The electrostatic energy of a lattice of point charges

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The electrostatic energy of a lattice of point charges

J. Olives

To cite this version:

J. Olives. The electrostatic energy of a lattice of point charges. Journal de Physique Lettres, Edp

sciences, 1985, 46 (24), pp.1143-1149. �10.1051/jphyslet:0198500460240114300�. �jpa-00232949�

L-1143

The electrostatic _{energy of} a lattice of point charges

J. Olives

CRMC2, Campus ^de Luminy, ¹³²⁸⁸ Marseille Cedex 9, ^France (Reçu ^le 13 juin 1985, accepte sous forme definitive le 21 octobre 1985 )

Résumé.

L’énergie électrostatique limite par maille ^E d’un réseau ionique ^de charges ponctuelles

est définie

la valeur limite de l’énergie d’un cristal fini divisée par

nombre de mailles

(lorsque

nombre tend

+ ~). ^{La valeur} explicite ^{de E est} calculée, ^{et reliée à} l’énergie ^d’Ewald.

Cette énergie E dépend de la suite des cristaux finis croissants (plus précisément, de la suite des ensembles finis croissants d’ions). ^En d’autres termes, et brièvement : E est

énergie volumique qui dépend ^{de la surface.}

Abstract

The limit electrostatic energy per cell E of a ionic lattioe of point charges is defined

the limit value of the energy of

finite crystal ^divided by its number of cells (when this number tends towards + ~). ^The explicit ^{value of E} is calculated and related to the Ewald energy. This energy E

depends

the sequence of the increasing ^finite crystals (more precisely,

the sequence of the

increasing ^{finite sets of} ions). ^{In other} words, ^and briefly : ^{E is}

volumic energy which depends

the surface.

LE JOURNAL DE PHYSIQUE - LETTRES

J. Physique ^{Lett. 46} (1985) L-1143 - L-1149 15 DÉCEMBRE 1985,

Classification

Physics ^Abstracts

41.10D - 61.50L

1. Introduction.

The electrostatic energy of ^an ionic lattice, in which the ions are considered as point charges (with electrical neutrality ^{of the} cell), may be written as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:0198500460240114300

L-1144 JOURNAL DE PHYSIQUE - ^LETTRES

where n

n, 81 + n2 a2 ⁺ n3 a3 and p

pl 81 + P2 a2 ⁺ P3 a3 (81’ a2, a3 ^are cell vectors;

n1, n2, n3’ Pl, P2’ P3 ^are integers); ^{s, t are the} positions of the ions of the origin ^cell (n = 0); ~

is the charge of the ion s. Ewald [1] and ^Bertaut [2] considered that this energy may be written as

where m

n - p is supposed ^{to run over} all the lattice vectors; ^thus

where N is the number of cells, ^which leads to the electrostatic energy per cell :

Ewald [1] ^{and Bertaut} [2] obtained, ^{for this} energy, the value

with

where V is the volume of the cell; ^{h is a vector of the} reciprocal ^lattice; F(h) = L ^{qs e21tih.s;}

s

p and _cp are defined by

(p(u) depends only ^{on u} = 11 u II ^and qo) ^{on h}

II ^h II ), the function a being non-negative,

of spherical symmetry and such that

De Leeuw et al. [3] calculated the value of Eo, ^{for a cubic lattice, and with the} spherical ^mode

of summation :

they ^found

with the values of E1, E2 ^and E3 ^obtained by ^{Ewald and}

where a is the side length ^{of the} cubic cell and M = L ^{qs s} (dipole ^{moment of the} cell). ^Smith [4]

s

extended the preceding ^{result to} other modes of summation.

In this _paper, we present ^two results : in section 2, the calculation of Eo, and its relation to the Ewald _energy EE ; in section 3, ^the calculation of the limit electrostatic energy per cell E (defined

in section 3), and its relation to Eo (and ^{then to} EE). These results hold for an arbitrary ^lattice

of point charges - in which the cell is electrically ^{neutral, but its} dipole ^{moment may} be different from 0 - and for arbitrary modes of summation.

2. Calculation of Eo.

The mathematical proof corresponding ^to this section has been given in another _paper [5]. ^We

have observed that the result (3)-(4) of Ewald and Bertaut _may be considered as an incorrect

application ^of Poisson’s formula to the function f

since f ^{is not} continuous at h

0 (if ^the dipole ^moment of the cell M = L ^{qs s} is different from 0).

s

We have then replaced the function f by ^a sequence of continuous functions _fN ^{- to which}

Poisson’s formula _may be applied ^{- and we} have taken the limit of Poisson’s formula as

N -> + oo.

We obtain the following result, ifM =1= 0 :

Let (Bk) be any sequence of finite subsets of 7L3 such that

(0.0) ^{" 1}

-

3 cos2 Om

h fi

I.

k

( ) ^7L3

(iii) y 11 M 113 ^{has a finite} limit as k - + oo ; notations : m

(ml, m2, m3 ^E z3 ;

m e Bk~{0} II m II

and 0m ^{is the} angle ^{between M and m.}

L-1146 JOURNAL DE PHYSIQUE - ^LETTRES Then

exists and is equal ^to

where E1, E2 ^and E3 ^are given by (4) ^and

If M

0, ^{the sum}

is absolutely convergent ^and

The term E4 generalizes ^the expression (7) ^found by de Leeuw et ale [3] ^{for cubic} crystals (Fel-

derhof [6] obtained this term from a virtual uniform polarization M/ V). ^{The terms} E. ^and E~

represent ^{two sums of the same}

family

of numbers M’(1 - ^{3 cos2 0 )B} with the mode of

represent ^{two sums of the same} family of numbers 2 II m 113 m * 0’ ^with the mode of B 2~m~ 11 /~~~

summation (Bk) ^for E. (as ^for Eo), ^{and the} spherical ^{mode of summation for} ~. The contribution

E5 - ES ^{is generally} different from 0, if M =A 0 ^and (Bk) ^{is not the} spherical ^{mode of summation.}

3. The limit electrostatic energy per cell E.

We _may note that the considerations of the introduction leading ^to (2) ^are contradictory : ^the crystal ^is supposed ^{to be at the same} time infinite (m ^{runs over the whole} lattice) ^{and finite} (N cells).

The limit electrostatic energy per cell ^E may be correctly ^{defined as the} limit of the energy of ^a finite crystal (formed by ^a finite number of cells) ^divided by its number of cells (when this number tends towards + oo ) :

where (Ak) ^{is an} increasing sequence of finite subsets of7L3 such that U Ak = ^7L3 (each Ak represents

k

the shape ^{of a finite} crystal.).

3.1 RESULT. - Let (Ak) ^be any sequence of finite subsets of 7L3 such that

3) if M 96 0, ^we suppose that I - 3 cos’ 0. ^{has a} finite limit k _{+ oo;} notation 3 I M =1=

we suppose ^{t at}

~eB~B{0} ^{11 M 113}

~

as a Inlte 1IIllt as

+ 00; notatIons:

Bk

{ ~ 2013 ^{/?! ~ and} /? e A~ }; 0.

angle ^{between M and} m;

4) there exists (f(k)), a > 0, & ~ 0, ^oc > 0, ^{such that} 4a) ~.~) ~ f(k), ^{for all} k; notation : d(k)

sup 110 II; ;

neA~e

4b) 1 - c(A~ ~) ~ b . 11 f(k) m 11 " ^for all ~ and ~ e B~; ^notation: c(A~ ~) = card (Ak card n ~20132013~;

4b) ^I

c(A", ^{m) :::;:;} B ^{7 (~) /} for all k and mE B,,; ^notatIOn: c(A", m)

card A~ ^; 4c) if M :0 0, ^we suppose that for almost _every x e ~3, c(Ak, Mk) ^{has a} finite limit as k + oo,

which is denoted by c(x). Mk is defined by ^{x E} -77~ ^{7W (Mk + C); C}

^{[0, I [3 ;}

4d) if M =A 0, ^we suppose that for almost _{every x} e R3,

~ ~ ^(Bk

^{(~) has a} limit as k _{+ oo,}

y~(B~+C)

which is denoted by ~aM- ~A is a characteristic function of a subset A of ~3.

Then the limit electrostatic energy per cell E defined by (13) ^{exists and is} equal ^to

where

if M :A 0 (0. = angle ^{between M and} x); and, ^{if M = 0}

where

(the ^{sum on m is here} absolutely convergent). ^{In all cases, the value} of Eo ^is given in section 2.

3.2 EXAMPLES. - If

we have

All the above conditions 1) ^to 4) ^are satisfied : condition 3) ^{has been} proved ⁱⁿ [5], ^{and condition}

4) ^{is satisfied with} f(k) = d(k) = k ^{and oc}

^{1. Since} c(x) depends only ^on II x II, ^the integral E6 ^is

L-1148 JOURNAL DE PHYSIQUE - ^LETTRES

equal ^{to 0} (with ^{M as} polar axis, ^the integral ^{on 0 is} equal ^to 0), ^{so that}

with

if M ~ 0, and

if M

0.

If

(we note I n I

sup { ~ nl I, I n2 ^U I n3 I ~)~ We have

and the above condition 4) is satisfied ([(k) ^{= k and a}

1). ^{If condition} 3) ^{is also} satisfied, ^we

may then write where

if M ~ 0, or where

if M

0.

4. Numerical example.

As a numerical example, ^{we have} computed the different terms in a simple ^{case :} orthogonal

cell vectors aI, a2, a3 with II a1 II - a2 II

1J! a3 II = 2, qs = ⁺ 1 at s

0, _{qs - -} ^{1 at}

s 1

= 2" (a1 ^{+ a2 + a3).}

Calculation of Eo : ^we have found : El E2 ⁺ E3 EE ^{= - 0.74} (calculation ^{time - 5} s), E4

^1.57, E:5 = 1.33 with the « I n I mode of summation » (Bk = ^{f n I I n I}

sup(! n1 I, I n2 I~

I n3 I) k }), ES - ^1.82 (calculation ^{time - 20} s); ^{thus :} El - E2 ⁺ E3 ⁺ E4 - ^0.83 (calcula-

tion time - 5 s), El - E2 + E3 ⁺ E4 + ES - ES - ^0.34 (calculation ^{time 25} s). ^These

last two values have been also obtained by ^the direct calculation (8) ^of Eo, ^with, respectively, ^the spherical mode of summation and the I n mode of summation (with calculation times of ~ 15 s).

All the calculations were made on a 16 bit minicomputer (PDP 11 /45 ^of DEC).

Calculation of ^{E : With the} spherical shape (16) ^and according ^to (17) ^and (18), ^{we obtain}

E

0.83 (from ^the above value of Eo). ^With the I n shape (20) ^and according ^to (21) ^and (22), ^we

obtain : Eo = ^{0.34 (see} above), E6 = - ^0.17 (calculation ^{time 40} min), ^{so that E}

^{0.51. These}

two values E

0.83 (spherical shape) ^{and E}

^0.51 (I n shape) may be directly calculated with the help of (13), ^but the convergence is extremely slow. For the spherical shape, ^{we have obtained,}

E

0.66 at k

8 (calculation ^{time 5} h) ^{and E}

^{0.69 at k}

¹⁰ (calculation ^{time 15} h); and, ^for

the I n I shape, E

^{0.43 at k}

⁸ (calculation ^{time 20} h) ^{and E}

^{0.44 at k}

¹⁰ (calculation

time 62 h). These results show that the expression (14)-(15) ^{of E is much more} interesting ^for

calculation than the definition (13).

5. Conclusion.

The results presented in sections 2 and 3 show that - under the conditions 1) ^to 4) of section 3 .1 - the limit electrostatic energy per cell E (defined by (13)) exists and is related to the Ewald _energy

EE by

where the terms E4, ES ^and ES ^are given by (10) ^and E6 by ( 14).

The finite crystals ^{- used to define E - are formed} by ^{the sets of ions}

(n ^are the lattice vectors which give ^the positions ^{of the} cells; ^{s are the vectors which} give ^the positions ^{of the} ions of the cell n

0).

For a given crystal, each Sk depends ^on Ak ^{and on} the cell which has been chosen. The depen-

dence of all the terms E4, E5, E~ and E6 ^{on M} (which depends ^{on the} cell), clearly ^{shows the} dependence of E on the choice of the cell (for ^a given crystal). ^The dependence ^{of E on the} sequence

(Ak) appears in the terms Es ^and E6 ((Bk), ^{B and} c(x) depend ^on (Ak)).

References

[1] EWALD, ^P. P., ^Ann. Physik ⁶⁴ (1921) ^253.

[2] BERTAUT, F., ^J. Phys. ^{Rad. 13} (1952) ^499.

[3] ^DE LEEUW, ^S. W., PERRAM, ^{J. W. and} SMITH, ^E. R., Proc. R. Soc. London A 373 (1980) ^27.

[4] SMITH, ^E. R., Proc. R. Soc. London A 375 (1981) ^475.

[5] OLIVES, J., 1985, ^{submitted to J.} Phys. ^A.

[6] FELDERHOF, ^B. U., Physica ^{A 101} (1980) ^275.