page_title: Lindhard theory
section_title: Static limit
hierarchical_section_title: Lindhard theory / Analysis of the Lindhard formula / Three dimensions / Static limit
caption_attribution_description: English: statically screened potential and Coulomb potential in 3 dim
context_page_description: Lindhard theory, named after Danish professor Jens Lindhard, is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics and the random phase approximation.
  Thomas–Fermi screening can be derived as a special case of the more general Lindhard formula. In particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector is much smaller than the Fermi wavevector, i.e. the long-distance limit.
context_section_description: Second, consider the static limit ().
  The Lindhard formula becomes
  .
  Inserting the above equalities for the denominator and numerator, we obtain
  .
  Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get
  here, we used  and .
  Therefore, 
  Here,  is the 3D screening wave number (3D inverse screening length) defined as 
  .
  Then, the 3D statically screened Coulomb potential is given by
  .
  And the Fourier transformation of this result gives
  known as the Yukawa potential. Note that in this Fourier transformation, which is basically a sum over all , we used the expression for small  for every value of  which is not correct.
  For a degenerated Fermi gas (T=0), the Fermi energy is given by
  ,
  So the density is 
  .
  At T=0, , so .
  Inserting this into the above 3D screening wave number equation, we obtain
  This is the 3D Thomas–Fermi screening wave number.
  For reference, Debye–Hückel screening describes the nondegenerate limit case. The result is , the 3D Debye–Hückel screening wave number.
caption: Statically screened potential(upper curved surface) and Coulomb potential(lower curved surface) in three dimensions
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