Some Applications of Conditionally Convergent Lattice Sums

  • Edgar R. Smith
  • John W. Perram
Part of the NATO Advanced Science Institutes Series book series (volume 92)


We use some of the lattice sums developed in the previous lecture to study two systems. The first is the simple cubic lattice, with unit lattice spacing with a polarizable point of polarizability α at each lattice site. For this system we consider first a plane slab of the material in an external electric field. The response of the system to a small external field gives the dielectric constant of the system. We also calculate the response of the polarizable points close to the surface of the plane slab. Next we consider the response of a large sphere of the material to a charge or fixed dipole at the centre of the sphere and find an asymptotic expression for the polarization of points far from the “defect”. These expansions also allow calculation of the dielectric constant. All methods give the same static dielectric constant
$$ \varepsilon ~=~(1+8\pi \alpha /3)~(1-4\pi \alpha /3) $$
the standard Clausius-Mosotti result. The second system studied is a plane slab of ionic crystal responding to an applied electric field. We show how to calculate the distortion of the crystal produced by the field and thus obtain the dielectric constant of the crystal. Results for the dielectric constant for a CsCl structure are given and show how the dielectric constant depends on the repulsive potential between the ions.


Dielectric Constant Large Sphere Polarization Density Polarizable Point Shape Dependence 
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  1. 1.
    Böttcher, C.J.F., “Theory of Electric Polarization, vol. 1”, Elsevier, Masterdam (1973). See chapters 3, 4.Google Scholar
  2. 2.
    Lighthill, M.J., “Introduction to Fourier Analysis and Generalized Functions”, Cambridge (1958).Google Scholar
  3. 3.
    Smith, E.R., J. Phys. A. (Math. and Gen.) 13 (1980), L107 - L110.ADSCrossRefGoogle Scholar
  4. 4.
    Böttcher, C.J.F., ibid., chapter 6.Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • Edgar R. Smith
    • 1
  • John W. Perram
    • 2
  1. 1.Department of MathematicsUniversity of MelbourneParkvilleAustralia
  2. 2.Matematisk InstitutOdense UniversitetOdense MDenmark

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