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Madelung Constants.

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Madelung Constants
BY PROF. DR. R. HOPPE
INSTITUT FUR ANORGANISCHE U N D ANALYTISCHE CHEMIE DER UNIVERSITAT GIESSEN
(GERMANY)
Dedicated to Prof. Wilhelni Klernrn on the occasion of his 70th birthday
Madelung constants are simple numbers which depend on the type of structure investigated.
They are needed for the calculation (using the Born-Haber cycle) of lattice energies and
erithalpies of formation of ionic compounds. Each Madelung constant is the sum of partial
Madelung constants which represent the contributions of the individual ions to the total
lattice energy. The partial Madelung constants depend on the ionic charge and, clearly
though not stringently, on the coordination number. On the other hand, each Madelung
constant can be represented by a sum of Madelung constants for related simple primitive
A B structures. Surprisingly, these Madelung constants are numerically interrelated in a simple
manner, and are related to the partial Madelung constants of interstitial sites, - Madelung
constants of parameter-dependent structures (e.g. of the rutile or anatase type) and their
variations with the structure-determining quantities are of particular interest. Madelung
constants also yield information in the case of complex compounds and - surprisingly of non-metal compounds (e.g. XeF2, XeF4, XeFyXeF4).
For many problems both inside and outside the realm
of inorganic chemistry, it is necessary or desirable to
obtain reasonably reliable information about the free
energy of formation of unknown compounds or of
compounds that have not yet been subjected to thermochemical investigation. A very recent example of such a
problem is that of the thermodynamic stability of
xenon fluorides [ I , l a l .
Entropies can nearly always be found with a high degree
of accuracy from thermochemical data for analogous
compounds. The calculation of enthalpies of formation,
on the other hand, is difficult even today. An important
aid to the calculation of enthalpies of formation, and
one that is often used because of its fundamental
simplicity, is the Born-Haber cycle. It can be seen from
Fig. 1 that the two simple steps I and 11, i.e. the formation of single atoms of the gaseous substances and the
formation, from these atoms, of isolated gaseous ions,
are followed by a third, more complicated step, in which
all the ions combine to form the crystal structure
Even for compounds which can be considered as consisting largely of ions, it is difficult to calculate the
enthalpy of formation AHo corresponding to step IV
(Fig. 1) with sufficient accuracy. There are several
reasons for this. In the first place (and this is a fundamental weakness of the Born-Haber cycle), AH" is a
relatively small difference of values that are often very
large (ionization potential Isp, electron affinity E,,
enthalpy of sublimation AH:ubl, dissociation energy
Do, lattice energy U,), so that small individual errors
can lead to a considerable absolute error. Secondly, the
calculation of the lattice energy associated with step I11
[ I ] R. Hoppe, H . Mottorich, K . - M . Rodder, and W. Dihne, 2.
anorg. allg. Chern. 324, 214 (1963); N . Burrleft, Endeavour 23, 3
(1964).
[ l a ] R. Hoppe, Fortschr. chern. Forsch. 5, 213 (1965).
Angew. Cliem. ititerrrat. Edit.
Val. 5 (1966) / N o . I
(cf. Fig. 1) is not simple. This lattice energy can be
broken down into three terms as shown in Equation (I),
U,
=
+
[MF x f(zi) x NL x e 2 ] / r ~ ~Eric
+ Ecorr
(I)
where NL is the Avogadro number, e is the unit charge,
rKAis the shortest distance between cations and anions,
and f(zi) is a function of the ionic charge, which is often
written out in full for binary compounds [f(zi) = z1 x 221,
but is better included in the Madelung constant M F in
the case of more complicated compounds. (For the sake
of simplicity, it is always assumed that T = 0 OK.)
+6ClG,,
Fig. 1. Born-Haber cycle f o r NaCl (c) and K?[PtCI,] (b). The index f
denotes the solid state.
95
In order to calculate the first term in Equation (l), i.e.
the “Madelung component of the lattice energy”, we
must know the Madelung constant. This is determined
by the charge on the ions and by their positions in the
unit cell; in the remainder of this paper it will for binary
compounds always be based on the shortest distance between cations and anions. The calculation of the Madelung constants is laborious without the aid of modern
computers; consequently, Madelung factors have so far
been calculated only for about 50 types of structures
(most of which are very simple, often parameter-free
structures) (cf. list in 131).
The term E,c, which takes into account the Born repulsion
between the electron clouds that is due to non-Coulombic
forces, must be present to explain the finite value of the
equilibrium distances rKA. Thus if we were to consider only
the Madelung component of the lattice energy, this would
reach its “optimum” value at rKA = o! Further correction
takes into account the mutual polarization of the ions, the
zero-point energy, lattice-energy components due to additional bonding forces (e.g. bonds within complex ions),
etc. (cf. [2931); these correction terms are combined in the
in Equation (1). As can be seen from the review
term E,,
by Waddington [31, satisfactory accuracy in the calculation of
lattice energies by means of Equation (1) can at present be
achieved only for relatively few inorganic compounds, e.g.
the alkali metal halides and alkaline earth metal chalcogenides. Impressive calculations have recently also led to the
lattice energies of the higher alkali metal oxides “+I.
tions[**J of the same compound, so that differences in the
enthalpies of formation are essentially due to differences in
the Madelung components of the lattice energies. A similar
argument applies to the variation of c/a or of the oxygen coordinates in isotypic compounds of the same structural type,
such as NaSc02, NaY02, NaInOz, and NaTlO2 151.
11. Partial Madelung Constants
It follows implicitly from Ewald’s method [61 for the
calculation of Madelung constants that the Madelung
constant, and hence the Madelung component of the
lattice energy, can be expressed as a sum of terms which
are due to the different ions in the unit cell [71. We shall
call these terms partial Madelung constants (PMF) 18991.
If in a compound A,Bn all the A particles occupy
equivalent positions and if the same holds for all the B
particles, Equation (2a) is valid:
MF(AmBn) = m x PMF(A)
+ n x PMF(B)
(2a)
For example, for the NaCl type [Oi - Frn3m, Z = 4,
with Na+ in 4(a) and C1- in 4(b)], we haveEquation (3),
MF(NaC1) = PMF(Na+) f PMF(C1-)
(3)
and similarly, for the CaFz type [Oz - Fm3m, Z = 4,
with Ca2+ in 4(a) and F in S(c)], we have Equation (4).
A. General
+2x
MF(CaF2) = PMF(Ca*+)
I. The Madelung Constant and the Lattice Energy
Equation (1) can be used for the calculation of enthalpies
of formation only if sufficient Madelung constants are
known and if the uncertainty associated with the terms
En, and (in particular) E,,,, can be minimized. This
makes the calculation of more Madelung constants
desirable. The knowledge of Madelung constants is
even more important in cases where the Madelung component of the lattice energy can be used alone, e.g. for
investigating which modification of a compound (e.g.
Ti02) is thermodynamically stable under normal conditions, or why a parameter-dependent structure [*] (e.g.
the rutile type) favors certain parameters (e.g. c/a w
1/2112 and xo= 0.3), instead of other “adjacent”va1ues.
It can often be assumed in such cases that the terms ECn and
Ecorr are practically the same for “adjacent” modifica[2] M . Born: Atomtheorie des festen Zustandes. Teubner, Leipzig, 1923; Handbuch d. Physik, Springer, Berlin 1927, Vol. 24,
p. 420. M. Born and A . Lande, S.-B. preuB. Akad. Wiss., physik.math. kl. 45, 1048 (1918); M. Born and A . Lande, Verh. dtsch.
physik. Ges. 20, 210 (1918); F. Haber, ibid. 21, 750 (1919); M . F.
C. Ladd and W. H. Lee, Trans. Faraday SOC.54, 34 (1958); E.
Madelung, Physik. Z . 19, 524 (1918).
[3] T. C. Waddington: Advances in Inorganic Chemistry and
Radiochemistry. Academic Press, New York 1959, Vol. 1, p. 157.
[4] L . A . D’Orario and R . H . Wood, J . physic. Chem. 69, 2550
(1965); R . H . Woodand L . A . D’Orario, ibid. 69,2558 (1965); 69,
PMF(F-)
(4)
If, on the other hand, the particles A occupy j different
types of positions (multiplicity zj), while the particles B
occupy r different types of positions (multiplicity z,),
the general equation is (2b), where the factors mj = zj/Z
r
j
MF(AmBn)
=
x m j x PMF(Aj)
j=l
+ x n r x PMF(Br)
(2b)
r=l
and n, = z,/Z ensure that the individual PMF values are
correctly weighted ( Z = number of formula units per
unit cell). Thus for Mn2O3 [Tl - la3, Z 16, with MnI
in 8(a), MnII in 24(d), and 0 in 48(e)], we obtain
MF(Mn203)=
1/2
PMF(MnI)+
312 PMF(Mnn)+
3 PMF(0)
(5)
In contrast to the Madelung constants themselves, which
offer only an overall picture of the energy relationships
of a structural type, the partial Madelung constants
provide information about the potential energies of the
individual ions. For the calculation of partial Madelung
constants, see [7-91.
2562 (1965).
[*I In a parameter-dependent structure, the dimensions of the
unjt cell (e.g. the ratio cia for tetragonal or hexagonal crystals)
or the coordinates of the particles in the unit cell can be altered
[ * *] “Adjacent” modifications differonly slightly in the structuredetermining parameters.
[5] R . Hoppe, Bull. SOC.chim. France 1965, 1115; Chem. Zvesti
19, 172 (1 965).
[6] P. P. Ewald, Ann. Physik 64, 253 (1921).
[7] R . Hoppe, Dissertation, Universitat Miinster, 1954; 2. anorg.
allg. Chem. 283, 196 (1956).
[8] R . Hoppe, Habilitation Thesis, Universitxt Miinster, 1958.
[9] H. G. v . Schnering, Habilitation Thesis, Universitat Miinster,
without affecting the symmetry.
1963.
96
Angew. Chem. internat. Edit.
/
Vol. 5 (1966) 1 No. I
111. Partial Madelung Constants as Structural
Arguments [9,101
The partial Madelung constants of the various ions can
also be used as structural arguments:
Using modern Fourier methods, very accurate information about the position of the center of gravity of the
particles in a crystal can be obtained by X-ray analysis
of single crystals, often with an average error of only
h0.005 A. It is only in special cases, however, that
electron density distributions can be found from precision measurements and used as a basis for conclusions
regarding the type of chemical bonding [10al.
It is therefore still useful to employ concepts such as
ionic radius, sum of the radii, ratio of the radii, coordination number, etc., in the interpretation of crystal
structures (which are described only geometrically by
the positions of the particles). Of course, there is a
remarkable discrepancy between these concepts, which
are often only vaguely defined, and the accuracy of
modern methods of structural analysis. Thus in addition to attempts to obtain improved definitions of these
fundamental crystal-chemical quantities (cf. e.g. [ill),
every possibility of obtaining further information from
the structural geometry should be examined. Such a
possibility seems to be offered by the partial Madelung
constants: A three-dimensional description of the
potentials and potential energies of the individual particles corresponding to the three-dimensional network
of the centers of gravity is obtained by calculation of
the partial Madelung constants. This discontinuous
“grid” of partial Madelung constants may then be
made as “dense” as required by the calculation of
partial Madelung constants for interstitial sites.
It has been found that even in the case of compounds
with considerable non-ionic character, a better understanding of the reasons for the occurrence of certain
structures is obtained by comparison of the partial
Madelung constants with those of chemically related
compounds (e.g. BrF3, XeF4, and XeF2) [llal.
It has been known for some time that the Born-Haber
cycle can also be applied in the case of compounds that
do not consist of ions. Thus Kapustinskii[121 demonstrated that the lattice energies of non-ionic compounds
can often be calculated with reasonable accuracy with
the aid of the effective Madelung constants (cf. Section
A.IX), which were introduced by Klemm [131. (In favorable cases, the lattice energies of covalent compounds
are found with an accuracy of about +10 kcal/mole.)
[lo] R. Hoppe and H. G. v. Sclinering, unpublished work
(1962-1964).
[IOa] Cf. in particular E. W o f i l e t a/.,Z . Elektrochem., Ber. Bunsenges. physik. Chem. 63, 891 (1959); Z . physik. Chem. N.F. 3,
273, 296 (1955); 4, 36, 65 (1955); 10, 98 (1957).
[I 11 R. Hoppe, 18. IUPAC-Congress, Montreal (Abstracts of
Scientif. Papers, page 73).
[ I la ] Cf. for example [la].
[I21 K . F. Kapustinskii, Z . physik. Chem. B 22, 257 (1933); Zh.
fiz. Khim. 5 , 59 (1943); Acta physicochim. USSR 18, 370 (1943);
Quart. Rev. chem. SOC.(London) 10, 283 (1956); A. F. Kapustimkii and K. B. Yatsimirskii, Zh. ohshch. Khim. 19, 2191
(1949); 26, 941 (1956).
[I31 W. Klemm, Z . physik. Chem. B 12, I (1931).
Angc~w.Cliem. internat. Edit.
1 Vol. 5 (1966) / No. 1
IV. Madelung Constants of AB Types with Single
Primitive Unit Cells [8,10,13al
Simple relationships exist between Madelung constants
of some compounds [7,141, for example:
MF(CaF2) = MF(CsCI)
+2 x
MF(ZnS)
(6)
I n spite of reports to t h e contrary[ls], there is a n infinite
number of such relationships for a given structure, since t he
Madelung constant represents t h e s u m of all attractive and
repulsive interactions of t h e ions in t h e crystal a n d can be
broken d o w n i n a n infinite number of ways into partial sums.
The Madelung constants of AB compounds with simple
primitive unit cells (e.g. the CsCl type) are particularly
important, since the Madelung constant of any structural type can be expressed as a sum of Madelung constants of single primitive (not necessarily isometric)
AB types [*I. This follows from the fact that any structure can be derived by superposition of suitable, not
necessarily isometric single primitive AB types. The
Madelung constants of single primitive AB types are
therefore “building units” for all other Madelung
constants; they will be denoted in the remainder of this
article by “MF.
For t h e sake of simplicity, we shall discuss only Madelung
constants of single primitive AB types with cubic unit cells.
However, similar relationships exist for non-cubic single
primitive AB types.
For AB types with single primitive cubic unit cells, we
shall always use the arrangement A+ in O,O,O and Bin x,y,z, the corresponding Madelung constant being
denoted by oMF(x,y,z)k; for example
MF(CsC1)
“MF(1/2,1/2,1/2)k
When based on the shortest distance A-B, these Madelung constants all lie between 1 (the “Madelung constant” of the gaseous molecule AB) and MF(CsC1) =
1.76267. . . (cf. Table 1).
Table
1.
“MF(x, y, z)k for special values of x, y, z [8].
‘12
XI4
0
114
0
0
l/4
0
0
1,7626747730.
1,607425.. .
1.594367
1.412511...
1.388259
1.370681
1.141774...
1.087781...
1.035808...
The variations of “MF(x,O,O), “MF(x,l/2,0), ‘MF
(x,1/2,1/2) and OMF(x,x,x) in the range 0 x
112 are
shown in Figure2. It can be seen that the variations
follow roughly the same pattern, and are all characterized by the rapid fall-off from the maximum. The
< <
[l3a] R . Hoppe and H . G. Y . Schnering, unpublished work (1965).
[I41 F. Bertaut, J. Physique Radium 13, 499 (1952); C. R. hebd.
Seances Acad. Sci. 239, 234 (1954); P. Naor, 2. Kristallogr.,
Kristallgeometr., Kristallphysik, Kristallchem. 110, 2 (1958).
[I51 C. C. Benson, Canad. J. Physics 34, 888 (1956); G. C. Eenson and F. van Zeggeran, J. chem. Physics 26, 1083 (1957).
[ * ] The dimensions of the unit cells of these AB types need not
be the same as those of the structural type, thz Madelung constant of which is to he expressed as a sum.
97
Madelung constants of single primitive A+B- types
with non-cubic unit cells are also smaller than 1.76267.. .,
but some are smaller than 1.0, and they may even be
negative (cf. Section X).
1.1
1.i
1.E
/
1.5
m
+z
2 1.4
VI. Relationships between the Madelung Constants of
Single Primitive AB Types “71
The Madelung constants of more complex structural
types can all be expressed (cf. Section IV) as the sum
of the Madelung constants of single primitive cells of
the same size. It was assumed, therefore, that the
Madelung constants of primitive AB types are not
mutually related (“independent” Madelung constants) [*]; however, this assumption is wrong 1171:
Any single primitive AB structure, irrespective of the
size and symmetry of its unit cell, can be described by
a non-primitive pseudo-cell obtained by multiplication
of one, two, or all three of the principal axes; this
naturally does not change the Madelung constant, which
can therefore be expressed as the sum of the Madelung
constants of other single primitive AB types. If the nonprimitive pseudo-cell is based on m times the a-axis,
n times the b-axis, and p times the c-axis, we have
m-1
0-1
D-1
1.3
1.2
In this equation, the primes in C’, r’, s’, and t’ indicate that
the relationship r’ = s’ = t’ = 0 must not be valid. The factors arSt and ar’sy are simple numbers depending o n the
geometry of the structure, and ensuring that the values of
‘MF are correctly weighted and normalized in the summation.
1.1
1.0
0.2
I
I
0.4
0.6
For example, if we describe the CsCl structure by eight
times the primitive cell, we obtain
X-
Fig. 2. Madelung constants (“MF) of cubic AB types with single
primitive unit cells [arrangement: A+ in (O,O,O) and B- in (x,y,z)J.
V. Partial Madelung Constants of Single Primitive
AB Types
Single primitive AB types, irrespective of size and/or
symmetry of their unit cell have commutative sub-lattices [71. The partial Madelung constants of A+ and B- are
then numerically equal, each being half of the total
Madelung constant. Where the ions are monovalent,
P M F ( x , ~ , z )is~ between 0.5 and 0.88 (cf. Table 1).
Moreover, some non-primitive AB structures, such as
the rock salt, sphalerite and wurtzite types, also have
commutative sub-lattices. In these cases the partial
Madelung constants of the A+ and B- particles are
again numerically equal, each again being half of the
total Madelung constant. An important example of a
simple AB type with non-commutative sub-lattices is
the NiAs type, in which the Madelung constant, as
calculated by Zemannr161, is not a simple function of
the partial Madelung constants of A+ and B-, which
differ in their numerical values [*I.
The partial Madelung constants of polyvalent ions in single
primitive AB types are z; times as great as those of monovalent ions in the same lattice type (zi = number of unit
charges on the ion in question).
These unexpected and surprising relationships between the
Madelung constants of single primitive AB types, which exist
in every case irrespective of size and symmetry of the unit cell,
indicate that we are still far from a real understanding of the
Madelung constants. They also suggest that other numerical
relationships may exist between the Madelung constants of
single primitive AB types with isometric unit cells.
VII. Relationships between the Partial Madelung
Constants of Primitive AB Types and of Interstitial
Sites “71
If we consider an infinitely large crystal of a primitive
AB structure, and assume that only one interstitial site
is occupied by an additional ion, the partial Madelung
constant of this ion can be expressed as a sum of partial
Madelung constants of single primitive AB types, and
can then provide information about the energy of interstitial sites; defects can also be taken into account in
this calculation by the use of additional terms.
For example, the partial Madelung constant of a cation X+
occupying the position 1/2,1/2,0 in one unit cell only of a CsC1type crystal (A+ in O,O,O and B- in 1 / 2 , 1 / 2 , * / ~ > is:
PMF(X+)
=
[16] J. Zemann, Acta crystallogr. 11, 55 (1958).
[*] They were recently calculated as functions of c/a; see [13a].
98
P M F ( ~ ,0,0)
1 1
122 x PMF(2, z,0)
=
0.24329466
(9)
[I71 R. Hoppe, unpublished work.
Angew. Chem. internat. Edit. 1 Vol. 5 (1966) 1 No.
I
VIII. Reduced Partial Madelung Constants
In isotypic compounds of the same formula type, but
with different valences(e.g.Cu+Cl-, Znz+Sz-, Ga3+As3-),
the partial Madelung constants of the ‘‘cations” are
related as the squares of the valences zp, if the valence
factor f(zi) fromEquation (1) is included in the Madelung
constant. Similar relationships exist for more complex
structures. In addition to the partial Madelung constants PMF(Am’) and PMF(Bn- ), it is therefore convenient to define reduced partial Madelung constants [lo, 181 as the partial Madelung constants divided
by z,: i.e.
*PMF(Am+)
E
PMF(AmC’ and
m2
*PMF(Bn-)
E
PMF(Bn-)
___ n2
These quantities enable us to compare the potentials in
compounds of different types and with different ionic
charges.
The reduced partial Madelung constants are strongly
affected by the peculiarities of the crystal structures, and
(unfortunately) are by no means constant. A number
of typical values are given in Table 2, and Figure 3
shows how the reduced partial Madelung constants of
cations vary with the coordination number and the
ionic charge.
The reduced partial Madelung constants as found from Figure 3
are maximum values, and the values found for real structures
are usually lower. For the “coordination number” in such
cases, see Section 1X.
1X. The Madelung Constant as a Function of StructureDetermining Factors such as Composition and
Coordination Number
Many attempts have been made to estimate Madelung
constants. Klemm [131, in his simplified expression (10)
for the lattice energy, introduced an “effective” Madelung constant a’.
He showed that a’ is practically constant for chemically
similar compounds of analogous composition, but
depends strongly on the formula type and on the bonds
present. (The terms Enc and ECor,of Equation (1)
are here included in a‘).
Kapustinskii 1121 then showed that the lattice energies of
many compounds can be calculated from Equation (1 l),
often with good accuracy.
(zA and zB are the ionic charges of A and B, and Z is
the number of atoms per formula unit.) In this casethe
Madelung constant is considered as independent of the
coordination numbers of the particles A and B in the
binary compound AmBn, since Equation (11) is based
on Equation (12):
MF(AmBn)
=
z ~ z x~ Zxx const. 2
~
ZAX
ZB x Z XMF(NaC1)
~
2
(12)
Table 2. Reduced partial Madelung constants *PMF of the cations in
typical binary crystal structures.
Ion
Corn p ound
*PMF(Am+)
Na+
CaZ+
Bi3+
Sc’f
Sn4+
A4
NaCl
CaF2
BiFs
ScF3
SnF4
AB4
2 A in: 1/4,1/4,114
0.874.. .
0.819.. .
0.797.. .
0.685.. .
0.657.. .
0.701.. .
-
+
3/4.3/4, ’ 1 4
Nbs+
I
It is surprising to find that (contrary to the results of
calculations, which give a different Madelung constant
for each structural type) MF(NaC1) may be used for all
possible AB structures. For other binary structural types,
a simple calculation with the aid of Equation (12) gives
effective Madelung constants that should depend only on
the composition, i.e. on the formula type, and not on
structural pecularities! The explanation for this is as
follows:
In binary compounds, we have:
ZA
0.551.. .
NbCls
x
ZB
x Z
SE
EZ?.
(13)
Thus, according to Kapustinskii [Equations (1 1) and
(la]:
MF(AmBn) = 2
2
2 x
MF(NaC1)
2
=
zzt x PMF(NaLaCI).
(14)
\Kz=~
0
11\19131
1
-3
2
m
1,
5
Fig. 3. Maximum values of reduced partial Madelung constants [13al
of cations as a function of charge (m) and coordination number (Kz).
It can be seen that, although Kapustinskii himself failed
to notice it, this expression implicitly represents any
Madelung constant as the sum of partial Madelung
constants 1101.
Kapustinskii also assumed that all compounds can be
regarded as “NaCl derivatives” [in Equation (1 1) the
shortest interionic distance rKA is assumed for the coordination number 61, a view that has also been expressed by other authors (cf. e.g. 1191).
To understand this apparently bold simplification, it must be
remembered that a deviation from the coordination number 6
[I81 D . H . Templeton, J . chern. Physics 21, 2097 (1953); 23, 1826
(1955).
Angew. Chem. internnt. Edit. 1 Val. 5 (1966) No. 1
[I91 T. C . Waddington, Trans. Faraday SOC. 56, 305 (1960).
99
in a real structure rather frequently leads to lower coordination numbers, and nearly always results in lower partial
Madelung constants, and hence in smaller Madelung components of the lattice energy. However, such a deviation can
occur only if additional (e.g. covalent) bonds ensure that the
decrease in the Madelung component of the lattice energy is
overcompensated in some other way. If the overcompensation is comparatively slight, Kupustinskii’s equation (14) is
applicable, even for compounds that cannot be considered as
consisting predominantly of ions. For coordination numbers
above 6 , the Madelung component of the lattice energy again
changes only slightly, as can be seen from a comparison of
the Madelung constants of the NaCl and CsCl types. Moreover, the shortest distance dKA, too, depends on the coordination number and always changes in the same direction
as the Madelung constant. Thus alterations of the Madelung
constant are in part “internally” compensated.
MF=1.370681
d~-ildA.o=2
MF= 1386276
d,-,ldp.g = 2
n
n
n
n
If one does not wish to calculate lattice energies from
Equation (1 l), but to estimate Madelung constants from
Equation (12), it is more convenient in the case of
binary compounds [71 to use Equation (15)
in which
curate:
p + ($/P)Z
=
=
MF(NaC1) is replaced by the more ac(16)
1.81
In Equation (16), p is the “average” coordination number of A a n d B [i.e. 1/3(8+2x4)=5.33 for the CaF2 type].
Templetun 1181later also attempted to represent Madelung
constants of binary compounds as functions of the coordination number; he used the empirical equation (17)
M F = 1.89 - l/ph
(17)
in which Ph is the weighted harmonic mean of the coordination numbers of A and B. The fact that all the
straight lines in Fig. 3 meet at 1.8912 is probably closeIy
connected with Equation (17). The approximations used
by Huppe [Eq. (15)] and by Templeton give practically
the same results, as can be seen from the examples in
Table 3.
Table 3. Madelung constants MF of the CsCI, NaCI, and ZnS types.
_
_
_
~
1
Hoppe [7]
CsCl
NaCl
Zn+S-
I
Templeton 1181
I
I
1.761
1.727
1.638
I
I
1.765
1,723
1.640
Exact value
1.7626.. .
1.7475.. .
1.6380..
lei
Fig. 4. Simplz AB structures with the same “simple” coordination
number, and the corresponding Madelung constants (sizes of the particles chosen arbitrarily).
which are the “nearest” neighbors. To illustrate this,
Figure 4 shows some simple AB structures the coordination numbers of which, when counted on the basis of
the simple definition, are the same. The Madelung
constants1171 of these structures, on the other hand,
show the influence of the arrangements of the A and B
sub-lattices. From Figure 5 it can be seen how the
Madelung constant is affected by the nature of the A
and B sub-lattices (as a function of the coordination
number) (cf. also Table 4).
.
However, there is no strict relationship between the
Madelung constant and the coordination number! Such
a functional relationship does not exist even in the case
of the Madelung constants “MF of single primitive
AB structures. The reason is the simplicity of the concept of the coordination number, i.e. the fact that only
the nearest neighbors (with opposite charges) are counted for compounds considered as consisting of ions,
whereas the partial Madelung constants take into account the interactions of one ion with all the others,
however far away and whathever the sign of their
charge. Furthermore, in the case of more complex
structures (e.g. crystal structures of the NaSi type [201),
it is impossible to be completely objective in deciding
Table 4. Madelung constants of simple cubic AB types.
ordination number.
C.N.
100
co-
cubic
cubic
body-centered
face-centered
,ub lattices start in
~
2
A+: O,O,O
B-: ‘/2,0,0
M F = 1.37068.. .
o,o,0
o,o, 0
‘/4,1/4,1/4
‘14. l / 4 , 0
MF
MF = 1.41504.
=
1.40221.. .
A + : O,O,O
B-: ’ l z , 118. ‘Is
M F = 1.56968.. .
3
A+: O,O,O
B-: l / z , l / z 1 0
MF = 1.59436
MF = 1.63804.. .
A + : O,O,O
B-: 1 / z , O , O
MF = 1.74755.. .
8
[20] W. Klemm, J. Witte, and H. G. v. Sclznering, Z . anorg. allg.
Chem. 327, 260 (1964).
cubic
primitive
- C.N. =
A + : 0.0.0
B-: ’ 1 2 , ‘Iz, ‘/z
MF = 1.76267.. .
Angew. Chem. internut. Edit. 1 Vol. 5 (1966) 1 No. I
I. Madelung Constants of AB Types [ 10.171
Homogeneous affine transformations lead to changes in
the lengths of the axes and in the angles formed by them.
For the sake of simplicity, the following examples are
confined to special transformations in which the angle
remains unchanged, and in which the changes in the
principal axes can be expressed by one figure (the axial
ratio c/a).
10’
’
2
I
I
6
i
Kz
I
1. C h a n g e s i n t h e M a d e l u n g C o n s t a n t s on
T e t r a g o n a l C o m p r e s s i o n o r E l o n g a t i o n of
a C u b i c U n i t Cell[sl
L
8
Fig. 5. Madelung constants and “simple” coordination numbers of
simple cubic A B types with commutative sub-lattices. Primitive
face-centered (A), and body-centered (0)
sub-lattices. K z = coordination number.
(m),
The estimation of Madelung constants or the deduction of “effective coordination numbers” from calculated
partial Madelung constants will always raise difficulties,
since the precisely defined Madelung constant cannot
be readily related to the rather vague though useful
concept of the “coordination number”.
Estimations of this nature are best carried out with the aid of
Equation (18), in which *PMFi is the reduced partial Madelung constant and zi is the charge; pi corresponds to mj or
n, of Equation (2b), and the summation is carried out as in
Equation (2b). The * P M F i are found from comparable
structures, or from graphs as in Figure 7.
The following general rule is valid here, as for any
other structure:
Any affine transformation of a structural type with a
finite Madelung constant gives a new structure which
also has a finite Madelung constant if the shortest
distance dA-B remains smaller than the shortest distances dA-A and dB-B. On the other hand, when the
ratio dA-A/dA-B or dBPB/dApBapproaches zero, the
Madelung constant approaches --m. It is assumed here
that the Madelung constant is always based on the
shortest distance dA-B.
Table 5 contains some typical examples of Madelung
constants of AB structures with single primitive unit
cells derived from cubic unit cells by tetragonal compression or elongation. A characteristic feature is that
Table 5. Changes in the Madelung constants M F of cubic AB structures on
tetragonal affine transformation of the unit cell [81. A+ in: O,O,O;B- in:
X,Y,Z
@I.
B. Special Part
The Madelung constant of a structural type is of interest for the discussion of enthalpies of formation. However, it acquires a more general importance only when
considered in relation to other Madelung constants. An
“observed” crystal structure is only one of many, since
it is “surrounded” by a large number of other structures
(e.g. thermodynamically less stable modifications)
which are fundamentally possible, but which do not
occur at thermodynamic equilibrium or have not so far
been found.
It is particularly interesting in this connection to find
how the Madelung constant of a structural type changes:
(a) whengreater or smaller changes in the size and/or
in the symmetry of the unit cell occur, while the coordinates of the particles remain constant in relation
to the new axes, i.e. when the unit cell undergoes homogeneous affine transformations,
(b) when the coordinates of the particles change in
parameter-dependent structural types, while the size
of the unit cell remains unaltered, or
(c) when the changes described under (a) and (b) occur
simultaneously.
In the following sections, therefore, we shall discuss,
not only individual Madelung constants, but mainly
their dependence on the variations mentioned.
Augew. Chem. internot. Edit.
Vol. 5 (1966) No . I
1/2.0,0
1.386.. .
--m
1/2,1/2,0
--%
‘/2.1/2,1/2
--%.
O,0,lj2
~/2.0,~/2
--co
1/4,1/4.0
-cc
l/4,0,0
-a
1.386.. .
1.127.. .
1.095 . . .
1.158 . . .
1.244.. .
0 .9 3 4 .. .
0 .9 7 1 .. .
1.770.. .
1.370.. .
1.594 . . .
1.762 . . .
1.594.. .
1.087.. .
1.035.. .
0.774.. .
1.378.. .
1.615 . . .
0.911 . . .
0 .8 4 8 .. .
-a
1.615 . . .
--%
--5
the Madelung constant of the CsCl type falls off rapidly
towards --m on compression or elongation. That of the
NaCl structure, on the other hand, remains finite, as can
be seen from Table 6.
Table 6. The Madelung constant of the NaCl type o n tetragonal affine
transformation of the unit cell [8].
c/a
0
1
m
I MF
1.3562.. .
1.7475.. .
1.615.. .
2. C h a n g e s i n t h e M a d e l u n g C o n s t a n t s o n
H e x a g o n a l C o m p r e s s i o n o r E l o n g a t i o n of a
C u b i c U n i t Cell[lol
The CsCl and NaCl structures, which are fundamentally
different in their cubic arrangements, are found to be
101
affinely transformed in the hexagonal arrangement (cf.
Table 7).
Table 7. NaCl and CsCl types as affine transformations of the same
structure. The letters A, a, B, b, C, c denote the positions of the particles
with the coordinates given. The cation sites are denoted by capital letters
and the anion sites by small letters.
Table 8. Schematic representation of the ZnS structure. (For explanation
ofthe symbols, see Table 7.)
1
oooooooooo A
'1l12
++++i+++++
c
8/12
++++++++++ b
7/12
A (and a) :
0.0,z
B (and b):
1lz,2/3,z
C (and c):
2/3, 1 1 3 , ~
c/a =
: NaC1-type
c/a = 1/3/2 : CsCI-type
oooooo A
+++++ib
0 0 0 0 0 0
c
I/a
++++++ a
n
++++++ c
000000
0000000
o00 C
n
4/~2
0 0 0 0 0 0 0 0 0 0
3iI2
++++++++++a
oooooooooo A
0
A ( a n d a ) : O,O,z
B(and b): 1/3,2/3,z
C (and c): 2/3, lI3,z
c/a = 1/65 : Zinc blende type
Table 9. Schematic representation of the ZnO structure. (For explanation of the symbols, see Table 7.)
oooooo A
As we pass from the CsCl type (c/a = 1/6'/2)to theNaCl
type (cla = vc),the simple coordinationnumber changes
continuously from 8 to 6. If there were a direct relationship between the simple coordination number and the
Madelung constant, the latter should change in the
same direction as the coordination number, i.e.
a(MF)/a(c/a)+ 0 between the values given for c/a.
However, it can be seen from Figure 6 that this is not
the case. The two Madelung constants MF(CsC1) and
MF(NaC1) are, in fact, separated by a minimum.
1
718
4i8
oooooooooo A
A(anda): O,O,z
B (and b): 1/3,2/3,z
+++++++ ++t-b
oooooooooo B
3i8
++++++++++ a
0
oooooooooo A
c/a =
1813:ideal ZnO type
approaching 1 for c/a + 0, land --co for c/a + -a, (cf.
Figure 7). The Madelung constant of the wurzite type
shows a similar dependence on c/a, although (as can
be seen from Table 9) this type consists of hexagonally
close-packed sequences of A and B particles.
t
Y
\
\
10
rn
cla
-
Fig. 6. Dependence of the Madelung constants of the CsCl and NaCl
types on the ratio c/a in the hexagonal arrangement[lO].
The change in the Madelung constants is also affected by the
fact that during hexagonal elongation along [OOl], the shortest
distance dA-B in the c s c l type increases rapidly for two (or
six on analogous compression) of the original eight equally
distant neighbors. The steep fall in MF(CsC1) for both larger
and smaller values of c/a is directly related to this effect. Conversely, the relatively small decrease in MF(NaC1) on both
sidesof cia = V a i s due to the fact that the six nearest neighbors d o not split u p into non-equivalent groups.
The characteristic changes in the Madelung constants of
these two binary structural types are related to the fact
that, although the structures of many ternary compounds
(double oxides, double fluorides) are variants of the
NaCl type (e.g. the important cc-NaFeO2 type), no
corresponding variants of the CsCl type have so far
been found.
The sphalerite type, in which the A and B sub-lattices
are arranged somewhat differently (Table 8) also belongs
to this group. Here the Madelung constant MF(ZnS)
decreases monotonically on both sides of c/a = 1 / g ' / 3 ,
102
0
m
1.0
2.0
3.0
40
cla-
Fig. 7. Dependence of the Madelung constant of the sphalerite type
on the ratio c/a in the hexagonal arrangement.
An example of an AB structure with non-commutative
sub-lattices is the NiAs type (cf. Table lo), the Madelung
constant of which (Fig. 8) also passes through a maximurn[161, and approaches --a, for c/a + 0 and for
cia -+ 00.
4 Maximum
l.*O
1
601 20
140
160
cia
-
180
2 00
Fig. 8. Dependence of the Madelung constant of the NiAs type (noncommutative sub-lattices) o n the ratio cia [161.
Ordinate: Madelung constant.
Angew. Clietn. infernat. Edit. 1 Vol. 5 (1966) / No. I
we start with a body-centered arrangement of the anions
with regularly occupied quasi-octahedral holes (cf.
Figure 10) the significance of the ratio c/a = 1/2'/2 becomes immediately obvious: F o r this ratio the cubic
sub-lattice (as indicated in Figure 10) and the favorable
arrangement of the cations (with the charges +2 or
+4) remain unchanged.
Table 10. Schematic representation of the NiAs structure. ( N i : c ; As:
A, B; for explanation OF the symbols, see Table 7).
1
oaoooooooo A
A : 0.0,~
4-++ ! c
oooooooooO B
+f+i
i--I-+++i
c
ooooooooooA
B:
1/2,z/3,Z
C:
2/3,'/3,2
+ I-
3/4
'/I
'14
0
i--i:
+
T
11. Madelung Constants of Complicated Structural
Types
Recently calculated Madelung constants of some important
structural types are listed in Table 11 (see below).
n
n
n
0
0z = o
1
1. T h e M a d e l u n g C o n s t a n t of the R u t i l e
Structure
In the rutile type AB2, each cation A has only six
equally distant neighbors B if the axial ratio c/a and
the parameter x of the anion are related as
Bollnow [211 has calculated MF(rutile) for a number of values
of cia and x that satisfy Equation (19). A recent detailed
publication [221 suffers from the disadvantage that MF(ruti1e)
is based, not on the shortest A-B distance (or some other
simple difference), but on an undefined average of the distances to the six neighbors 4+ 2!), which are generally not all
equal; the nature of this average cannot be deduced from the
values given, but it is obviously not a simple mean. Moreover,
since the difference between the Madelung constant and the
Madelung component of the lattice energy is overlooked,
some of the conclusions drawn are incorrect.
559
056
m
583
0 60
607
629
0 6L
cla
-
652
674
Figure 9 shows MF(ruti1e) as a function of cia and x [lo].
The following points can be observed:
I . The Madelung constant has a maximum for c/a :-=
1/2'/2 and x - = 5/16 [in accordance with Equation (19)].
It became known only recently that this maximum is due
to the structural geometry of the rutile type [171. Thus if
Anyew. Chem. internot. Edit.
1 C'ol. 5 11966) / N o . 1
Fig. 10. Structural geometry of the rutile type. T h e lengths and directions of the arrows show the displacement of the anions t o the true
positions (sizes of the particles chosen arbitrarily).
2. No known compound with a rutile structure corresponds t o this ideal case. This is at least partly due t o
the fact that the "grid" of Madelung factors (Figure 9)
is calculated for point-shaped ions and does not take
into account finite and even fixed ionic radii.
In typical rutile-type compounds AB2, such as difluorides
(e.g. ZnFz) and dioxides (e.g. SnOn), the ratio of the distances dA-B (1.9 to 2.1 8,) and dB-B (2.6 to 2.7 8,) is expected to be between 0.7 and 0.8. From the maximum of the
Madelung constant at cia = 1212 and x = 5/16, it follows
697
720
0 68
Fig. 9. Two-dimensional diagram of a three-dimensional representation
of the Madelung constants of rutile as a function of xanion and cla in
accordance with Equation (19). The figures beside the curves are
10,.
[MF(rutile)-4]
[21] 0. F. Bo//~oM',
Z . Physik 33, 741 (1925).
[22] W. H . Buur, Acta crystallogr. 14, 209 (1961).
11491101
that dA-B/dB-B = 0.83 [by comparison, for cia = 0.6 and
0.295, we find dA-B/dB-B = 0.721; the deviations towards cia < 1212 are therefore understandable to some
extent.
3. Even with a knowledge of the Madelung constants, it
is not possible t o discuss the question of why the parameters of a given rutile-type compound (e.g. ZnF2)
have the observed values (c/a = 0.666, x = 0.303:
cf. [239 instead of other values close to these.
X =
Thus if these parameters are slightly changed, it can be assumed that the shortest distance dA-B [2.015 8, ( 2 x ) for
ZnFz] will remain constant as is required by the concept of
[23] W. H . Buur, Acta
crystallogr. 9, 515 (1956); I / , 488
(1958).
103
fixed ionic radii, or that the molar cell volume [20.87 cm3 for
ZnFz] will remain unchanged, as is to be expected if the
volume increments of the various ions are constant. Examples are known which show that neither of these two concepts (which are mutually exclusive, cf. 1241) is strictly correct.
This means that in the Madelung component of the lattice
energy [Equation (l)] of “adjacent” rutile-type compounds,
the numerator can be given accurately, but not the denominator (the shortest distance dA-B). This difficulty is encountered with all parameter-dependent structures.
4. Finally it should be borne in mind that abnormally
short distances dBWBare found in dioxides with rutile
structures, such as SnO2 and Ti02 (the shortest 0-0
distance in Ti02 is 2.53 A). The only other cases in
which such short distances are found are complex
anions, e.g. SO:-, PO:-, or where strong hydrogen
bonding occurs. These short distances are due to the
strong polarization of the 0 2 - particles by the highly
charged cations (Ti4f etc.); they show that the picture of
a compound consisting of rigid ions must be modified
for these dioxides.
2. T h e M a d e l u n g C o n s t a n t of t h e A n a t a s e
Structure
The anatase structure also depends on two parameters,
namely the ratio c/a of the tetragonal unit cells (cf.
Figure 11) and the parameter zo of the anions. For c/a =
2.0 and zo = 1/4, we have a variant of the NaCl type in
which half of the cation sub-lattice has been removed
n
systematically. Figure 12 shows the variation of the
Madelung constant MF(anatase), and Figure 13 shows
the variation of the partial Madelung constants for
two limiting cases: in the first case, c/a varies while
zo (=1/4) remains constant [six originally equally distant
B neighbors of each A particle become unequally distant (4 + 2 ) as c/a increases], and in the second case zo
changes with c/a in accordance with Equation (20)
z*,
=
+ a*/2c2
(20)
(in this case, each A particle remains surrounded by six
equally distant B particles). It is interesting to note that
in compounds of the “filled-up’’ anatase type (or-LiFeO2
type), that have been studied so far, zo = zO*, i.e. for
any given c/a the arrangement of the 0 particles is such
as to give the optimum Madelung constant [5,24a1.
3. M a d e l u n g C o n s t a n t s of C o m p l e x S t r u c t u r e s
It can be seen from Figure l b that in the case of coordination compounds, the Born-Haber cycle can be
applied in various ways; the term “complexing energy”
is also used with various meanings (it can be defined by
steps IIIa or IIIb’ in Figure 1b [251). In stating the Madelung constants of complex structures, therefore, it is
particularly important to specify the reference distance.
The values in Table 11 are based on the shortest central
ion-ligand distance in each case.
n
Table 11. Madelung constants of other structural types.
Compound
Type
Type
Si02
High cristobalite
Low cristobalite
Rutile type
(ZnFz parameter)
individual type
individual type
individual type
Li3Bi type
individual type
individual type
individual type
individual type
individual type
individual type
AZfB;
AZ+B;
Az+B,
ZnFz
NbC14
SnF4
SnI4
BiF3
FeCI3
T~IJ
MoCli
NbOI?
NbOCI?
MoOClz
Ref.
4.4534. .
4.428
4.7656..
191
[91
[lo, 13al
12.650..
A4+Bi
14.084..
A4+Bi
12.439..
A4+Bi
9.573..
A3+B,
8.261. .
A3+B;7.738..
A3+B,
7.673..
A3+B,
A4 +Bz-CT 13.542..
A 4 i B 2 - c ~ 14.311..
14.099..
A4+B2-C;
191
[lo, 13al
1101
171
[91
[91
191
191
individual type
Fig. 11. Unit cell of the anatase type (sizes of the particles chosen
arbitrarily).
MF
individual type
individual type
PI
[91
10.093..
7.867..
7.508..
16.199..
11,154..
individual type
individual type
individual type
individual type
individual type
individual type
individual type
individual type
16.18,. .
11.553..
14.591..
11.791..
15.873
11.792..
15.981..
11.031.
.
Madelung Constants of Molecular Lattices
20
lrlL9112)
22
24
26
cla
28
30
Fig. 12. Madelung constant M F of the anatase type as a function of cla
and zo [13al.
[24] W’. Biltr: Raurnchernie der festen Stoffe. W. de Gruyter,
Leipzig 1934.
104
Compounds such as BrF5, SiF4, SbC13, and XeF2 form
solids with structures in which the building units are
individual molecules. Typical covalent bonds seem to
[24a] R . Hoppe, B. Schepers, H . J. Rohrbortr, and E. Vielhnber,
Z . anorg. allg. Chem. 339, 130 (1965).
[25] R . Hoppe, Z . anorg. allg. Chem. 291, 4 (1957).
Atrgew. Chem. internat. Edit.
Vol. 5 (1966) No. 1
z0.z;
090 B-
I
distinctly polar, in agreement with spectroscopic findings (cf. [la]). This explains the apparently large enthalpy
of sublimation (12.3 kcal/mole).
I.
Maximum
080t
2. T h e M a d e l u n g C o n s t a n t of t h e XeF2.XeF4
Structure
o“20
22
26
2L
c/a
28
30
Fig. 13. Partial Madelung constants P M F of the anatase type as functions of c/a and zo [13al.
be present here. With the exception of Klemnz’s effective
Madelung constants 1131, little was previously known
about the Madelung constants of such structures. It is
of interest, however, how these Madelung constants are
related to those of typically ionic compounds. The
Madelung constants of XeF2, XeF2.XeF4, and XeF4 are
therefore briefly discussed in the following sections.
1 . T h e M a d e l u n g C o n s t a n t of t h e XeF2
S t r u c t u r e [la]
Linear F-Xe-F molecules are present in the tetragonal
unit cell of xenon difluoride (cf. Figure 14). The PMF
values and the Madelung constant of this structure are
given in Table 12. PMF(Xe2+) for the coordination
number 2 fits readily into the pattern of PMF(A2f)
values for higher coordination numbers, as does the
Madelung constant MF(XeF2) itself [*I. Thus XeF2 is
I
I
lk
-
@ XeD,@Fin<<XeF2y>
Xe”,OFinc<XeF,.>
Fig. 15. Crystal structure of XeF2.XeF4 (a = 6.64 A, b = 7.33 A,
c = 6.40 A, B = 92 “ 4 0 ) (sizes of the particles chosen arbitrarily).
Table 12. Comparison of Madelung constants and partial Madelung constants of XeF2, XeFrXeFd, and
XeF4 [la].
I
I
I
I
I
I
1 -1
I
1.932
9.4138
Reference distance Xe-F[A]
P M F (Xe*+)
P M F (Xe4+)
P M F (FI-)
P M F (FII-)
P M F (FIII-)
MF
0.60535
0.6749
0.86372
0.8 6209
16.200
0.63250
Si-F
9.03938
PMF(Si4’)
8.8262
0.86372
0.9157
PMF(F-)
0.8233
1 3.7992
I
1.909
=*
-=
0.8184
-~
--
3.87550
12.6972
12.4910
12.4889
3. T h e M a d e l u n g C o n s t a n t of t h e XeF4
Structure
The monoclinic unit cell of XeF4 contains discrete,
practically square XeF4 groups (cf. Figure 16); the
PMF values and the Madelung constant are given in
Table 13. Comparison with the data for XeF2.XeF4
Table 13. Madelung constants and partial Madelung constants of a
number of tetrahalides with respect to the shortest distance A-B=I
[la, 9, 101.
I
Fig. 14. Tetragonal unit cell of XeFz (sizes of the particles chosen
arbitrarily).
[*] For details, see [la].
Atigew. Clienr. internat. Edit.
Vol. 5 (1966)
1 No. I
PMF(A4+)
PMF (FI-)
PMF(FII-)
P M F (FIII-)
_
MF
[Xe[4lF41
I
I
&“bI6lC141
I
1 1 1 1
I
9.4138
0.8233
I
0.8184
_
&ISn[6lF41
12.6972
~
10.5153
1.0364
0.7480
14.0841
I
8.9515
1.1102
;:;;I:
12.6505
1
[Si[4lF41
8.8262
C.9157
12.4889
105
0F
I
shows that the PMF(F-) values and the Madelung
constant are of the expected magnitude. Comparison
with the data for SiF4 shows that the PMF(F-) values
for XeF4 are smaller than those for SiF4. The reason is
the electrostatically unfavorable planar shape of the
XeF4 unit. The smaller value of PMF(Si4+) as compared
with PMF(Xe4+) indicates the more intimate entanglement of the XeF4 groups. From the heats of sublimation (SiF4: 6.15 kcal/mole, XeF4: 15.3 kcal/mole) the
Madelung constant of the XeF4 type would be expected
to be larger than that of the SiF4 type. Table 13 shows
that this is indeed the case. However, in considerations
of this kind one has to be satisfied with modest accuracy.
..xeN
Fig. 16. Crystal structure of XeF4 (a = 5.05 A, b = 5.92 A, c =
5.77 k, , fi = 99.6 ”). For the sake of clarity, only some of the F atoms are
shown (sizes of the particles chosen arbitrarily).
Received: October 5th, 1965 [A 491/275 IE]
German version: Angew. Chem. 78, 52 (1966)
Translated by Express Translation Service, London
Structure Formation and Molecular Mobility in Water and in Aqueous Solutions
BY PROF. DR. E. WICKE
INSTITUT FOR PHYSIKALISCHE CHEMIE DER UNIVERSITAT MtfNSTER (GERMANY)
Dedicated to Prof. W. Klemm on the occasion of his 70th birthday
The study of the structure of water and of aqueous solutions has recently received new impetus from the efforts at commercial desalineation of sea water and from developments in
molecular biology. The current view that, apart from single molecules, water contains only
one type of structural element, namely ,,flickering ”network structures with tetrahedrally
hydrogen-bonded water molecules (two-states model) is proving inadequate in the interpretation of new experimental data and in the calculation of fhermodynamic functions. After
a critical discussion of the basis of this model and of the concept of hydrogen bonds, a second
kind of structural element, i.e. a third state, is suggested: small aggregates of molecules containing mainly non-tetrahedral hydrogen bonds as well as some tetrahedral ones, and
packed more densely than allowed by the lattice-like structure. These aggregates - dimers to
hexamers - can be regarded as the primary products of disruption of the network structures,
and displace the latter as structural components in water with increasing temperature or
concentration of solutes. This “combined” model allows a consistent interpretation of the
properties of water and of the various effects of dissolved substances.
Introduction
Intermolecular structures have long been held responsible for the anomalies in the physicochemical properties
of water, and are also used to explain its behavior as a
solvent. The most important method up to a few years
ago of detecting these structures was based on the radial
distribution function of the intermolecular distances as
found from X-ray diffraction photographs of liquid
water. The development of microwave and in particular
NMR spectroscopy now enables us to take a closer look
at the interactions between neighboring H20 molecules
and between dissolved particles and H20 molecules.
Information about the mobility of H20 molecules,
particularly with regard to rotary motion (reorientation),
106
provides a better picture of the “structure”, as well as a
more sensitive means of determining the effect of solute
particles on the short-range order of the H20 molecules.
In the following pages the development of current views
will be discussed systematically and critically. It appears that we have been too ready to assume that the
liquid structure of water is due to an intermolecular
linkage of the same type as in ordinary ice (this will be
referred to below as linkage by “tetrahedral” hydrogen
bonds). Thus in addition to the larger structural units,
i.e. the fluctuating networks or clusters, in which linkage
is in fact mainly due to tetrahedral H bonds, too little attention has been paid to the possibility of a second,
smaller type of aggregate, in which the molecules are
held together by more dipolar non-tetrahedral hydrogen
bonds.
Atigew. Cheiir. intermit. Edit.
1 Vol. 5 (1966)
No. 1
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