragg Law



       Just as Diffraction-Gratings can diffract ordinary light, crystals can diffract X-rays , provided their inter-planar distances are comparable to the wavelength of the incident X-rays.
Image bragg_anim

The above figure shows a two-dimensional cross-sectional view of a crystal, the black horizontal lines depicting a set of parallel planes (Miller-planes).
When a beam of X-rays strike one of the planes, they undergo specular (mirror-like) reflection or scattering from the regularly spaced scattering centres (denoted by red circles). However, reflection from successive parallel planes interfere giving rise to a diffraction pattern. Bragg gave us a simple law which relates the wavelength of the incident X-rays, the inter-planar distance (AC) of the reflecting planes of the crystal and the glancing angle of reflection (acute angle made by the beam with the black horizontal lines depicting the planes). :

  lambda = lambda = The wavelength of the X-ray
  theta = theta = Glancing angle of the incident X-ray with the reflecting plane
  d = Inter-planar spacing of the reflecting planes
  n = 1,2,3,.....  = order of diffraction

The figure also depicts the motion of two points on the two parallel beams which originate in the same wave-front but suffer a path-difference of one-wavelength due to reflection by the respective planes. This path difference equates to a phase difference of 360 degrees or equivalently zero degrees. This results in a constructive interference and the two beams are reinforced to give a strong diffraction maximum. Simple trigonometry tells us that the path difference is equal to BC plus BD which equals 2 d sin theta. This proves Bragg's law for first order diffraction maximum. We may note that if the path difference between the two beams equals an integral number of wavelengths, then too the phase difference would be zero and we would have diffraction maxima. This would give rise to higher order diffraction, represented by n=2,3,4..etc. Bragg's law tells us that for a particular set of miller (h,k,l) planes having a fixed value of inter-planar spacing (d), X-rays of fixed wavelength (lambda) would underg Bragg's diffraction only for a fixed glancing angle theta. However, since there are many sets of planes in the same crystal, Bragg's law is satisfied for different glancing angles. The order of diffraction may be different.

  Below is presented a java applet through which you can see for yourself how, for Bragg's Law to be satisfied for a first order (n=1) diffraction-maximum, the three parameters, lambda, theta and d, need to be appropriately related. Click on any one of the three buttons at the top and input the values of any pair of parameters to view the picture of the diffraction phenomenon.

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Abhijit Poddar 2008-01-07