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Interionic repulsive force and compressibility of ions
Physics of the Earth and Planetary Interiors, 13 (1976) 97—104 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
97
INTERIONIC REPULSIVE FORCE AND COMPRESSIBILITY OF IONS YOSHIAKI IDA The Institute for Solid State Physics, The University of Tokyo, Tokyo (Japan)
1
(Received June 21, 1976;accepted for publication July 22, 1976)
Ida, Y., 1976. Interionic repulsive force and compressibility of ions. Phys. Earth Planet. Inter., 13: 97—104.
a
The compressibility of an individual ion is examined, in comparison with known set of data for the alkali halides. A simple extrapolation of ionic radius to high pressure is not acceptable, because the pressure derivative of ionic radius changes for different salts. According to the classical concept of an elastic ion, the repulsive potential energy between the ions i and j is specified by the nature of each ion as: (~j+Pj)exp[(q~4~qJ—r)/(pi+ Pj)] as a function of the interionic distance r. In this expression, q
1 and Pi are the ionic radius and ionic compressibility, respectively, in a suitably modified meaning. Such a form of the repulsive potential fits well to the data of lattice constants and bulk moduli. The parameters q~ and p~are evaluated for alkali and halogen ions, and an anion turns out to be much more compressible than a cation. The present treatment may be usefully applied to the minerals in the Earth’s mantle, which contain only a few major ions.
1. Introduction The concept of ionic radius is understood most simply on the basis of the empirical additivity rule which states that observed interionic distances are well approximated by a sum of characteristic radii of ions. Applying the empirical rule to accumulated data of lattice constants, Shannon and Prewitt (1969) recently re-evaluated the values of ionic radii. The Earth scientists who study the materials of the deep interior of the earth often encounter the question of how the ionic radius changes with pressure. Unfortunately, the answer of this question has not yet been given quantitatively, even if we may conceive a qualitative idea of the compression of ions. A quantitative formulation of ionic compressibility is not a simple problem, because the pressure derivative of an ionic radius is not only a function of the ion species, but changes from salt to salt. This is shown in the next 1
Address: The Institute for Solid State Physics, The University of Tokyo, Roppongi, Minato-ku, Tokyo-106, Japan.
section, before we proceed to a more sophisticated treatment. It is believed that an ionic radius is related to the nature of interionic repulsive force (Pauling, 1928; Born and Huang, 1954; Tosi, 1964). Assuming an explicit dependence of the repulsive interaction on ionic radii, Tosi (1964) actually obtained a set of ionic radii for alkali and halogen ions. It is shown in the present paper that ionic compressibility is also connected with interionic repulsion. A simple formula, which describes the repulsive potential in terms of the ionic radii and the ionic compressibilities, succesfully interprets the change in lattice constant, bulk modulus and cohesive energy of the NaC1 type of alkali halides. It is expected that this treatment can be usefully applied to geophysical or geochemical problems, even if the present paper is only concerned with the justification of the idea itself using the abundant and refined data for alkali halides. The application of lattice theory to geophysical problems has been attempted, especially for the prediction of elastic properties (Anderson and Liebermann, 1970; Demarest, 1972; Weidner and
98
Simmons, 1972). The present treatment points out that the repulsive potential is determined by the ionic species without regard to a particular crystal. Therefore the number of arbitrary parameters is substantially reduced in the formulation of lattice theory.
2. Additivity rules According to an empirical additivity rule, the cation— anion separation r~ is approximately written in terms of the ionic radii rc and ra of respectively cation and anion as: r +r c a ca ‘- ~ It is conversely said that ionic radii can be evaluated so as to satisfy consistently a group of eq. 1 for various observations of interionic distances. It is noted, however, that no change occurs in eq. 1, even if all of r~ and ra are replaced by rc + ~ and ra respectively, introducing an arbitrary constant This means that the difference of ionic radii between positive and negative ions are not uniquely determined by the additivity rule alone. To eliminate such ambiguity, we usually assume an ionic radius for a standard ion, and calculate the difference of other radii from the standard. It is a more complicated problem to determine the standard radius itself. The additivity rule (1) was actually applied to the NaCl type of alkali-halide crystals, and the ionic radii =,.
—
~,
~.
r1 for alkali and hologen ions were calculated in a leastsquares method in which all the equations for r~ were equally weighted. The result is given in Table I. In this calculation, the nearest-neighbor interionic distance r = rca in Table II (Wyckoff, 1963) were used, and the standard radius was taken to be 0.74 A for Li~.Some traditional ionic radii are also given in Table I. It is not intended that our values evaluated here are better than the traditional ones. The calculation was made simply for the convenience of the comparison with other results obtained below. How the values of r~in Table I reproduce the original interionic distances r0~in Table II is demonstrated by the solid circles in Fig. 1. Here rC~means rc + ra. It is found that all the sixteen interionic distances are specified by the eight ionic radii as accurate as 2%. If the additivity rule (1) also holds under pressure, the ionic radii may be obtainable as a function of the pressure F, repeating at each pressure the same calculation as above. Under the same assumption, we may differentiate eq. 1 with respect to F, and obtain: K
c +K a
K
(2)
ca
where K Ka and K~ are the pressure derivatives of re, ra and rca, respectively, with inverse sign. The value of K~at atmospheric pressure is evaluated from the observed bulk modulus K, as: ‘r/3K
K
where r
=
(3)
r~.
TABLE I Traditional ionic radii and calculated values of ri and
of solution A
Ion
Ionic radii (A)
4 Li Na~ K~ Rb~
Pauling (1928) 0.60 0.95 1.33 1.48
Tosi (1964) 0.90 1.21 1.51 1.65
Shannon and Prewitt (1969) 0.74 1.02 1.38 1.49
1.36 1.81 1.95 2.16
1.19 1.65 1.80 2.01
1.33 1.81 1.96 2.20
F— Br I *
A
Values by Ahrens (1952) for anions.
*
r1 (A) 0.74 1.00 1.32 1.46
(A/Mbar) 0 1.12 3.26 4.29
1.32 1.83 1.99 2.23
0.04 2.58 3.82 6.12
99 0
TABLE II
~t~0
~
_____________________________________
-
Data of interionic distance r and bulk modulus K MaterialrK’< (A)
(Mbar)
(A/Mbar)
UP UC1 LiBr UI NaF NaCI NaBr
2.0087 2.56477 2.7507 3.000 2.310 2.82028 2.98662
0.682 0.318 0.257 0.188 0.485 0.250 0.199
0.98 2.69 3.57 5.32 1.59 3.76 5.00
NaI KF KC1
3.2364 2.674 3.14647 353278
0.161 0.318 0.178 0:120
6.70 2.80 5.89 9.81
RbF RbCl RbBr RbI
2.82 3.2905 3.427 3.671
0.27 3 0.165 0.138 0.108
3.44 6.65 8.28 11.33
~
-
II
Ti
I
I
•
A B
c c2 —1
-
______________________________________________ 2 r (A) Fig. 2. The pressure derivative KCAL of interionic distance calculated for solutions A (Table I), B, C~and C2 (Table III), compared with the observation ROBS in Table II.
were avoided. It was also considered important to use as many measurements of the same author(s) as possible, and to keep the internal consistency of the data. The values of LiF, LiBr, Lil, Nal, KBr and RbF are from Alexandrov and Ryzhova (1961), those of LiC1, NaF, NaCl, NaBr, KF, RbCl, RbBr and RbI from Lewis et al. (1967), and KC1 and KI from Norwood and Briscoe (1958). The applicability of eq. 2 is examined in the same manner as eq. 1. A least-squares determination gives the values of K~in Table I, in which ~i = 0 for Li~is assumed as a standard. The deviation of KCAL = Kc + Ka from SOBS = K~ is plotted by solid circles in Fig. 2. In this case, however, the additivity rule (2) does not succesfully reproduce the observations, and the difference becomes even as large as 100% for LiF. Therefore we have to conclude that the pressure derivative of an ionic radius is not a property determined by the ionic species alone. In other words, eq. 1 does not hold so accurately that the pressure derivative is meaningful. The result of together with r~inTable I is referred to solution A in this paper.
~ I
—~
-~...
The data of K in Table II are adopted from ultrasonic measurements made at room temperature and atmospheric pressure by various authors. No correction is made to convert adiabatic into isothermal bulk modulus, since the correction is usually smaller than the scatter of data in different measurements. When several data were available for the same material, the measurements with large deviations from the average
I A
0.02 ‘~‘
0
g~
~
002
$
_______
0
-
-
2
3. Repulsive potential
rCA)
Fig. 1. The interionic distance CAL calculated for solutions in Table A (Table tOBS I),B, C1 and C2II.(Table III), compared with the observation
In Figs. 1 and 2, the deviations from the additivity of the(1) interionic r. Furthermore tendency rules and (2) distance change rather regularly asthea function
100
shown in the data of the two figures is similar. It is
u
thus suggested that we have to make some correction that depends on r, before we define a quantity that suitably represents the compressibility of an ion. Among such effects to be corrected, the electrostatic Coulomb energy first comes mind. We may suppose that the ionic radius and its pressure derivative are related not to the total but to the repulsive interaction between ions. We shall now study this idea in more detail. Strictly speaking, the repulsive potential is of quanturn-mechanical origine. The rigorous treatment based on quantum mechanics is, however, too elaborate to allow a simple analysis that is of importance practical. We would rather attempt to formulate the repulsive potential, keeping a classical concept of ions in mind. The formulation in this section should thus be regarded as a phenomenological one to be examined in corn-
where the parameters q~and p~are considered to represent the property of the ion i. The same form is assumed for the ion j with q1 and p1 replaced by q~
parison with observations. We assume that the interionic interaction is analogous to an elastic repulsion between the bodies that are in contact with each other. In this picture, the interionic distance ~ is divided int~the separate regions rj and rj that belong to the ions i and j, respectively, i.e.: =
ri
+ 1~~
+ u~(r~)
—f
f0p1
exp [(q1 r~)/p~] —
(7)
and ions, i1. The quantityf0 is a common for all the which is introduced to adjustconstant the dimension of eq. 7. Eq. 6 coupled with eq. 7 yields:
f f0 exp [(q1 r~)/p1I
(8)
—
We may rewrite eq. 5, using eqs. 7 and 8, as u~= (p~ +~ Coupling eq. 8 with eq. 4,f is expressed as a function of r1~.Finally we obtain the desired expression of as:
u~~(r~~) =~ exp [(q~~ r~~)/p~~] —
where the parameters ~
(9) and P~are introduced by:
=q1+q~ (10) Pi~= P1 + P1 (11) It is interesting that eq. 9 has the same form as ~q. 7. The exponential type of repulsive potential has been widely employed for various analyses, because it is acceptable from both theoretical and experimental
viewpoints (Tosi, 1964). New relations (10) and (11) derived here reduce the interionic repulsion to the nature of individual ions. In principle, the value offo can be chosen arbitrarily.
(5)
All the relations obtained here are unaffected by the change of fo to .f0’, if it is accompanied by the corresponding shift of q1 to q~’as:
Here each term in eq. 5 is considered to be a function of the corresponding dimension. In a more realistic picture, u1 will correspond to the kinetic energies of electrons and electrostatic repulsions between electrons in the same ion, both of which depend on the volume confining the electrons, and thus on r1. In addition to eqs. 4 and 5, we have a subsidiary relation from the condition that the repulsive force f should balance between the two ions, as: u1’(r~)= u~’(rj)=
=
(4)
and the repulsive energy u1~is written as a sum of “elastic energies” u~and u~of both the regions, as: u~~(r11) = u1(r1)
1(r1)
(6)
In eq. 6, u1’ means the derivative of u~with respect to etc. We can obtain the same relation from the assumption that should be minimized under a fixed r11. The three eqs. 4—6 completely determine as a function of r1~,if U1(r~)and u1(r1) are prescribed from the “elastic property” of each ion. Let us give an explicit form of u1(r~),as:
q~’=
—
p1 lnf0’/fo
(12)
Here we specify fo so as to make the argument of exponential function in eq. 9 relatively small at atmospheric pressure. Namely, we use for the calculation of fo the relation: (13) with the observed interionic distances rm in Table II (Note that q~itself is determined by eq. 10, not by eq. 13; eq. 13 is simply used to eliminate the ambiguity off0.) Under such a choice off0, the parameter q~is regarded as an ionic radius. From eq. 8, we have: f=f0 at r1 = q1 (14) We may thus say that f0 gives the interionic repulsive
101
force at atmospheric pressure, which is approximately the same for different ions. It is not a new idea to connect the ionic radius with a parameter of the repulsive potential. Tosi’s (1964) calculation of asionic based on essentially the the same relation eq. radii 10. Inis his treatment, however, hardness parameter p~is not split into P~but assumed either to be a constant or to change from salt to salt. The, quantity Ki in eq. 2 is related to as: = (p~/fo)(df/~) (15) which holds at r~= ~ Eq. 15 means that a relative cornpressibility of an anion is specified by p 1 at atmospheric pressure. From this equation, we have the relations:
Ki
6ra
= =
~PcI(Pc [PaI(Pc ++ Pa)l~ Pa)] 6r
(16) (17)
which give the changes of rc and ra under pressure, in terms of the observable decrease of r. In this meaning, we may call the parameter i~an ionic compressibility,
4. Lattice theory
ni
=
6, n2
= 12, and a = \/2. We here assume a complete ionicity for alkali halides, and thus zc = 1 and Za = —1. For all the negative ions treated here, 13~is 0.75, while ~cc = 2 for and except 1.25 forforNat, and 3m isLi~ unity the K~ salts inRb TheLi~ quantity ! ~ = 1.375. cluding in which In addition to the terms included in eq. 18, one sometimes considers the contributions from thermal and Van der Waals energies. In this paper, we neglect both the effects, which are of secondary importance. In fact, it is not always clear that those contributions are completely negligible in the present analysis. Un~.
fortunately we are able to estimate the effects only approximately, because of the restricted validity of the formulation and the uncertainty of experimental data. Themight considerations thermal and Van of derambiWaals energies thus cause of additional sources guity and make it impracticable to test the applicability of the treatment in comparison with the simple additivity rules (1) and (2). For this reason, it seems better at least in the first step to take only the Coulomb and repulsive interactions into account. The differentiations of Uwith respect to the volume yield the following expressions of the pressure P and the bulk modulus K:
In the Born type of lattice theory (Born and Huang, 1954; Tosi, 1964), interionic interactions are described by a superposition of two-body central forces. The po-
P = —(1 /3hr2) [—Mzczae2/r2 + n iI3cau~(r)
tential energy U is written:
2a/2) {$cc’~cc(°~) + 13~u~(a,) }] (20) 2 /r3 + n i ‘3m {—(2/r)u’~(r) K = (1 /9hr) [4Mzczae + u~(r)}+ (nact,t3cc/2){—(2/r)z4c(ar) -f au~c(ar)}
UMzczae2lr + ni13cau~(r)+ ~ +L3~u~(ar)]
+
(n
(18) + (n2
as a function of the nearest-neighbor distance r. The first term gives the electrostatic energy, in whichM is the Madelung constant, z~and za are the valences of cation and anion, and e is the electronic charge. The other terms corn~from first and second nearest-neighbor repulsions. Here n 1 and n2 are the number of first and second nearest-neighbor ions, respectively, and a is ratio of rcc = r~to r. The factor ~ gives a quantum-mechanical correction to a classical picture of the repulsive interaction (Pauling, 1928), and is explicitly specified by: =
1
+
+ z~INj
(19)
3~/2){—(2/r)U~(ar) + au~(w-)}]
where h is the volume per ion pair divided by r3. We put h = 2 for the present problem. 5. Estimate of the parameters
The formulation based on the lattice theory connects the parameters of the repulsive potential with the observed values of r and K. First we consider only the nearest neighbor interaction, neglecting the terms involving Ucc and u~.Substituting eq. 9 for u~,and solving eqs. 20 and 21 in whichPis put equal to zero, we obtam the following expressions: 3(KM K) 2~MI (22) = ~ [ q~= r + ~m ln [3hr2 ~M’ /fon ii3~] (23) -
—
where N1 isi. the number of outer electrons contained in the ion For NaCl type of lattice, we have M = 1.747558,
(21)
—
102
Here we denote by ~M and KM the contributions from the electrostatic interaction to pressure and bulk
that eqs. 9—11 are also applicable to Ucc and u~with the same parameters q1 and ~ as in urn.
modulus, i.e., the first terms of eqs. 20 and 21, respec-
Unlike the previous calculation, the problem cannot be linearized when the second nearest-neighbor interactions are taken into account. In this case, the calcu-
tively.
In this case, the calculation is straigtforward. First qca and Pca are obtained from eqs. 22 and 23 for the data in Table II. The ionic parameters q~and p~are
lation must be made by an iterative method. We may still use eqs. 22 and 23 ifPM and KM are understood to include the contributions from ~ and u~in eqs.
evaluated separately from the two additivity rules (10)
and (11) in the same manner as was previously mentioned. Before the calculation, we fix the quantity f0 at a tentative value. After the calculation, we shift f0 and q1 according to eq. 12, so as to meet eq. 13 best. The result, which is referred. to solution B, is given in Table III. The standard values to eliminate the ambiguities in the additivity rules were chosen rather arbitrarily
without a significant reason, Once all of q~and ,o~are determined, we can cornpute rCAL from eq. 20 with P = 0, and KCAL from eqs. 21 and 3. The comparisons between calculation and observation are given in the same figure (Figs. 1 and 2) as was discussed before. It is remarkable that solution B matches the observations considerably better than solution A, especially for i<. Clearly the
r~1ations(10) and (11) hold more successfully than eqs. I and 2. According to Figs. 1 and 2, solution B still con-
20 and 21, in addition to the Madelung terms. The right-hand sides of eqs. 22 and 23 thus depend on the unknown parameters q1 and p~.These parameters are, however, roughly estimated, as in solution B. Starting from approximate values of q~and ~ we can re-evaluate q~and Pca from eqs. 22 and 23 and obtain a better solution of q1 and p1 from eqs. 10 and 11. Repeating this calculation until a sufficient convergence is achieved, we have a final solution that satisfies all the relations self-consistently. During the above-mentioned iteration, the parameters for a standard ion are kept unchanged. The solution obtained by the calculation thus depends on the assumed standard values. In the case of solution A and B, a change of the standard parameter simply causes a constant shift of the other parameters, and does not influence rCAL or K CAL• In contrast to those cases,
tains a systematic deviation from the observations,
the second nearest-neighbor interactions depend on the absolute thethus parameters through of q~a ~ 2Pa’scale etc.,of and the calculations U,= K, r and and
This motivates us to extend the treatment the second nearest-neighbor repulsions. Let us to assume
Paa = K change with the selected values of the standard ion.
TABLE III Values of ionic radius q~,ionic compressibility p~,parameter fo, and standard deviations 0p and °Kfor solutions B, C 1 and C2 Ion
B
(A) Li~ Na~ K~ Rb~
Br
F 0p (106 f~ (10~dyn) Mbar) °K(10~ Mbar)
C1
(A)
(A)
C2
(A)
(M
(A)
0.80 1.09 1.35 1.46
0.054 0.050 0.056 0.054
0.95 1.28 1.57 1.69
0.070 0.070 0.080 0.080
0.57 1.00 1.33 1.46
0.006 0.033 0.054 0.054
1.39 1.78 1.91 2.10
0.234 0.261 0.272 0.284
1.21 1.59 1.71 1.88
0.218 0.237 0.246 0.257
1.52 1.87 1.99 2.15
0.238 0.255 0.262 0.269
71.93 9.87 31.1
61.80 1.36 5.50
46.03 3.14 5.68
103
Such dependence enables us to evaluate the standard
(24)
The whole parameters for solutions C~and C2 are listed in Table III. In comparison with the traditional. ionic radii (see Table I), solution C1 gives the parameters q~close to those of Tosi (1964), while the values of in C2 resemble Pauling (1928). Theq~ calculated resultsthe ofradii r andbyK are shown in Figs. 1 and 2 along with all the previously obtained solutions. The agreement between calculation and observation is further improved, becoming quite satisfactory.
(25)
6. Discussions
parameters themselves, so as to give a best fit for the observations. Along the same line, Tosi (1964) obtained his values of ionic radii, even if the concept of ionic compressibility did not appear in his treatment. 0K by:Let us introduce the standard deviations up and =
~ ~1~(Pc~i)~ ~
~
—
KOBS)2
where the summations extend over all the sixteen salts in Table II. Generally 0K is much larger than 0p~and
we here try to minimize
UK. A contour map of the distribution of UK.is given in Fig. 3, in which Li~
The cohesive energy U was evaluated with use of eq. 18 for solutions B, C 1 and C2. The results, which are given in Table IV, agree reasonably well with the
is chosen as the standard ion. There is a long and 0K approximately slender valley that has a value of equal to 6 l0~Mbar. It is certain that the desired solution is found in the vicinity of this valley. It is a delicate problem, however, to determine the solution more precisely. The position of the minimum is probably
observation (Tosi, 1964) in all the three solutions. The test for K inout Figs. along with the agreement ofrUand points that1 and the 2additivity relations (10) and (11) are applicable to the repulsive potential, and that the parameter ~ as well as q~is defined by the nature of an individual ion. A pressure derivative K~ of ionic radius is related to p 1 by eq. 15. This relation
influenced by various sources of ambiguity, such as the
now indicates that the failure of the additivity rule
.
accuracy
of the data of r and K, and the effects of ther-
mal and Van der Waals energies, etc. Here we tentatively pick up two local minima C1 and C2 as the representa-
tive solutions. These two points happen to be situated at two extreme ends of the valley.
1
0
~~
stant but changes from salt to salt.
TABLE IV Calculated and observed cohesive energies (kcal/mole) Material
Calculated
UF
—243.8
—246.9
—248.9
‘:~
I~
I~
KC1 KEr
—165.8 —158.1
—165.7 —158.2
—165.7 —158.3
—165.8 —158.5
KI
—147.9
—148.4
—148.8
—149.9
RbF RbC1 RbBr
—186.1 —1.59.7 —152.6
—184.6 —159.5 —152.7
—183.8 —159.5 —152.9
—181.4 —159.3 —152.6
RbI
—143.1
—143.6
—144.2
—144.9
LT;~~::ii~ii//,// /)
T’
0.5 0
(2)is caused by the factor df/dF, which is not a con-
0.05
p
~j
Fig. 3. The distribution of the standard deviation °Kof eq. 25 as a function of the standard parameters .°~and qc for Li~.
The number attached to each countour gives °Kin the unit of iO~ Mbar. The two local minima C1 and C2 correspond to solutions C1 and C2 in Table III
Ob-
—242.3
~IO~ ~1O~ ~F~!
104 Although there is more or less ambiguity in the parameters determined here, it is remarkable that both solutions C 1 and C2 give substantially higher. values of p~to the anions than to the cations. In other words, positive ions are very hard to deform, compared with negative ions. Usually it is said that an anion will be more compressible because of its larger size. However, this does not give a sufficient interpretation. 4, In in fact, ~ of F is still much larger than that of Rb spite of their having almost the same characteristic sizes. A probable reason may be given as follows. As a result of the electrostatic interaction, the electrons
will be drawn much strongly toward the center in a positive ion than in a negative ion. Therefore, a cation should have higher electron density and be less deformable than an anion. It may be noticed that p 4 is exceptionally 1 of Li small in solution C 2, while no such anomaly is found in solution C1. This cannot always be the reason, however, to reject C2, nor to regard4 has C1 asa being morestrucrealdistinctive istic among than C2.alkali Actually only Li ions; the number of ture and halogen outer electrons is two for Li~but eight for the other ions..
According to the present treatment, the energy and other quantities derived from it are written in terms of the properties of the constituent ions without the need for information about the composite particular material. This suggests useful applications to geophysical and geochemical problems. We are often obliged to estimate energy, volume and elastic property, etc. for
such a material or under such a condition that is very hard to realize practically. The application of lattice theory has not made a remarkable contribution to the Earth sciences, mainly because too many parameters appearing in the formulation made the prediction al-
most arbitrary. This difficulty is considerably reduced by the present treatment. The niaterials composing the Earth’s mantle are mainly composed of four ions, Mg2~,Fe2~,Si4~and 02_. If the contrast of p~between positive and negative ions is due to the ionic charge itself, as has been suggested above, the difference between p~and ~a
will be more dominant for those ions because of their higher valences. This means that p~is practically negli-
24, Fe2~and Si44, and that only Pa of 02 gible for Mg is effective. It is thus suggested that lattice theory is made very simple and tractable for mantle minerals.
Acknowledgments
The ideawith of this study was derived continued discussions Prof. Y. Matsui. I alsofrom thank Dr. N. Onishi for his valuable comments. The numerical calculation was made with use of the computer FACOM 230-48 in the Institute for Solid State Physics.
References Ahrens, LII., 1952. The use of ionization potentials, Part 1. Ionic radii of elements. Geochim. Cosmochim. Acta, 2:
155—169.K.S. and Ryzhova, T.V., 1961. The elasticpropAlexandrov, erties of crystals. Soy. Phys. — Crystallogr., 6: 228—252. Anderson, O.L. and Liebermann, R.C., 1970. Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications. Phys. Earth Planet. Inter., 3: 61—85. Born, M. and Huang, K., 1954. Dynamical Theory of Crystal Lattices. Oxford University Press, London, 432 pp. Demarest, Jr., H.H., 1972. Extrapolation of elastic properties to high pressure in the alkali halides. J. Geophys. Res., 77: 848—856.
Lewis, J.T., Lehoczky, A. and Briscoe, C.V., 1967. Elastic constants of the alkali halides at 4.2°K. Phys. Rev., 161: 877887. Norwood, M.H. and Briscoe, C.V., 1958. Elastic constants of potassium iodide and potassium chloride. Phys. Rev., 112: 45—48. Pauling, L., 1928. The sizes of ions and their influence on the properties of salt-like compounds. Z. Kristallogr., 67:
377404.
Shannon, R.D. and Prewitt, C.T., 1969. Effective ionic radii in oxides and fluorides. Acta Crystallogr., B25: 925—946. Tosi, M.P., 1964. Cohesion of ionic solids in the Born model. In: F. Seitz and D. Turnbuil (Editors), Solid State Physics, Vol. 16, Academic Press, New York, N.Y., pp. 1—120. Weidner, D.J. and Simmons, G., 1972. Elastic properties of
alpha quartz and the alkaliRes., halides on an interatomic force model. J. Geophys. 77:based 826—847. Wyckoff, W.G., 1963. Crystal Structures, Vol. 1. Wiley, New York, N.Y., pp. 86—91.
97
INTERIONIC REPULSIVE FORCE AND COMPRESSIBILITY OF IONS YOSHIAKI IDA The Institute for Solid State Physics, The University of Tokyo, Tokyo (Japan)
1
(Received June 21, 1976;accepted for publication July 22, 1976)
Ida, Y., 1976. Interionic repulsive force and compressibility of ions. Phys. Earth Planet. Inter., 13: 97—104.
a
The compressibility of an individual ion is examined, in comparison with known set of data for the alkali halides. A simple extrapolation of ionic radius to high pressure is not acceptable, because the pressure derivative of ionic radius changes for different salts. According to the classical concept of an elastic ion, the repulsive potential energy between the ions i and j is specified by the nature of each ion as: (~j+Pj)exp[(q~4~qJ—r)/(pi+ Pj)] as a function of the interionic distance r. In this expression, q
1 and Pi are the ionic radius and ionic compressibility, respectively, in a suitably modified meaning. Such a form of the repulsive potential fits well to the data of lattice constants and bulk moduli. The parameters q~ and p~are evaluated for alkali and halogen ions, and an anion turns out to be much more compressible than a cation. The present treatment may be usefully applied to the minerals in the Earth’s mantle, which contain only a few major ions.
1. Introduction The concept of ionic radius is understood most simply on the basis of the empirical additivity rule which states that observed interionic distances are well approximated by a sum of characteristic radii of ions. Applying the empirical rule to accumulated data of lattice constants, Shannon and Prewitt (1969) recently re-evaluated the values of ionic radii. The Earth scientists who study the materials of the deep interior of the earth often encounter the question of how the ionic radius changes with pressure. Unfortunately, the answer of this question has not yet been given quantitatively, even if we may conceive a qualitative idea of the compression of ions. A quantitative formulation of ionic compressibility is not a simple problem, because the pressure derivative of an ionic radius is not only a function of the ion species, but changes from salt to salt. This is shown in the next 1
Address: The Institute for Solid State Physics, The University of Tokyo, Roppongi, Minato-ku, Tokyo-106, Japan.
section, before we proceed to a more sophisticated treatment. It is believed that an ionic radius is related to the nature of interionic repulsive force (Pauling, 1928; Born and Huang, 1954; Tosi, 1964). Assuming an explicit dependence of the repulsive interaction on ionic radii, Tosi (1964) actually obtained a set of ionic radii for alkali and halogen ions. It is shown in the present paper that ionic compressibility is also connected with interionic repulsion. A simple formula, which describes the repulsive potential in terms of the ionic radii and the ionic compressibilities, succesfully interprets the change in lattice constant, bulk modulus and cohesive energy of the NaC1 type of alkali halides. It is expected that this treatment can be usefully applied to geophysical or geochemical problems, even if the present paper is only concerned with the justification of the idea itself using the abundant and refined data for alkali halides. The application of lattice theory to geophysical problems has been attempted, especially for the prediction of elastic properties (Anderson and Liebermann, 1970; Demarest, 1972; Weidner and
98
Simmons, 1972). The present treatment points out that the repulsive potential is determined by the ionic species without regard to a particular crystal. Therefore the number of arbitrary parameters is substantially reduced in the formulation of lattice theory.
2. Additivity rules According to an empirical additivity rule, the cation— anion separation r~ is approximately written in terms of the ionic radii rc and ra of respectively cation and anion as: r +r c a ca ‘- ~ It is conversely said that ionic radii can be evaluated so as to satisfy consistently a group of eq. 1 for various observations of interionic distances. It is noted, however, that no change occurs in eq. 1, even if all of r~ and ra are replaced by rc + ~ and ra respectively, introducing an arbitrary constant This means that the difference of ionic radii between positive and negative ions are not uniquely determined by the additivity rule alone. To eliminate such ambiguity, we usually assume an ionic radius for a standard ion, and calculate the difference of other radii from the standard. It is a more complicated problem to determine the standard radius itself. The additivity rule (1) was actually applied to the NaCl type of alkali-halide crystals, and the ionic radii =,.
—
~,
~.
r1 for alkali and hologen ions were calculated in a leastsquares method in which all the equations for r~ were equally weighted. The result is given in Table I. In this calculation, the nearest-neighbor interionic distance r = rca in Table II (Wyckoff, 1963) were used, and the standard radius was taken to be 0.74 A for Li~.Some traditional ionic radii are also given in Table I. It is not intended that our values evaluated here are better than the traditional ones. The calculation was made simply for the convenience of the comparison with other results obtained below. How the values of r~in Table I reproduce the original interionic distances r0~in Table II is demonstrated by the solid circles in Fig. 1. Here rC~means rc + ra. It is found that all the sixteen interionic distances are specified by the eight ionic radii as accurate as 2%. If the additivity rule (1) also holds under pressure, the ionic radii may be obtainable as a function of the pressure F, repeating at each pressure the same calculation as above. Under the same assumption, we may differentiate eq. 1 with respect to F, and obtain: K
c +K a
K
(2)
ca
where K Ka and K~ are the pressure derivatives of re, ra and rca, respectively, with inverse sign. The value of K~at atmospheric pressure is evaluated from the observed bulk modulus K, as: ‘r/3K
K
where r
=
(3)
r~.
TABLE I Traditional ionic radii and calculated values of ri and
of solution A
Ion
Ionic radii (A)
4 Li Na~ K~ Rb~
Pauling (1928) 0.60 0.95 1.33 1.48
Tosi (1964) 0.90 1.21 1.51 1.65
Shannon and Prewitt (1969) 0.74 1.02 1.38 1.49
1.36 1.81 1.95 2.16
1.19 1.65 1.80 2.01
1.33 1.81 1.96 2.20
F— Br I *
A
Values by Ahrens (1952) for anions.
*
r1 (A) 0.74 1.00 1.32 1.46
(A/Mbar) 0 1.12 3.26 4.29
1.32 1.83 1.99 2.23
0.04 2.58 3.82 6.12
99 0
TABLE II
~t~0
~
_____________________________________
-
Data of interionic distance r and bulk modulus K MaterialrK’< (A)
(Mbar)
(A/Mbar)
UP UC1 LiBr UI NaF NaCI NaBr
2.0087 2.56477 2.7507 3.000 2.310 2.82028 2.98662
0.682 0.318 0.257 0.188 0.485 0.250 0.199
0.98 2.69 3.57 5.32 1.59 3.76 5.00
NaI KF KC1
3.2364 2.674 3.14647 353278
0.161 0.318 0.178 0:120
6.70 2.80 5.89 9.81
RbF RbCl RbBr RbI
2.82 3.2905 3.427 3.671
0.27 3 0.165 0.138 0.108
3.44 6.65 8.28 11.33
~
-
II
Ti
I
I
•
A B
c c2 —1
-
______________________________________________ 2 r (A) Fig. 2. The pressure derivative KCAL of interionic distance calculated for solutions A (Table I), B, C~and C2 (Table III), compared with the observation ROBS in Table II.
were avoided. It was also considered important to use as many measurements of the same author(s) as possible, and to keep the internal consistency of the data. The values of LiF, LiBr, Lil, Nal, KBr and RbF are from Alexandrov and Ryzhova (1961), those of LiC1, NaF, NaCl, NaBr, KF, RbCl, RbBr and RbI from Lewis et al. (1967), and KC1 and KI from Norwood and Briscoe (1958). The applicability of eq. 2 is examined in the same manner as eq. 1. A least-squares determination gives the values of K~in Table I, in which ~i = 0 for Li~is assumed as a standard. The deviation of KCAL = Kc + Ka from SOBS = K~ is plotted by solid circles in Fig. 2. In this case, however, the additivity rule (2) does not succesfully reproduce the observations, and the difference becomes even as large as 100% for LiF. Therefore we have to conclude that the pressure derivative of an ionic radius is not a property determined by the ionic species alone. In other words, eq. 1 does not hold so accurately that the pressure derivative is meaningful. The result of together with r~inTable I is referred to solution A in this paper.
~ I
—~
-~...
The data of K in Table II are adopted from ultrasonic measurements made at room temperature and atmospheric pressure by various authors. No correction is made to convert adiabatic into isothermal bulk modulus, since the correction is usually smaller than the scatter of data in different measurements. When several data were available for the same material, the measurements with large deviations from the average
I A
0.02 ‘~‘
0
g~
~
002
$
_______
0
-
-
2
3. Repulsive potential
rCA)
Fig. 1. The interionic distance CAL calculated for solutions in Table A (Table tOBS I),B, C1 and C2II.(Table III), compared with the observation
In Figs. 1 and 2, the deviations from the additivity of the(1) interionic r. Furthermore tendency rules and (2) distance change rather regularly asthea function
100
shown in the data of the two figures is similar. It is
u
thus suggested that we have to make some correction that depends on r, before we define a quantity that suitably represents the compressibility of an ion. Among such effects to be corrected, the electrostatic Coulomb energy first comes mind. We may suppose that the ionic radius and its pressure derivative are related not to the total but to the repulsive interaction between ions. We shall now study this idea in more detail. Strictly speaking, the repulsive potential is of quanturn-mechanical origine. The rigorous treatment based on quantum mechanics is, however, too elaborate to allow a simple analysis that is of importance practical. We would rather attempt to formulate the repulsive potential, keeping a classical concept of ions in mind. The formulation in this section should thus be regarded as a phenomenological one to be examined in corn-
where the parameters q~and p~are considered to represent the property of the ion i. The same form is assumed for the ion j with q1 and p1 replaced by q~
parison with observations. We assume that the interionic interaction is analogous to an elastic repulsion between the bodies that are in contact with each other. In this picture, the interionic distance ~ is divided int~the separate regions rj and rj that belong to the ions i and j, respectively, i.e.: =
ri
+ 1~~
+ u~(r~)
—f
f0p1
exp [(q1 r~)/p~] —
(7)
and ions, i1. The quantityf0 is a common for all the which is introduced to adjustconstant the dimension of eq. 7. Eq. 6 coupled with eq. 7 yields:
f f0 exp [(q1 r~)/p1I
(8)
—
We may rewrite eq. 5, using eqs. 7 and 8, as u~= (p~ +~ Coupling eq. 8 with eq. 4,f is expressed as a function of r1~.Finally we obtain the desired expression of as:
u~~(r~~) =~ exp [(q~~ r~~)/p~~] —
where the parameters ~
(9) and P~are introduced by:
=q1+q~ (10) Pi~= P1 + P1 (11) It is interesting that eq. 9 has the same form as ~q. 7. The exponential type of repulsive potential has been widely employed for various analyses, because it is acceptable from both theoretical and experimental
viewpoints (Tosi, 1964). New relations (10) and (11) derived here reduce the interionic repulsion to the nature of individual ions. In principle, the value offo can be chosen arbitrarily.
(5)
All the relations obtained here are unaffected by the change of fo to .f0’, if it is accompanied by the corresponding shift of q1 to q~’as:
Here each term in eq. 5 is considered to be a function of the corresponding dimension. In a more realistic picture, u1 will correspond to the kinetic energies of electrons and electrostatic repulsions between electrons in the same ion, both of which depend on the volume confining the electrons, and thus on r1. In addition to eqs. 4 and 5, we have a subsidiary relation from the condition that the repulsive force f should balance between the two ions, as: u1’(r~)= u~’(rj)=
=
(4)
and the repulsive energy u1~is written as a sum of “elastic energies” u~and u~of both the regions, as: u~~(r11) = u1(r1)
1(r1)
(6)
In eq. 6, u1’ means the derivative of u~with respect to etc. We can obtain the same relation from the assumption that should be minimized under a fixed r11. The three eqs. 4—6 completely determine as a function of r1~,if U1(r~)and u1(r1) are prescribed from the “elastic property” of each ion. Let us give an explicit form of u1(r~),as:
q~’=
—
p1 lnf0’/fo
(12)
Here we specify fo so as to make the argument of exponential function in eq. 9 relatively small at atmospheric pressure. Namely, we use for the calculation of fo the relation: (13) with the observed interionic distances rm in Table II (Note that q~itself is determined by eq. 10, not by eq. 13; eq. 13 is simply used to eliminate the ambiguity off0.) Under such a choice off0, the parameter q~is regarded as an ionic radius. From eq. 8, we have: f=f0 at r1 = q1 (14) We may thus say that f0 gives the interionic repulsive
101
force at atmospheric pressure, which is approximately the same for different ions. It is not a new idea to connect the ionic radius with a parameter of the repulsive potential. Tosi’s (1964) calculation of asionic based on essentially the the same relation eq. radii 10. Inis his treatment, however, hardness parameter p~is not split into P~but assumed either to be a constant or to change from salt to salt. The, quantity Ki in eq. 2 is related to as: = (p~/fo)(df/~) (15) which holds at r~= ~ Eq. 15 means that a relative cornpressibility of an anion is specified by p 1 at atmospheric pressure. From this equation, we have the relations:
Ki
6ra
= =
~PcI(Pc [PaI(Pc ++ Pa)l~ Pa)] 6r
(16) (17)
which give the changes of rc and ra under pressure, in terms of the observable decrease of r. In this meaning, we may call the parameter i~an ionic compressibility,
4. Lattice theory
ni
=
6, n2
= 12, and a = \/2. We here assume a complete ionicity for alkali halides, and thus zc = 1 and Za = —1. For all the negative ions treated here, 13~is 0.75, while ~cc = 2 for and except 1.25 forforNat, and 3m isLi~ unity the K~ salts inRb TheLi~ quantity ! ~ = 1.375. cluding in which In addition to the terms included in eq. 18, one sometimes considers the contributions from thermal and Van der Waals energies. In this paper, we neglect both the effects, which are of secondary importance. In fact, it is not always clear that those contributions are completely negligible in the present analysis. Un~.
fortunately we are able to estimate the effects only approximately, because of the restricted validity of the formulation and the uncertainty of experimental data. Themight considerations thermal and Van of derambiWaals energies thus cause of additional sources guity and make it impracticable to test the applicability of the treatment in comparison with the simple additivity rules (1) and (2). For this reason, it seems better at least in the first step to take only the Coulomb and repulsive interactions into account. The differentiations of Uwith respect to the volume yield the following expressions of the pressure P and the bulk modulus K:
In the Born type of lattice theory (Born and Huang, 1954; Tosi, 1964), interionic interactions are described by a superposition of two-body central forces. The po-
P = —(1 /3hr2) [—Mzczae2/r2 + n iI3cau~(r)
tential energy U is written:
2a/2) {$cc’~cc(°~) + 13~u~(a,) }] (20) 2 /r3 + n i ‘3m {—(2/r)u’~(r) K = (1 /9hr) [4Mzczae + u~(r)}+ (nact,t3cc/2){—(2/r)z4c(ar) -f au~c(ar)}
UMzczae2lr + ni13cau~(r)+ ~ +L3~u~(ar)]
+
(n
(18) + (n2
as a function of the nearest-neighbor distance r. The first term gives the electrostatic energy, in whichM is the Madelung constant, z~and za are the valences of cation and anion, and e is the electronic charge. The other terms corn~from first and second nearest-neighbor repulsions. Here n 1 and n2 are the number of first and second nearest-neighbor ions, respectively, and a is ratio of rcc = r~to r. The factor ~ gives a quantum-mechanical correction to a classical picture of the repulsive interaction (Pauling, 1928), and is explicitly specified by: =
1
+
+ z~INj
(19)
3~/2){—(2/r)U~(ar) + au~(w-)}]
where h is the volume per ion pair divided by r3. We put h = 2 for the present problem. 5. Estimate of the parameters
The formulation based on the lattice theory connects the parameters of the repulsive potential with the observed values of r and K. First we consider only the nearest neighbor interaction, neglecting the terms involving Ucc and u~.Substituting eq. 9 for u~,and solving eqs. 20 and 21 in whichPis put equal to zero, we obtam the following expressions: 3(KM K) 2~MI (22) = ~ [ q~= r + ~m ln [3hr2 ~M’ /fon ii3~] (23) -
—
where N1 isi. the number of outer electrons contained in the ion For NaCl type of lattice, we have M = 1.747558,
(21)
—
102
Here we denote by ~M and KM the contributions from the electrostatic interaction to pressure and bulk
that eqs. 9—11 are also applicable to Ucc and u~with the same parameters q1 and ~ as in urn.
modulus, i.e., the first terms of eqs. 20 and 21, respec-
Unlike the previous calculation, the problem cannot be linearized when the second nearest-neighbor interactions are taken into account. In this case, the calcu-
tively.
In this case, the calculation is straigtforward. First qca and Pca are obtained from eqs. 22 and 23 for the data in Table II. The ionic parameters q~and p~are
lation must be made by an iterative method. We may still use eqs. 22 and 23 ifPM and KM are understood to include the contributions from ~ and u~in eqs.
evaluated separately from the two additivity rules (10)
and (11) in the same manner as was previously mentioned. Before the calculation, we fix the quantity f0 at a tentative value. After the calculation, we shift f0 and q1 according to eq. 12, so as to meet eq. 13 best. The result, which is referred. to solution B, is given in Table III. The standard values to eliminate the ambiguities in the additivity rules were chosen rather arbitrarily
without a significant reason, Once all of q~and ,o~are determined, we can cornpute rCAL from eq. 20 with P = 0, and KCAL from eqs. 21 and 3. The comparisons between calculation and observation are given in the same figure (Figs. 1 and 2) as was discussed before. It is remarkable that solution B matches the observations considerably better than solution A, especially for i<. Clearly the
r~1ations(10) and (11) hold more successfully than eqs. I and 2. According to Figs. 1 and 2, solution B still con-
20 and 21, in addition to the Madelung terms. The right-hand sides of eqs. 22 and 23 thus depend on the unknown parameters q1 and p~.These parameters are, however, roughly estimated, as in solution B. Starting from approximate values of q~and ~ we can re-evaluate q~and Pca from eqs. 22 and 23 and obtain a better solution of q1 and p1 from eqs. 10 and 11. Repeating this calculation until a sufficient convergence is achieved, we have a final solution that satisfies all the relations self-consistently. During the above-mentioned iteration, the parameters for a standard ion are kept unchanged. The solution obtained by the calculation thus depends on the assumed standard values. In the case of solution A and B, a change of the standard parameter simply causes a constant shift of the other parameters, and does not influence rCAL or K CAL• In contrast to those cases,
tains a systematic deviation from the observations,
the second nearest-neighbor interactions depend on the absolute thethus parameters through of q~a ~ 2Pa’scale etc.,of and the calculations U,= K, r and and
This motivates us to extend the treatment the second nearest-neighbor repulsions. Let us to assume
Paa = K change with the selected values of the standard ion.
TABLE III Values of ionic radius q~,ionic compressibility p~,parameter fo, and standard deviations 0p and °Kfor solutions B, C 1 and C2 Ion
B
(A) Li~ Na~ K~ Rb~
Br
F 0p (106 f~ (10~dyn) Mbar) °K(10~ Mbar)
C1
(A)
(A)
C2
(A)
(M
(A)
0.80 1.09 1.35 1.46
0.054 0.050 0.056 0.054
0.95 1.28 1.57 1.69
0.070 0.070 0.080 0.080
0.57 1.00 1.33 1.46
0.006 0.033 0.054 0.054
1.39 1.78 1.91 2.10
0.234 0.261 0.272 0.284
1.21 1.59 1.71 1.88
0.218 0.237 0.246 0.257
1.52 1.87 1.99 2.15
0.238 0.255 0.262 0.269
71.93 9.87 31.1
61.80 1.36 5.50
46.03 3.14 5.68
103
Such dependence enables us to evaluate the standard
(24)
The whole parameters for solutions C~and C2 are listed in Table III. In comparison with the traditional. ionic radii (see Table I), solution C1 gives the parameters q~close to those of Tosi (1964), while the values of in C2 resemble Pauling (1928). Theq~ calculated resultsthe ofradii r andbyK are shown in Figs. 1 and 2 along with all the previously obtained solutions. The agreement between calculation and observation is further improved, becoming quite satisfactory.
(25)
6. Discussions
parameters themselves, so as to give a best fit for the observations. Along the same line, Tosi (1964) obtained his values of ionic radii, even if the concept of ionic compressibility did not appear in his treatment. 0K by:Let us introduce the standard deviations up and =
~ ~1~(Pc~i)~ ~
~
—
KOBS)2
where the summations extend over all the sixteen salts in Table II. Generally 0K is much larger than 0p~and
we here try to minimize
UK. A contour map of the distribution of UK.is given in Fig. 3, in which Li~
The cohesive energy U was evaluated with use of eq. 18 for solutions B, C 1 and C2. The results, which are given in Table IV, agree reasonably well with the
is chosen as the standard ion. There is a long and 0K approximately slender valley that has a value of equal to 6 l0~Mbar. It is certain that the desired solution is found in the vicinity of this valley. It is a delicate problem, however, to determine the solution more precisely. The position of the minimum is probably
observation (Tosi, 1964) in all the three solutions. The test for K inout Figs. along with the agreement ofrUand points that1 and the 2additivity relations (10) and (11) are applicable to the repulsive potential, and that the parameter ~ as well as q~is defined by the nature of an individual ion. A pressure derivative K~ of ionic radius is related to p 1 by eq. 15. This relation
influenced by various sources of ambiguity, such as the
now indicates that the failure of the additivity rule
.
accuracy
of the data of r and K, and the effects of ther-
mal and Van der Waals energies, etc. Here we tentatively pick up two local minima C1 and C2 as the representa-
tive solutions. These two points happen to be situated at two extreme ends of the valley.
1
0
~~
stant but changes from salt to salt.
TABLE IV Calculated and observed cohesive energies (kcal/mole) Material
Calculated
UF
—243.8
—246.9
—248.9
‘:~
I~
I~
KC1 KEr
—165.8 —158.1
—165.7 —158.2
—165.7 —158.3
—165.8 —158.5
KI
—147.9
—148.4
—148.8
—149.9
RbF RbC1 RbBr
—186.1 —1.59.7 —152.6
—184.6 —159.5 —152.7
—183.8 —159.5 —152.9
—181.4 —159.3 —152.6
RbI
—143.1
—143.6
—144.2
—144.9
LT;~~::ii~ii//,// /)
T’
0.5 0
(2)is caused by the factor df/dF, which is not a con-
0.05
p
~j
Fig. 3. The distribution of the standard deviation °Kof eq. 25 as a function of the standard parameters .°~and qc for Li~.
The number attached to each countour gives °Kin the unit of iO~ Mbar. The two local minima C1 and C2 correspond to solutions C1 and C2 in Table III
Ob-
—242.3
~IO~ ~1O~ ~F~!
104 Although there is more or less ambiguity in the parameters determined here, it is remarkable that both solutions C 1 and C2 give substantially higher. values of p~to the anions than to the cations. In other words, positive ions are very hard to deform, compared with negative ions. Usually it is said that an anion will be more compressible because of its larger size. However, this does not give a sufficient interpretation. 4, In in fact, ~ of F is still much larger than that of Rb spite of their having almost the same characteristic sizes. A probable reason may be given as follows. As a result of the electrostatic interaction, the electrons
will be drawn much strongly toward the center in a positive ion than in a negative ion. Therefore, a cation should have higher electron density and be less deformable than an anion. It may be noticed that p 4 is exceptionally 1 of Li small in solution C 2, while no such anomaly is found in solution C1. This cannot always be the reason, however, to reject C2, nor to regard4 has C1 asa being morestrucrealdistinctive istic among than C2.alkali Actually only Li ions; the number of ture and halogen outer electrons is two for Li~but eight for the other ions..
According to the present treatment, the energy and other quantities derived from it are written in terms of the properties of the constituent ions without the need for information about the composite particular material. This suggests useful applications to geophysical and geochemical problems. We are often obliged to estimate energy, volume and elastic property, etc. for
such a material or under such a condition that is very hard to realize practically. The application of lattice theory has not made a remarkable contribution to the Earth sciences, mainly because too many parameters appearing in the formulation made the prediction al-
most arbitrary. This difficulty is considerably reduced by the present treatment. The niaterials composing the Earth’s mantle are mainly composed of four ions, Mg2~,Fe2~,Si4~and 02_. If the contrast of p~between positive and negative ions is due to the ionic charge itself, as has been suggested above, the difference between p~and ~a
will be more dominant for those ions because of their higher valences. This means that p~is practically negli-
24, Fe2~and Si44, and that only Pa of 02 gible for Mg is effective. It is thus suggested that lattice theory is made very simple and tractable for mantle minerals.
Acknowledgments
The ideawith of this study was derived continued discussions Prof. Y. Matsui. I alsofrom thank Dr. N. Onishi for his valuable comments. The numerical calculation was made with use of the computer FACOM 230-48 in the Institute for Solid State Physics.
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155—169.K.S. and Ryzhova, T.V., 1961. The elasticpropAlexandrov, erties of crystals. Soy. Phys. — Crystallogr., 6: 228—252. Anderson, O.L. and Liebermann, R.C., 1970. Equations for the elastic constants and their pressure derivatives for three cubic lattices and some geophysical applications. Phys. Earth Planet. Inter., 3: 61—85. Born, M. and Huang, K., 1954. Dynamical Theory of Crystal Lattices. Oxford University Press, London, 432 pp. Demarest, Jr., H.H., 1972. Extrapolation of elastic properties to high pressure in the alkali halides. J. Geophys. Res., 77: 848—856.
Lewis, J.T., Lehoczky, A. and Briscoe, C.V., 1967. Elastic constants of the alkali halides at 4.2°K. Phys. Rev., 161: 877887. Norwood, M.H. and Briscoe, C.V., 1958. Elastic constants of potassium iodide and potassium chloride. Phys. Rev., 112: 45—48. Pauling, L., 1928. The sizes of ions and their influence on the properties of salt-like compounds. Z. Kristallogr., 67:
377404.
Shannon, R.D. and Prewitt, C.T., 1969. Effective ionic radii in oxides and fluorides. Acta Crystallogr., B25: 925—946. Tosi, M.P., 1964. Cohesion of ionic solids in the Born model. In: F. Seitz and D. Turnbuil (Editors), Solid State Physics, Vol. 16, Academic Press, New York, N.Y., pp. 1—120. Weidner, D.J. and Simmons, G., 1972. Elastic properties of
alpha quartz and the alkaliRes., halides on an interatomic force model. J. Geophys. 77:based 826—847. Wyckoff, W.G., 1963. Crystal Structures, Vol. 1. Wiley, New York, N.Y., pp. 86—91.