The quantities obtained from cubic, hexagonal, etc., lattice sums, evaluated at 
, are called Madelung constants.
For cubic lattice sums
 |
(1)
|
the Madelung constants expressible in closed form for even indices 
,
a few examples of which are summarized in the following table, where 
is the Dirichlet
beta function and 
is the Dirichlet
eta function.
To obtain the closed form for 
, break up the double sum into pieces that do not include
,
 |
(2)
|
 |
(3)
|
 |
(4)
|
where the negative sums have been reindexed to run over positive quantities. But 
,
so all the above terms can be combined into
![b_2(2s)=4[sum_(i,j=1)^infty((-1)^(i+j))/((i^2+j^2)^s)+sum_(i=1)^infty((-1)^i)/(i^(2s))].](/images/equations/MadelungConstants/NumberedEquation2.svg) |
(5)
|
The second of these sums can be done analytically as
 |
(6)
|
which in the case 
reduces to
 |
(7)
|
The first sum is more difficult, but in the case 
can be written
 |
(8)
|
Combining these then gives the original sum as
 |
(9)
|
is given by Benson's formula (Borwein and Bailey
2003, p. 24)
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
(OEIS A085469), where the prime indicates that summation over (0, 0, 0) is excluded.
is sometimes called "the" Madelung constant, corresponds to the Madelung
constant for a three-dimensional NaCl crystal. Crandall (1999) gave the expression
![M=-2pi+(Gamma(1/8)Gamma(3/8)sqrt(2))/(pi^(3/2))
+2sum^'_(m, n, p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(4pisqrt(m^2+n^2+p^2))+1]).](/images/equations/MadelungConstants/NumberedEquation7.svg) |
(13)
|
Similar results were found by Tyagi (2004),
 |  | ![-1/2-(ln2)/pi-pi/3-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(2pisqrt(m^2+n^2+p^2))-1])](/images/equations/MadelungConstants/Inline32.svg) |
(14)
|
 |  | ![sqrt(2)-pi+2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(2pisqrt(m^2+n^2+p^2))+1])](/images/equations/MadelungConstants/Inline35.svg) |
(15)
|
 |  | ![-1/4-(ln2)/(2pi)-(2pi)/3+1/(sqrt(2))-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(4pisqrt(m^2+n^2+p^2))-1])](/images/equations/MadelungConstants/Inline38.svg) |
(16)
|
the last of which converges rapidly. Averaging (16) and (13) then gives the beautiful equation
![M=-1/8-(ln2)/(4pi)-(4pi)/3+1/(2sqrt(2))+(Gamma(1/8)Gamma(3/8))/(pi^(3/2)sqrt(2))
-2sum^'_(m,n,p=-infty)^infty((-1)^(m+n+p))/(sqrt(m^2+n^2+p^2)[exp(8pisqrt(m^2+n^2+p^2))-1]),](/images/equations/MadelungConstants/NumberedEquation8.svg) |
(17)
|
which is correct to 10 decimal digits even if the sum is completely omitted (Tyagi 2004).
However, no closed form for 
is known (Bailey et al. 2006).
For hexagonal lattice sums, 
is expressible in closed form as
 |  |  |
(18)
|
 |  |  |
(19)
|
(OEIS A086055).
