text: In 1928, LennardJones and Dent published two papers, "" and "", that for the first time, outlined a calculation of the potential of the electric field in a vacuum, produced by a thin sodium chloride crystal surface. They gave an expression for the electric potential produced by a system of point charges in vacuum (although not a real cubic sodium chloride ionic lattice). The expression for the potential in vacuum, , at the point r = {x, y, z}, near the cubic lattice of point ions with different signs, the charge , and the period a (a crystalline solid is distinguished by the fact that the atoms making up the crystal are arranged in a periodic fashion), can be represented in the form:
  : is the lateral vector that fixes the observation point coordinates in the sample plane.
  : is the reciprocal lattice vector.
  :s is the number of planes to be calculated inside the crystal; s set to zero would calculate the surface plane.
  The expression sums the set of potential static charges for the surface and lower planes of the crystal lattice. LennardJones and Dent showed that this expression forms a rapidly convergent Fourier series. Harold Eugene Buckley, a crystallographic researcher at the University of Manchester until his death in 1959, had suggested that their results should be treated with caution. For example, the contraction a crystal plane would suffer under the conditions prescribed would not be the same as that of a similar plane with a solid mass of crystal behind it. Another difficulty arises because calculation of crystal surface field force fields are so great that simplifying assumptions have to be made to render them capable of a solution.
  Michael Jaycock and Geoffrey Parfitt, then respectively senior lecturer in surface and colloid chemistry at Loughborough University of Technology and professor of chemical engineering at Carnegie Mellon University, concurred with Buckley, noting that "an ideal crystal, in which the ionic positions at the surface were identical to those achieved in the bulk crystal... is obviously extremely improbable." However, they acknowledged that the LennardJones and Dent model was singularly elegant, and like most researchers working before the advent of modern computers, they were limited in what could be attempted computationally. Nonetheless, LennardJones and Dent demonstrated that the force exerted on a single ion, by a surface with evenly distributed positive and negative ions, decreases very rapidly with increasing distance. Later work by Jason Cleveland, Manfred Radmacher, and Paul Hansma, has shown that this result has direct application to atomic force microscopy by predicting that noncontact imaging is possible only at small tipsample separations.