Chapt-2-prob - Homework#10 1 Square lattice free electron energies(a Show for a simple square lattice(two dimensions that the kinetic energy of a free

Chapt-2-prob - Homework#10 1 Square lattice free electron...

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Homework #10 1. Square lattice, free electron energies. (a) Show for a simple square lattice (two dimensions) that the kinetic energy of a free electron at a corner of the first Brillouin zone is higher than that of an electron at the midpoint of a side face of the zone by a factor of 2. (b) What is the corresponding factor for a simple cubic lattice (three dimensions)? (c) What bearing might the result of (b) have on the conductivity of divalent metals? Solution: (a) At the corner of the first Brillouin zone, = a a k π π , r , thus 2 2 2 2 2 2 2 2 2 2 ) 1 , 1 ( ) 1 1 ( 2 2 ma ma m k E π π h h h = + = = . At the midpoint of a side face, = 0 , a k π r , thus 2 2 2 2 2 2 2 2 ) 0 , 1 ( 2 ) 0 1 ( 2 ma ma E π π h h = + = . Therefore 2 / ) 0 , 1 ( ) 1 , 1 ( = E E . (b) For a simple cubic lattice, at the corner of the first Brillouin zone, = a a a k π π π , , r , thus 2 2 2 2 2 2 2 2 2 ) 1 , 1 , 1 ( 2 3 ) 1 1 1 ( 2 ma ma E π π h h = + + = . At the midpoint of a side face, = 0 , 0 , a k π r , thus 2 2 2 2 2 2 2 2 ) 0 , 0 , 1 ( 2 ) 0 1 ( 2 ma ma E π π h h = + = . Therefore 3 / ) 0 , 0 , 1 ( ) 1 , 1 , 1 ( = E E . (c) For a divalent metal, the electron density is 3 / 2 a n = . Then the Fermi energy is 2 2 3 / 2 2 2 2 3 / 2 2 2 2 53 . 1 ) / 6 ( 2 ) 3 ( 2 ma ma n m E F h h h = = = π π π . Thus ) 1 , 1 , 1 ( ) 0 , 0 , 1 ( E E E F < < . The Fermi sphere is greater than the first Brillouin zone in the [100] direction, but smaller than the first Brillouin zone in the [111] direction.
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