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A general method for the calculation of Madelung constants for intermetallic compounds
A GENERAL
METHOD
FOR THE
CALCULATION
FOR INTERMETALLIC G.
V.
RAYNOR,
OF MADELUNG
CONSTANTS
COMPOUNDS*
P. NAOR,
and C. TYZACKt
The crystal structures of many intermetallic compounds may be expressed by a simple notation baaed upon the sequence of hexagonal layers of atoms in some simple crystallographic direction. Using this notation, a general method for the calculation of Madelung constants for such structures has been developed, in which the electrostatic interaction of a central ion with various layers of ions above and below that in which the central ion lies is expressed in terms of six functions. These functions have been evaluated and tabulated. The use of the new method, which appears to be the simplest yet proposed, is illustrated by reference to the wurtzite and nickel arsenide structures; it is suggested that, by means of such calculations, quantitative expreseion may be given to the qualitative concept of heteropolarity in the discussion of intermetallic compounds. UNE
MBTHODE
GfiNfiRALE POUR LES
DE CALCUL DES CONSTANTES COMPOSES INTERMBTALLIQUES
DE
MADELUNG
Les structures cristallines de nombreux compos6s interm6talliques peuvent s’exprimer par une notation simple reposant sur la succession des couches atomiques hexagonales dans une direction cristallographique simple. En utilisant cette notation, les auteurs ont d6velopp6 une mbthode g&&ale de calcul des constantes de Madelung pour des structures oti l’interaction Blectrostatique entre un ion central d’une couche atomique et les ions des couches adjacentes est exprimbe au moyen de six fonctions que l’on a calcul6es et dress&es en tableaux. L’emploi de cette nouvelle m&hode qui semble 6tre la plus simple de celles proposires jusqu’h prbsent, est illustrb par des exemples se rapportant aux structures de la wurtzite et de l’arseniure de nickel. Les auteurs pensent qu’8 l’aide de tels calculs, on pourrait trouver une expression quantitative du concept qualitatif d’h6t6ropolarit6 dans les problbmes interessant les compo&s interm&alliques. EINE
ALLGEMEINE
METHODE ZUR BERECHNUNG DER MADELUNG-KONSTANTEN INTERMETALLISCHER VERB.tNDUNGEN
Die Kristallstrukturen vieler intermetallischer Verbindungen lassen sich in einem einfachen Schema beschreiben, welches auf der Folgeordnung hexagonaler Atomschichten in einfachen kristallographischen Richtungen beruht. Unter Benutzung dieses Schemas wurde eine allgemeine Methods fiir die Berechnung der Madelung-Konstanten solcher Strukturen entwickelt, bei der die elektrostatische Wechselwirkung eines Zentral-Ions mit verschiedenen Schichten von Ionen oberhalb und unterhalb der Schicht des Zentralions durch sechs Funktionen ausgedriickt wird. Diese Funktionen wurden berechnet und tabelliert. Der Gebrauch der neuen Methode, die zur Zeit die einfachste zu sein scheint, wird am Beispiel der Wurtzit- und NiAs-Struktur erliiutert; es wird ferner darauf hingewiesen, dass mit Hilfe derartiger Berechnungen dem Begriff der HeteropolaritBt bei der Diskussion intermetallischer Verbindungen eine quantitative Grundlage gegeben werden kann.
1. INTRODUCTION Binary
compounds
in the
solid
phases may state
may
be
classified into the following general types: (i) Homopolar
be considered
ably in electrochemical
compounds,
as similar to ionic com-
pounds, in so far as the components
differ consider-
characteristics.
Compounds
of type (iv) are of interest, since the valency of inorganic chemistry are not fulfilled, and
rules their
(ii) Ionic compounds, (iii) Normal valency intermetallic compounds, (iv) Abnormal valency intermetallic compounds, compounds” and other including “electron
properties vary from those of the metallic phases to those exhibited by some compounds classified as
structures of predominantly metallic type. The factors affecting the formation of types (i) and (ii) are well understood. Class (iii) includes such
of the ra,nge of properties electron compounds such these are formed between comas y-brass occur; ponents which differ little in electrochemical character,
compounds as Mg,Sn, in which the normal valencies of the components are satisfied;
and their stability at characteristic valency electron/ atom ratios is understood in terms of Brillouin zone theories.(l) With increasing electrochemical
* Received January 18, 1957. 7 Department of Physical Metallurgy, Edgbaston, Birmingham 15. ACTA
METALLURGICA,
VOL.
6,
The
group these
University,
SEPTEMBER
1967 433
normal
valency
compounds.
At
the
metallic
end
difference between the components, compounds of more complex types occur, and it is with the factors
484
ACTA
METALLURGICA,
affecting their formation that we a.re concerned in this paper. On general grounds these phases appear to represent a transition between ionic and metallic binding. In a typical ionic structure the ions may be considered to be hard impenetrable spheres, the packing of which is determined by electrostatic energy considerations and ionic size factors. The interaction between anions, other than electrostatic, is generally limited to closed-shell repulsions and slight van der Waals attractions, while interaction between cations is, in general, negligible. Considering anions of Group VIB of the Periodic Table, the degree of polarizability increases in the order 0, S, Se, Te, thus increasing the possibility of van der Waals attractions between anions contributing to the lowering of the energy. For anions of earlier groups of the Periodic Table, these tendencies are enhanced, and relatively strong van der Waals binding between large, polarizable anions may transform gradually into metallic binding because of the redistribution of electron density more uniformly over the volume occupied by the compound. Thus, in the transitional range between ionic and metallic bonding, the electronegative components tend to have large, easily polarizable ions. We may thus expect the energy of the structure to be lowest for the maximum number of anion-anion contacts, and that compounds typical of the transition between the ionic and the metallic states will crystallize so that the electronegative ions, viewed alone, constitute a close-packed structure. A survey of those structures in which the electronegative component is a metal or semi-metal of Groups IVB, VB, and VIB of the Periodic Table, and t.he electropositive component is a metal of Groups IB, IIB, IA, IIA, or a transitional metal, shows that the great majority of the compounds included in the survey crystallize in structures in which the electronegative components are in close-packed arrangements. This tendency is lost for purely ionic or purely metallic structures. Thus, in the caesium chloride structure the anions form a simple cubic arrangement. No compound in the transition range should crystallize in this structure, since there is no opportunityfor close-packmgof electronegative components, and in fact only one pha,se (BiTl) of the transition type does adopt this structure. All other known representatives are either purely metallic (e.g. NiAl) or ionic. In the ionic calcium fluoride structure, the fluorine anions constitute a simple cubic lattice; the intermetallic phases which are isomorphous with this structure are of the purely metallic type (e.g., PtSn,, AuAl, PtIn,). In the transition range
VOL.
5,
1957
(e.g. Mg,Sn, Mg,Pb) structures appear which are not isomorphous, but anti-isomorphous, with calcium fluoride, and it is the electronegative component which takes up a close-packed structure in agreement with the above generalization. For the compounds which fall into the transition cla.ssification, therefore, we may infer that electrostatic binding forces are of considerable importance; it would be helpful to be able to assess the contribution made by such forces. The electrostatic energy E of a binary compound of equiatomic composition is given by the expression: E = -MNe2z2/r
(1)
where N is the number of neutral groups of anions and cations, e is the electronic charge, and r the closest anion-cation distance in the structure; M is the Madelung constant, which is a function of the geometry of the lattice, and z is the charge on a single ion. The lowest electrostatic potential energy is attained by a structure of maximum Madelung constant; any deviation from a structure satisfying this condition, for a given binary compound, is interpreted as due to the operation of forces of a different nature. The magnitude of the ionic cont,ribution to binding may therefore be assessed from the magnitude of M. The calculation of M, however, is laborious in all but the simplest cases. The observation that in a large number of intermetallic compounds the electronegative components take up close-packed arrangements suggests the possibility of a more general approach to the calculation. 2. THE
“STRUCTURE
PLAN”
NOTATION
As noted above, the structures of many intermetallic compounds with properties intermediate between those of purely ionic and purely metallic materials contain close-packed planes of atoms in a hexagonal array. Such planes appear in t,he facecentred cubic, close-packed hexagonal, body-centred cubic, simple cubic, and rhombohedral structures. Some tetragonal or orthorhombic crystals also exhibit pseudo-hexagonal symmetry and contain slightly distorted hexagonal planes of atoms. Such structures, therefore, contain a sequence of hexagonal atomic layers along some crystal direction, and it is upon the existence of these layers that the “structure plan” notation, as outlined below, is based. A simple example is afforded by the face-centred structure (Fig. 1). If the atoms are projected onto the (111) plane, then atoms of the type I form a hexagonal arrangement ; atoms marked II project into the centres of alternate triangles formed by
G.
V.
RAYNOR,
P.
NAOR,
C.
AND
TYZACK:
CALCULATION
OF
MADELUNG
COh’STANTS
structure becomes simple cubic; layers disappear, and, denoting
485
alternate hexagonal these missing layers
by dots, the layer sequence is: A,.O,,I.A,,.O,.A,,,.O,,.A, Omitting
the
layers
corresponding
sites, the face-centred
A,. . .A,,. FIG. 1. Unit cell of the face-centred cubic The crosses mark octahedral interstices, and black circles mark tetrahedral interstices.
structure. the small
If atoms
in all possible
following
sequences
into
of type the
centres
projection sional
I, while
atoms
of the
marked
remaining
(Fig. 2) takes the form
lattice,
dimensional
which
sublattices
to this projection
in a direction
of a two-dimeninto
to
the
of
structures layers,
(111)
planes.
This
define the face-centred
as may of
interstices
be
appreciated
Fig.
1,
which
between
from
to
further
the
original
(small
positions
are
face-centred
T, the sequence becomes:
ATI This atoms
II0 IIITA I
structure are
Face-centred
removed
from
IIIA I
cubic:
cubic.
. .A,
tures, the relative heights [ 1111 direction, considered by
reference
to
Fig.
of the layers along the vertical, being denoted
2 by
the
subscript
Roman
Assuming equal cell dimensions, numerals. interlayer distances are in the proportions:
Structure
plans
geometrical
of this type
relationships
to be concisely
enable
between
expressed
many simple
by addition
This is to be described
elsewhere,
of the lattices
or subtraction. but it may
be
noted here that where no layers are missing the subscripts for adjacent planes must differ; an identithe
atoms
for adjacent
planes would
in one plane
sites,
the
which is forbidden by packing considerations. restriction does not necessarily apply when missing
from
the
structure
interpenetration
of
in
the
structure
(Fig.
adjacent
plan.
face-centred
lie
plane, This layers
Thus,
3) is represented
two
that
would
immediately
diamond
those
imply
vertical)
the
are
above
([ll l]
If now
tetrahedral
the
b.c.c. : s.c . : f.c.c. = & : 4 : 1.
cal subscript
IIT IIIOT I IIA IIITO I IIT IIIA I
is body-centred
IIIA4A IA II
Such sequences will be referred to as “structure plans,” a,nd interpret the fact that the same sequence of hexagonal layers occurs in all three crystal struc-
con-
cubic sites, then, denoting octahedral positions by the symbol 0 and tetrahedral positions by the symbol
the
Simple cubic:
octahedral
interstices
If all these interstitial
in addition
these
the hexagonal
contains
(crosses) and tetrahedral
circles).
occupied,
difference
lies in the spacing between
sideration black
The
4AA JIIAAA I [I IIII II
A 1. #. A,,. . .A,,,.
since the same sequence is obtained,
mercury.
are identical,
By reference
along the (111) direction, for body-centred cubic, simple cubic, and rhombohedral structures such as that
positions
of layers in the [ll l] direction,
three two-
layer sequence
normal
. .AII,. . ,A,
cubic:
=IAA I II
The
the structure is defined
does not uniquely
cubic structure,
project
triangles.
I, II, and III.
or “plan,”
by the well-known
sequence
is subdivided
III
octahedral
may be written down: Body-centred
atoms
to
cubic structure is regenerated:
the
as the
cubic
sub-
lattices :
A,. . .A,,.
. .A,,,.
. .A, = A,A,, . . A,,A,,,
+ .A,,.
FIQ. 2. Projection of atoms in face-centred structure on to (111) plane.
cubic
. .A,,,.
The hexagonal different type,
. .A,. . .
I
..
AIIIA,.
*AI
layers which approach closely are of while those with identical subscripts
are well separated.
ACTA
486
METALLURGICA,
VOL.
5,
anion-cation
1957
distance.
For
the
binary
compound
A B, (3)
The sum zj ?-
FIG. 3. The diamond
Similar
structure
hexagonal layers
in
the
interstices
may
be
representing
[OOO,l] direction.
developed
for
the sequence
of
The
close-packed
structure plan:
III
TAT0 I II
between
I
III
and tetrahedral
with the sequence:
TAT0 II I
III TAT I II
II
structures
which
valuable
in the case of complex
To describe
plan notation
structures
may
structures,
is also applic-
such as that of MgZn,,
sublattices, instead of three as in Fig. 2; the intimate relationship of this structure with Ohose of MgCu, is then immediately apparent. For the method of calculating Madelung
constants, however, above suffices. 3. THE
the
simple
For the simpler E,
outlined
MADELUNG
of intermetallic
of interest to know the contribution
compounds
I
structures,
the interaction
energy
of an A ion with all other ions is equal to the energy E,
of a B ion with all other ions.
structures
this is not necessarily
so, and the total electrostatic
energy then becomes +
fve2(zAzB) B M
r
(5)
from which we may write: M = Pf, where M,
and M,
constants. The Madelung
-I- M13)/2
(61
may be called partial Madelung constant,
the sum of the electrostatic
therefore, interactions
represents of a central
ion, bearing unit charge, with all other ions in the lattice, which also bear unit charges. has been studied
The summation
by Apell,(2) Madelung,t3) Ewald,c4)
and Born;c5) their different types of solution involve advanced mathematics and much computational work even for structures of high symmetry. Slaterc6) and Frank(‘) as pointed
CALCULATION OF CONSTANTS
In the discussion
acheme
(4) \
M
A
be
or neutral units,
Ne2(zA%)_
M
structure plans may be referred to a two-dimensional hexagonal network which is subdivided into four
and MgNi, illustrating
_
I
is capable of revealing relation-
different
over all pairs of ions,
r
in which all octahedral
to which the structure able.
=
For more complex
obscured by the more usual crystallographic notations, such as the Space Group notation. This is particularly
E
interaction
are filled corresponds II
by summing
so that if there are N “molecules”
A,. . .A,,. . .A,. . .A,,. . .A,
This type of notation ships
E is computed
structure.
structure has the following
A structure
AT0 I
plans
structures,
hexagonal
M, and depends only on the geometry of the crystal lattice. The total electrostatic energy of the crystal
-----u
u-------
is known as the Madelung constant,
have
proposed
simpler methods,
but,
out by Fisher,(*) the Slater method does
not lead to accurate results in several cases, while for certain lattice types the method of Frank requires the it is
to the binding
evaluation
of a large number of terms.
The principle of the Slater and the Frank methods
energy which may be regarded as arising from electro-
is to sum the interactions
static
We may consider a binary compound AB as an array of positive and negative charges, which as a whole is electrically neutral.
finite lattice,
The electrostatic interaction energy with all other ions is given by:
just outside the lattice. These fractional charges are such as to preserve the electrical neutrality of the finite lattice. The methods differ in the positioning of the fractional charges. Slater -locates them on
interactions.
of the ith ion
(2) where zi and zj are the charges on the ith and jth ions, e is the electronic charge, and pirr is the distance between the interacting ions, r being the shortest
of a central
ion with a
and to correct for the finite nature of
the lattice by calculating the interaction of the central ion with certain artificial fractional charges placed
lattice sites, while Frank places them on central positions between lattice sites. In the development to be described, the technique adopted by Frank is retained,
but the method
is generalized,
by using
G.
V.
RAYNOR,
the description hexagonal
P.
AND C.
CALCULATION
TYZACK:
of structures in terms of sequences of
layers.
interactions
NAOR,
may
exist
between
the
ion and various layers of ions considered and the calculation
central
interaction
CONSTANTS
48’7
of the central ion with a whole with a layer of
type III) is:
as a whole,
for a specific structure
MADELUNG
layer of type II (or, by symmetry,
We may then specify the different
which
Similarly,
OF
ml =
becomes
P2 +
p1+
P3 +
P4 +
. . . + Pm
a matter of choosing the correct types of interaction by reference to the structure plan. The finite lattice
where the terms pl, p2, etc. are the interactions of the central ion with successive rings of atoms in the
takes the form of a hexagonal
layer
prism, made up of a
sequence of hexagonal layers. The central atom is defined as lying in a layer of type I. This may interact with ions in layers of types I, II, and III. Above and below
each layer
a hexagonal
layer
of positive
of
artificial
negative
For
interaction
of
the
central
two cases must be distinguished:
ions we imagine
fractional
considered.
ion with layers of fractional charges between ionic layers of types I and II (or, by symmetry, I and III), (i) The layer
of fractional
layer just outside
charges forms the top
the finite lattice.
Here the whole
charges, at a normal distance
of d/2, where d is the
layer reacts with the central ion, and the interaction
spacing between ionic layers.
The charges lie at the
is represented by the sum
centres
of lines joining
Similarly,
the ions in the real layers.
each layer of negative
artificial
positive
charges
above
cir + ‘1s + 0s -1 ‘14 + . . . urn = s,,
ions has layers of a.nd below.
Inside
(ii) Layers of ions exist above the layer of fractional
the lattice the fractional charges cancel, and only those outside the finite lattice need to be considered.
the finite lattice
The
a hexagonal
central
ion thus interacts
with
the
whole
or
charges.
In this case all the fractional are neutralized,
fringe surrounding
charges within
and only
those in
the finite lattice
at
a part of layers of fractional charges pla,ced between ionic layers of types I and II, I and III, and II and III. If the fractional charges are between
a certain height intera.ct with the central ion. This interaction is given by the term onl+i. Interaction
layers of type I, they fall completely within the finite lattice, unless they form an external layer at the top
between ionic layers of types II and III may be expressed by a similar function; this type of inter-
or bottom
of the prism.
We assume each ion to have
unit charge, and for computational fractional
convenience
each
charge is l/3.
of the central ion with a layer of fractional
action
action
of
the
central
ion
(Fig. 4). with
that
successive
hexagonal
of atoms
The possible interactions To
the
functions
the whole
the
are summarized in Table 1.
functions
we
must
know
the
fractiona,l charges in other layers. The function q is a trivial case and equals l/d, where d is the distance
in the
with upper
between
upper
layer
is
between
layers.
the central
ion and the ions or
In the hexagonal
array
I of Fig.
2, the square of the distances of ions on the sublattice from the central ion may be written: l2 = ,rn2 -
rrz, 7~a, 7~~’. . . , VT,,. Interaction
of the central ion with represented by:
evaluate
immediately
layer, counting up to the mth ring, are then described by
possible
distances
x be VT~; interactions
rings
and is only
First, inter-
above it (atom 2) is expressed as a term q ( = l/d). Let the interaction between the central ion and the six atoms surrounding
encountered,
when the layer of fractional charges forms the top la.yer just above the finite lattice.
The interaction of the chosen central ion in a layer of type I with another layer also of type I is expressed as a sum of simple functions
is rarely
charges
mk +
k2
(m > 0; 0 < k < m).
Thus for the six ions surrounding
the central
l2 = 1, the unit
of distance
the interatomic
_.
~___
I
being
ion,
TABLE 1
Interacting with layer of type Centml ion in leyer of type
0 Central ion
Pm. 4. Dirtgram illustrating interaction of layer of type I with central ion in another type I layer.
I
II
III
P.Q
r Pdl T
T T P.P
I’I
T
=I
T
Layer of fractional charges between I and II
Layer of fractional charges between I and III
Layer of fractional charges between II and III
488
ACTA
METALLURGICA,
VOL.
5,
1957
Where m is even o, = di-2 -
k=l
distance in the layer, while ions in the next hexagonal shell lie at distances Z2= 3, Z2= 4, as shown in Fig. 5. The rrnzfunction is thus: rr,=6&m2-mk+
k2 + cP)-*
k=l
where the summation indicates the addition of all values of (m2 - mk + k2 + d2)-4 which are produced when m takes one of the values 1,2, 3 . . . , and k, ior any given value of m, takes the values 1, 2,3, . . . , m. The squares of the distances from a central ion on sublattice I to ions on sublattices II and III of Fig. 2 follow the sequence 113,413, 7/3, 13/3, 16/3, 19f3, 2513,2813 . . .
The numerators are numbers, half of which are represented by the totality of numbers 3ma 3k(m - k + 1) + 1, while the other half are represented by the totality of numbers 3m2 + 3m - 3k(m - k + 1) + 1. Thus we derive: p,=31/3
[
m-l +
2
k=l
(3m2-3km+3k2+3m-3k+1$-3d2)-t
For the t function, the corresponding algebraical expression is complicated; this function is considered below in relation to a finite lattice. The remaining fictions, may, however, be evaluated to any desired value of m by the expressions given. In the initial development of the method, some experiment was necessary to determine the size of the finite lattice required for accurate results. This exploratory work indicated that evaluations of the Madelung constant accurate to 0.001 were obtainable by considering the central ion to be surrounded, in its own plane, by two hexagonal shells of ions of its own type (Fig. 6); the finite lattice is then constructed by erecting further layers above and below this plane, as dictated by the particular crystal structure considered. To this approximation, the functions take the following values, where d denotes the normal distance between the central plane and the particular layer involved: q(d) = l/d p(d) = S[(l + d2)-” + (3 + d2)-’ + (4 + d2)“] r(d) = 3%‘5[(1 + 3d2)-” + (4 + 3d2)-”
+ 2(7 + 3d2)-t] o(d) = ~%[2(43
+ 12d2)-* + 3(49 + 12d2)-$
1
s(d) = v’%[(l
+ 12d2)-” + 2(7 + 12dz)-” ’
+ 2(13 -+ Bdz)-9
+ 2(19 + 12d2)-”
+ (25 + 12d2)-a-+ 2(31 + 12d2)-*
+ 2(37 + l%P-“1 t(d) = 2/4[2(1 + 4d2)-” + 2(3 + 4d)-* + 4(7 -/- 4dd2)-f
+ 2(9 + 4d2)-8 + 2(13 f 4d2)-“J
f8k + 7 + 12da)-6 m-1
+
1
3k + 1 + 12d2)-*
+ 2(61 + 12dV$1
for interaction of the central ion with layers of type II only, or with layers of type III only. Considering layers upon which fractional charges are placed, the fractional charges occur either between I- and II-type ions, or I- and III-type ions. By projection of both types of fractional charge onto Fig. 2, it may be shown that the following equations hold: Where m is odd
-
6km + 12k2
k=l
(3m2 - 3km + 3k2 -
5 (3mz-3~+3kz-3k+l+~z)-~ k=l
m’F1(3m2 -
3m + 1 + 12d2)-*
+ 2
FIG. 5. Diagram representing the squ-s of the distance from central ion to ions in successive hexagont61shells.
[
&
3*m2 -
3km + 3k2 -
3m + 1 + 12d2)-*
1
FIU. 6. Central ion surrounded by two hexagonal shells of atoms, which forms basis of finite lattice.
G.
V.
RAYNOR,
P.
NAOR, TABLE
Interplanl w distance d
TYZACK:
2
r
2,
C.
AND
CALCULATION
___ 8
(T
0.2 0.3 0.4 0.5 0.6 ;:;: 0.9 1.0 1.1 1.2 1.3 1.4 1.5 ::; 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2,5
MADELUNG
The Madelung
CONSTANTS
constant
t
489
may now be written
down in terms of p, q, r, Q, and s functions, in mind that the values of these functions
bearing tabulated
in Table 2 refer to charges of unity on the ions, and fractional charges of l/3. If the axial ratio of the
-. -
3.4282 3.4241 3.4117 3.3913 3.3635 3.3287 3.2876 3.2409 3.1896 3.1342 3.0758 3.0147 2.9519 2.8879 2.8234 2.7587 2.6943 2.6305 2.5675 2.5056 2.4451 2.3859 2.3283 2.2723 2.2179 2.1651
11.7221 11.6278 11.3645 LO.9797 LO.5260 LO.0451 9.5643 9.0988 8.6560 8.2390 7.8484 7.4835 7.1430 6.8257 6.5298 6.2539 5.9965 5.7561 5.5314 5.3212 5.1244 4.9398 4.7666 4.6039 4.4508 4.3066
12.4641 12.4248 12.3098 12.1270 11.8879 11.6052 11.2917 LO.9587 LO.6155 LO.2694 9.9259 9.5889 9.2611 8.9441 8.6392 8.3468 8.0672 7.8002 7.5457 7.3032 7.0724 6.8527 6.6437 6.4448 6.2555 6.0752
lattice.
OF
-
-
structure
is /I, and the distance
between
ions in a
given layer is unity:*
M=
-
-
7.7960 7.4333 7.0957 6.7815 6.4888 6.2159 5.9613 5.7237 5.5015 5.2936 5.0989 4.9164 4.7450 4.5839 4.4324 4.2896
7.7923 7.4310 7.0942 6.7803 6.4881 6.2152 5.9607 5.7233 5.5011 5.2933 5.0987 4.9163 4.7449 4.5838 4.4323 4.2895
-P(O)
+P(;)+P(Z)-2N3)
4:)
+
+q(y) +G!(T) - aiv) + i!lg!j +~(3 -24jj +r(y) +to(Aj - to(;) - fo($j+ tq;j + fcT($j + 25(Z)
-
Inserting the appropriate numerical values for the calculated functions, M = 1.6399, in excellent agreement with the value quoted above. Another structure of interest in the study of intermetallic compounds is that of nickel arsenide;
Tables of the functions p, r, G, s, and t as functions In the tables prepared of d may now be constructed.
many compounds between transitional electronegative metals or metalloids
in the authors’ laboratory,
general structure type, but the axial ratio p is variable. It is therefore of interest to know the varia-
the variable d was changed
in steps of 0.01 over a range 0 < d < 2.5. The figures in Table 2 show values for steps of O-1 in d,
tion
and may be used for interpolation.
so that the electrostatic
Madelung
for
any
structure which may be represented in terms structure plan of the type discussed above.
of a
4.
constants
may
APPLICATION
now
Using this table
TO
be delived
SPECIFIC
The method of use of the functions may be illustrated
by a calculation
CASES
discussed above of the Madelung
constant
of wurtzite, for which a value of M = 1.641 h 0.002 for c/n = 1/S/3 is given in the literature. The “structure plan” for wurtzite is: Wn
- * B,,A, . . B,A,, . . B,,
forces
of
the
may
Madelung
constant
with
contribution
be assessed.
The
metals adopt
axial
and this
ratio,
to the binding
“structure
plan”
for
nickel arsenide is: (i) A,.%.A,.Bn.A, (ii) Bnl.A1.Bn.A~.Bn~ From the point of view of electrostatic two formulations it is necessary
energy, these
are not quite equivalent,
so that
to take the mean of calculations
for
the two sequences (see equation 5). For the interactions between integral charges, we may write down:
in the [OOO,l] direction
The unit cell is chosen so that the top and bottom Then ion layers occur at layers are A,, layers.
(i) -P(O)
-
2~ i 0
-
2ip(B) -2qp
heights 0, 3~18, c/2, 7c/8, and c, and layers of fractional charges are placed at heights 3c/l6, l&/16. The layers of fractional
7c/16, llc/16, and charges at 3c/16
and llc/lB are of types I or II, and cancel, with no fringe charges; the layers of fractional charges at 7~116 and l&/16 are of type III, and cancel within the finite lattice, but leave marginal charges round the outside. The fractional charges must take the value l/4 to preserve the neutrality of the finite
0
(fi) -P(O) - 2r i
2q i 0 +
- %3(B)- WB) +
* The symbols p(z), p, q, etc. for interplanar in the brackets.
2r(i)+2r(y)
q( z
), etc.,
spacings
2r(zj+4:)
denote
the
values
of the magnitudes
of
given
ACTA
490
Adding and dividing
0
-p(O)-p
f
METALLURGICA,
0
-2ppB-q
;
of fractional
charge
(i)
1957
by components
+ 2r(Y)
equal
to
l/6
are The
g(E)+ o(z)+O(i)+‘(:I +
+ o(T)
+ o(y)
occurs,
not at
which differ widely in electrochemical
character are greater than 1.633. In phases such as TiSe (fi = 1.75) the binding forces must be predominantly ionic, and it is unnecessary bonding
between
assumed
by
ratios.
the
Ehrlich,@)
As /3 decreases
cations to
to postulate
in cation
account
strong
layers,
for
high
(Table 3) there is a decrease
electrostatic cqg)+sgjthan important part.
interactions
+ o(T)
has frequently
play an increasingly
of the nickel arsenide phases
been discussed,(lO~ll) and one of the
chief features is that atoms of the transitional Adding and dividing
as
axial
in the Madelung constant; this indicates that in nickel arsenide phases of low axial ratio forces other
The crystallography (ii) O(i)
constant
approximately 1.78; this interprets the observation that the axial ratios of nickel arsenide phases formed
+ 2r p
5118, and 7Q/S.
Madelung
the ideal axial ratio of 1.633, but at an axial ratio of
0
-2q(f!q
each
placed at heights /3/S, 3g/S, interactions therefore are:
5,
The maximum
by 2 gives:
-r(f) Layers
VOL.
metal
(electropositive component) form chains parallel to the c axis of the crystal; the decrease in axial ratio
by 2 gives
which occurs as the difference in the electrochemical character
of
attributed
to metallic
the
components forces,
decreases
between
these chains, to which the partially the
The Madehmg constant M’ relative to unit distance between two ions in a layer is therefore: M’ =
-P(O)
-P
+2$!)
; 0
-$)
-2P(B)
+
-
243
!J ; 0
+ o(b)
-
transition
metal
probably
+ o(;)
axial ratio in a given nickel arsenide phase may be associated
with
departure
from
value
of variable
by multiplying
packed arrangement
+s(T)
axial ratio we correct this
by
;
the
corrected
constant
M then refers to the closest anion-cation distance taken as unity. Putting in numerical values from
Table
2, the results
summarized
in Table
are obtained. TABLE
~ 1.20 1.28 1.36 1.44 1.52 1.60 1.68 1.70 1.72 _
--
____~ 1.025 1.659 I.680 1.700 1.714 1.723 1.729 1.730 1.730
1.74 1.76 1.78 1.80 1.82 1.84 1.92 2.00
1.730 1.731 1.731 1.731 1.730 1.729 1.725 1.717
composition
3
of
all available
in voids
the electroThe extra in the close-
of metalloid atoms;
constant
atoms randomly but the structure
a calculation
for the case of extra
-4
distributed would be difficult, corresponding to complete filling voids
at A,B
may
be
relatively
simply treated. Completely “filled in” structures, such as that of Cu.&, have the extra A atoms in trigonal voids in the B atom layers. Considering the electrostatic interactions along the vertical rows of
3
the
AB towards A,B, where A represents positive (transitional metal) component.
+ :a(;)
of the Madelung In a structure
filled d-shells of The
contribute.
axial ratio of the nickel arsenide structure is variable
%@)
+ ogj
be
not only from system to system, but also within a given system as a function of composition. A low
A atoms are accommodated + fqg
may
the atoms in
atoms
only,
the
summation
:[-29(;) -29(P) + 2. %gj +;_;+f-p The introduction
takes
the
form:
- 2.2!7(/%]
“1=*[-;I
of the extra A atoms
thus tends
to bring about a mutual cancellation of the repulsive and attract#ive electrostatic forces parallel to the hexagonal axis. The electrostatic repulsions present
G.
V.
RAYNOR,
P.
in the AB structme axial
ratio
NAOR,
C.
AND
TYZACK:
are relaxed, and the decrease of
as the composition
tends
towards
may be understood
in terms of metallic
between A atoms. The examination
of the Madelung
nickel arsenide static
forces
ponents
structure
thus suggests that electrosince, where the com-
appreciably
structure,
the
be preferred
electrochemically,
the
of the Madelung
sodium
chloride
on the grounds
energy considerations
structure,
of electrostatic
alone, as the Madelung constant
(l-748) is larger than that of the hexagonal From
of the
This cannot be the only important factor, since the cubic analogue of the nickel
arsenide would
interactions
constant
axial ratio corresponds to a maximum constant. however,
A,B
the geometry
and the nature
represents
forces
forces operative
structure.
of the nickel arsenide
of the A metal,
that the structure electrostatic
and
structure
a compromise
for which
cellent
were
nonelectrostatic
attractive
calculations
have
also
been
the maximum l-55
agreement
and with
l-75
values
of M were
respectively,
experimentally
in exobserved
values. simple
description
notation
has
been
of crystal structures
developed
for
the
which correspond
to
layers of hexagonally arranged atoms perpendicular to a certain crystallographic direction. This class of crystal those may purely
structures
adopted
by
be classified
includes
described
a large proportion
intermetallic
compounds
as in the transition
ionic to purely
of this notation
for a given structure may be simply evaluated.
Calcu-
lations carried out emphasize
of the
electrostatic
metallic
range
bonding.
of
which from
By means
it is possible to generalize the method
by Frank(‘)
details
so that the Madelung
constant
MADELUNG
contribution compounds,
and
compounds,
by
in certain
comparing
the
such as the axial
ratios,
from electrostatic
energy
with those to be expected considerations
the importance
to the bonding
of the structures,
the crystal chemistry of intermetallic and especially the development of
forces of a metallic nature
may be better understood.
The method of approach to the problem adopted in the present paper gives a quantitative interpretation to the qualitative which is frequently of the bonding consistent
concept
of heteropolarity
used in discussions
in intermetallic
use of “Structure
on the nature
compounds,
Plans”
and the
is an important
aid in the necessary numerical evaluations. ACKNOWLEDGMENTS
The work described gram
on
the
crystal
forms part of a general prochemistry
of
phases in progress at the University
intermetallic
of Birmingham,
which has received generous financial assistance from the Royal Society, the Department of and Industrial Research, and Imperial Industries
Lt.d.
Grateful
Scientific Chemical
acknowledgment
is also
made of the award of a British
Council scholarship
to one of the authors
and of
maintenance
5. CONCLUSION
A
491
between
along A atom chains.
Madelung-constant
calculated
CONSTANTS
it may be inferred
carried out by the method described above for the Mg,Bi, and Ma,As structures. The axial ratio values
OF
intermetallic
are important,
differ
CALCULATION
(P.
N.),
a D.S.I.R.
allowance to another (C. T.). REFERENCES
N. F. and JONES H. The Theory of the Properties of Metals and Alloys Clarendon Press, Oxford (1936). 2. APELL P. Acta Math. Stockh. 4, 313 (1884). 1. MOTT
3. 4. 5. 6. 7. 8. 9. 10. II.
MADELUNG E. PhyS. 2. 19, 524 (1918). EWALD P. P. Ann. Phys. Lpz. 64, 253 (1921). BORN M. 2. Phys. 7, 124 (1921). SLATER J. C. Introduction to Chemical Physics Wiley, New York (1939). FRANK F. C. Phil. Mug. VII, 41, 1287 (1950). FISHER E. J. Chem. Phys., 19,1284 (1951). ERRLICH P. 2. Anorg. Chem. 266, 1 (1949). LAVES F. and WALLBAUM H. I. 2. Anger. M~wT. 4, 17 (1942). CASTELLIZ L. and HALL F. 2. Metallk. 35, 222 (1943).
METHOD
FOR THE
CALCULATION
FOR INTERMETALLIC G.
V.
RAYNOR,
OF MADELUNG
CONSTANTS
COMPOUNDS*
P. NAOR,
and C. TYZACKt
The crystal structures of many intermetallic compounds may be expressed by a simple notation baaed upon the sequence of hexagonal layers of atoms in some simple crystallographic direction. Using this notation, a general method for the calculation of Madelung constants for such structures has been developed, in which the electrostatic interaction of a central ion with various layers of ions above and below that in which the central ion lies is expressed in terms of six functions. These functions have been evaluated and tabulated. The use of the new method, which appears to be the simplest yet proposed, is illustrated by reference to the wurtzite and nickel arsenide structures; it is suggested that, by means of such calculations, quantitative expreseion may be given to the qualitative concept of heteropolarity in the discussion of intermetallic compounds. UNE
MBTHODE
GfiNfiRALE POUR LES
DE CALCUL DES CONSTANTES COMPOSES INTERMBTALLIQUES
DE
MADELUNG
Les structures cristallines de nombreux compos6s interm6talliques peuvent s’exprimer par une notation simple reposant sur la succession des couches atomiques hexagonales dans une direction cristallographique simple. En utilisant cette notation, les auteurs ont d6velopp6 une mbthode g&&ale de calcul des constantes de Madelung pour des structures oti l’interaction Blectrostatique entre un ion central d’une couche atomique et les ions des couches adjacentes est exprimbe au moyen de six fonctions que l’on a calcul6es et dress&es en tableaux. L’emploi de cette nouvelle m&hode qui semble 6tre la plus simple de celles proposires jusqu’h prbsent, est illustrb par des exemples se rapportant aux structures de la wurtzite et de l’arseniure de nickel. Les auteurs pensent qu’8 l’aide de tels calculs, on pourrait trouver une expression quantitative du concept qualitatif d’h6t6ropolarit6 dans les problbmes interessant les compo&s interm&alliques. EINE
ALLGEMEINE
METHODE ZUR BERECHNUNG DER MADELUNG-KONSTANTEN INTERMETALLISCHER VERB.tNDUNGEN
Die Kristallstrukturen vieler intermetallischer Verbindungen lassen sich in einem einfachen Schema beschreiben, welches auf der Folgeordnung hexagonaler Atomschichten in einfachen kristallographischen Richtungen beruht. Unter Benutzung dieses Schemas wurde eine allgemeine Methods fiir die Berechnung der Madelung-Konstanten solcher Strukturen entwickelt, bei der die elektrostatische Wechselwirkung eines Zentral-Ions mit verschiedenen Schichten von Ionen oberhalb und unterhalb der Schicht des Zentralions durch sechs Funktionen ausgedriickt wird. Diese Funktionen wurden berechnet und tabelliert. Der Gebrauch der neuen Methode, die zur Zeit die einfachste zu sein scheint, wird am Beispiel der Wurtzit- und NiAs-Struktur erliiutert; es wird ferner darauf hingewiesen, dass mit Hilfe derartiger Berechnungen dem Begriff der HeteropolaritBt bei der Diskussion intermetallischer Verbindungen eine quantitative Grundlage gegeben werden kann.
1. INTRODUCTION Binary
compounds
in the
solid
phases may state
may
be
classified into the following general types: (i) Homopolar
be considered
ably in electrochemical
compounds,
as similar to ionic com-
pounds, in so far as the components
differ consider-
characteristics.
Compounds
of type (iv) are of interest, since the valency of inorganic chemistry are not fulfilled, and
rules their
(ii) Ionic compounds, (iii) Normal valency intermetallic compounds, (iv) Abnormal valency intermetallic compounds, compounds” and other including “electron
properties vary from those of the metallic phases to those exhibited by some compounds classified as
structures of predominantly metallic type. The factors affecting the formation of types (i) and (ii) are well understood. Class (iii) includes such
of the ra,nge of properties electron compounds such these are formed between comas y-brass occur; ponents which differ little in electrochemical character,
compounds as Mg,Sn, in which the normal valencies of the components are satisfied;
and their stability at characteristic valency electron/ atom ratios is understood in terms of Brillouin zone theories.(l) With increasing electrochemical
* Received January 18, 1957. 7 Department of Physical Metallurgy, Edgbaston, Birmingham 15. ACTA
METALLURGICA,
VOL.
6,
The
group these
University,
SEPTEMBER
1967 433
normal
valency
compounds.
At
the
metallic
end
difference between the components, compounds of more complex types occur, and it is with the factors
484
ACTA
METALLURGICA,
affecting their formation that we a.re concerned in this paper. On general grounds these phases appear to represent a transition between ionic and metallic binding. In a typical ionic structure the ions may be considered to be hard impenetrable spheres, the packing of which is determined by electrostatic energy considerations and ionic size factors. The interaction between anions, other than electrostatic, is generally limited to closed-shell repulsions and slight van der Waals attractions, while interaction between cations is, in general, negligible. Considering anions of Group VIB of the Periodic Table, the degree of polarizability increases in the order 0, S, Se, Te, thus increasing the possibility of van der Waals attractions between anions contributing to the lowering of the energy. For anions of earlier groups of the Periodic Table, these tendencies are enhanced, and relatively strong van der Waals binding between large, polarizable anions may transform gradually into metallic binding because of the redistribution of electron density more uniformly over the volume occupied by the compound. Thus, in the transitional range between ionic and metallic bonding, the electronegative components tend to have large, easily polarizable ions. We may thus expect the energy of the structure to be lowest for the maximum number of anion-anion contacts, and that compounds typical of the transition between the ionic and the metallic states will crystallize so that the electronegative ions, viewed alone, constitute a close-packed structure. A survey of those structures in which the electronegative component is a metal or semi-metal of Groups IVB, VB, and VIB of the Periodic Table, and t.he electropositive component is a metal of Groups IB, IIB, IA, IIA, or a transitional metal, shows that the great majority of the compounds included in the survey crystallize in structures in which the electronegative components are in close-packed arrangements. This tendency is lost for purely ionic or purely metallic structures. Thus, in the caesium chloride structure the anions form a simple cubic arrangement. No compound in the transition range should crystallize in this structure, since there is no opportunityfor close-packmgof electronegative components, and in fact only one pha,se (BiTl) of the transition type does adopt this structure. All other known representatives are either purely metallic (e.g. NiAl) or ionic. In the ionic calcium fluoride structure, the fluorine anions constitute a simple cubic lattice; the intermetallic phases which are isomorphous with this structure are of the purely metallic type (e.g., PtSn,, AuAl, PtIn,). In the transition range
VOL.
5,
1957
(e.g. Mg,Sn, Mg,Pb) structures appear which are not isomorphous, but anti-isomorphous, with calcium fluoride, and it is the electronegative component which takes up a close-packed structure in agreement with the above generalization. For the compounds which fall into the transition cla.ssification, therefore, we may infer that electrostatic binding forces are of considerable importance; it would be helpful to be able to assess the contribution made by such forces. The electrostatic energy E of a binary compound of equiatomic composition is given by the expression: E = -MNe2z2/r
(1)
where N is the number of neutral groups of anions and cations, e is the electronic charge, and r the closest anion-cation distance in the structure; M is the Madelung constant, which is a function of the geometry of the lattice, and z is the charge on a single ion. The lowest electrostatic potential energy is attained by a structure of maximum Madelung constant; any deviation from a structure satisfying this condition, for a given binary compound, is interpreted as due to the operation of forces of a different nature. The magnitude of the ionic cont,ribution to binding may therefore be assessed from the magnitude of M. The calculation of M, however, is laborious in all but the simplest cases. The observation that in a large number of intermetallic compounds the electronegative components take up close-packed arrangements suggests the possibility of a more general approach to the calculation. 2. THE
“STRUCTURE
PLAN”
NOTATION
As noted above, the structures of many intermetallic compounds with properties intermediate between those of purely ionic and purely metallic materials contain close-packed planes of atoms in a hexagonal array. Such planes appear in t,he facecentred cubic, close-packed hexagonal, body-centred cubic, simple cubic, and rhombohedral structures. Some tetragonal or orthorhombic crystals also exhibit pseudo-hexagonal symmetry and contain slightly distorted hexagonal planes of atoms. Such structures, therefore, contain a sequence of hexagonal atomic layers along some crystal direction, and it is upon the existence of these layers that the “structure plan” notation, as outlined below, is based. A simple example is afforded by the face-centred structure (Fig. 1). If the atoms are projected onto the (111) plane, then atoms of the type I form a hexagonal arrangement ; atoms marked II project into the centres of alternate triangles formed by
G.
V.
RAYNOR,
P.
NAOR,
C.
AND
TYZACK:
CALCULATION
OF
MADELUNG
COh’STANTS
structure becomes simple cubic; layers disappear, and, denoting
485
alternate hexagonal these missing layers
by dots, the layer sequence is: A,.O,,I.A,,.O,.A,,,.O,,.A, Omitting
the
layers
corresponding
sites, the face-centred
A,. . .A,,. FIG. 1. Unit cell of the face-centred cubic The crosses mark octahedral interstices, and black circles mark tetrahedral interstices.
structure. the small
If atoms
in all possible
following
sequences
into
of type the
centres
projection sional
I, while
atoms
of the
marked
remaining
(Fig. 2) takes the form
lattice,
dimensional
which
sublattices
to this projection
in a direction
of a two-dimeninto
to
the
of
structures layers,
(111)
planes.
This
define the face-centred
as may of
interstices
be
appreciated
Fig.
1,
which
between
from
to
further
the
original
(small
positions
are
face-centred
T, the sequence becomes:
ATI This atoms
II0 IIITA I
structure are
Face-centred
removed
from
IIIA I
cubic:
cubic.
. .A,
tures, the relative heights [ 1111 direction, considered by
reference
to
Fig.
of the layers along the vertical, being denoted
2 by
the
subscript
Roman
Assuming equal cell dimensions, numerals. interlayer distances are in the proportions:
Structure
plans
geometrical
of this type
relationships
to be concisely
enable
between
expressed
many simple
by addition
This is to be described
elsewhere,
of the lattices
or subtraction. but it may
be
noted here that where no layers are missing the subscripts for adjacent planes must differ; an identithe
atoms
for adjacent
planes would
in one plane
sites,
the
which is forbidden by packing considerations. restriction does not necessarily apply when missing
from
the
structure
interpenetration
of
in
the
structure
(Fig.
adjacent
plan.
face-centred
lie
plane, This layers
Thus,
3) is represented
two
that
would
immediately
diamond
those
imply
vertical)
the
are
above
([ll l]
If now
tetrahedral
the
b.c.c. : s.c . : f.c.c. = & : 4 : 1.
cal subscript
IIT IIIOT I IIA IIITO I IIT IIIA I
is body-centred
IIIA4A IA II
Such sequences will be referred to as “structure plans,” a,nd interpret the fact that the same sequence of hexagonal layers occurs in all three crystal struc-
con-
cubic sites, then, denoting octahedral positions by the symbol 0 and tetrahedral positions by the symbol
the
Simple cubic:
octahedral
interstices
If all these interstitial
in addition
these
the hexagonal
contains
(crosses) and tetrahedral
circles).
occupied,
difference
lies in the spacing between
sideration black
The
4AA JIIAAA I [I IIII II
A 1. #. A,,. . .A,,,.
since the same sequence is obtained,
mercury.
are identical,
By reference
along the (111) direction, for body-centred cubic, simple cubic, and rhombohedral structures such as that
positions
of layers in the [ll l] direction,
three two-
layer sequence
normal
. .AII,. . ,A,
cubic:
=IAA I II
The
the structure is defined
does not uniquely
cubic structure,
project
triangles.
I, II, and III.
or “plan,”
by the well-known
sequence
is subdivided
III
octahedral
may be written down: Body-centred
atoms
to
cubic structure is regenerated:
the
as the
cubic
sub-
lattices :
A,. . .A,,.
. .A,,,.
. .A, = A,A,, . . A,,A,,,
+ .A,,.
FIQ. 2. Projection of atoms in face-centred structure on to (111) plane.
cubic
. .A,,,.
The hexagonal different type,
. .A,. . .
I
..
AIIIA,.
*AI
layers which approach closely are of while those with identical subscripts
are well separated.
ACTA
486
METALLURGICA,
VOL.
5,
anion-cation
1957
distance.
For
the
binary
compound
A B, (3)
The sum zj ?-
FIG. 3. The diamond
Similar
structure
hexagonal layers
in
the
interstices
may
be
representing
[OOO,l] direction.
developed
for
the sequence
of
The
close-packed
structure plan:
III
TAT0 I II
between
I
III
and tetrahedral
with the sequence:
TAT0 II I
III TAT I II
II
structures
which
valuable
in the case of complex
To describe
plan notation
structures
may
structures,
is also applic-
such as that of MgZn,,
sublattices, instead of three as in Fig. 2; the intimate relationship of this structure with Ohose of MgCu, is then immediately apparent. For the method of calculating Madelung
constants, however, above suffices. 3. THE
the
simple
For the simpler E,
outlined
MADELUNG
of intermetallic
of interest to know the contribution
compounds
I
structures,
the interaction
energy
of an A ion with all other ions is equal to the energy E,
of a B ion with all other ions.
structures
this is not necessarily
so, and the total electrostatic
energy then becomes +
fve2(zAzB) B M
r
(5)
from which we may write: M = Pf, where M,
and M,
constants. The Madelung
-I- M13)/2
(61
may be called partial Madelung constant,
the sum of the electrostatic
therefore, interactions
represents of a central
ion, bearing unit charge, with all other ions in the lattice, which also bear unit charges. has been studied
The summation
by Apell,(2) Madelung,t3) Ewald,c4)
and Born;c5) their different types of solution involve advanced mathematics and much computational work even for structures of high symmetry. Slaterc6) and Frank(‘) as pointed
CALCULATION OF CONSTANTS
In the discussion
acheme
(4) \
M
A
be
or neutral units,
Ne2(zA%)_
M
structure plans may be referred to a two-dimensional hexagonal network which is subdivided into four
and MgNi, illustrating
_
I
is capable of revealing relation-
different
over all pairs of ions,
r
in which all octahedral
to which the structure able.
=
For more complex
obscured by the more usual crystallographic notations, such as the Space Group notation. This is particularly
E
interaction
are filled corresponds II
by summing
so that if there are N “molecules”
A,. . .A,,. . .A,. . .A,,. . .A,
This type of notation ships
E is computed
structure.
structure has the following
A structure
AT0 I
plans
structures,
hexagonal
M, and depends only on the geometry of the crystal lattice. The total electrostatic energy of the crystal
-----u
u-------
is known as the Madelung constant,
have
proposed
simpler methods,
but,
out by Fisher,(*) the Slater method does
not lead to accurate results in several cases, while for certain lattice types the method of Frank requires the it is
to the binding
evaluation
of a large number of terms.
The principle of the Slater and the Frank methods
energy which may be regarded as arising from electro-
is to sum the interactions
static
We may consider a binary compound AB as an array of positive and negative charges, which as a whole is electrically neutral.
finite lattice,
The electrostatic interaction energy with all other ions is given by:
just outside the lattice. These fractional charges are such as to preserve the electrical neutrality of the finite lattice. The methods differ in the positioning of the fractional charges. Slater -locates them on
interactions.
of the ith ion
(2) where zi and zj are the charges on the ith and jth ions, e is the electronic charge, and pirr is the distance between the interacting ions, r being the shortest
of a central
ion with a
and to correct for the finite nature of
the lattice by calculating the interaction of the central ion with certain artificial fractional charges placed
lattice sites, while Frank places them on central positions between lattice sites. In the development to be described, the technique adopted by Frank is retained,
but the method
is generalized,
by using
G.
V.
RAYNOR,
the description hexagonal
P.
AND C.
CALCULATION
TYZACK:
of structures in terms of sequences of
layers.
interactions
NAOR,
may
exist
between
the
ion and various layers of ions considered and the calculation
central
interaction
CONSTANTS
48’7
of the central ion with a whole with a layer of
type III) is:
as a whole,
for a specific structure
MADELUNG
layer of type II (or, by symmetry,
We may then specify the different
which
Similarly,
OF
ml =
becomes
P2 +
p1+
P3 +
P4 +
. . . + Pm
a matter of choosing the correct types of interaction by reference to the structure plan. The finite lattice
where the terms pl, p2, etc. are the interactions of the central ion with successive rings of atoms in the
takes the form of a hexagonal
layer
prism, made up of a
sequence of hexagonal layers. The central atom is defined as lying in a layer of type I. This may interact with ions in layers of types I, II, and III. Above and below
each layer
a hexagonal
layer
of positive
of
artificial
negative
For
interaction
of
the
central
two cases must be distinguished:
ions we imagine
fractional
considered.
ion with layers of fractional charges between ionic layers of types I and II (or, by symmetry, I and III), (i) The layer
of fractional
layer just outside
charges forms the top
the finite lattice.
Here the whole
charges, at a normal distance
of d/2, where d is the
layer reacts with the central ion, and the interaction
spacing between ionic layers.
The charges lie at the
is represented by the sum
centres
of lines joining
Similarly,
the ions in the real layers.
each layer of negative
artificial
positive
charges
above
cir + ‘1s + 0s -1 ‘14 + . . . urn = s,,
ions has layers of a.nd below.
Inside
(ii) Layers of ions exist above the layer of fractional
the lattice the fractional charges cancel, and only those outside the finite lattice need to be considered.
the finite lattice
The
a hexagonal
central
ion thus interacts
with
the
whole
or
charges.
In this case all the fractional are neutralized,
fringe surrounding
charges within
and only
those in
the finite lattice
at
a part of layers of fractional charges pla,ced between ionic layers of types I and II, I and III, and II and III. If the fractional charges are between
a certain height intera.ct with the central ion. This interaction is given by the term onl+i. Interaction
layers of type I, they fall completely within the finite lattice, unless they form an external layer at the top
between ionic layers of types II and III may be expressed by a similar function; this type of inter-
or bottom
of the prism.
We assume each ion to have
unit charge, and for computational fractional
convenience
each
charge is l/3.
of the central ion with a layer of fractional
action
action
of
the
central
ion
(Fig. 4). with
that
successive
hexagonal
of atoms
The possible interactions To
the
functions
the whole
the
are summarized in Table 1.
functions
we
must
know
the
fractiona,l charges in other layers. The function q is a trivial case and equals l/d, where d is the distance
in the
with upper
between
upper
layer
is
between
layers.
the central
ion and the ions or
In the hexagonal
array
I of Fig.
2, the square of the distances of ions on the sublattice from the central ion may be written: l2 = ,rn2 -
rrz, 7~a, 7~~’. . . , VT,,. Interaction
of the central ion with represented by:
evaluate
immediately
layer, counting up to the mth ring, are then described by
possible
distances
x be VT~; interactions
rings
and is only
First, inter-
above it (atom 2) is expressed as a term q ( = l/d). Let the interaction between the central ion and the six atoms surrounding
encountered,
when the layer of fractional charges forms the top la.yer just above the finite lattice.
The interaction of the chosen central ion in a layer of type I with another layer also of type I is expressed as a sum of simple functions
is rarely
charges
mk +
k2
(m > 0; 0 < k < m).
Thus for the six ions surrounding
the central
l2 = 1, the unit
of distance
the interatomic
_.
~___
I
being
ion,
TABLE 1
Interacting with layer of type Centml ion in leyer of type
0 Central ion
Pm. 4. Dirtgram illustrating interaction of layer of type I with central ion in another type I layer.
I
II
III
P.Q
r Pdl T
T T P.P
I’I
T
=I
T
Layer of fractional charges between I and II
Layer of fractional charges between I and III
Layer of fractional charges between II and III
488
ACTA
METALLURGICA,
VOL.
5,
1957
Where m is even o, = di-2 -
k=l
distance in the layer, while ions in the next hexagonal shell lie at distances Z2= 3, Z2= 4, as shown in Fig. 5. The rrnzfunction is thus: rr,=6&m2-mk+
k2 + cP)-*
k=l
where the summation indicates the addition of all values of (m2 - mk + k2 + d2)-4 which are produced when m takes one of the values 1,2, 3 . . . , and k, ior any given value of m, takes the values 1, 2,3, . . . , m. The squares of the distances from a central ion on sublattice I to ions on sublattices II and III of Fig. 2 follow the sequence 113,413, 7/3, 13/3, 16/3, 19f3, 2513,2813 . . .
The numerators are numbers, half of which are represented by the totality of numbers 3ma 3k(m - k + 1) + 1, while the other half are represented by the totality of numbers 3m2 + 3m - 3k(m - k + 1) + 1. Thus we derive: p,=31/3
[
m-l +
2
k=l
(3m2-3km+3k2+3m-3k+1$-3d2)-t
For the t function, the corresponding algebraical expression is complicated; this function is considered below in relation to a finite lattice. The remaining fictions, may, however, be evaluated to any desired value of m by the expressions given. In the initial development of the method, some experiment was necessary to determine the size of the finite lattice required for accurate results. This exploratory work indicated that evaluations of the Madelung constant accurate to 0.001 were obtainable by considering the central ion to be surrounded, in its own plane, by two hexagonal shells of ions of its own type (Fig. 6); the finite lattice is then constructed by erecting further layers above and below this plane, as dictated by the particular crystal structure considered. To this approximation, the functions take the following values, where d denotes the normal distance between the central plane and the particular layer involved: q(d) = l/d p(d) = S[(l + d2)-” + (3 + d2)-’ + (4 + d2)“] r(d) = 3%‘5[(1 + 3d2)-” + (4 + 3d2)-”
+ 2(7 + 3d2)-t] o(d) = ~%[2(43
+ 12d2)-* + 3(49 + 12d2)-$
1
s(d) = v’%[(l
+ 12d2)-” + 2(7 + 12dz)-” ’
+ 2(13 -+ Bdz)-9
+ 2(19 + 12d2)-”
+ (25 + 12d2)-a-+ 2(31 + 12d2)-*
+ 2(37 + l%P-“1 t(d) = 2/4[2(1 + 4d2)-” + 2(3 + 4d)-* + 4(7 -/- 4dd2)-f
+ 2(9 + 4d2)-8 + 2(13 f 4d2)-“J
f8k + 7 + 12da)-6 m-1
+
1
3k + 1 + 12d2)-*
+ 2(61 + 12dV$1
for interaction of the central ion with layers of type II only, or with layers of type III only. Considering layers upon which fractional charges are placed, the fractional charges occur either between I- and II-type ions, or I- and III-type ions. By projection of both types of fractional charge onto Fig. 2, it may be shown that the following equations hold: Where m is odd
-
6km + 12k2
k=l
(3m2 - 3km + 3k2 -
5 (3mz-3~+3kz-3k+l+~z)-~ k=l
m’F1(3m2 -
3m + 1 + 12d2)-*
+ 2
FIG. 5. Diagram representing the squ-s of the distance from central ion to ions in successive hexagont61shells.
[
&
3*m2 -
3km + 3k2 -
3m + 1 + 12d2)-*
1
FIU. 6. Central ion surrounded by two hexagonal shells of atoms, which forms basis of finite lattice.
G.
V.
RAYNOR,
P.
NAOR, TABLE
Interplanl w distance d
TYZACK:
2
r
2,
C.
AND
CALCULATION
___ 8
(T
0.2 0.3 0.4 0.5 0.6 ;:;: 0.9 1.0 1.1 1.2 1.3 1.4 1.5 ::; 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2,5
MADELUNG
The Madelung
CONSTANTS
constant
t
489
may now be written
down in terms of p, q, r, Q, and s functions, in mind that the values of these functions
bearing tabulated
in Table 2 refer to charges of unity on the ions, and fractional charges of l/3. If the axial ratio of the
-. -
3.4282 3.4241 3.4117 3.3913 3.3635 3.3287 3.2876 3.2409 3.1896 3.1342 3.0758 3.0147 2.9519 2.8879 2.8234 2.7587 2.6943 2.6305 2.5675 2.5056 2.4451 2.3859 2.3283 2.2723 2.2179 2.1651
11.7221 11.6278 11.3645 LO.9797 LO.5260 LO.0451 9.5643 9.0988 8.6560 8.2390 7.8484 7.4835 7.1430 6.8257 6.5298 6.2539 5.9965 5.7561 5.5314 5.3212 5.1244 4.9398 4.7666 4.6039 4.4508 4.3066
12.4641 12.4248 12.3098 12.1270 11.8879 11.6052 11.2917 LO.9587 LO.6155 LO.2694 9.9259 9.5889 9.2611 8.9441 8.6392 8.3468 8.0672 7.8002 7.5457 7.3032 7.0724 6.8527 6.6437 6.4448 6.2555 6.0752
lattice.
OF
-
-
structure
is /I, and the distance
between
ions in a
given layer is unity:*
M=
-
-
7.7960 7.4333 7.0957 6.7815 6.4888 6.2159 5.9613 5.7237 5.5015 5.2936 5.0989 4.9164 4.7450 4.5839 4.4324 4.2896
7.7923 7.4310 7.0942 6.7803 6.4881 6.2152 5.9607 5.7233 5.5011 5.2933 5.0987 4.9163 4.7449 4.5838 4.4323 4.2895
-P(O)
+P(;)+P(Z)-2N3)
4:)
+
+q(y) +G!(T) - aiv) + i!lg!j +~(3 -24jj +r(y) +to(Aj - to(;) - fo($j+ tq;j + fcT($j + 25(Z)
-
Inserting the appropriate numerical values for the calculated functions, M = 1.6399, in excellent agreement with the value quoted above. Another structure of interest in the study of intermetallic compounds is that of nickel arsenide;
Tables of the functions p, r, G, s, and t as functions In the tables prepared of d may now be constructed.
many compounds between transitional electronegative metals or metalloids
in the authors’ laboratory,
general structure type, but the axial ratio p is variable. It is therefore of interest to know the varia-
the variable d was changed
in steps of 0.01 over a range 0 < d < 2.5. The figures in Table 2 show values for steps of O-1 in d,
tion
and may be used for interpolation.
so that the electrostatic
Madelung
for
any
structure which may be represented in terms structure plan of the type discussed above.
of a
4.
constants
may
APPLICATION
now
Using this table
TO
be delived
SPECIFIC
The method of use of the functions may be illustrated
by a calculation
CASES
discussed above of the Madelung
constant
of wurtzite, for which a value of M = 1.641 h 0.002 for c/n = 1/S/3 is given in the literature. The “structure plan” for wurtzite is: Wn
- * B,,A, . . B,A,, . . B,,
forces
of
the
may
Madelung
constant
with
contribution
be assessed.
The
metals adopt
axial
and this
ratio,
to the binding
“structure
plan”
for
nickel arsenide is: (i) A,.%.A,.Bn.A, (ii) Bnl.A1.Bn.A~.Bn~ From the point of view of electrostatic two formulations it is necessary
energy, these
are not quite equivalent,
so that
to take the mean of calculations
for
the two sequences (see equation 5). For the interactions between integral charges, we may write down:
in the [OOO,l] direction
The unit cell is chosen so that the top and bottom Then ion layers occur at layers are A,, layers.
(i) -P(O)
-
2~ i 0
-
2ip(B) -2qp
heights 0, 3~18, c/2, 7c/8, and c, and layers of fractional charges are placed at heights 3c/l6, l&/16. The layers of fractional
7c/16, llc/16, and charges at 3c/16
and llc/lB are of types I or II, and cancel, with no fringe charges; the layers of fractional charges at 7~116 and l&/16 are of type III, and cancel within the finite lattice, but leave marginal charges round the outside. The fractional charges must take the value l/4 to preserve the neutrality of the finite
0
(fi) -P(O) - 2r i
2q i 0 +
- %3(B)- WB) +
* The symbols p(z), p, q, etc. for interplanar in the brackets.
2r(i)+2r(y)
q( z
), etc.,
spacings
2r(zj+4:)
denote
the
values
of the magnitudes
of
given
ACTA
490
Adding and dividing
0
-p(O)-p
f
METALLURGICA,
0
-2ppB-q
;
of fractional
charge
(i)
1957
by components
+ 2r(Y)
equal
to
l/6
are The
g(E)+ o(z)+O(i)+‘(:I +
+ o(T)
+ o(y)
occurs,
not at
which differ widely in electrochemical
character are greater than 1.633. In phases such as TiSe (fi = 1.75) the binding forces must be predominantly ionic, and it is unnecessary bonding
between
assumed
by
ratios.
the
Ehrlich,@)
As /3 decreases
cations to
to postulate
in cation
account
strong
layers,
for
high
(Table 3) there is a decrease
electrostatic cqg)+sgjthan important part.
interactions
+ o(T)
has frequently
play an increasingly
of the nickel arsenide phases
been discussed,(lO~ll) and one of the
chief features is that atoms of the transitional Adding and dividing
as
axial
in the Madelung constant; this indicates that in nickel arsenide phases of low axial ratio forces other
The crystallography (ii) O(i)
constant
approximately 1.78; this interprets the observation that the axial ratios of nickel arsenide phases formed
+ 2r p
5118, and 7Q/S.
Madelung
the ideal axial ratio of 1.633, but at an axial ratio of
0
-2q(f!q
each
placed at heights /3/S, 3g/S, interactions therefore are:
5,
The maximum
by 2 gives:
-r(f) Layers
VOL.
metal
(electropositive component) form chains parallel to the c axis of the crystal; the decrease in axial ratio
by 2 gives
which occurs as the difference in the electrochemical character
of
attributed
to metallic
the
components forces,
decreases
between
these chains, to which the partially the
The Madehmg constant M’ relative to unit distance between two ions in a layer is therefore: M’ =
-P(O)
-P
+2$!)
; 0
-$)
-2P(B)
+
-
243
!J ; 0
+ o(b)
-
transition
metal
probably
+ o(;)
axial ratio in a given nickel arsenide phase may be associated
with
departure
from
value
of variable
by multiplying
packed arrangement
+s(T)
axial ratio we correct this
by
;
the
corrected
constant
M then refers to the closest anion-cation distance taken as unity. Putting in numerical values from
Table
2, the results
summarized
in Table
are obtained. TABLE
~ 1.20 1.28 1.36 1.44 1.52 1.60 1.68 1.70 1.72 _
--
____~ 1.025 1.659 I.680 1.700 1.714 1.723 1.729 1.730 1.730
1.74 1.76 1.78 1.80 1.82 1.84 1.92 2.00
1.730 1.731 1.731 1.731 1.730 1.729 1.725 1.717
composition
3
of
all available
in voids
the electroThe extra in the close-
of metalloid atoms;
constant
atoms randomly but the structure
a calculation
for the case of extra
-4
distributed would be difficult, corresponding to complete filling voids
at A,B
may
be
relatively
simply treated. Completely “filled in” structures, such as that of Cu.&, have the extra A atoms in trigonal voids in the B atom layers. Considering the electrostatic interactions along the vertical rows of
3
the
AB towards A,B, where A represents positive (transitional metal) component.
+ :a(;)
of the Madelung In a structure
filled d-shells of The
contribute.
axial ratio of the nickel arsenide structure is variable
%@)
+ ogj
be
not only from system to system, but also within a given system as a function of composition. A low
A atoms are accommodated + fqg
may
the atoms in
atoms
only,
the
summation
:[-29(;) -29(P) + 2. %gj +;_;+f-p The introduction
takes
the
form:
- 2.2!7(/%]
“1=*[-;I
of the extra A atoms
thus tends
to bring about a mutual cancellation of the repulsive and attract#ive electrostatic forces parallel to the hexagonal axis. The electrostatic repulsions present
G.
V.
RAYNOR,
P.
in the AB structme axial
ratio
NAOR,
C.
AND
TYZACK:
are relaxed, and the decrease of
as the composition
tends
towards
may be understood
in terms of metallic
between A atoms. The examination
of the Madelung
nickel arsenide static
forces
ponents
structure
thus suggests that electrosince, where the com-
appreciably
structure,
the
be preferred
electrochemically,
the
of the Madelung
sodium
chloride
on the grounds
energy considerations
structure,
of electrostatic
alone, as the Madelung constant
(l-748) is larger than that of the hexagonal From
of the
This cannot be the only important factor, since the cubic analogue of the nickel
arsenide would
interactions
constant
axial ratio corresponds to a maximum constant. however,
A,B
the geometry
and the nature
represents
forces
forces operative
structure.
of the nickel arsenide
of the A metal,
that the structure electrostatic
and
structure
a compromise
for which
cellent
were
nonelectrostatic
attractive
calculations
have
also
been
the maximum l-55
agreement
and with
l-75
values
of M were
respectively,
experimentally
in exobserved
values. simple
description
notation
has
been
of crystal structures
developed
for
the
which correspond
to
layers of hexagonally arranged atoms perpendicular to a certain crystallographic direction. This class of crystal those may purely
structures
adopted
by
be classified
includes
described
a large proportion
intermetallic
compounds
as in the transition
ionic to purely
of this notation
for a given structure may be simply evaluated.
Calcu-
lations carried out emphasize
of the
electrostatic
metallic
range
bonding.
of
which from
By means
it is possible to generalize the method
by Frank(‘)
details
so that the Madelung
constant
MADELUNG
contribution compounds,
and
compounds,
by
in certain
comparing
the
such as the axial
ratios,
from electrostatic
energy
with those to be expected considerations
the importance
to the bonding
of the structures,
the crystal chemistry of intermetallic and especially the development of
forces of a metallic nature
may be better understood.
The method of approach to the problem adopted in the present paper gives a quantitative interpretation to the qualitative which is frequently of the bonding consistent
concept
of heteropolarity
used in discussions
in intermetallic
use of “Structure
on the nature
compounds,
Plans”
and the
is an important
aid in the necessary numerical evaluations. ACKNOWLEDGMENTS
The work described gram
on
the
crystal
forms part of a general prochemistry
of
phases in progress at the University
intermetallic
of Birmingham,
which has received generous financial assistance from the Royal Society, the Department of and Industrial Research, and Imperial Industries
Lt.d.
Grateful
Scientific Chemical
acknowledgment
is also
made of the award of a British
Council scholarship
to one of the authors
and of
maintenance
5. CONCLUSION
A
491
between
along A atom chains.
Madelung-constant
calculated
CONSTANTS
it may be inferred
carried out by the method described above for the Mg,Bi, and Ma,As structures. The axial ratio values
OF
intermetallic
are important,
differ
CALCULATION
(P.
N.),
a D.S.I.R.
allowance to another (C. T.). REFERENCES
N. F. and JONES H. The Theory of the Properties of Metals and Alloys Clarendon Press, Oxford (1936). 2. APELL P. Acta Math. Stockh. 4, 313 (1884). 1. MOTT
3. 4. 5. 6. 7. 8. 9. 10. II.
MADELUNG E. PhyS. 2. 19, 524 (1918). EWALD P. P. Ann. Phys. Lpz. 64, 253 (1921). BORN M. 2. Phys. 7, 124 (1921). SLATER J. C. Introduction to Chemical Physics Wiley, New York (1939). FRANK F. C. Phil. Mug. VII, 41, 1287 (1950). FISHER E. J. Chem. Phys., 19,1284 (1951). ERRLICH P. 2. Anorg. Chem. 266, 1 (1949). LAVES F. and WALLBAUM H. I. 2. Anger. M~wT. 4, 17 (1942). CASTELLIZ L. and HALL F. 2. Metallk. 35, 222 (1943).