Brillouin zone in physics : defination, explanation

Brillouin zone in physics 

Defination of brillouin zone : A region bounded by the Wigner seitz plane in reciprocal lattice is a brillouin zone. 

 Brillouin  zone in Linear lattice 

            Let us first discuss about the formation of brillouin zone in linear lattice. In one dimensional lattice the energy discontinuities occur when k=n(pi/a)
    Where n is a positive integer 
                  k is a wave vector 
                  a is the lattice spacing 
 Kronig and penny gives the concept of periodically varying potential. 
  They said that the energy spectrum consist of an infinite number of allowed energy bands separated by intervals in which there are no energy levels. These are called as forbidden energy gap
       The boundary of allowed energy levels correspond to the value of cos(ka)= (+-1)
   ka=n(pi)
   k= n(pi/a)
This value of k can also obtain from Bragg's reflection condition in reciprocal lattice, which states that if G is the translational vector in reciprocal lattice 
            ( k+G/2).G=0
             k= n(pi/a)
So, a line representing k values is divided by discontinuities in the energy. These discontinuities can be represented by segment of length +-(pi/a) . These line segments are referred as brillouin zone. 
        First reflection occurs at k=+-(pi/a) 
This reflection is occurs because the wave reflected from our atom in a linear lattice interferes constructively with the wave from nearest surrounding atom in the neighborhood. 
Brillouin zone in physics
   The phase difference for the value of k=+pi/a and k=-pi/a between two reflected wave is 2pi/a. The region in k space is known as first brillouin zone. 
       When the electrons have k value between pi/a and 2pi/a along X axis direction, then the region is known as second brillouin zone. In this zone, the reflection and energy gap occurs at 2pi/a in both positive and negative direction. 
         In this way we can form third, fourth etc Brillouin zone. 
   This is all about brillouin zone in physics. We can brillouin zone for two dimensional lattice also but here we discuss only about one dimensional lattice brillouin zone in physics. 


If you have any queries about brillouin zone in physics, you can ask in comment box. 
Thanks a lot 😊.
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