Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES
Lecture 31 : Electron in a Periodic Potential
Reciprocal Lattice
  We have seen that Bragg diffraction occurs for values of $ k-$ vectors which are multiples of $ \pi/d$. Bragg diffraction is a consequence of periodicity of the lattice and diffraction pattern forms a geometric image which bears an inverse relationship with lattice periodicity. Because the scale associated with the points at which Bragg condition is satisfied has the dimensions of inverse length (as seen in the one dimensional relation $ k=\pi/d$), the geometrical structure defined by diffraction peaks is called the reciprocal lattice corresponding to the real space lattice which is also referred to as the direct lattice .
  An equivalent way of describing diffraction in crystal is known as von-Laue formulation, according to which the condition of constructive interference is satisfied if
 
$\displaystyle \vec R\cdot(\vec k -\vec k^\prime) = 2\pi n$
 

where $ \vec R$ is a direct lattice vector, $ \vec k$ is the wave vector of the incoming wave, $ \vec k^\prime$ that of outgoing wave and $ n$ is any integer.

  One can define reciprocal lattice vector $ \vec G$
 
$\displaystyle \vec G\cdot\vec R = 2\pi n$
  where $ n$ is an integer.
 
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