Download CRYSTALS *)

De W e t t e , F . W .
1959
P h y s i c a 25
1225-1240
EFFECTS OF ATOMIC POLARIZATION IN IONIC
CRYSTALS *)
by F. W. DE W E T T E
Instituut voor theoretische Fysica, Rijksuniversiteit te Utrecht, Nederland
Synopsis
T h e effect of h i g h e r o r d e r a t o m i c p o l a r i z a t i o n s on t h e cohesive e n e r g y of ionic
crystals is t r e a t e d o n t h e basis of a m o d e l a c c o r d i n g to w h i c h t h e c r y s t a l c o n s i s t s - o f
polarizable n o n - o v e r l a p p i n g e l e c t r o s t a t i c c h a r g e d i s t r i b u t i o n s . B y m e a n s of m e t h o d s
d e v e l o p e d in p r e v i o u s papers, t h e m u l t i p o l e s i n d u c e d in t h e a t o m s b y t h e c r y s t a l l i n e
field are c a l c u l a t e d in a s e l f - c o n s i s t e n t way. A general f o r m u l a for t h e effect of t h e s e
polarizations o n t h e cohesive energy" of ionic c r y s t a l s is d e r i v e d a n d n u m e r i c a l l y
e v a l u a t e d for t h e c r y s t a l s of KC1, K I , CsC1 a n d CsI.
§ 1. Introduction. The bulk of the cohesive energy of ionic crystals is a
result of the Coulomb attraction between the positive and the negative ions.
This Coulomb energy is usually calculated on the basis of a point-ion model
of the crystal. In a similar fashion, in the theory of macroscopic polarization of matter by an external electric field, the atoms are always considered
as point dipoles. Actually, the ions and atoms are charge distributions
of finite extent which are deformable (polarizable) in the crystalline field.
In the present paper we will investigate the effect of this ionic deformation
on the electrostatic binding energy of ionic crystals. The deformation
correction to the Madelung energy (the cohesive energy of the point charges)
turns out, for the examples calculated, to be of the same order of magnitude
as the zero-point energy of the crystal.
Another effect which was studied in I is the influence of the atomic
deformation on the dielectric properties of a crystal. This effect turns out
to be of little practical importance for the above mentioned ionic crystals
and even less so for the noble-gas crystals of A and Xe. We will therefore
not discuss these effects.
Our considerations will be based on a classical model of a crystal as
consisting of non-overlapping electrostatic charge distributions. A convenient way of describing the deformations of such charge distributions
is in terms of the higher multipole moments which are induced in them by
*) Based on part of a-doctoral dissertation entitled: Electrostatic Fields in Ionic Crystals t) (m
the following to be quoted as I). For many details we will refer to this dissertation.
-- 1225 -Physica 25
1226
F.w.
DE WETTE
the crystalline electric field. We have laid the foundation for such a description in a previous publication 2) (to be quoted as II), in which we derived
expressions for the crystalline potential in multipole lattices.
In § 2 we will derive and discuss the self-consistent equations for the
self-induced higher order multipole moments in an ionic lattice. In § 3 we
derive an expression for the electrostatic cohesive energy of such a crystal
and in § 4 we discuss the numerical calculation of this energy for the crystals
of KC1, KI, CsC1 and CsI. After a discussion of the method and the results
in § 5, we conclude by mentioning some relations existing between lattice
sums and b y quoting values for some of the lattice sums for cubic lattices
(appendix).
§ 2. The sell-induced multipoles in the cha~e distributions o/ an ionic
lattice. We first derive the equations for the higher multipole moments
which are induced in the charge distributions of an ionic lattice by the
crystalline electric field, if these charge distributions are polarizable. In II,
§ 6 we derived an expression for the electrostatic potential in a lattice with
arbitrary charge distributions in the unit cell (forming a neutral basis), as an
expansion in spherical harmonics around a lattice point (cf. eq.(1) below).
If a polarizable charge distribution, ~dth no permanent multipole moments,
is placed at the origin in such a potential field, then the Y*k,n-components
of the potential will induce corresponding multipole moments Qk,n in the
charge distribution. In an ionic crystal all lattices sites are occupied by
such polarizable charge distributions. While the net charge and all the
higher multipoles of a particular charge distribution contribute to the
crystalline potential everywhere else in the lattice, these same higher
multipoles themselves are induced b y the potential around that particular
site, which in turn is a result of the net charges and the multipoles in the
rest of the lattice. It is therefore possible to determine, by a self-consistent
procedure, the multipoles which are in this way induced in all the charge
distributions of the lattice. The same procedure can of course be followed
when an external potential field is present.
When a polarizable charge distribution is placed in an external potential
field, the induced multipole moments are completely determined by the
polarizabilities of the charge distribution. These polarizabilities depend
solely on the internal structure of the charge distribution. For the moment
it suffices to introduce these polarizabilities as the proportionality factors
between the inducing potential field components and the induced multipole
moments. For the following discussion we assume the charge distributions
to be of such a structure that a certain k, n-component .of the potential is
only effective in inducing a multipole moment of the same k and n. I.e. we
assume the polarizabilities to be scalars.
As in II, § 6, we consider a lattice with y different types of charge distri-
1227
E F F E C T S OF ATOMIC P O L A R I Z A T I O N IN IO N IC CRYSTALS
butions (labelled b y i, / = 1,2 ..... y). The inducing potential around a
lattice site of the t y p e i is then according to II, (35)
V")(R) = Y~lc,nA(i)k,nRkY*k,n(O, q)),
(1)
where A")k,n is given b y II, (38), viz.
A(Ok,n :
(4zt/2k + 1) ~'k,n(0[0, ½) +
+ ~Ll>o,,,,F t,,,,;~.,,,~Q
~ ") t,m S")'k+,,,,-m(OlO, .~) +
+ Y,~i~ Qwt,,,~ S(J)k+l,~--,,i (R¢,I0, ½)}.
(2)
®' is a lattice sum over a composite lattice, the basis of which contains
the charges q¢,
~'k,,, (0 !0, ½) = Z,{ E j q j r a , f "
Y~,m(Oa,j, epa,j).
(3)
r,~,j is the length of the lattice vector• 2, standing for X1, ;t2, 2a, is the ceUindex, 1 is the base index• The prime on the s u m m a t i o n sign means t h a t
the term ),~ = 22 = 2a = 0 is excluded from the summation. S (') indicates
a lattice sum over a simple Bravais lattice, viz.
S")~,,,(R',O, ½) = E(')alra -- RI - k - t Ye,~(O,a, ~r,)i).
(4)
R indicates the point at which the sum is evaluated. The quantities Qt.m
are the multipole moments of the ionic charge distributions. For a charge
distribution with density p(r, O, do) t h e y are defined b y
Qt,m = (4~/21 + 1)fp(r, O, dp)rt Yt,m(O, dO) dar.
(5)
Finally, Fz,m :e,), is a numerical factor defined b y
Fz,,,,..~., ~ = ( _ l ) t - m 2 ~ / ~ •
2l+
• (2k+l)(2k+21+l)
1
(k+l+n--m)!(k+l--n+
m)!l -~
(l + m) ! (Z -- m) ! (k + n) ! (k -- n) ! ] "
(6)
For further details see II.
The induction equations for the multipole moments which are induced
by the potential (1) in a polarizable charge distribution centered around
the i th lattice site reads
Q")k,,, = -- a")k 4")k--,n.
(k > 0)
(7)
The coefficients c~(i)k are the polarizabilities of the charge distribution of the
type i. For charge distributions which are spherically symmetric when not
in an external field (which is the only type t h a t will be considered here),
does not depend on n. (The minus sign in (7) assures agreement (in the
case k = 1) with the usual definition of the dipole polarizability a by means
of ~t ---- ~E, where/~ is the induced dipole m o m e n t and E the electric field
strength).
1228
F . W . DE.WET'rE
The sell-consistent equations determining the multipole moments Q(l~k,n,
which are induced in the charge distributions b y the crystalline potential,
now follow from the insertion of (2) into (7) which leads to
(llota'k)Qal~,n + (4~r/2k + 1) ~'~,n(O[O, ½) +
+ El>o,,,~Fl,,,~;k,,,{Q'*h,mSCi~'tc+t,n-re(O I 0, ½) +
+ ZJ¢{ Q'Jh,mS~J~k+~,n-m(Rj*10, ½)} = 0.
(k > 0 ; i , i =
(8)
1,2 . . . . . r)
Subject to the condition of non-overlapping charge distributions (cf. II,
§ 6) this infinite set of equations is exact. I.e., given the values of the
polarizabilities a~*lk and of the lattice sums ~'k,n and Sk+l, m-n, the sets of
self-induced multipole moments Q'i~k,n can, in principle, be determined
exactly. In actual calculations, of course, one v~ill have to break the set off
after some finite value of k.
In the following we will restrict ourselves to lattices such that each
lattice point is a center of inversion. For such lattices all the lattice sums
~'~,v and S~, v vanish for odd #,. By inspection of (8) it is thus seen that in
these lattices there exists no coupling between the sets of odd and even
multipoles (odd and even k respectively).
This means t h a t in a lattice containing polarizable charge distributions with only
a net charge but no permanent higher multipoles (e.g. permanent dipole moments ),
only even multipoles will be induced, since the existence of the crystalline potential is
after all due only to the presence of these net charges (permanent monopoles). In the
case where permanent dipoles are also present, an additional set of odd multipoles
will be induced, but there will exist no coupling between both sets of multipoles. If
finally, only dipoles are present (no net charges), then only the set of odd multipoles
will be induced.
Symmetry
considerations.
So far we have not mentioned the
lattice symmetry except for the inversion symmetry which is a property
of all Bravais lattices. Obviously, the special symmetry of the particular
lattice that is considered will cause further simplifications of the selfconsistent equations. A first simplification is due to the fact that the
crystalline potential must exhibit the symmetry of the environment of the
lattice site around which it is taken. This means that only certain multipoles
can have values different from zero. A second simplification, related to the
first, results from the fact that for a lattice of a given symmetry, only
certain lattice sums will have values different from zero. This can cause a
considerable simplification of the self-consistent equations for the remaining
non-vanishing multipole moments.
In this connection two considerations are of interest:
1. The 2k + 1 functions YA-,. form an orthogonal basis in the (2k + l)-dimensional
representation space of the three dimensio~ml rotation-reflection ~roup R-* (3). The
cO,stallo¢raphic point groups are finite subgroups of R± (3). Only those YA.,n's will
EFFECTS OF ATOMIC POLARIZATION IN IONIC CRYSTALS
1229
a p p e a r in a n e x p a n s i o n of t h e t y p e ( 1), w h i c h b e l o n g t o a l i n e a r c o m b i n a t i o n (for fixed
k) i n v a r i a n t u n d e r o p e r a t i o n s of t h e p a r t i c u l a r s y m m e t l y g r o u p p e r t a i n i n g t o t h e
e n v i r o n m e n t of t h e l a t t i c e p o i n t u n d e r c o n s i d e r a t i o n . C o m b i n i n g (1) a n d (7), i t is
o b v i o u s t h a t o n l y m u l t i p o l e s for t h e s e s a m e k, n - v a l u e s will b e i n d u c e d in t h e c h a r g e
d i s t r i b u t i o n a t t h e l a t t i c e site. Moreover, t h e r a t i o of t h e m a g n i t u d e s of t w o s u c h
m u l t i p o l e s (for fixed k) will be t h e s a m e as t h e r a t i o w i t h w h i c h t h e c o r r e s p o n d i n g
YA,,n's a p p e a r in t h e i n v a r i a n t l i n e a r c o m b i n a t i o n , p r o v i d e d t h e r e exists a t m o s t one
such linear c o m b i n a t i o n for a g i v e n k.
2. I t c a n be s h o w n t h a t for a g i v e n l a t t i c e o n l y t h o s e s u m s S'k,n(0[0, v), S'k,n(0[k½, v)
or Sg,~z(Rt]0, v) are d i f f e r e n t f r o m zero s u c h t h a t t h e c o r r e s p o n d i n g Yk, n belongs to a
l i n e a r c o m b i n a t i o n of Yk, n'S (for fixed k), w h i c h is i n v a r i a n t u n d e r t h e o p e r a t i o n s of
t h e c r y s t a l l o g r a p h i c p o i n t g r o u p of t h a t lattice. W e will call s u c h a n i n v a r i a n t l i n e a r
c o m b i n a t i o n a n invariant lattice harmonic Yk *). F u r t h e r m o r e , t h e r a t i o of c e r t a i n
n o n - v a n i s h i n g Sk,n's w h o s e c o r r e s p o n d i n g Yk,n'S c o n t r i b u t e to t h e s a m e vk is t h e
c o m p l e x c o n j u g a t e of t h e r a t i o w i t h w h i c h t h e s e Yg,n's a p p e a r in ¥~, p r o v i d e d t h e r e
exists a t m o s t one YA: for g i v e n k. W e r e s t r i c t o u r c o n s i d e r a t i o n s t o cases w h e r e . t h i s
c o n d i t i o n is fulfilled.
T h e s e p r o p e r t i e s h o l d for l a t t i c e s u m s w h i c h are a b s o l u t e l y c o n v e r g e n t , i.e. for
sums w h o s e v a l u e s are u n i q u e l y d e t e r m i n e d . (For v = ½, w h i c h is m o s t c o m m o n ,
this is t h e case for k > 2). C o n d i t i o n a l l y c o n v e r g e n t s u m s (for k = 2 if v = ½) d e p e n d
on t h e o r d e r of s u m m a t i o n or, as one m a y say, o n t h e " s h a p e " of t h e infinite c r y s t a l
(cf. ref. 12). A g i v e n ordel of s u m m a t i o n (shape of crystal) defines for a g i v e n l a t t i c e
a n e w s y m m e t r y g r o u p (in general of lower s y m m e t r y t h a n t h e original one) w h i c h
a g a i n m a y give rise to c e r t a i n Yk's. N o w t h e a b o v e s t a t e m e n t s c a n also b e applied t o
t h e c o n d i t i o n a l l y c o n v e r g e n t sums, p r o v i d e d t h e y are all s u m m e d in t h e p r e s c r i b e d
order.
T h e i n v a r i a n t l a t t i c e h a r m o n i c s c a n in c e r t a i n cases b e d e t e r m i n e d in a r e l a t i v e l y
simple w a y f r o m a c o m p a r i s o n of t h e c o r r e s p o n d i n g l a t t i c e sums. F o r a p r o o f of t h e s e
s t a t e m e n t s we refer to I (p. 52 ff).
It should be remarked that the considerations given above only apply
to crystals whose charge distributions have net charges, i.e. crystals whose
binding energy is (at least partly) of ionic character. (They do not apply
to van der Waals crystals such as the crystals of the noble gases). The net
charges of the ions give rise to the initial crystalline electric field which is
the main cause of the induction of all the higher multipole moments. The
contribution which these higher multipoles in turn make to the crystalline
field is necessarily of smaller order (considering realistic values of the
polarizabilities) and their effect will therefore manifest itself as a correction
to the case of the point-charge lattice.
Externally induced multipoles in a lattice. When a crystal is brought
into an external electric field then this field will, like the crystalline field
itself, induce a set of multipoles in the atoms (ions). A homogeneous external
field, as is usually considered, induces only dipoles. But these dipoles in the
lattice give rise to an additional crystalline field which in turn induces a
*) Since Yk depends on the polar angles vq' and q0, the form of Yg depends on the choice of the
coordinate system.
1230
F.W.
DE WETTE
whole set of higher moments. Due to the fact that they result from dipoles
and because of the inversion symmetry, these are all odd multipoles (odd k).
In the case of ionic crystals this means that an additional set of electric
moments is induced. These moments are completely independent of the set
of even multipoles which, as we have seen above, results from the net charges
of the ions.
Due to the fact that the odd multipoles are a result of the external field
they will also appear in crystals consisting of neutral units such as molecular
crystals or the crystals of the noble gases. These considerations, therefore,
also apply to such crystals.
One should expect the induction of these odd multipoles b y the external
field to have an effect on the macroscopic polarization of the crystal, i.e.
on its dielectric properties. An analysis of such an effect shows that it
exists but that it is negligible compared to other effects which may cause
deviations from the Clausius-Mossotti equation. (For more details we refer
to I, p. 44, 48, 62).
§ 3. The cohesive energy o/ an ionic crystal with sell-induced multipoles.
We will now derive an expression for the electrostatic cohesive energy of
a lattice consisting of polarizable charge distributions. These charge distributions have net charges (which are the cause of the crystalline potential)
b u t for simplicity we again assume that they have no permanent multipole
moments, i.e. that the free charge distributions are spherically symmetric.
The total electrostatic energy of the lattice is given bv
E = { f p ( R ) V(R)d3R,
(9)
where p(R) is the charge density in the lattice and V(R) the potential.
However, in the case of crystals one is interested in the electrostatic cohesive
energy Ecoh, which is the energy required to build up the crystal from its
constituent units, viz. the free ions. One may picture Ecoh by first placing
all the free ions in a lattice of similar form but with infinite lattice distance
and then allowing the lattice distance to shrink to its real value. The energy
which is released during this process is minus the cohesive energy of the
crystal. Since the higher multipoles grow from zero to their ultimate values
during this shrinking process, their formation energy, (i.e. the work done
against the internal forces of the charge distributions) is automatically
taken into account.
In the case of non-overlapping charge distributions, which we are considering throughout, we may express the charge density p as follows
p(R) = ~A~i
P'i'A(R),
(10)
where p"l~(R) is the charge density of the charge distribution (2, i) of
the type i in the unit cell ), P"IA is zero outside the region of (2, i). The
E F F E C T S OF ATOMIC POLARIZATION IN IONIC CRYSTALS
1231
potential V in this region is the sum of the potential due to the charge
distribution (2, i) itself" VC*)sal, and the potential due to all the other charge
distributions in the lattice, which we call V "~ (cf. II, (31)),
+ V",(R).
V(R) =
(11)
Substitution of (10) and (11) into (9) gives
E : XA,i{½fp'i'A(R)Y¿i'saf(R) dsR + ½fp"'A(R)V"'(R) d3R}.
]'he first term on the right is the self-energy of the deformed charge distribution (2, i). Clearly, this term m a y be considered as the sum of the selfenergy of the undeformed (free) charge distribution (2, i) and the energy
required to bring (2, i) into the deformed state which it has in the crystal.
Since the cohesive energy Econ equals E minus the self-energy of the undeformed (free) charge distributions, we have
Ecoh = ]~,i{de/ormation energy o/(2, i) + ½f p(i~(R) V"~(R) dzR}. (12)
The deformation energy is the total work done against the internal forces
of the charge distributions in building up the multipoles. The expression for
this energy may be determined by the following reasoning" The energy of a
given charge distribution of density p placed in an external potential V is
E1 = f p V d3R. If, however, the charge distribution is induced by the
external potential V, its energy is just Ez = ½El, as is well-known. If the
charge distribution is first built up, which requires the energy Eaef, and
then brought into the potential V we must have
E d e f + E1----E2
Eaef:
or
-- ½ f p V d 3 R .
(13)
In the case under consideration ,the p of (13) represents the induced part
of the density of the charge distribution (i). We may write
p'O(R, O, 9) = P(°o,o(R) Yo.o(O, ~o) + Xt>0,m P(i)t,m(R) Yt,m(#, of) =
= pei~net + p~Oina.
(14)
1e f P(i~aina(R)V(*~( R) d3R"
(15)
Hence
de/ormation energy o/ (2, i) =
-
-
Substitution of (15) into (12) leads to
Ecoh
=
~ , i { ~ ~f P
(i) ~a( R
)V "~ (R )dzR}A.
(16)
Writing V ") : V"~na + V")ind, where Y(i~ne t is the potential resulting from
the net charges of the other charge distributions and V"~ina that resulting
from their induced charge distributions, we find
E~ol, :
rl f p . ~ ,et( R )V(i~net(R) d3R q-- ~
l f p (~),m(R)V {~}i,~d(R ) d3R}~
.T,a,it:.,
= E M a d + Eco,.~.
(17)
1232
F.W.
DE WETTE
Although written down for a crystal, this expression is valid for the cohesive
energy of any assembly of non-overlapping polarizable charge distributions.
The first term gives the energy of mutual interaction of all the net charges:
it is the usual energy expression for an assembly of point charges. (For an
ionic crystal this is the Madelung energy EM,~). The second term gives the
extra cohesive energy (E,o~) which is due to the fact that the charge
distributions polarize each other. This term turns out to have a very simple
form, viz. half the energy of all the net charges in the fields of all the induced
charge distributions (which of course is equal to half the energy of all
induced charge distributions in the fields of all the net charges). Using (14),
(1) and (5) we find for E~o~
Ecorr = (1/8~) ZA,i A(i';~o,o iua Q(f}o,o,
(18)
with
A~t~0,0 i,a = Z~>0,mFz,m;0,0 {Q(*}t,mS~l}'z,-m (0 [0, ½) +
+ ZJ~ Q%,m S%,-m(Rj, I0, ½)}. (19)
(cf. (2)). Strictly speaking (18) is valid for crystals of infinite extension (in
which case E,o~r is infinite) or for finite crystals in the approximation in
which surface effects are neglected. (18) m a y be considered to express an
energy per mole if one extends the 2-summation over the number of unit
cells that form a mole of the substance. In that case the coefficients
A (Oo,olna are still the ones that are computed for an infinite crystal. We may
then write
Eeorr = (N/8~r) Z* A"~o,oina Q"~o,o,
(20)
where N is the number of unit cells per mole.
§ 4. Corrections to the binding energy/or some alkali halide crystals(NaC1and CsCl-type)*). We will now briefly discuss, for some examples, the
calculation of the corrections to the electrostatic binding energy, which
arise from the self-induced multipoles in the lattice. To find this correction
for a particular crystal from (20), one first has to solve the self-consistent
equations (8) for that crystal. For this one has to know, first the numerical
values of the polarizabilities ~li~k for the ions under consideration, which
characterize these particular ions in (8), and secondly the numerical values
of the lattice sums ~'k,n and S(~('~k~l, n-m for the relevant lattices, which
introduce the specific lattice symmetry into (8).
Calculation o/ the higher polarizabilities o~k t ). Although the self-consistent
equations (8) have been derived with a classical model of" a lattice of non*) F o r a m o r e d e t a i l e d d i s c u s s i o n cf. I, p. 58ff.
t) F o r a m o r e d e t a i l e d d i s c u s s i o n cf. I, p. 55ff.
EFFECTS
OF ATOMIC POLARIZATION
IN IONIC CRYSTALS
1233
overlapping electrostatic charge distributions, they are (apart from this
condition of no overlap, which is not a serious restriction for ions with
completed electron shells) completely general. In order to have a realistic
model of the crystal, the polarizabilities ~k should be calculated quantummechanically. For spherically symmetric atoms or ions (i.e. with completed
electron shells) one can easily derive an approximate expression for ak.
The derivation which is based on the variation method, is a generalization
to higher multipoles of the derivation of K i r k w o o d ' s expression for the
dipole polarizability 3). We find
~
=
16~(2k -- 1)
k2(2k + 1) 2 a0
{~,i r~2k Yk,o2(O~, ~b~)}oo2
{E,r, 2k-2 Ye-l,o2(0*, 4'*)}oo
.
(21)
{ }o0 denotes the matrix element f # o * { } #0 d~-, where ~o is the ground
state wave function of the atom (ion). i numbers the electrons and a0 =
= h2/me 2. For k ---- I (9.1) reduces to Kirkwoods' expression for the dipole
polarizability.
For the numerical evaluation of ~4 for CI-, K +, I - and Cs ÷ we used
simple Slater orbitals 4)5). In table I we give the results of these calculations
together with the experimental values of the dipole polarizabilities of
F a j a n s and J o o s 6). It is seen that the calculated values for ~1 agree
reasonably well with the experimental values.
TABLE
experimental I
~t" 1024
CIA
K+
IXe
Cs +
3.53
1.65
0.88
7.55'
4.10
2.56
I
I*)
calculated w i t h (5)
:ct" 102"
3.40
1.79
1.03
6.37
4.06
2.73
I
~a' 1058
7.94
2.20
0.729
53.9
19.1
7.60
I
a4" 107~
18.1
4.32
0.915
178
48.7
15.5
The lattice sums S('~k,n which occur in the self-consistent equations have
been treated in greater detail in a previous publication 7). In the appendix
we mention the relations which exist between lattice sums for cubic lattices
and we quote numerical values for some of these lattice sums.
The solution o/the sel/-consistent equations. The NaCl-type crystal consists
of a simple cubic (sc) lattice with lattice distance (1.d.) a, say, and alternately
positive and negative ions at the lattice sites. The lattice consists of two
interpenetrating face centered cubic (/cc) sublattices with l.d. ax/2, one
carrying the positive, the other the negative charges. Since the positive
*) The v a l u e s of ~,1 and ~a w e r e c a l c u l a t e d for the e v a l u a t i o n of the multipole effect
dielectric c o n s t a n t .
on the
1234
F.W.
DE WETTE
and negative charged ions have equal but opposite net charges (magnitude q)
the composite lattice sum in (8) is in this case =t= qS'k,nSC(Olkt, ½), taken
over a sc lattice. The lattice sums in the third term of (8) are to be taken
over t h e / c c sublattices. Similarly the CsCl-type crystal is a body centered
cubic (bcc) lattice (1.d. a) consisting of two sc sublattices (1.d. 2a/v/3).
As was mentioned in § 2, the lattice symmetry can considerably simplify
the self-consistent equations. This is particularly true for these cubic
lattices, for which the rotation-reflection s y m m e t r y of the surroundings
of a lattice point (determining which multipole moments Qtc,n can occur)
and that of the entire lattice (determining which lattice sums S'~,n have
values different from zero) are the same, viz. that of the full cubic group
Oh. For this symmetry group the invariant lattice harmonics Yk can be
determined from the ratio of the related lattice sums SZc,n sc for a cubic
lattice (cf. I p. 53 and II, § 3). Inversely, if Y~ = c n Y * k , ~ + c n ' Y * k , n ' + ...
then Q~,n : Q~,n, = S*k,n : S*k,n" = c , : c~,. This allows for a simplification of the self-consistent equations since only one Qk,n and one Sk,,
need to be retained for each k.
In these calculations we have broken off the self-consistent equations
after k = 4. This is justified by the smallness of the corrections due to the
hexadecapoles (k = 4; these are the first non-vanishing multipoles in a cubic
lattice). The convergence of the multipole expansion can be judged from
the percentage increase of the energy correction, when the next higher
order (non-vanishing) multipoles are also taken into account. For these
cubic crystals the convergence was found to be quite rapid *).
For the alkali halide crystals one can choose a unit cell containing only
two particles with equal but opposite charges. The Madelung energy EM,~z
gives the binding energy of all positive-negative point charge pairs in a
mole. Eco,.r (20) now consists of two terms, one representing the total
correction to the binding energy of all the positive charges in a mole, the
other the correction to that of all the negative charges. In table II we
1
compare both these terms, numerically (for various examples), to ~EM,,,
t
and their sum to EMo d. For comparison we also include the results for lattices
of the same structure but build up from conducting spheres with alternately
pos. and neg. net unit charges. In one case the radii of the spheres is r = ½a
(touching spheres) in the other r = ½a. All the energy corrections due to
the higher multipoles are negative, which means that they increase the
binding of the system.
*) T h e c o n v e r g e n c e can be m o s t easily c h e c k e d for a model of a c r y s t a l in which the c h a r g e
d i s t r i b u t i e n s are r e p l a c e d b y c o n d u c t i n g spheres with a l t e r n a t e l y positive and n e g a t i v e net charges.
(A collection of u n c h a r g e d c o n d u c t i n g s p h e r e s as a model for an e x t e r n a l l y polarizable m e d i u m was
first used b y M o s s o t t i in 1836). T h e a d v a n t a g e of this nlodel is the simple fornl of its polarizabilities.
A s p h e r e with r a d i u s r has the polarizabilities ~l,- = r 2z'*l. T h e c o n v e r g e n c e of our m e t h o d was found
to be q u i t e r a p i d for a realistic choice of the radii of the spheres (as c o m p a r e d to the n e a r e s t neighI)our distance), viz. such t h a t the resulting polarizabilities are c o m p a r a b l e to the ionic polarizabilities.
1235
EFFECTS OF ATOMIC POLARIZATION IN IONIC CRYSTALS
The last column of table II gives the percentage corrections to the Madelung energy for the entire crystal. These are rather small: less than 0,5%
for KC1 and CsC1 and about 1~/o for KI and CsI. However, relative to other
corrections to the Madelung energy which are usually taken into account
(e.g.v.d. Waals energy, zero-point energy) they are not small. The fact
that the corrections for KC1 and KI are slightly larger than those for CsC1
and CsI, respectively, can be understood as follows. The corrections are
due to the energy of the net charges in the field of the higher multipoles.
The strength of these multipoles is determined by the reduced polarizabilities,
which means that the neighbour distance plays a significant role. It is
obvious, that the smaller the distance is, the larger the multipoles will be.
Since the neighbour distance in KC1 is smaller than that in CsC1 (3.14 A
versus 3.56 A) the hexadecapoles of C1- in KC1 will be larger than those
of C1- in CsC1. Hence K + finds itself in a stronger multipole field than Cs+,
which thus leads" to a larger correction. The same holds for KI and CsI.
That the difference in lattice arrangement between the sc and the bcc lattices
plays a minor role in these differences can be seen from the comparable
results obtained for lattices with conducting spheres.
"FABLE I I
Eeorr for pos. ion
lattice
structure
KCI
I¢.I
CsCl
CsI
Spheres r =
,,
r =
j,
r~
,,
r =
½a
~(t
½a
½a
sc
sc
bcc
bcc
sc
sc
bcc
bcc
(in % of ½EM)
J
i
I
i
0.71
2.4
0.38
1.0 I
4.5*
0.065
4.1 *
O. 1
( K +)
(K +)
(Cs +)
(Cs +)
Eeor for neg. ion
(in % of ½EM)
0.09
0.2
0.36
0.58
4.5*
0.065
4. l *
0.1
(Cl-)
(t-)
(Cl-)
(i-)
Eeorrfor ion pair*)
(in % of E M )
0.4
1.3
0.37
0.8
9*
0.13
8.2*
0.2
*) T h e s e n u n l b e r s m a y be increased b y a b o u t 5% of their value if multipoles with k > 4 are
i n c l u d e d in the c a l c u l a t i o n . Since the percentage corrections to the Madelung energy are small, i t
did not seem w o r t h w h i l e to include these higher multipoles.
T h e values indicated by all asterisk are much less accurate than the others, because for these
large ( " t o u c h i n g " ) c o n d u c t i n g spheres the convergence of the multipole e x p a n s i o n is not too g o o d .
These n u m b e r s m a y be increased b y as much as 20% of their value, if multipoles for k > 4 are
included.
It should be remarked that the relative smallness of the corrections to
the Madelung energy for these crystals is partly due to the high symmetry
of the cubic lattices. In less symmetric lattices quadrupoles will play a role
and the resulting corrections could very well be an order of magnitude
larger than those found for cubic lattices.
§ 5. Discussion. The classical theory of ionic crystals has been very
succesful in giving a qualitative as well as a quantitative explanation of
the main properties of these crystals, such as the cohesive energy and the
1236
F.W.
DE W E T T E
elastic and dielectric properties. In view of these achievements it seemed
worthwhile to try to improve upon this theory in such a way that a number
of effects related to the finite .size of the ions could be taken into account.
Accordingly, our considerations have been based on a classical model of a
crystal as consisting of a rigid lattice with non-overlapping charge distributions. A description like the present one is a model since it strictly limits
its attention to a certain aspect, viz. the electrostatic polarization. But this
limitation enables us to treat this aspect (within the limits of the model) in
grealer detail than would be possible in a more realistic theory. This situation
m a y be illustrated by the example of the intra-crystalline polarization.
Although this phenomenon can, of course, in principle also be incorporated
in a quantum-mechanical theory it is very hard to carry it through in actual
calculations, as m a y be seen from L ~ w d i n's exfensive quantum-mechanical
treatment of ionic crystals 8). In order t o be able to calculate crystal
properties from first principles, he had to resort to free-ion wave functions
to keep the calculations manageable, i.e. he had to neglect polarization
effects. This example clearly shows that it might be rather difficult to
fully incorporate polarization effects in a thorough quantum-mechanical
treatment.
In this light our classical considerations can be justified as follows. It
is known that the bulk of the cohesive energy of ionic crystals originates from
the Coulomb interaction between the ions. This energy is quite well represented b y the electrostatic Madelung energy of a crystal of point charges.
Even a quantum-mechanical theory like that of L 6 w d i n , relies on the
classically computed Madelung energy for the main part of the cohesive
energy. What we have done, then, is to refine this classical calculation
of the electrostatic part of the cohesive energy, viz. by taking account of
the finite size of the ions. It is reasonable to assume that if the polarization
effects could be calculated quantum-mechanically, they would turn out
to be of the same order of magnitude as is found classically.
The relative importance o/ the polarization energy. We have restricted
our considerations to the polarization correction to the cohesive energy of
a cubic crystal. The other contributions to the cohesive energy, viz. the
overlap and exchange energies (together forming a ,,repulsion" energy),
the van der Waals energy and the zero-point energy can be derived either
from a semi-empirical theory (B o r n-M a y e r) or from a quantum-mechanical
theory ( L 6 w d i n ) . In table III we compare these energy contributions for
the crystals considered (numbers derived from S e i t z 9)) with the polarization energy as calculated in § 4. According to our sign convention (cf. § 3)
a positive energy weakens the cohesion of the crystal an6 a negative energy
strengthens it. The numbers express percentages of the Madelung energy.
It is seen that the polarization energy is of the same order of magnitude
as the zero-point energy. The smallness of the polarization energy for these
EFFECTS
OF ATOMIC POLARIZATION
1237
IN IONIC CRYSTALS
crystals is partly a result of the high symmetry of the cubic lattices, which
causes the quadrupoles to vanish. In lattices of lower symmetry the quadrupoles will be different from zero and the polarization energy might
conceivably be of the same order of magnitude as the van der Waals energy
or even larger. So far we have not carried out calculations for such lattices,
since that would require the knowledge of lattice sums which are not as
easy to evaluate as those for the cubic lattices.
TABLE
KC1
KI
CsCI
CsI
III
Repulsive
(positive)
Van der W a a l s
(negative)
Zero p o i n t
(positive)
Polarization
(negative)
12
10
II
10
4
4.4
7
7.6
0.76
0.6
0.6
0.5
0.4
1.3
0.4
0.8
The fact that the polarization energy is relatively small, does not imply
that its calculation is only of minor interest. As has already been suggested
by M a y 10), the polarization energy could play an important role in considerations about the relative stability of the various lattice types. This
conjecture is supported not only by the fact that the polarisation energy is
comparable to the van der Waals and zero-point energies, which are usually
both taken into account in stability considerations, but also by the consideration that it is probably more dependent on the lattice structure than
these other energies. On these grounds it must be concluded that stability
considerations in which the polarization energy is neglected are not reliable
N e u g e b a u e r and G o m b ~ s 11) have long ago calculated the polarization
energy of the KC1 crystal. They found an energy term which is of the same
order of magnitude as the van der Waals energy and which also varies with
the inverse sixth power of the lattice distance. This indicates that quadrupoles are responsible for this effect. However, we have shown that quadrupoles cannot occur in a cubic crystal like that of KC1. It is apparent from
their analysis that these authors have not properly taken the cubic symmetry
into account. The first non-vanishing multipoles, the hexadecapoles (k -~ 4),
give rise to an energy correction which varies with the inverse tenth power
of the lattice distance.
Other polarization e[[ects. Contributions to the cohesive energy and
the dielectric constant are not the only polarization effects tllat will be
present in crystals .The interactions of the multipoles with one another as
well as with the net charges, will influence all phenomena in which the
inter-ionic forces play a role. We may, therefore, also expect some influence
on the elastic properties and the dynamic behaviour of the crystal. We have
not investigated these eifects, but in particular, those concerned with the
elastic properties may be worth looking into. Since the multipole forces
are short range, their influence on the elastic properties may be relatively
1238
F.W. DE WETTE
more important than on lhe cohesive energy, the bulk of which is due
to the long range Coulomb force. For unstrained cubic crystals the first
non-vanishing multipoles are the hexadecapoles for which the force varies
as r -6. These forces will give a correction to the compressibility as well as
to the other elastic constants. In the latter case also quadrupole forces may
be important (~-~r -4) since quadrupoles can be induced in strained cubic
crystals. Moreover, since all .these forces are non-central, they will also
effect deviations from the Cauchy relations.
Overlap de/ormation. The electrostatic atomic (ionic)deformation is
not the only type of deformation that will be present in a real crystal.
The overlap of neighbouring particles will also give rise to deformation
since the total overlap of a certain particle by its neighbours is direction
dependent. This overlap deformation (which" is of quantum-mechanical
nature) can either enhance or reduce the electrostatic deformation. For
positive ions both deformations will enhance each other since the electrons
will be repelled from the regions of the negative nearest neighbours electrostatically as well as by virtue of the Pauli principle. For negative ions these
effects act in opposite directions. For crystals with ions of different size
like KC1 and KI it m a y be concluded that the overlap deformation reduces
the total effect of the electrostatic deformation. If we consider the figures
for KC1 and KI in table I we see that the bulk of the polarization effect
is due to the energy of the net charge of K + in the field of the induced
multipoles of C1- and I-, respectively. Since these multipoles are reduced
in strength by the overlap deformation, the total effect will be smaller.
Acknowledgements. The author is greatly indebted to Professor B. R. A.
N ijboer, who suggested the present investigation, for his most valuable
advice and criticism. He also profited from many stimulating discussions
with Professor N. G. v an K a m p e n.
APPENDIX
Relations between lattice sums /or cubic lattices. We first mention some
simple relations which exist between lattice sums for cubic lattices (sc, /cc
and bcc). Such relations can be useful in deriving the numerical values of
certain lattice sums from the known values of others, and they may be
used for purposes of computational cross-checking.
The lattice sums considered here are of the form
S'z,m(R [k, n) = ~'~ ]R -- r~ [-~n-zYt,m(OA, c~A) exp(2~zik.r~).
In a previous paper 7) we have dealt with the problem of bringing these
sums into a rapidly converging form which is well-suited for numerical
computations.
Where S'z,m denotes the vaht~ of the sum, as in these relations, we mean
EFFECTS OF ATOMIC POLARIZATION IN IONIC CRYSTALS
1239
its value for a lattice with nearest neighbour distance 1. Moreover, since
these values depend on the orientation of the crystal axes with respect
to the coordinate axes, the lattice sums in each single relation should be
referred to the same coordinate system. For cubic lattices it is most natural
to choose the x-, y- and z-axis along the three four-fold axes of the lattice.
Consider the sum
S'l,mSC(OIk~, n)
Y,Z(-a "1)z'+
' a*+ rA-2n-tYl,mkt0A, ¢,0
:
for a sc lattice. The lattice points are alternately positive and negative and
the sum is taken with respect to a positive lattice site (which is excluded
from the summation). This sc lattice consists of two interpenetrating /cc
lattices, one containing all the positive the other all the negative points. If
the sc lattice has the nearest neighbour distance l / v ' 2 then the/cc lattices
have the nearest neighbour distance I. It is now easily seen that we have
the relation.
2 n+~z
s%.:(Olk,,
n)
=
S'z,mf~(OlO, n)
-
St,mfCC(R~]O, ½),
(A.1)
R~ = ½(al + a2 + a3). Note that in the second sum in the right hand side
the origin should be included in the summation.
In this fashion a total of nine relations can be written down immediately,
combining lattice sums of the sc, /cc and bcc lattices (cf. I, p. 64). Similar
relations also hold for Bravais lattices of lower symmetry for which one can
define simple, face centered and body centered unit cells.
S o m e values o / l a t t i c e s u m s / o r cubic lattices. Not all of the lattice sums
quoted here have been used in the calculations of this paper but it might
be useful to list everything that has been computed.
We have listed the values of
5:'1,0 = ( 4 ~ / 2 / + 1)t St,o,
i.e. of the sums containing the unnormalized spherical harmonics. The
values of the sums St,+4 (for 1 = 4 , 6, 8, 10) and St, _+8 (for 1 = 8 , 10),
which do not vanish for cubic lattices, can be derived from the values
presented here by making use of the invariant harmonics Yx(cf. § 2 and II, § 3).
All values were computed for lattices with nearest neighbour distance 1.
The z-axis is chosen along one of the four-fold axes of the lattice. The values
for the sc lattice are listed in table IV, those for t h e / c c and bcc lattices in
TABLE
IV
Silnple cubic lattice
t
2
4
6
8
10
[
5:',,o.qofo, ½)
-s*~ t
3.1083
0.5698
3.27t3
1.0096
[ 5~'t,o~qOlk½,½) ] 5~t,o~(R.~ I O, ½)
ot
--3.5789
--0.9895
--2.9329
--1.0114
--~ t
--6.2147
4.8732
4 3890
1240
E F F E C T S OF ATOMIC P O L A R I Z A T I O N IN I O N I C C R Y S T A L S
table V. Most values were computed directly and checked by the use of
one of the relations mentioned above. The asterisk indicates values not
computed directly, t indicates values obtained by ,,plane wise" summation
in the x-y-planes (cf. reference 12).
TABLE V
1
2
4
6
8
10
Face centered cubic lattice
--~=V2t
--1.331
--2.374*
3.59
--0.043
---~=V'2 t
18.914
8,821 *
70.20
45.732"
Body centered cubic lattice
--=Vat
--1.513
1.989
2.099 *
Ot
4.542
--1.572
--0.3063 *
Received 17-10-59.
REFERENCES
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
I1)
12)
De W e t t e , F. W., Electrostatic Fields in Ionic Crystals. Doctoral dissertation, Utrecht 1959.
De W e t t e , F. W. and N i j b o e r , B. R. A., Physica 24 (1958) 1105.
K i r k w o o d , J. G., Phys. Z. 33 (1932) 57.
S l a t e r , J. C., Phys. Rev. 36 (1930) 57.
E y r i n g , H., W a l t e r , J. and K i m b a l l , G. E., Quantum Chemistry, Wiley, New York 1944,
p. 162.
F a j a n s , K. and J o o s , G., Z.f. Physik 23 (1924) 1. Cf. also Hdb. d. Physik Vol. 24, Part 2,
Springer, Berlin 1933, p. 942.
N i j b o e r , B. R. A. and De W e t t e , F. W., Physica 23 (1957) 309.
L~Swdin, P e r - O l o v , A theoretical Investigation into some properties o/io~ic Crystals. Inaugural
dissertation, Uppsala 1948.
S e i t z , F., The modern Theory o/ Solids. McGraw-Hill, New York 1940, p. 88.
M a y , A., Phys. Rev. 52 (1937) 339.
N e u g e b a u e r , T h . and Gomb~is, P., Z. f. Physik 89 (1934) 480.
N i j b o e r , B. R. A. and De W e t t e , F. W., Physiea 24 (1958) 422.