On the Exact Evaluation of the Face-Centred Cubic Lattice Green Function

The mathematical properties of the lattice Green function

Open image in new windoww=w ₁+iw ₂ lies in a complex plane which is cut from w=?1 to w=3, and {? ₁,? ₂,? ₃} is a set of integers with ? ₁+? ₂+? ₃ equal to an even integer. In particular, it is proved that G(2n,0,0;w), where n=0,1,2,…, is a solution of a fourth-order linear differential equation of the Fuchsian type with four regular singular points at w=?1,0,3 and ∞. It is also shown that G(2n,0,0;w) satisfies a five-term recurrence relation with respect to the integer variable n. The limiting function

$G^{-}(2n,0,0;w_1)\equiv\lim_{\epsilon\rightarrow0+}G(2n,0,0;w_1-\mathrm{i}\epsilon) =G_{\mathrm{R}}(2n,0,0;w_1)+\mathrm{i}G_{\mathrm {I}}(2n,0,0;w_1) ,\nonumber $

where w ₁∈(?1,3), is evaluated exactly in terms of ₂ F ₁ hypergeometric functions and the special cases G ^?(2n,0,0;0), G ^?(2n,0,0;1) and G(2n,0,0;3) are analysed using singular value theory. More generally, it is demonstrated that G(? ₁,? ₂,? ₃;w) can be written in the form

Open image in new window Open image in new windowξ, K(k _?) and E(k _?) are complete elliptic integrals of the first and second kind, respectively, with

$k_{-}^2\equiv k_{-}^2(w)={1\over2}- {2\over w} \biggl(1+{1\over w} \biggr)^{-{3\over2}}- {1\over2} \biggl(1-{1\over w} \biggr ) \biggl(1+{1\over w} \biggr)^{-{3\over2}} \biggl(1-{3\over w} \biggr)^{1\over2}\nonumber $

and the parameter ξ is defined as

$\xi\equiv\xi(w)= \biggl(1+\sqrt{1-{3\over w}} \,\biggr)^{-1} \biggl(-1+\sqrt{1+{1\over w}} \,\biggr) .\nonumber $

This result is valid for all values of w which lie in the cut plane. The asymptotic behaviour of G ^?(2n,0,0;w ₁) and G(2n,0,0;w ₁) as n→∞ is also determined. In the final section of the paper a new ₂ F ₁ product form for the anisotropic face-centred cubic lattice Green function is given.