Ionic sizes and born repulsive parameters in the NaCl-type alkali halides—I

J. Phye. Chem. Solids

IONIC

Pergamon

SIZES AND THE THE

Printed in Great Britain.

Press 1964. Vol. 25, pp. 31-43.

BORN

NaCl-TYPE

REPULSIVE ALKALI

HUGGINS-MAYER F. G. FUMI~

PARAMETERS

IN

HALIDES-I

AND PAULING

FORMS*

and M. P. TOSI

Argonne National Laboratory, (Received 10 June 1963

Argonne,

Illinois

; revised 29 July 1963)

Abstract-The

Born model of ionic solids is shown to permit the complete determination from solid-state data of crystal radii for the alkali and halogen ions in the individual alkali halides with the NaCl structure, which are directly comparable with the experimental crystal radii deducible from the X-ray maps for the electron distribution in the crystal. The determination is based on the direct relationships which are established, by a physical interpretation of the observed approximate additivity of the interionic distance, between the crystal radii and the parameters entering plausible expressions for the first and second neighbor contributions to the Born repulsive energy. Two different forms of the repulsive energy are considered: the Huggins-Mayer farm, having an exponential dependence on the ionic distance with a hardness parameter common to the family of salts, and the Pauling form, having an inverse-power dependence on the ionic distance, again with a common hardness parameter. The parameters for these forms are determined from solid-state data by a Huggins-type procedure. The crystal radii of the alkali and halogen ions in the NaCl structure obtained with the Huggins-Mayer form are, respectively, larger and smalier than the traditional crystal radii by about two-tenths of an angstrom. Their uncertainty is estimated to be about 0% A by comparison with the crystal radii obtained with the Pauling form. The crystal radii of the ions in NaCl and LiF agree rather closely with those deduced from the X-ray maps of the electron distribution available for these crystals. The value of the determination of electron maps for other alkali halide crystals is emphasized, to teat in particular the significance of the small regular variations of the crystal radius of an ion from crystal to crystal suggested by the calculations. The applicability of the Born model procedure for the determination of crystal radii to other families of ionic solids is pointed out.

1. PRODUCTION THE approximate additivity of the interionic distance in the alkali halides with the NaCl structure led long ago to the notion of crystal radius,(s) as a measure of the average size of each alkali and halogen ion in this structure. The knowledge of the

interionic distances in these crystals is not, however, sufficient by itself to determine the absolute vaIues of the crystal radii. One needs an additional criterion to fix the absolute size of one ion or the relative size of a pair of alkali and halogen ions. LAND&~) proposed o~ginaily to assume that the lattice parameter in Lif is determined by the direct contact of the I- ions. Later authors(%Q chose instead to fix the ratio of the radii of isoelectronic alkali and halogen ions by using the polarizabilities of free ions and of ions in solution. WASASTJE~A(~) takes this ratio as equal to the fourth root of the ratio of the corresponding ionic refractivities in solution, determined by assuming that hydrogen does not contribute to the molar refractivity of the hydrogen halides in sofution.

* Based on work undertaken at the Universitv of Pavia, Italy and largely performed at the Argonne National Laboratory under the ausoices of the U.S. Atomic Energy Commission. Prelimit&y reports of this work were presented at the 1959 International Symposium on Alkali Halides at the Oregon State Collene. at the 1959 Meeting of the Italian Phy&al Society (RGf: I) and at the 1961 March Meetinn of the American Phvsical . Society (Ref. 2). t Ako: Department of Physics, Northwestern University, Evanston, Illinois. B

31

32

F.

G.

FUMI

PAULING(@ elaborates considerably on Wasastjerna’s criterion. However, he still fixes the ratios in question from the polarizabilities of ions in solution, determined by adopting for the polarizability of the potassium ion a value computed for the free ion. In fact, the ratios of the Pauling crystal radii for K+ and Cl- ions, and for Nat and Fions, are only slightly larger than the ratios of the mean radii for the outer electron orbits in the free ions, as computed by HARTREEt7) from selfconsistent field wave functions with exchange. The PAULING set of crystal radii for the alkali and halogen ions in the NaCl structure(s) is based directly on the Wasastjerna-Pauling criterion, The sets of GOLDSCHMIDT,@) of ZAcHARrAsEN(s) and of AHRENS,(~~) on the other hand, are based on this criterion indirectly, since they involve respectively the adoption of the Wasastjerna radius for the Fion and of the Goldschmidt radii for the Cl- ion and for the halogen ions. These various sets are in fact essentially identical, except for the Li+ radius which is determined from different salts by the different authors. KORDE#) has pointed out recently that these traditional crystal radii satisfy approximately simple empirical relationships with the nuclear charges of the ions and with the molar refractivities of the salts. He has then used these relationships, and the interionic distances, to calculate the crystal radii, finding of necessity values practically identical to the traditional crystal radii. It is apparent that both the Land& criterion and the Wasastjerna-Pauling criterion have objectionable features as criteria to determine crystal radii, a fact which has not generally been sufficiently emphasized. Indeed, while the LandP criterion is simply an assumption, the WasastjernaPauling criterion fixes the relative size of the alkali and halogen ions with reference to a condition in which the ions are nearly free. A first attempt to determine crystal radii in the NaCl-type alkali halides entirely from crystal data, without invoking the Land& assumption, was made in a classic paper by PAULING.(~~) The paper uses the Born model of ionic solids and introduces characteristic lengths of the ions (basic radii*) as parameters entering a plausible expression for the nearest neighbor and next-nearest neighbor con* We adopt the term basic radii, first introduced by HUGGINS and MAYER,W instead of the term standard radii originally used by PAULINCA

and

M.

P.

TOSI

tributions to the Born repulsive energy. The expression adopted has an inverse-power dependence on the distance between the ions. The basic radii are rendered commensurable with the size of the ions by fixing an arbitrary constant in the Born repulsive energy so that the sum of the basic radii for the positive and the negative ion in a properly chosen reference crystal is equal to the interionic distance. They are then determined by a fit of the model expression of the lattice energy to the observed interionic distances in the alkali halides with the NaCl structure through the equilibrium condition. The resulting values of the basic radii for the alkali and halogen ions turned out to be essentially identical to their traditional crystal radii, and this numerical correspondence was taken by Pauling as a definite confirmation of these crystal radii. As we will see in detail in Section 4, Pauling’s determination of basic radii has a number of objectionable features, and his results do not provide in our view a valid confirmation of the traditional crystal radii. HUGGINS and MAYER~~,~~) have carried out later more refined determinations of the basic radii of the alkali and halogen ions in the alkali halides for a different plausible form of the nearest and nextnearest neighbor contributions to the Born repulsive energy, having an exponential dependence on the distance between the ions. The determination involves a fit of both the interionic distance and the compressibility of the various alkali halides, and yields also rather accurate values for the cohesive energy of these crystals. The connection between these basic radii and the crystal radii of the ions, however, was never examined. Our paper consists of two main parts. On the one hand, we establish direct relationships between the differences of the basic radii of the alkali and halogen ions in the NaCl-type alkali halides for the and the corresponding Huggins-Mayer form, ratios of the basic radii for the Pauling form, and the differences or ratios of the crystal radii of the alkali and halogen ions in the NaCl structure. These relationships permit the complete determination of crystal radii for the alkali and halogen ions in the individual NaCl-type alkali halides such that the sum of the cationic and anionic radius in each crystal reproduces exactly the observed interionic distance. These crystal radii, which will, of course, vary slightly from crystal to

IONIC SIZES AND BORN REPULSIVE

PARAMETERS

crystal for each ion, are directly comparable with the crystal radii that can be deduced from the experimental maps of the electron distribution in the crystal. On the other hand, we carry out parallel determinations of the basic radii of the ions in the NaCl-type alkali halides for the Huggi~s-payer and Pauling forms of the Born repulsive energy through a fit of crystal data, analogous to the HUGGINS treatmen+) of the Huggins-Mayer form. Our only motivation in considering the Pauling form was to refine the early treatment by Pauling to obtain results which could be compared with those for the Huggins-Mayer form. In fact the inverse-power form has no theoretical justification@@ and actually yields systematic disagreement with the experimental cohesive energies of the alkali halides in standard thermodynamic conditions.(l@ The values that we obtain for the crystal radii of the alkali ions are significantly larger than the traditional crystal radii, and the values of the crystal radii of the halogen ions are correspondingly smaller. This is physically reasonable since the ratios of the traditional radii for the alkali and halogen ions refer to nearly free ions. Moreover the values that we obtain for the crystal radii of the ions in NaCl and LiF agree rather closely with the experimental crystal radii deduced from the maps of the electron distribution available for these crystals.(17-lg) We should emphasize that the new crystal radii are based entirely on crystal data but involve directly the assumptions of the Born model. Both the Born model and the concept of crystal radius imply the assumptions that the ions retain an identity in the crystal and that the interionic forces in different salts be similar. However, the adoption of a specific form for the dependence of the Born repulsion on the distance between the ions, and the explicit consideration of the nextnearest neighbor contribution to the repulsive energy, are essential elements in the solution of the problem. In this connection it is thus pertinent to stress that the crystal radii that we obtain with the Huggins-Mayer form and with the Pauling form are practically the same within their respective uncertainties. 2. CRYSTAL RADII AND BORN RRPULSIVE PARAMETERS

The lattice energy WL of an alkali halide crystal

IN THE NaCl-TYPE ALKALI HALIDES-I

33

under hydrostatic pressure, a function only of the interionic distance r, is composed of an attractive and a repulsive part. The Born model assumes that the attractive part -A(r) is given by the Madelung energy and by the dipole-dipole and dipolequadrupoIe van der Waals interaction energies, and postulates a plausibIe form for the dependence of the repulsive part R(r) on the interionic distance. The parameters entering a particular form of R(r) are then tradition~ly determined from crystal data in standard thermodynamic conditions by means of the equation of state and its volume derivative at constant temperature in the vibrational Hildebrand form: dw;, -=dV

13T K

(1)

Here K and /3 are the isothermal compressibility and the coefficient of volume thermal expansion of the crystal. These equations are strictly correct for a quasi-harmonic solid at zero pressure and at classical temperatures(zs) and are reasonably accurate for the present purpose. We will consider two different forms for the Born repulsive energy, the Huggins-Mayer form and the Pauling form. These assume respectively an exponential and an inverse-power dependence of the Born repulsion on the distance between the ions, with a hardness parameter common to a family of salts, and consider explicitly the first and second neighbor contributions. As is shown in the Appendix, these assumptions together with the observed approximate additivity of the interionic distance within a family of salts permit one to justify the specific forms of the repulsive energy adopted by HUGGINS and MAYER(~~J~) and by PAULING.(~~) The Huggins-Mayer form for the repulsive energy in an alkali halide crystal with the NaCl structure can be written as folfows (see Appendix) : R(r) = 4c+b+b_ exp( -r/p) + 6(c++bz + c___@.) x exp( - 2/Vlrlp)+ Here c++, c-- and c+- are numerical

(3) coefficients of

F.

34

G.

FUMI

and

the order of unity, introduced and determined by PAULING,(‘S) which give a rough expression of the dependence of the Born repulsion between two ions on their net charges and outer electronic configurations. The complete set of Born repulsive parameters for the family of salts consists of the five parameters b+ and of the four parameters b-, characteristic of the alkali and halogen ions within the family, and of the parameter p, common to the entire family. As we will see in Section 3, these parameters are completely determined by a simultaneous fit to crystal data for the various salts by means of equations (1) and (2). We propose to show that the ratios b-/b+ bear a direct relationship to the corresponding differences of crystal radii. To this end we consider the pertinent equation for the equilibrium interionic distance. This equation has the form (see equation (11)): b+b_ r=pln-,

B

(4)

where the quantity B changes from crystal to crystal and determines the small deviations from exact additivity of the interionic distance. On the one hand, one can look upon equation (4) as expressing the interionic distance as the sum of two characteristic lengths Y+ and I- for the positive and the negative ion (basic radii) plus a length which varies slightly from crystal to crystal. The basic radii are defined by writing b+ = bljz exp(r+/p), b- = b1/2 exp(r_/p) with b an arbitrary constant. If the arbitrary constant b is picked so that the equilibrium interionic distance in a reference crystal is equal to the sum of the pertinent Y+ and I_, the basic radii become commensurable with crystal radii and can be interpreted as such. This is in fact the approach adopted by PAULINC~) in his determination of crystal radii by the Born model with the Pauling form of the repulsive energy. The differences Y--Y+ satisfy by definition the equation :

On the other hand, one can look upon equation (4) as expressing the interionic distance as the sum of two crystal radii R+ and R- for the positive and the negative ion which vary slightly from crystal to crystal for each ion. If the deviations from

M.

P. TOSI

additivity are attributed in equal parts ions, -one has then at once: bR--R+

to the two

(6)

= p In ii-+

This equation, when combined v.ith the observed interionic distance, permits a complete determination of the crystal radii for the alkali and halogen ions in the individual salts. An entirely analogous discussion can be carried out for the Pauling form of the Born repulsive energy, which reads as follows (see Appendix): R(Y) = 6c+_(bm,+ b_)“v +6[c,~+(2b+)m+c__(2b_)m](2

[Z]v)-“.

(7)

The complete set of Born repulsive parameters for the family of salts consists here of the nine parameters b+ and b_, characteristic of the alkali and halogen ions within the family, and of the parameters n and m, that we assume as common to the entire family. This assumption, which was adopted also in the original treatment of P.wLIxc,(~~)is analogous to the use of a unique value for the parameter p in the Huggins-Mayer form. The various parameters are again determined by a fit to crystal data by means of equations (1) and (2), as we will see in Section 4. To show that in this case the ratios b-/b+ are directly related to the corresponding ratios of crystal radii we consider the pertinent equation for the equilibrium interionic distance (see equation (15)). This can be written in the form: Y = (b, + b_)B,

(8)

where the quantity B changes slightly from crystal to crystal and is responsible for the deviations from additivity. If one attributes these deviations in proportional parts to the positive and the negative ion, one obtains at once the equation: R-

b-

R,

=.iY+

(9)

This equation, together with the observed interionic distance, permits again a complete determination of the crystal radii for the alkali and halogen ions in the individual salts. The analogous relationship : Y_ b= -, (10) Y+ b+

IONIC SIZES AND BORN REPULSIVE

PARAMETERS

is satisfied by definition by the basic radii of the ions, which are introduced by writing b+ = br+, b= br- with b an arbitrary constant. By fixing the arbitrary constant b so that the sum of the basic radii be equal to the interionic distance in a reference crystal, the basic radii become commensurable with crystal radii and can be interpreted as such. 3. HUGGINS-MAYER

FORM

If one adopts the Huggins-Mayer form for the repulsive energy, equations (1) and (2) for each NaCl-type alkali halide can be rewritten as an equation for the characteristic parameters b+ and band as an equation for the parameter p:

IN THE NaCI-TYPE ALKALI HALIDES-I

35

least-squares procedure, i.e., by imposing that the sum of the squares of the differences between the two members of equation (11) in the seventeen salts be a minimum. The procedure is repeated till consistency is achieved. The calculations were carried out on an IBM 704 computer. The variable metric minimization program, prepared by the Applied Mathematics Division of the Argonne National Laboratory [704 Share Library Routine AN2 0131, was used to search for the location of the minimum over a wide range of physically significant values for the repulsive parameters, covering in particular the values estimated from the traditional crystal radii. Refined values of the repulsive parameters at the minimum were then obtained by --I

Y = p ln

I

1+1/2

&+_b+b._T

z:+ +

Here

- 1 = - 2/2 ( F? + a,$ 7~_+f-_f-I

+!

exp[(l-y’2)rlp-J

i

J

7

P

WP)(d4- 1 1/2P/PX7/5)

>” +

6C,.

+ >;)exp[(* + + 8L&

- d2)r/p].

I I

1 @I)

j

I

(12)

4Yaj3T (13)

o Z!!!Z 2”5 2 + -42cr +72Dr Y 76 -G--

! and ar, Cr and D, are the Madelung constant and the van der Waals coefficients, referred to the nearest-neighbor distance. The values of the coefficients Cr and D, are taken from MAYER.@~) For the various crystal data entering the equations we have adopted two separate sets of values, to assess the sensitivity of the Born repulsive parameters to uncertainties in the data. The two sets of data are reported in Table 1. The system of equations (11) and (12) for the alkali halides with the NaCl structure is solved as follows. One adopts a set of initial values for the parameters b, for the rive alkali ions and for the parameters b- for the four halogen ions and one uses equation (12) to compute a value for the parameter p in each crystal. The average value of p is then introduced into equation (11) for each salt of the family. The parameters b, and b- are then determined from the system of equations (11) by a

solving the least-squares problem exactly by an iterative procedure.Table 2 reports the final values of the Born repulsive parameters for the Huggins-Mayer form obtained from the two sets of crystal data. The basic radii for the individual ions in the two sets differ only by about 0.02 A. The root mean square deviation of the values of the interionic distance given by the right-hand side of equation (11) from the experimental values is 0.006 A for the first set of data and 0.003 A for the second set of data. The parameters reported by H~TGGINS(~“) for the second set of data do not quite correspond to the actuaf minimum of the least-squares problem. The basic radii for the alkali ions are too small by about 0.1 A and the basic radii for the halogen ions are too large by the same amount. The adoption of acommonvalue for the parameter p in the entire family of the NaCl-type alkali

F.

36

Table 1. Crystal data for NaCl-type

G.

FUMI

M.

P.

TOSI

alkali halides at room temperature and at atmospheric pressure.* - 1/K(aKlaP) T K 1IK@K/aT)p

(10-5 i3deg-l)

A

and

(10-l” ems/dyne)

(lo-is

( 1O-4 deg-l)

LiF LiCl LiBr LiI

2.014 2.570 2.751 3.000

2.010 2.572 2.745 3 .ooo

9.2 12.2 14.0 16.7

1.49 3.36 4.20 5.83

1.17 3.41 4.31 6.01

.(!)

NaF NaCI NaBr NaI

2.317 2.820 2.989 3.237

2.310 2.814 2.981 3.231

9.8 11.0 11.9 13.5

2.15 4.17 5.02 6.64

2.11 4.26 5.08 7.07

(2) 6.2 4.1 1

KF KC1 KBr KI

2.674 3.147 3.298 3.533

2.665 3.139 3.293 3.526

10.0 10.1 11.0 12.5

3.28 5.73 6.75 8.55

3.30 5.63 6.70 8.54

1 3.5 4.8 3.4

RbF RbCl RbBr RbI

2.815 3.291 3 445 3.671

2.815 3.270 3.427 3.663

(95) 9.85 10.4 11.9

3.81 6.40 7.69 9.48

4.10 6.65 7.94 9.57

(2)

CsF

3.004

3.004

9.5

4.25

4.25

10.2

5 20 24 37.3

15.3 19.8 24.3 37.3

18 21 25 28

17.5 24.6 25.5 40.0

1 5 6 6

20 30 31 32

20.2 27.2 31.9 36.7

(5) 7.8

(17) 17 21 21

$;:E; 35.0 43 .o

2 7

4 3.8 4.6

cm2/dyne)

2 7 (“s)

::; 10.2

28.4

28.4

* The first set of data is largely based on the recent compilation by CUBICCIOTTI D. (J. them. Phys. 31, 1646 (1959)), but the Spangenberg values for the adiabatic compressibility (SPANGENBERG K., Naturwissenschaften 43, 394 (1956)) have been converted into values for the isothermal compressibility, and the values estimated by Cubicciotti for the temperature and pressure coefficients of the compressibility have been replaced, whenever possible, by the experimental values given by HUGGINS and by BRIDCMAN P. W. (Proc. Amer. Acad. Arts Sci. 74, 21 (1940); 77, 187 (1949)). The second set of data, and the values of the coefficient of volume thermal expansion, are taken from HUGGINS. Values in parentheses are estimated.

-_

Table 2. Born repulsive parameters for the Huggins-Mayer

form in the NaCl-type alkali halides.*

Lif

Na+

Kf

Rb+

Cs+

F-

Cl-

Br-

I-

1st set of data

7.455

20.26

46.92

68.39

102.5

17.23

57.57

83.87

145.9

0.3394

2nd set of data

8.245

22.16

51.45

74.93

114.9

16.86

58.13

85.18

151.2

0.3347

-__

P

__ ._

* The characteristic parameters b+ and b_ are in units of 1O-s (erg/molecule)l/s and the parameter p is in units of angstrom. An alternative expression of the parameters obtained from the first set of data in terms of basic radii is given in Table 8.

halides, taken equal to the average of the values in the various crystals, needs further discussion. The mean deviation of the values of p in the individual crystals, computed with equation (12), from their average value is admittedly small, amounting in particular to 6 per cent for the first set of data. However, the values of the compressibility of the various crystals computed from equations (12) and (14), using the average value of p and the

experimental values of T and of the temperature and pressure coefficients of the compressibility, deviate sizeably, in some cases, from the experimental values. Table 3 reports the discrepancies for the first set of data. This points to the advisability of allowing the value of the parameter p to vary from crystal to crystal. We will discuss the determination of the parameters for the generalized Huggins-Mayer form in a subsequent paper.

IONIC SIZES AND BORN REPULSIVE PARAMETERS IN THE NaCI-TYPE ALKALI HALIDES-I

37

T&e 3. Ratios between the experimental compressibility and the compressibility computed with the

Li

Na

K

Rb

Cs

0.97 0.91 0.99

1.01 0.99 0.98

0.79

1.18

1.06

0.97 0.91 0.98 0.99

F

0.83

Cl

0.99

Br I

1.03 l-40

The values of the cohesive energy of the alkali halides with the NaCl structure in standard thermodynamic conditions computed with the Huggins-Mayer form, using the repulsive parameters obtained from the first set of data, are reported in Table 4, together with,their deviations from the experimental values. The experimental values of the cohesive energy and the values of the vibrational internal energy entering the computed cohesive energy are those adopted by Tosr.(ls) The over-all agreement with experiment is good, as expected. 4. PAULlNG FORM If one adopts the Pauling form for the repulsive T&k 4, Cohesive enugy at room t~~ffat~re ~

_--Li F Cl Br I

-

==: ____.~ *

241*4(0.9) 197.0(1’9) 186.7(3,1) 173.9(3 -8)

-

--_~__l-.

For the pertinent crystal data we use the first set of values reported in Table 1. The values of n in the individual salts are computed by equation (16), their average value is introduced into equation (15), and the system of these equations for the family’of salts is solved by a least-squares procedure, using again the variable metric minimization program on an IBM 704 computer. The calculations reveal that the root mean square deviation of the values of the interionic distance given by the right-hand side of equation (15) from the experimental values reaches its minimum value when m,ln is as large as l-5, but remains between 0.003 A and 0.004 A as m/n ranges from 1 to 2. The corresponding variations in the parameters b+ and b- are significant, particularly as m/n varies from 1 to 1.5. The variations are such that the ratios b.+/b-. for the isoelectronic ions increase somewhat as m/n departs from the value l-5 in either direction. Table 5 reports the values of the repulsive parameters for two values of m/n. One of these is

and at atmos~h~~c~ressure com~ted with the Hugged-payer ~K~altrnole~.~ -__I --_C_X

fm

Na -214*7(-0.3) - 181 .l(l -5) - 172*7(0-9) -162.1(1.1) -=zzzz1_1__1

K - 190.5( -0.7) - 165.5(0.3) -158~8(-0~3) - 149.9~0.0) --

Rb -182.0(-06) -159*9(-O*@ -153.7(-1.1) -145q-0.7) -~ -_-.I__~_-

cs -174*4(-1.9)

The values in parentheses give the differences between the computed and the experimental cohesive energy.

energy, equations (1) and (2) for each NaCl-type alkali halide yield an equation for the characteristic parameters b, and b- and the parameter m, and an equation for the parameter n:

the rather large value at the minimum. The other is the type of value that one anticipates from a comparison of equation (15) with equation (1 l), by requiring that the causes for the deviations from additivity for the Pauling form be physically analogous to those for the Huggins-Mayer form. It is apparent that the discrepancies between the values of the parameters 6+ and b_ in the tsvo sets are generally around 30 per cent and that the ratios b+,@- for the isoelectronic ions are larger in the second set by almost 20 per cent for the sodiumfluorine pair and by almost 10 per cent for the other pairs.

38

F.

G.

FUMI

and

Other disadvantages of the Pauling form relative to the Huggins-Mayer form become apparent when one considers the over-all fit of the compressibilities of the various crystals and the over-all agreement of the computed cohesive energies with the experimental values. The mean deviation of the values of the parameter n in the individual crystals from their average value is 12 per cent as compared to 6 per cent for the parameter p. The cohesive energies in standard thermodynamic conditions computed with the first set of repulsive parameters for the Pauling form are reported in Table 6, and it is apparent that their over-all deviation from the experimental values is significantly larger than that for the Huggins-Mayer form. This is true also if one uses the second set of repulsive parameters reported in Table 5. The smallest values that we obtain for the ratios b+/b_ for isoelectronic ions, namely those corresponding to m/n = 1.5, are still more than 10 per cent larger than the values obtained by PAULING(‘2) with his simple treatment. The origin of the discrepancy must lie in the significant differences between Pauling’s treatment and our own. Pauling determines the repulsive parameters characteristic of the ions from room temperature

M.

P.

TOSI

values of the interionic distance by means of the equation of state of the static crystal, thus neglecting the vibrational pressure in the crystal. He neglects also the van der Waals interactions. Both of these effects lead to an underestimate of the cohesive energy. Finally Pauling fixes a priori the parameter n as equal to 9 and the parameter m as equal to 8 or 9. The cohesive energies in standard thermodynamic conditions computed with Pauling’s expression for the lattice energy, using his values for the repulsive parameters, deviate from the experimental values by amounts which exceed the deviations reported in Table 6 by about 2 Kcal/mole and the mean deviation of the indiv-idual values of n (computed with the equation appropriate to the Pauling treatment) from the value 9 is 19 per cent. 5. CRYSTAL

RADII

Equations (5) and (lo), together with the values determined in Sections 3 and 4 for the Born repulsive parameters of the Huggins-Mayer form and of the Pauling form in the NaCl-type alkali halides, yield values for the differences and for the ratios of the crystal radii of alkali and halogen ions in the NaCl structure. These values, and the values

Table 5. Alternative sets of Born repulsive parameters for the Paul& ___~ ~~~~__~~_ .____ p-p;

form in the NaCl-type

alkali halides.* -

Li+

Na+

K’-

Rb+

Cs+

F-

Cl-

Br-

I-

n

m/n

0.53

0.72

0.88

0.94

1 .Ol

0.88

1.05

1.10

1.18

8.28

1.5

0.67

0.94

1.17

1.27

1.38

0.97

1.29

1.39

1.53

8.28

1.1 --._ __~_

._______

* The characteristic parameters b+ and b- are in units of [lOAiz erg/molecule x An]i/m. The values of these parameters can be interpreted also as values of basic radii in angstroms if the arbitrary constant b is assigned the value unity in the appropriate units.

Table 6. Cohesive energy at room temperature and at atmospheric pressure computed with the Pauling form (Kcal/mole).* Li F Cl Br I

-251.6(-9.3) - 198.3(0.6) - 185.7(4.1) - 170.2(7.5)

* The

values

in parentheses

Na

K

-219.2( -4.8) - 180.3(2.3) -17oq3.1) -158.2(5.0)

- 191 .O( - 1.2) -162.2(3.6) - 154.6(3 ‘9) - 144.7(5.2)

give the differences

between

the computed

Rb - 181.2(0.2) -155.4(3.9) - 1485(4.1) - 139.6(5.3)

and the experimental

cs - 170.8(1’7)

cohesive

energy.

IONIC

SIZES

AND BORN REPULSIVE

PARAMETERS

computed from the traditional crystal radii, are reported in Table 7 for the four pairs of isoelectronic ions. Clearly both the Huggins-Mayer and the Pauling form of the repulsive energy point to the conclusion that the traditional radii overestimate the sizes of the halogen ions relative to those of the alkali ions. A set of basic radii of the ions in the NaCl-type alkali halides for the Huggins-Mayer form, having the significance of crystal radii, is given in Table 8. It is referred to KC1 as the crystal in which the sum of the basic radii for the alkali and the halogen ion is equal to the interionic distance, and is derived from the first set of repulsive parameters given in Table 2. The choice of KC1 as reference crystal gives a rough balance between positive and negative deviations from additivity of the interionic distance in the entire family of crystals. The crystal radii of the alkali and halogen ions in the individual crystals computed by means of equation (6) from the same set of repulsive parameters for the Huggins-Mayer form are reported in Table 9. The average value of the crystal radius of each ion agrees with the value of the pseudocrystalline basic radius reported in Table 8 within 0.05 A. The use of the second set of repulsive parameters for the Huggins-Mayer form reported in Table 2 leads to values of the Table 7. Differences and ratios of crystal radiifor

Huggins-Mayer form, 1st set of data r+-r_ (A) Huggins-Mayer form, 2nd set of data Traditional radii Pauling form, m/n = 1.5, 1 .l Traditional radii

r+lr_

Table 8. Pseudocrystalline

_

IN THE

NaCI-TYPE

ALK4LI

HALIDES-I

39

ionic radii which differ from those given in Table 8 and Table 9 only by 0.02 A. On the other hand, the replacement of the Pauling values for the coefficients c+--, c++ and c__ by the value unity would cause the crystal radii of the alkali and the halogen ions to increase and to decrease, respectively, by about 0.05 A (0.15 .% for the lithium ion). The crystal radii for the alkali and halogen ions in the individual NaCl-type alkali halides that one calculates from the Pauling form by means of equation (9) show some spread owing to the spread in the values of the repulsive parameters for this form illustrated in Table 5. Over the entire spread the radii of the alkali ions are slightly smaller than those reported in Table 9, and the radii of the halogen ions are correspondingly larger. However, at one end of the spread, the discrepancy is typically only 0.05 A and, at the other end, it generally does not exceed 0.1 A. Only in the fluorides of potassium, rubidium and cesium the maximum discrepancy reaches 0.2 A, thus leading to an overlap with the traditional crystal radii. In conclusion, it appears that the value 0.05 A represents a reasonable measure of the typical uncertainty of the crystal radii reported in Table 9. The crystal radii of the Lif ion could be underestimated by up to 0.1 A if Pauling’s adoption of sizeably higher values for the coefficients cc- and

isoelectronic alkali and halogen ions in the NaCl structure. ~_____.___

Na+-F-

K+-Cl-

Rb+-Br-

Csf-I-

0.055

- 0.069

- 0.069

-0.120

- 0.041 -0.48

- 0.043 - 0.47

- 0.092 -(0.47-0.X)

0.84-0~91 0.73

0+5-0.91 0.76

0.86-0.90 0.75-0.78

0.091 -(0~35-0~41) 0.82-0.97 0.70-0.74

basic radii for the Huggins-Mayer form halides referred to KC1 (A).*

in the NaCl-type

alkali

Li+

Na+

K+

Rb+

Csf

F-

Cl-

Br-

I-

0.914

1.254

1.539

1.667

1.804

1.199

1.608

1.736

1.924

* The pertinent value of b is 0.254 x lo-i2 erg/molecule.

The value of p is 0.3394 A, as given in Table 2.

F.

40

Table 9.

Crystal

G.

FUMI

and

M. P. TOSI

radii for the alkali and halogen ions in the NaCl-type derived from the Huggins-Mayer form (A).*

alkali halides

Li

Na

K

Rb

CS

F

0.86 1.15

1.19 1.13

1.51 1.17

1.64 1.17

1.80 1.20

Cl

0.94 1.63

1.59

1.23

1.54 1.61

1.67 1.62

( 1.83 >t 1.64

Br

0.96 1.79

1.25 1.74

1.55 1,75

1.69 1 a76

( 1.84 >t 1.78

I

1 .oo 2 .oo

1.28 1.95

1.57 1.96

1.71 1.96

( 1.86 >t 1.97

-

___1

* For each alkali halide the upper number gives the radius of the alkali ion and the lower number gives the radius of the halogen ion. t These crystal radii are obtained using the interionic distance in the NaCl-type phase observed bv Schulz uDon deposition from the vapor onto cleavage surfaces of mica or certain other crystals (i,. G. SCHULZ,J. &em. Phys. 18, 996 (i950)). c++ in the lithium halides than in the other salts is incorrect. 6. DISCUSSION The way in which the Born model approach described in this paper eliminates the difficulty inherent in the crystallographic determination of crystal radii in the family of the alkali halides with the NaCl structure is in essence as follows. The observed approximate additivity of the interionic distance effectively means that the interionic distance can be written either as: I = R++R_

(17)

Y = r++r_+A.

(18)

or as

Here R+ and R_ are crystal radii of the ions which vary slightly from crystal to crystal, whereas Y+ and Y.- are characteristic lengths of the ions within the family and the length A varies only slightly from crystal to crystal. The crystallographic approach uses equation (17) and interprets R, and R- as characteristic lengths of the ions to be determined from the observed interionic distances by an averaging procedure. The differences between the crystal radii of alkali and halogen ions remain then intrinsically undetermined by an additive constant. The Born model approach uses

instead equation (18). The form of the equation is used to justify the way in which the characteristic lengths of the ions r+ and Y_ enter the expression of the Born repulsion between nearest neighbors, for an assumed dependence on the ionic distance whose hardness is determined by the compressibility of the crystal. The specification of the physical causes for the deviations from additivity, and an extrapolation to the Born repulsion between next-nearest neighbours, lead one then to express the interionic distance in terms of a limited number of parameters. This number is smaller than the number of equations for the interionic distance, which are all independent, and thus the characteristic lengths of the ions can be determined from crystal data. The crystal radii R, and R- are finally obtained from the characteristic lengths, for comparison with the experimental crystal radii, by splitting the length A in each salt between the two ions in an appropriate way. The specification and the extrapolation involved in the procedure are justified a posteriori by the good fit of the observed interionic distances provided by the model with this limited number of parameters. On the other hand, the effect of the form assumed for the repulsive energy can be investigated by adopting different forms. The determination of the Born repulsive parameters in the NaCl-type alkali halides from crystal

IONIC SIZES AND BORN REPULSIVE

PARAMETERS

data has been carried out both for an exponential and an inverse-power dependence of the repulsive energy on the distance between the ions, under the assumption that the hardness parameter be constant within the family of salts. The HugginsMayer form provides a better over-all fit of the compressibilities of the crystals and a better overall agreement of the computed cohesive energies with the experimental values. The crystal radii of the alkali and halogen ions in the NaCl-type alkali halides derived from the Born model with the Huggins-Mayer form are, respectively, systematically larger and systematically smaller than the traditional crystal radii by about two-tenths of an angstrom. Their uncertainty is estimated to be about O-05 A by comparison with the crystal radii derived with the Pauling form. Table 10 compares the average values of the new crystal radii with the traditional radii. The deviations of the new crystal radii from the traditional crystal radii are physically reasonable since the ratios of the traditional radii for the alkali and halogen ions are determined from the ratios of the polarizabilities of nearly free ions. In effect, the depression of the tail of the wave function for an ion in a crystal caused by the Pauli principle should be more pronounced for a halogen ion than for the isoelectronic alkali ion, and moreover the Madelung potential at the anionic (cationic) site will contract (expand) the electron cloud of the free

ion. This last effect is illustrated e.g. by the qu~tum-mech~ic~ calculations carried out recently by PETRASWEN, ABARENKOV and KRISTOFEL@~) with a point-ion approximation to the crystal potential. A “tightening” of the halogen ions, and a “loosening” of the alkali ions, in passing from the free-ion state to the crystal, was first suggested by FAJANSand Joos(ss) in their paper on molar refraction and reemphasized by Fajans in later publications,(s4) and is rather directly confirmed by the comparison of the free-ion polarizabilities of PAULINC$) with the polarizabilities of ions in the alkali halide crystals derived by TESSMAN, KAHN and SHOCKLEY.(~~)In fact, while the ionic deformations caused by the Madelung potential are most relevant in determining the polarizabilities of ions in crystals, as the polarizability is essentially connected with the mean radius of the outer electronic orbit, the ionic contractions caused by the Pauli principle should be directly relevant to the crystal radii of the ions, which are largely determined by the behavior of the wave functions of the ions near the ions’ boundaries. Thus the connection between polarizability and crystal radius can be only qualitative. The crystal radii of the ions in NaCl and LiF obtained with the Born model can be compared directly with the experimental crystal radii which are available for these crystals.(r7-1s) These are deduced from the location of the minimum of the

Table 10. Average crystal radii of the alkali and halogen ions in the NaCl-type II_Li+ Huggins-Mayer form Traditional

Na+

K+

41

IN THE NaCi-TYPE ALKALI HALIDES-I

alkati halides (in). --

Rb+

cs+

F-

Cl-

Br-

I1.97

0.94

1.24

1.54

1.68

1.83

1.16

1.62

l-76

0*60-0*78

0*95-0+8

1.33

l-48-1*49

1.65-1~69

1.33-1.36

l-81

1.95-1.96

2.16-2.20

F---z==

Table 11. Crystal radii in NaCl and LiF (A) -~ Exper~en~l Wrrra et al. ‘&XiOKNECHT Na+ CILi+ F-

1.17 l-65 0.9 1-l

1.18 1.64 -

Higgins-gayer form

l-23 l-59 0.86 1.15

Pauling form

Traditionaf

1*15-1.18 1.67-l 64 0*76-0*82 1~25-1~19

0.95-0.98 l-81 0.60-0*78 l-33-1-36

42

F. G. FUMI

and

electron density in the crystal at room temperature along the nearest-neighbor line, as determined from X-ray measurements. The electron density between neighboring ions drops to a very low value (< 0.1 cl/As) in NaCl and to a rather low value (< 0.2 e1/A3) in LiF. The comparison is illustrated in Table 11. The agreement is well within the combined uncertainty of the crystal radii derived from the Born model and of the experimental crystal radii. This last uncertainty can be estimated from Hosemann’s analysis of SCHOKNECHT’S data for NaCl(l*) and is found to be less than 0.1 A. The experimental crystal radii in LiF reported by WITTE et aZ.(ls) can be expected to overestimate slightly the cation-anion radius ratio since no correction was made for the bulging out of the Lif ion in the [loo] direction and of the F- ion in the [110] direction. It would be most valuable to have accurate electron maps of other alkali halide crystals with the NaCl structure. These would provide a direct test of the significance of the small regular variations of the crystal radius of an ion from crystal to crystal indicated by our calculations. Some support to the significance of these variations is given by the fact that they are entirely similar for the Huggins-Mayer and Pauling forms. The procedure described in this paper for the determination of pseudocrystalline basic radii and of crystal radii in individual salts from crystal data by the Born model is not limited, of course, to the NaCl-type alkali halides. Other families of “ionic” solids with a given structure in which the necessary Born repulsive parameters could in principle be completely determined from crystal data are the cesium and thallium chlorides, bromides and iodides and the NaCl-type oxides and sulphides of the alkaline-earth metals. On the other hand, the availability of average crystal radii for the alkali and halogen ions in the NaCl-type alkali halides is sufficient by itself to determine average crystal radii for other ions in the same coordination through the traditional crystallographic procedures, based on the additivity of the interionic distance and on the coordination correction. Acknozeledgenrents-It is a pleasure to acknowledge the helpful comments of M. L. HUGGINS and L. PAULING, the stimulating interest of K. FAJANS, R. HOSEMANNand E. W~~LFEL and a critical reading of the manuscript by L. GUTTMAN. We wish also to thank T. C. CHEN and

M. P. TOSI

D. P. HENRY (IBM Research Center) and M. BUTLER (Argonne National Laboratory) for advice on the calculational procedures, and we are especially indebted to A. STRECOK(Argonne National Laboratory) for programming the calculations on the IBM 704 computer.

REFERENCES 1. FUME F. G. and TOSI M. P., Suppl. Nuooo Cim. 16, 123 (1960). 2. FUMI F. G. and TOSI M. P., Bull. Amer. phys. Sot.

6, 170 (1961). 3. BRAGGW. L., Phil. Mug. 40,169 (1920); FAJANS K. and GRIMM H., 2. Phys. 2, 299 (1920). 4. LANDI?A., Z. Phys. 1, 191 (1920). 5. WA~ASTJERNAJ. A., Sot. Sci. Fenn. Comm. Phys. Math. I, no. 37 and no. 38 (1923). 6. PAULING L., Proc. Roy. Sot. A114, 181 (1927); J. Amer. them. Sot. 49,765 (1927). See also PAULINC L., The Nature of the Chemical Bond Section 13.2. Cornell University Press, Ithaca (1960). 7. HARTREED. R., The Calculation of Atomic Structures, Chap. 7 and Appendix 2, Table Al. Wiley, New York (1957). 8. GOLDSCHMIDTV. M., Norske Vid. Akad. Oslo Skr., Math. Nat. Kl. no. 2 (1926). 9. ZACHARIASEN W. H., reported by KITTEL C., Introduction to Solid State Physics, Table 3.5. Wiley, New York (1956). 10. AHRENS L. H., Geochim. Cosmochim. Acta 2, 155 (1952). 11. KORDE~E., 2. Kristallogr. 115, 169 (1961). 12. PAULINC L., Z. Kristallogr. 67,377 (1928); J. Amer. them. Sot. 50, 1036 (1928). See also PAULING L., The Nature of the Chemical Bond, pp. 523-526. Cornell University Press, Ithaca (1960). 13. HUGGINSM. L. and MAYER J. E., J. them. Phys. 1, 643 (1933). 14. HUCGINS M. L., J. them. Phys. 5, 143 (1937); 15, 212 (1947). 15. See e.g. SE~TZ F., The Modern Theory of Solids Section 11. McGraw-Hill, New York (1940). 16. Tosr M. P., J. Phys. Chem. Solids 24, 965 (1963). phys. Chem. 3, 296 17. WITTE H. and WGLFEL E., Z. _(1955). 18. SCHOKNECHT G., 2. Naturf. 12a, 983 (1957). See also HOSEMANN R., Freiburger Forschung. B37, 99 (1959). 19. KRUG J., WITTE H. and W~~LFELE., Z. phys. Chem. 4, 36 (1955). 20. FUMI F. G. and TOSI M. P., J. Phys. Chem. Solids 23, 395 (1962). 21. MAYER J. E., J. them. Phyi 1, 270 (1933). 22. PETRASHENM. I., ABARENKOVL. V. and KRISTOFEL N. N., Opt. Spectrosc. 9, 276 (1960); Vest. Leningrad Univ. 16, 7 (1960). 23. FAJANS K. and JOOS G., 2. Phys. 23, 1 (1924). 24. See e.g. BAUER N. and FAJANS K., J. Amer. them. Sot. 64, 3023 (1942). 25. TESSMANJ. R., KAHN A. H. and SHOCKLEYW., Phys. Reu. 92, 890 (1953).

IONIC

SIZES

AND BORN

REPULSIVE

PARAMETERS

IN THB

NaCI-TYPE

ALKALI

HALIDES-I

43

We shall refer to this expression as the generalized E&&m-Mayer form. The complete set of Born repulsive parameters for this form in the entire family of salts consists of the parameter b, of the nine basic radii for the Let us write the Born repulsive energy in an alkali alkali and halogen ions and of a parameter p for each halide crystal with the NaCl structure in the form: salt. Thus the number of parameters is now less than the number of independent equations available for their A(r)= 6& exp( - r/p){ 1 -l- BZ exp[( 1 - 1/2)r/p]} determination, which are two for each salt. In this paper we restrict ourselves to assume that the parameter p is common to the entire family of salts, where 6B1 exp( - r/p) and 6B1Bs exp( - -1/[Z]r/p represent following the earlier work of HUCCINS and MAYER.(‘~-“) the nearest-neighbor and next nearest-neighbor conWith this assumption, the parameter 6 becomes an tributions. The three independent parameters BI, Bz arbitrary constant since the r-independent factors in and p which enter equation (A.l) for each crystal cannot, equation (A.3) are now completely determined by the of course, be determined as such from the equation of nine parameters b+ = b’@ exp(r+jp), b_ = bl@ exp(r_/p), state of the crystal and its volume derivative in standard characteristic of the alkali and halogen ions within the thermodynamic conditions. To eliminate this difficulty, family of salts. The expression of the Born repulsive one elaborates somewhat equation (A.l), making use of energy reduces thus to the Huggins-Mayer form the observed approximate additivity of the interionic (equation (3)). distance. The equation of state of the crystal at negliIn an analogous fashion one can see that if one writes gible pressure reads as follows: the Born repulsive energy in the form:

A~~o~~te

u~~tivity

APPENDIX of the interionic distance and Born rej%dsive energy

(A.11

R(Y) = 6&r91

- dA/dr - 6r2/lT/K

(A.4 The approximate additivity of the interionic distance, I = r++ r_+ 4 with r+ and Y_ characteristic lengths of the ions within the family of salts, implies that BI contains a factor exp[(r++r_)/pJ. The physical causes for the deviations from additivity include the variation in the Born repulsion between next-nearest neighbors, and the variations in the attractive force and in the vibrational pressure, from crystal to crystal. The hardness parameter p of the Born repulsion causes additional deviations from additivity if it is allowed to vary from crystal to crystal. If one assumes now, in harmony with the previous literature, that B1 exp[(r++r_)/p] = c+-b, with b a parameter common to the family of salts, one is in fact assuming that the only additional cause for deviations from additivitv is the Pauling coefficient c+-. This differs from unity only for the lithium salts, A plausible extension of the resulting expression for the nearestneighbor repulsion to the next-nearest neighbor repulsion leads to the following expression for the Born repulsive energy :

R(r) = 6c+_bexp(y)exp(-

+6b [c++exp(:)

5) +h_exp(:)]

+Bz(1/2)-'5(A.4)

the approximate additivity of the interionic distance implies that B1 contains a factor (b++ b_)n, where b+ and 6_ are parameters characteristic of the ions within the family of salts. If one assumes now with PAIJLINC(‘2~ that B1 = c+-(b++b_)m, where nt is a new parameter which need not coincide with n, one is in fact introducing with m a new factor inf3uencing the deviations from additivity. One expects that rn will not differ greatly from n. The extension of the expression for the nearest-neighbor repulsion to the next-nearest neighbor repulsion yields then directly the Pauling form (equation (7)). The number of parameters for this form in the family of salts can now be further reduced to allow their determination by assuming, for instance, a simple relationship between the values of nz and n in each crystal. PAULING(~~) chose to adopt the more stringent assumption that the parameters n and m be common to the entire family of salts. Clearly the arguments that we have used to justify the expressions commonly adopted for the parameters B1 and Ba in the exponential and inverse-power forms for the Born repulsive energy are merely plausibility arguments. We refer specifically to the detailed identification of the causes for the deviations from additivity and to the extrapolation of the expression for the first neighbor repulsion to the second neighbor repulsion. A justification for these arguments is provided only a posteriori by the attainment with the various forms of a good fit of the observed interionic distances.