Nijboer, B. 1R. A.
De W e t t e , F. W.
1957
Physica XXIII
309-321
ON T H E CALCULATION OF LATTICE SUMS
by B. R. A. N I J B O E R and F. W. DE W E T T E
Instituut voor theoretische Fysica, Rijksuniversiteit te Utrecht, Nederland
Synopsis
A s h o r t a n d s t r a i g h t f o r w a r d m e t h o d for t h e c o n v e r s i o n of slowly c o n v e r g i n g l a t t i c e
s u m s i n t o e x p r e s s i o n s w i t h good c o n v e r g e n c e is p r e s e n t e d . As a n i l l u s t r a t i o n a wellk n o w n e x p r e s s i o n for t h e M a d e l u n g c o n s t a n t is r e d e r i v e d . T h e m e t h o d is t h e n a p p l i e d
t o t w o general t y p e s of l a t t i c e sums, t h e l a t t e r of w h i c h h a s n o t been t r e a t e d before.
§ 1. Introduction. Lattice sums, i.e. summations over the sites of an
infinite perfect lattice of some potential energy function or force interaction,
occur in m a n y branches of crystal physics. The appearance of lattice sums
in calculations of the lattice energy of ionic crystals and in considerations on
the stability of the various lattice types are the best known and also oldest
examples. Other cases where the evaluation of these sums is of some importance are investigations of the electromagnetic, optical, or elastic properties
of crystals.
A difficulty met with in the evaluation of lattice sums always has been
the question of their convergence. It presents itself under two aspects,
firstly the convergence of the series as such and secondly the rapidity of
convergence, the latter question being mainly of practical importance.
It is a well-known fact t h a t interactions decreasing slowly with distance
(e.g. the Coulomb interaction) can give rise to lattice sums which exhibit
only a conditional convergence. Since in this case the sum of the series is
not uniquely determined, physical arguments will have to be invoked in
order to arrive at physically meaningful answers. But even in the case of
unconditional convergence, the actual rapidity of convergence is generally
so poor that the series as they stand, are not very useful for computations.
In the existing literature on lattice sums a great deal of attention has been
devoted to these problems. The classic examples in this connection are the
treatments of M a d e l u n g 1) .) (for the Madelung-constant) and of E w a l d s)
(for more general types of lattice sums), in which the application of a summation recipe, based on physical arguments, and the transformation of the
lattice sum into a more rapidly converging form are closely connected. Still
*) An equivalent method has been developed by O r n s t e i n and Z e r n i k e o).
-
-
309.-
310
B. R. A. N I J B O E R AND F. V¢. DE W E T T E
another method has been proposed b y E v j en 4). Most of the later developments are generalizations or modifications of these methods. The present
paper is not different in this respect. It gives a method for the evaluation
of lattice sums which seems to be more straightforward and less complicated
than the usual procedures. Although in some respects this method is quite
similar to that of M i s r a 5), it was in its present simple form first used for
a special case b y P l a c z e k , N i j b o e r and V a n H o v e 6) in connection with
the theory of neutron scattering in dense systems *). It can be readily
extended to apply to a wide class of lattice sums. In this paper we will,
after some preliminaries, first illustrate the method b y rederiving a familiar
expression for the Madelung-constant (§ 3). Then, after a short exposition
of the principle involved in this summation method (§ 4), we will proceed to
apply the scheme to two more general types of lattice sums (§§ 5, 6), the
latter of which as far as we know, has not been treated before.
§ 2. Preliminaries. We will consider lattice sums of perfect simple lattices
(Bravais lattices); the method can easily be generalized for composite lattices. The usual notation for Bravais lattices will be used.
The unit cell of the lattice is specified b y the basic vectors al, a~, a3.
The volume of the unit cell is
v~ - - a l " (a~ A a3).
(1)
The space vector r will be expressed in terms of the basic vectors
r = r l a l + r2ae + r3a3.
(2)
The origin is chosen at the origin of one of the unit ceils, the zero cell. The
position vector of any lattice point (lattice vector) is
rA = 41al + 42a~ + 43a3,
(3)
where 4 denotes the set it1, 42, 48 (pos. and neg. integers).
The unit cell of the reciprocal lattice is mapped out b y the reciprocal
lattice vectors bl, b2, bs, defined b y the relations
a , . bj = 6 0 .
(4)
A vector in reciprocal space will be expressed in terms of the b's
h = h l b l + h2b2 + h3ba.
(5)
Similarly the reciprocal lattice vector is written as
h A -- 41bl + 4~b2 +
~3b3;
again 41, 4~, 43 are positive and negative integers.
*) An equivalent treatment for the Madelung constant was also given by B e r t a u t 7).
(6)
ON T H E C A L C U L A T I O N OF L A T T I C E SUMS
~l 1
The lattice sums that will be considered here are summations over the
sites of an infinite perfect simple lattice of a potential energy function of the
general form ~b(r -- R) exp 2ni(k. r) (in general a noncentral potential).
R is the field point, k an arbitrary vector in reciprocal space. Such sums are
$(rh -- R) e 2rrfk'rh
S(R]k) ---- ~Al,h2,Za=-oo
+~
(7)
S'(RIk) ---- ~o+oo,
z-a,,;~,,;~s=-0o ~b(r;~ __ R) e2~k'r,x .
(8)
and
~ ' means that the term ~1 = 22 ---- ;ta ---- 0 is excluded from the summation.
~Al,;~2.Aa will be abbreviated as ~ . One has obviously
S(R]k) = S'(RIk ) + ~b(- R).
(9)
Since
S(R + r~,lk) = exp (2zdk, rz, ) S(R[k)
only vectors R lying within the zero cell need to be considered. Therefore,
as we consider simple lattices only, R will not be equal to any lattice vector except perhaps R = 0 (In the latter case S usually has no meaning for the
common choices of ~b). S gives the potential field in a lattice where the origin
is occupied, S' gives the field in the neighbourhood of an empty site. For
R = 0 S' is called the sel/potential of the lattice.
§ 3. The Madelung constant ~M. In this section we will illustrate the method
b y applying it to a very simple and well-known case, namely the calculation
of the Madelung constant. This constant is defined as the self potential in
a lattice where the lattice sites are occupied b y alternately positive and
negative unit charges
=
(10)
ra
The transformation of =M into a more rapidly converging expression
((l 0) is only conditionally convergent) takes place in four steps.
1. (lO) is formally written in terms of an integral
~M = f - ~
dr
where
w(r) ---- ~',~ (--1) Ax+A~+A85(r -- ra) = ~
kt=½(bl+b~+ba)
(11)
exp (2~ik½. r) 5(r -- rA); (12)
(cf. e n d o f § 6 ) .
~ 5 ( r - rA) is a function which has 6-singularities at the lattice points
(except at the origin) and which is zero elsewhere. The effect of the function
w(r) in the integral (11) is to give the sum (10), b y virtue of the definition
of the 6-function.
312
B. R. A. NIJBOER
AND F. W. DE WETTE
2. (11) is split into two integrals
=,, =
¢
dr.
(la)
• (x) is the complement of the errorfunction
¢(x) = (2/~/~r)f$ e - " dt ---- Erfc (x).
(14)
3. The next step consists in applying Parseval's/ormula s) to the second
integral of (13). This formula states that if F(h) and G(h) are the threedimensional Fourier transforms (FT3) o f / ( r ) and g(r) respectively, then
f F(h) 6*(h) dh = f l(r)g* (r) dr.
(15)
Using now
FT3{w(r)} = (1/va) •a b{h -- (h a -- k~)} -- 1
(cf. appendix 3, (A. 12))
(16)
and
FT3
-= ~
(cf. appendix4, (A. 19))
(17)
we find with (15) for the second integral of (13)
{1 -- q} (v'~r)} dr =
[Y, a b { h - - ( h a - k ½ ) } -
1] ~ - ~ dh.
(18)
4. Finally, the integrals in (13) are "evaluated", leading back to summatjons. One obtains
aM
=
Z~ (-- 1)a*+a'+;h ¢(V~ra) +
ra
1 Xa e-'lha-k-~l'
-=va
]ha --
k_~12
2
(19)
where we used f { e x p (--azh2)/z~h 2} dh = 2.
This expression is a very convenient one for the evaluation of aM; the functions appearing in it have been tabulated extensively 9). Only very few
terms of each of both series are needed to give a very accurate value. This
expression for aM is equivalent to the one that has been derived b y E w a 1d
and S h o c k l e y * ) for a cubic lattice.
The expression (19) is a rapidly converging one as contrasted to (10) which
exhibits a slow conditional convergence only. Yet, going from (10) to (19),
no explicit use has been made of any physical argument, leading to a
uniquely determined value of aM with the proper physical meaning. It can
be shown, however, that the present method is equivalent to a procedure in
which the potential field due to all lattice points within a sphere around
• ) See K i t t e l lo). If o n e c h o o s e s "t~ = ~ in t h e E w a l d - S h o c k l e y f o r m u l a it b e c o m e s i d e n t i c a l w i t h
(19) for t h e c a s e Va = 1. ( T h e f i r s t s u m m a t i o n in t h e E - - S e x p r e s s i o n , as q u o t e d b y K i t t e l , s h o u l d
b e t a k e n o v e r o d d v a l u e s of k o n l y ) .
ON T H E CALCULATION OF LATTICE SUMS
313
the origin is calculated, and where afterwards the limit for infinite ~cadius of
the sphere is taken. This means that the terms are added in order of increasing distance of the lattice points from the origin, which from a physical
point of view seems in m a n y cases to be a sound procedure~
§ 4. Principle o/the summation method. Before applying the summation
method to more general types of lattice sums, we will first, by a qualitative
reasoning, illustrate the underlying idea of it for the case of a one dimensional
series.
Suppose that we have a smooth function [(x) which appl:oaches zero
slowly for x --+ co;/(o) m a y be either finite or infinite. Then the sum
s = E T l(n)
(20)
if it does exists, will converge slowly.
The proposed method consists in introducing an auxiliary function
~(x) for which ~(o) is finite and which approaches zero quickly as x -+ co
(a third condition follows below). We m a y write
S = X~ l(n) ~(n) + X~/(n) {1 -- ~(n)}.
(21)
The first series in (20) will have good convergence whereas the second series
will have the same rate of convergence as the original expression (20).
Since the Fourier transform of a smooth function is a function which
approaches zero rapidly for increasing argument, we m a y expect that the
"flatter" the function [(x) {1 -- ~(x)} is, the better the convergence of the
Fourier transform of Y,/(n) { 1 - ~(n)} will be. (Other examples of this
property of the Fourier transform are the uncertainty relation in quantum
mechanics and the fact that FT{cS(x)} = 1 and vice versa). The third
condition to be imposed on ~(x), therefore, is t h a t /(x) {1 -- ~(x)} should
be a smooth function in the neighbourhood of x = 0 (the smoothness further
away from the origin being assured by that of [(x)).
It is clear t h a t the conditions to be fulfilled by ~ by no means determine
its choice uniquely but leave room for the choice of a "convenient" form
for it, i.e. such a form that conveniently summable series are obtained.
This can for instance require that the relevant functions or parts thereof
are available in tabulated form.
The above method m a y be used for the evaluation of one-, two- as well
as three-dimensional series (cf. appendix 4).
§ 5. Treatment o/a more general type o/lattice sum. In this section we will
treat a more general type of lattice sum, of which the Madelung constant
is a special case. The derivation is carried out along exactly the same lines
as in § 3, but the single steps in it will be considered in a little more detail.
314
B. R. A. N I J B O E R AND F. XV. DE W E T T E
The shortness of the particular m e t h o d of this paper is due to the use of
b-functions in the intermediate steps of the derivation. These b-functions
will be employed in the w a v that is c u s t o m a r y in physics, without any
claim of mathematical rigour.
The lattice sum to be considered in this section is
e2~r/k-r~
S'(RIk, n) :
~-ir
~
_ RI2"
(22)
In principle n can be any real n u m b e r for which (22) exists (cf., however,
the end of this section). E x a m p l e s of lattice sums of this t y p e have been
treated extensively in the literature (e.g. the cases k ---- 0 and kt). It will
be considered here to illustrate the general m e t h o d which has been outlined
in the preceding section.
The first step again consists in expressing S' formally in an integral
S ' ( R ] k , n) =
f
w(r, k)
(23)
[r - - R[ 2~ d r
with
w(r, k) ---- Y,~ b(r -- ra) exp (2~rik. r).
(24)
The next step is to introduce a suitable auxiliary function ~1 and to split
(23) into two parts according to (21). A convenient choice is
~l(r
- - R) = F ( n , = Ir - - R t 2 ) / F ( n ).
(25)
Then
1 - - ~ l ( r -- R) = 7(n, ~ [r -- RI2)/F(n ).
(26)
T'(n, x) is the incomplete gamma~unction 11) defined b y
P(n, x) = f ~ e-t t n-1
dr;
(27)
7(n, x) = F(n) -- P(n, x) = f ~ e-~ t n-1 dt.
(28)
F r o m (25), (26) and (27) it is easily verified that this ~1 satisfies the three
conditions mentioned in § 4.
We can now write (23)
S ' ( R Ik, n) =
1 Ef
- - _P(n)
w(r, k)
F(n,n,r -- R, 2)
[r - - RI 2n
dr +
f
w(r, k)
7(n,n,r--
lr~lZnRt2 , d r
. (29)
It will be convenient to write in the second integral
w(r, k) = ~(r, k) -- b(r) exp (2zdk • r)
(30)
~ ( r , k) = ]~z b(r -- r~) exp (2~ik • r).
(31)
hence
ON THE CALCULATION OF LATTICE SUMS
31
At this stage we a p p l y Parseval's formula to the second integraLof (29).
Using
ETa [~(r, k)] ---- (1/va) {o(h + k) + b(h + k)},
(32)
~(h + k) ---- X~ &{h -- (h A -- k)}
(33)
where
(cf. appendix (A. 12)), and making use of (30), we can write for (29)
S'(R
l [ f w(r, k) F(n, :z [r -- R[2)
]k, n) -- F(n)
Ir _ Rig"
d r --- f b ( r ) e°'~'" 7(n' = [r -- R[ g)
Ir -- R]2dr +
+ Va' {(r(h + k) + b(h + k)} FTa L
[r -- RI2n
A
A"
F r o m applying (A. 4) to (A. 17) (see appendix) it follows that
FT3 [7(n' ~ ]r -- Rig ) ,] = 2,,-31, h~n-3 r ( •
{r -- Rig"
n + ~, ~h2)e ~ " ' "
(35)
f a ( r ) 62w'/k.r 7(n, :~ }r -- Rig)
1 7(n, ~Rg)/R g" for R =# 0
[r -- R[ 2n
d r ---- [ z~n/n for R : 0, n > 0
(36a)
(36b)
Furthermore
and
f 6(h + k) hZn-3 F ( - - n + ~,
a zrh2) e2,~h.R dh
3 :rk2 ) exp (-- Z-rik .R) for k :/: 0 (37a)
_ J k ~"-a F ( - - n + ~,
-- [ :~-~+%/(n -- {)
for k=O,
n >~-.
(37b)
Making use of these relations and carrying out step four, i.e. replacing the
integrals b y the summations which t h e y represent, we finally find as the
general expression for (22)
1 I
S'(RIk, .)- r(.)
F(n, :r IrA -- R[ 2) e2w/k.rA
r(n,
~rR 2)
Rg-
+
4 v~2a-~ a]2ZA [hA -- kl s~-3 F ( - - n + ~, ~ [h A -- klg)e 2'a<"A-k)'R1 . (38)
The expression in this form holds for R :# 0 and k =# 0. For the case R = 0
(k :/: 0) the second term in the right hand side should be replaced b y its
limit value :rn/n; the series then only converges for n > 0. For k = 0 the
second s u m m a t i o n should carry a prime and the term ~n/Va(n -- ~) should
3
be added. In that case the sum only exists for n > ~.
316
B. R. A. N I J B O E R AND F. W. DE W E T T E
Instead of = as a factor in the a r g u m e n t of the incomplete g a m m a
function (cf. (25)), a n y a r b i t r a r y n u m b e r = could of course have been
choosen. However, the particular choice = = = leads, as can be verified
from the a s y m p t o t i c expansion for the incomplete g a m m a function, to the
same r a p i d i t y of convergence for both series in (38).
The incomplete g a m m a function F(n, x) has been t a b u l a t e d for various
values of n (cf. ref. 12 p. 206 and ref. 13). In the most c o m m o n l y occurring
case n is either integer or half integer. Because of the recurrence relation
/'(n -+- 1, x) = n F(n, x) + x n e-°~,
F(n, x) then can always be related to
F(1, x) =
e-x or F(O, x) = -- El(-- x)
and
v(~, x) = V ~ Erfc (V~)
respectively. These functions have been t a b u l a t e d extensively 9) 12).
§ 6. Treatment o/the lattice sum
Szm(o]k, n) = El Yz,,(o, ¢) e'-''~kra
r~ ~ + z
= 0, 1, 2 . . . . .
----- 0, 4 -
1 ......
(39)
~
l
'
where we employed the usual notation for spherical harmonics"
• Yam(O, ~) = (2•) -½ P-~(cos 0) eim$. ~ is the normalized associated Legendre
function. 0 and ¢ are the polar angles of r a with respect to an arbitrarily
chosen cartesian coordinate system.
The sum (39) is the self potential corresponding to Sz'r~(R]k, n) of which
S'(R]k, n) (22) is the special case l = 0. For simplicity we will restrict our
considerations here to sums of the t y p e (39). The extension to the case
R :# 0 is straightforward.
We will not carry out the derivation, which is completely analogous to
t h a t of § 5, in detail but only write down the most i m p o r t a n t formulae. A
convenient form for the auxiliary function ~2 is
a2(r) ---- F(n
q- l, =r2)/F(n q- l).
(40)
It is shown in the appendix (A. 13) t h a t
FTa ~ ~,(n + l, ~r2)2n+zYtm(O'
r
¢) ~ =
= il~z~n+~-31o-h9"'+l-a F(-- I* + ~, ~zh2) Yzm (*9,~p), (41)
where/~ and ~ are the polar angles of the vector h, taken with respect to the
ON THE
CALCULATION
OF LATTICE
317
SUMS
same coordinate system as 0 and $. Furthermore it is easily verified that
f~(r)
~27r/k.r
y(n + l,
~y2)
r2n+ l
Yzm(O,$)dr = 0
for l > 0 , n + l
>0
(42)
and that
f~(h + k) h 2 n + l - 3 F(-- n + 3, nh 2) Yzm(O, cp) dh =
k 2n+1-3 F(-- n + 3~, ~rk2) ( _ 1 ) Z Ytm(Ok, ~k) for k ~ 0 (43a)
0 fork=
0,1 > 0 , n > { ;
(43b)
0k and 9k are the polar angles of k.
Using (41), (42) and (43a) we have for the case k ~ 0
,
1
[ - , , F(n -}- l, ~rr'~) Ylm(O, dp) e2;'r/k'rA
S t,,,~(olk, n) -- F(n + l) L ~
il~z2n + l - a / 2
+--
Va
r~ n+'
- -
+
-I
/
3
Z~ lh~--kl2n+l-a F(-- n + ~,,'z
[hA--kl2) Ym~(Oha_k, %,A_k) . (44)
(/>0,
n+!>0)
For k ~- 0 the second summation should be read with a prime ; the sum then
only exists for n > :~ (cf. (43b)). This is also the case when k is equal to any
other reciprocal lattice vector, as is easily verified.
Knowledge of lattice sums of the type (39) is of interest for various
problems. If, for instance, one expands the potential field around an arbitrary lattice point of an ionic lattice in terms of spherical harmonics, the
coefficients of this expansion can be expressed in terms of the sums
t
S~m(olk.~, ½), as can be shown very easily. Of course this potential is also
simply S'(R[k v ½), but as this sum involves the distances r A - R, it is
sometimes less convenient. This is just a simple example of a general
connection, existing between the lattice sums S'(R[k, n), S;,~(RIk, n) and
t
Stm(o[k, n), to which we hope to come back in a subsequent paper.
In some cases the sums (39) can, without, any calculation, be shown to
vanish on account of the symmetry of the lattice considered. This holds
t
in particular for the sums Sl,~(olo, n). Since, for example, in a Bravais
lattice the origin is a center of inversion and since further Y~m(O', ~') =
---- (--1)~ Yzm(O, ~) (0' ---- ~r -- 0 and ~' ---- ~r + $ being the inversion angles
t
of 0 and ~) all S~,~(o[o, n) will vanish for odd l, because of a complete cancellation of terms. This is true independent of any additional symmetry of the
lattice. An other example is that of the cubic lattice, where the question
t
which sums Sl~(olo, n) vanish and which do not, is obviously closely related
to the well-known problem of the cubic spherical harmonics, i.e. the problem
which linear combinations of Ytm's have the cubic symmetry.
318
B.R.A.
N I J B O E R AND F. W. D E W E T T E
Remark on the use o/the vector k (cf. (7), (8)).
of the reciprocal space vector k, appearing in the
general lattice sums (22) and (39), allows for
variety of situations b y these two expressions.
Using r a = Y~,2ia~ and k = Y~ikibi (i = 1, 2, 3)
that
k • r~ = E, ki2,.
The freedom in the choice
factor exp (2~i k - ra) in the
the description of a wide
it follows from a,. b, = 6,j
(45)
The special choice k ---- ½(bl + b2 + b3) leads, as follows i m m e d i a t e l y
from (45) to exp ( 2 ~ i k . rA) ---- (--1);h+;~2+kL which can be applied to the
case of an ionic lattice (cf. § 3). More generally k can be chosen in such a
w a y as to describe lattices in which the charges or multipoles v a r y in a
periodic w a y as regards magnitude or orientation.
APPENDICES
1. Lattice sums and Fourier trans/orm. The three-dimensional
trans/orm of a f u n c t i o n / ( r ) is defined as
FT3 {/(r)} = f / ( r )
exp (2~ih. r) d r ~- F(h),
Fourier
(A. 1)
the inversion is
FT3 ~F(h)} = f F(h) exp (-- 2.~tih. r) dh = / ( r ) .
(A. 2)
From (A. 1) it follows in particular that
FTa {/(r) exp (2~ik • r)} = F ( h + k)
(A. 3)
FT3 {/(r - R)} = F(h) exp (2nih. R).
(A. 4)
and
A well-known formal expression for the 6-function is
6(r -- ~) ----f exp {-- 2~zi h - (r -- ~)} dh,
for which we clearly can write
6(r -- ~) ---- FT3 {exp (2nih • ~)}.
(A. 5)
FT3 {6(r -- ~)} = exp (2zdh • ~).
(A. 6)
B y inversion
In a similar fashion one can derive the Fourier transform of the function
~ 6 ( r - r~), which has 6-singularities at the lattice points and which is
zero elsewhere. One finds
FT3 {~;~ 6(r -- r~)} ----- ~,~ exp (2~ih • rA).
(A. 7)
The sum in the right hand side of (A. 7) is, except for the multiplying factor 1/va, presicely a s u m m a t i o n of 6-functions over the reciprocal lattice
points.
ON T H E CALCULATION OF LATTICE SUMS
319
2. Proo/ o/ the relation
~A exp (2~ih • rA) = (1/Va) Y4 6(h -- ha).
(A. 8)
We start out from Poisson's sum/ormula 14)
+oo
~,~=-oo/(x
-- n) = Y,+Z-oo exp (2zdvx) f_+~ [(~) exp (-- 2hive) d~. (A. 9)
Inserting for/(x) the function b(x) we find
+co
X,~=-oo 5(x -- n) = X+_°°_ooexp (2zdvx).
Using r A -~ Yn~,ai
(A. 10)
(i = 1, 2, 3) we can write for the left h a n d side of (A. 8)
•A exp (2zdh • rA) ---- XA exp {2~i •, ~t,(h • a,)},
which, b y virtue of (A. 10), leads to
~a exp (2~ih • rA) = Y~z {H* 5(h- a, -- ~,)}.
The integers R, (i ---- 1, 2, 3) m a y be written as ~, ~ - h a . a , ,
i m m e d i a t e l y from (4) and (6). We t h u s can write
as follows
lEA exp (2zdh • rA) ---- Y4 (1-[i 8{(h -- ha) • ai}].
Let the basisvector a, have the cartesian components a O, then the 9 numbers
ait (i, i = 1. 2, 3) determine a transformation, the m a t r i x of which we will
denote b y A. Then the numbers h • a, (i = 1, 2, 3) are the components of the
vector Ah which results from applying the transformation A to h. Using
5(r) -----5(x) 8(y) 5(z) we can writte, therefore,
Y,A exp (2zdh • rA) = Y~A[~{A(h
-
-
hA)}].
If we now use 5(Ax) = (1/]det A[) b(x) and Idet A] ---- va we arrive at
]~A exp (2zdh • rA) = (1/va) ]~A 8(h -- ha).
This completes the proof of (A. 8). Finally, from (A. 7) and (A. 8)
FT3 {Y~A6(r -- rz)} = (1/Va) 5",A 5(h -- ha).
(A. 11)
Applying (A. 3) to (A. 11) one finds
FT3 lEA 6(r -- rA) exp (2~ik • r)] ---- (1/va) Y~A6(h + k -- ha). (A. 12)
3. Proo/ o/ the relation
FTa ~ Y(n + l, ~zr2
~'n+zYlm (Or' ~r) ] =
= izn~n+l-~/: h2n+Z-a V(-- n + ~, nh 2) Y~m(~gk, 9h). (A. 13)
We will here give the proof of the reverse statement, viz. t h a t the FTa of
the right h a n d side is equal to the bracket expression on the left h a n d side.
320
B. R. A. N I J B O E R
A N D F. W. D E W E T T E
One can easily show t h a t 15)
F T 3 [g(h)Y~m(~gh, ~Ph)] = 2z~(-- i) l Ylm(O r, Cr) r - ' / ' ( ~ g ( h ) Jz+~I: (2,xhr)h']2 dh.
a0
Applying this to the left h a n d side of (A. 13) we have
F T 3 [it~ 2n+t-~'-" h 2n+l-3 F ( - - n + ~., ~h 2) Ylm(t9 b, q~h)] =
h 2n+t-U°- F ( - - n + ~, ~h 2) Jt+,/~ (2rehr) dh -- Ia
= Br-%
where
B = 2~ 2n+L-'/3 Yzm(O,, ¢,).
Inserting the expression for the incomplete g a m m a f u n c t i o n (cf. (27)) we
have
I1 = B r-'/'-
_
__
Z
dh h 2n+L-~/°-Jz+,:° (2~hr
"
B
(2~) 2n+l-'1"- r 2"+z
j-i °
,fl° e- x x - n d x =
h2
e - x x -n+'/+ dx
+-+,++
~o
t 2n+l-"l" Jr+,~, (t) dt. (A. 14)
This must be shown to be equal to
12
?(n + l, 7~r2) Ytm(O r, $r)
r2n+l
--
Ylm(Or, $r) (+r2
r2n+l
Jo
e -~
x n+~-I
(A. 15)
dx.
In other words we must prove the equality
(A 16).
r 2n+t 11 = r 2n+l 12.
It is obvious t h a t for r = 0 the equality holds. If we now can prove t h a t the
derivatives of both sides of (A. 16) with respect to r are equal t h e n relation
(A. 13) is established.
One finds after a short derivation
e -u2 uZ+3/, f~+,/.. (2rv/~u) du =
d (r2n+z I m ) = Brr -n-'l'z+'l' r 2"+z-'/'dr
J0
=
2~ n+t r 2n+2t-1 Ytm (0,, Cr) e--Trr2,
(l ~> - - 3).
For the value of the integral see for instance reference 1 ! p. 177. It is easily
seen t h a t (d/dr)(r 2n+t 12) leads to the same expression. This completes the
proof of (A. 13).
Since the proof holds for all l > -- ~3 we have for l = 0
F T 3 [~(n,z~r2)/r 2n] = ~d"n-''' h2n-3"F( -
n + 3 ~zh2).
(A. 17)
4. Note on evaluation o/ one- and two-dimensional series. The method, employed in this paper for the evaluation of three-dimensional series, m a y as
ON T H E C A L C U L A T I O N OF L A T T I C E SUMS
321
well be used for the evaluation of one- and two- dimensional series. For
sums of the type (22) but now one- or two-dimensional, the same auxiliary
function as was used in the three-dimensional case, viz. ~(r) = I'(n, :~rg')/F(n)
can be employed. For reference we give the one- and two-dimensional
Fourier transforms of the relevant functions
FT1 {~(n, z~r2)/r 2n} = zc2n-'l', [hi 2n-1/'(--n + ½, ~h 2)
FT2 {7(n, 7~r2)/r 2n} = ~2n-1 hg.n-2 F ( - - n + 1, zrhg").
The derivation of these relations is completely analogous to that of (A. 17),
with which t h e y should be compared.
Received 26-1-57.
REFERENCES
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
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14)
15)
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E w a l d , P. P., Ann. Physik 64 (1921) 253.
E v j e n , H. M., Phys. Rev. 89 (1932) 675.
M i s r a , R. D., Proc. Camb. Phil. Soc. 36 (1940) 173.
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Cf. for instance: T i t c h m a r s h , E. C., Introduction to the Theory o! Fourier Integrals, 2ud. ed.
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K i t t e l , C., Introduction to solid State Physics, John. WHey, New York 1953, p. 347 ff.
Cf. for instance, M a g n u s , W., O b e r h e t t i n g e r ,
F., Formeln und Satze etc. Springer, Berlin
1948, p. 125.
F l e t c h e r , A., M i l l e r , J. C. P., R o s e n h e a d , L., An Index o[ mathematical Tables, McGrawHill, New-York 1946.
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Cf. for instance: M a d e l u n g , E., Die math. Hil]smittel des Physikers, Springer, Berlin 1950,
p. 69.
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Physica X X I I I
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