SIMULTANEOUS DIFFUSION OF CALCIUM AND STRONTIUM IN NaCl SINGLE CRYSTALS

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SIMULTANEOUS DIFFUSION OF CALCIUM AND

STRONTIUM IN NaCl SINGLE CRYSTALS

H. Machida, W. Fredericks

To cite this version:

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JOURNAL D E PHYSIQUE Colloque CI, supplement au n° 12, Tome 37, Decembre 1976, page C7-385

SIMULTANEOUS DIFFUSION OF CALCIUM AND STRONTIUM

IN NaCl SINGLE CRYSTALS (*)

H . M A C H I D A ( * * ) a n d W. J. F R E D E R I C K S D e p a r t m e n t of Chemistry, Oregon State University,

Corvallis, Oregon 97331, U . S. A.

Résumé. — Des coefficients de diffusion en fonction de la concentration, pour le 45Ca2+ et le 85Sr2+ dans du NaCl purifié furent mesurés avec une méthode de sectionnement. Le NaCl fut

purifié par un processus d'échange d'ions CI2-HCI, et les cristaux formés dans 1/6 d'atmosphère de HC1. Les traceurs furent purifiés sur de petites colonnes d'échanges d'ions pouvant être écartées, afin d'éliminer les impuretés précédentes et consécutives avant l'usage dans un recuit de diffusion. Des recuits isothermiques de diffusion furent faits dans une gamme de température de 448 °C à 683 °C. Aux températures supérieures à 500 °C (le point de fusion eutectique minimum dans ce système), la diffusion se produisit à partir d'une source de vapeur ; en dessous de 500 °C des sources déposées en surface furent utilisées. Les coefficients de diffusion à saturation, enthalpies et entropies de rapports impureté-lacune furent calculés d'après le modèle ionique commun (0 pour la diffusion simultanée d'ions bivalents dans des halogénures alcalins. Dans le NaCl les coefficients de diffusion à saturation Ds(Ca) et £>s(Sr) sont donnés par les formules :

Z>s(Ca) = 1,14 x IO-3 exp(— 0,851 eV/&T)cm2/s

et

Ps(Sr) = 2,30 x IO-3 exp(— 0,925 eV/kT) cm^/s

pour le calcium et le strontium, respectivement. L'énergie libre Gibbs d'association du complexe impureté-lacune dans le NaCl pour le calcium peut être représentée par

Ag(Ca) = — 0,752 eV + (2,69 x 10-" eV/K) T

et celle pour le strontium par

Ag-(Sr) = —0,671 eV + (3,35 x 10-" eV/K) T.

Abstract. — Concentration dependent diffusion coefficients for 4 5Ca2 + and 85Sr2+ in purified

NaCl were measured using a sectioning method. NaCl was purified by an ion exchange — CI2-HCI process and the crystals grown under 1/6 atmosphere of HC1. The tracers were purified on small disposable ion exchange columns to remove precessor and daughter impurities prior to use in a diffusion anneal. Isothermal diffusion anneals were made in the temperature range from 448 °C to 683 °C. At temperatures above 500 °C (the lowest melting eutectic in this system) diffusion was from a vapor source ; below 500 °C surface deposited sources were used. The saturation diffusion coefficients, enthalpies and entropies of impurity-vacancy associations were calculated using the common ion model (*) for simultaneous diffusion of divalent ions in alkali halides. In NaCl the saturation diffusion coefficients Ds(Ca) and Z)s(Sr) are given by

Ds(Ca) = 1.14 x 10-3 exp(—0.851 eV/&r)cm2/s

and

Z)s(Sr) = 2.30 x 10-3 exp(—0.925 eV/kT) cm2/s

for calcium and strontium, respectively. The Gibbs free energy of association of the impurity vacancy complex in NaCl for calcium can be represented by

A^(Ca) = — 0.752 eV + (2.69 X 10"4 eV/K) T

and that for strontium by

Ag-(Sr) = — 0.671 eV + (3.35 x 10~4 eV/K) T.

(*) Research supported by the National Science Foundation under grant GP-6893 and based on a portion of a thesis submitt-ed in partial fulfillment of the requirements for the degree Doctor of Philosophy.

(**) Present address : Kyoto Ceramic Co., Kyoto, Japan. (!) KRAUSE., J. L. et FREDERICKS, W. J., J. Phys. & Chem.

Solids 32 (1971) 2673 ; / . Physique Colloq. 34 (1973) C9.

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C7-386 H. MACHIDA AND W. J. FREDERICKS Introduction.

-

The results reported here are part

of a program to obtain a consistent collection of diffusion and association parameters for divalent cations in alkali halides. The divalent ions 210Pb2+ and 109Cd2+ w ere chosen for the initial studies in both NaCl and KC1 13, 41. In this paper we report on 45Ca2f and '*Sr2+ in NaCl. All the above atoms have a completely filled orbital exposed when in the divalent state.

These experiments used thz method of simultaneous diffusion of two divalent cations into purified NaCl crystals [I, 21. Now when two divalent cations diffuse simultaneously into a pure crystal the major source of vacancies affecting one diffusant (in addition to those it introduces itself) is the other diffusant. Because the concentration of both diffusants can be measured the total vacancy concentration affecting each diffusant is more accurately known than in single ion diffu-

sion experiments.

Theory.

-

The interpretation of simultaneous diffusion of impurities in ionic crystals has been dis- cussed by Krause and Fredericks [3, 41. The divalent

impurities diffuse only when adjacent to a vacancy, therefore Fick's law must be applied to fluxes of impurity-vacancy complexes. If

x

denotes the mole fraction and i, j, C and A as subscripts denote the species Ca2+, Sr2+, cation vacancy and anion vacancy, respectively, the two fluxes of impurity-vacancy complexes are given by

and

Here t is time, x the distance into the crystal, and Ds the diffusion coefficient when every impurity i or j has a vacancy adjacent to it.

The two flux equations are coupled because of the interdependence of

xi,

and

xjc

through their common component C. Assumin glocal equilibrium everywhere along the diffusion profile their magnitude is set by the four mass action relations and the electroneutrality condition that describe the impurity defect system. These are

and

where k is the Boltzman constant, T the absolute temperature and Ag the Gibbs free energy.

The quantities obtained in the laboratory from the analysis of diffusion experiments are the total con-

centrations of diffusants ci and cj as a function of

x , t, and T. It is convenient to introduce these mea- surable concentrations into the above equations by defining degrees of association pi = xiC/ci and

pj = xic/cj. The flux equations were transformed

into finite difference equations by the Schmidt [3] method for computer solution.

Each diffusion profile was fitted using a nonlinear regression program and a CDC-3300 computer. Allnatt's et al. [4] expression was used for the Schottky product Ks(T). Examples of calculated diffusion profiles are shown in figures 1 and 2.

0 45 90 135 180 225 270 315 360 405

DISTANCE (MICRONS)

FIG. 1. - Penetration profiles of Ca2+ and Srz+ in NaCl at 601 OC. Vapor source. tt = 9.446 40 x

lo5

s. X : Crystal A.

: Crystal B.

+

: Errors. Solid curves are calculated profiles generated using the following parameters :

Ag(Ca) =

-

0.340 eV, Ds(Ca) = 1.30 x 10-8 cmZ/s ;

Ag(Sr) =

-

0.378 eV, Ds(Sr) = 1.09 x 10-8 cm2/s.

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SIMULTANEOUS DIFFUSION O F CALCIUM AND STRONTIUM I N NaCl SINGLE CRYSTALS C7-387

0 65 130 195 260 325 390 455 520 585

DISTANCE (MICRONS )

FIG. 2. -Penetration profiles of Ca2+ and Sr2+ in NaCl at 683 OC. Vapor source. t t = 5.623 80 x 105 s. X : Crystal A.

: Crystal B.

+

: Errors. Solid curves are calculated profiles generated using the following parameters :

Ag(Ca) = - 0.309 eV, Ds(Ca) = 3.84 x 10-8 cmz/s ;

Ag(Sr) = - 0.354 eV, Ds(Sr) = 3.11 x 10-8 cm2/s. Monovalent Ions

-

Ag Br BO2 K OH Rb T1

The radioisotopes used as tracers were 45Ca (tl,, = 165 d ) and 85Sr (tIr2 = 64 d ) . The 45Ca decays to the stable isotope 4 5 S ~ by beta emission and "Sr decays to the stable isotope 84Rb by electron capture [6]. The latter, Rb, is monovalent and causes little perturbation of the tracer diffusion. Scandium's usual oxidation number is

+

3 and thus could cause a significant perturbation in the vacancy concentration if present in amounts comparable to that of 45Ca. Solutions of

4 5 c a 2 + _{and of 85Sr2+ in HCl were obtained from}

New England Nuclear Corp. and ICN in a radiometric purity of 99

+

%.

The specific activities of 15.5 m Cu/mg Ca and 3.88 mCu/mg Sr were given by the former supplier and 20.1 mCu/mg Ca and 15.2 m Cu/mg Sr by the latter. Neither supplier claimed chemical purity.

The isotopes were purified by nitric acid elution from an ion-exchange resin. A similar method has been given by Rane and Bhatki [7]. Bio-Rad Labo- ratories AG-50W-X8 (200-400 mesh) cation exchange resin packed in disposable polypropylene columns (7 mm I. D. by 4 cm high with a resin bed of approximately 3 ml) was used in this work. The same procedure was used to obtain pure *%r. The Rb' elutes initially along with the other alkali ions. A complete description will be given elsewhere.

Because of the low vapor pressure of CaCl, and SrCl, both vapor phase and surface layer experiments were used to obtain a reasonable temperature range. Menge [8] reported a eutectic in the NaCl-CaCl,

Analysis of OSU purzjied NaCl single crystals

Conc. (PPM) - < 0.01 (1) 0.95 < 0.01 (1) < 1 0.23 0.012 < 0.01 (I) Method

-

OAS NAA ( 2 ) OAS NAA (') OAS

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NAA (3*4) OAS

Methods : OAS is optical absorption spectroscopy. NAA is nuclear activation analysis. AAS is atomic absorption spectroscopy.

Divalent Ions - Ca Co Cr Fe Mn Pb Sc Sr Zn Conc. (PPM) - 0.025 0.000 05 0.000 5 0.127 f 4 < 0.003 (I) < 0.01 (1) 0.000 04 0.037 0.000 3 Method

-

NAA (3) NAA (3) NAA (3) NAA (3,4) NAA (2) OAS NAA (3) NAA (3) NAA (3)

(1) Set by detection limit of analytical procedure.

(2) Activation with OSU TRIGA Reactor. Analysis by OSU Radiation Center.

(3) Activation with reactor at Missouri. Analysis by OSU Radiation Center.

(4) Determined by comparator technique. All other determinations by 3~ calculations, and thus is an upper limit.

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CI-388 H. MACHIDA AND W. J. FREDERICKS system melting at 5000C. The NaC1-SrCl, system

has a eutectic which melts at 570 OC [9]. A eutectic exists in the CaC1,-SrC1, system with a melting point at 6460C [lo]. Unfortunately the ternary system has not been studied. To avoid any possible violation of boundary conditions the surface layer experiments were used only at temperatures well below the lowest melting of the binary eutectics. Two diffusion anneals at 448 OC and 481 OC were from surface layer sources. A mixture of carrier and tracer solutions (0.100 mCi/mg for CaCI, and 0.079 mCi/mg for SrCl,) was evaporated to dryness. The residue was dissolved in six 0.1 ml portions of NaC1-saturated 95

%

ethyl alcohol. These solutions were deposited on one of the larger faces of a crystal sample and evaporated to dryness with a heat lamp. A second uncoated face was placed against the coated face, thus forming a sandwich with the thin tracer layer between two pure crystals. This crystal sandwich was placed on a quartz pedestal in a quartz diffusion ampoule with a second quartz pedestal placed on top of the crystals as a weight. The warmed ampoule was flushed with C1, several times, then sealed under 113 atmosphere of CI,.

All other diffusion anneals were from vapor sources. The tracer-carrier mixtures were prepared in a similar way, but the activities were increased to 0.398 mCilmg for the CaCl, and 0.127 mCi/mg for the SrC1,. Sufficient solution was evaporated in the bottom of the diffusion ampoule to provide a large excess of source materials to insure that the vapor was in equilibrium with solid CaCl, and SrCl, throughout the entire diffusion. The ampoule was flushed with C1, several times, then sealed under 116 to 113 atmosphere C1,.

The diffusions were carried out in exactly the same manner described by Krause and Fredericks [I, 21.

At the end of the diffusion the ampoule was immer- sed in ice water, the crystals removed and two-milli- meter sections cleaved from the edges so that the sample remaining contained a one-dimensional diffusion profile. An American Optical Co. Model 960 sledge microtome was used to section the crystals. The surface of the crystal was aligned paralled to the microtome knife by observing the interference pattern produced by a lamp behind the blade and crystal with a binocular microscope.

The "Sr decays to the stable isotope "Rb by electron capture with the emission of a gamma photon with an energy of 514 key, and the 45Ca decays to the stable isotope 4%c by emitting beta particles with the maximum energy of 252 keV. The gamma radiation of "Sr at the 514 keV photopeak was counted by placing the vials in a 3 in. x 3 in. NaI well-type scintillation detector with a multichannel analyzer. A 10 keV/channel window was used in the gamma radiation counting. Samples having activities well above the background were counted long enough to give a standard deviation no greater than 3

%.

Because of lower impurity concentrations, samples in the vapor source diffusion needed to be counted for much longer times than those in the surface layer diffusion. The beta of 45Ca was counted with a Pac- kard Co. Tri-Carb Liquid Scintillation Spectrometer Model 3375. The sample was dissolved in 1 ml deionized water and the vial was placed in a large beta counting glass vial, and then 20 ml New England Nuclear Aquasol as a scintillation liquid was added to it. The counting vial with the small vial, the sample and the fluor solution in it was shaken well to insure that the resulting solution was uniform. It was observed that the resulting solution was trans- parent. The blank for the background counting was prepared by dissolving 5 mg of the host crystal with deionized water and adding the fluor solution. The standard solution was made in the same way except adding either 0.002 ml, 0.005 ml or 0.010 ml of 45CaC12 stock solution. In the liquid scintillation counting the quench corrections were made by the channels ratio method [1 11.

All the activities were corrected for the decay by using the half-life periods of 64 days for "Sr and 165 days for the 45Ca. After all the corrections had been made, the mole fractions of each ion were calculated.

Experimental results. - Diffusion profiles of Ca2+ and Sr2+ in NaCl were measured at eight temperatures ranging from 448 0C to 683 0C. Each profile was determined from two separate measurements of the penetration of Ca2+ and Sr2+ into NaCl. These are typical of the other six. The curves shown as solid lines in these figures are the diffusion profiles calculated as described above using the Ds and Ag values for each ion given in the figure caption.

Each experimental point contains an error cross that represents the 1

%

confidence limit. The errors were determined by thc propagation of errors method [12]. The limit of error in activity required for that calculation was obtained using the method of Wang and Willis [l I]. Figure 1 was chosen because it represents the maximum uncertainty in any of the data obtained in these experiments. The 1

%

uncertainty bars were large in figure 1 because the counting time of each sample was much shorter than that nor- mally used.

Figure 2 is the profile of the highest temperature diffusion anneal. This data should exhibit the largest surface anomaly due to back diffusion on quenching [I]. The technique of quenching seems adequate in these experiments to reduce any perturbation from back diffusion on cooling of the sample to a negligible quantity.

The results of all eight experiments are summarized in table II which gives the free energy of association,

Ag, and the saturation diffusion coefficient, Ds,

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SIMULTANEOUS DIFFUSION OF CALCIUM AND STRONTIUM IN NaCl SINGLE CRYSTALS C7-389

Values of Ag and Ds used to generate dzfusion of Ca2+ and Sr2+ in NaCl

( I ) S is surface layer diffusion. V is vapor phase diffusion.

rimental data. In all cases the calculated profiles fit the experimental data extremely well as shown in the two examples shown here. The saturation diffusion coefficient for Ca2+ in NaCl is given by

Ds(Ca) = 1.14 x loe3 exp(- 0.851 eV/kT) cm2/s and that of Sr2+ in NaCl is given by

Ds(Sr) = 2.30 x exp(- 0.925 eV/kT) cm2/s

.

These two expressions and the experimental values of Ds for both Ca2+ and Sr2+ are shown in figure 3.

FIG. 3. - Log Ds vs. 1/T from diffusion in NaCl. X : Ca2+.

: Srz+. Solid lines with Do(Ca) = 1.14 x 10-3 crnzls,

U(Ca) = 0.851 eV ; Do(Sr) = 2.30 x 10-3 cm2/s, U(Sr) = 0.925 eV. Expt. (I)

-

S S

v

Figure 4 shows the Gibbs free energies of association of the two complexes,

-

Ag(Ca) and

-

Ag(Sr), as a function of temperature. The straight lines are

FIG. 4. - Gibbs free energies of association in NaCI. X : Ca2+.

: Sr2+. Solid lines with

Aha(&) =

-

0.572 eV, Asa(Ca) = - 2.69 x 10-4 eV/K ;

Aha(Sr) = - 0.671 eV, Asa(Sr) = - 3.35 x 10-4 eV/K.

the least squares fit to the experimental data and are given by

Ag(Ca) =

-

0.572 eV

+

(2.69 x eV/K)T and

Ag(Sr) =

-

0.671 eV

+

(3.35 x loe4 eV/K)T, expressed in the form Ag = Ah,

-

As, T. Here

-

0.572 eV and - 0.671 eV are the values of the enthalpy of complex formation, Ah,, for Ca and Sr, respectively, and

-

2.69 x eV/K and

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(3-390 H. MACHIDA AND W. J. FREDERICKS

Discussion.

-

Many values for parameters similar

_Do

₌_1.22_x _{exp(0.325 J%),} to those obtained in these experiments have been reoort-

ed. Some care must be used in making direct compa- rison of the quantities listed. Slifkin and Brebec 1131 as well as Allnatt and Pantelis [14] report the sum of migration, association and half the Schottky enthalpy. Taking B6niCre1s 1151 value for the Schottky enthalpy (2.5 eV), and Slifkin's and Brebec's association enthalpies, the migration enthalpy lies in the range 0.82 to 0.87 eV. Using the association enthalpy from the present work and 2.5 eV for Ah,, Allnatt's and Pan- telis' results give 0.78 eV for the migration enthalpy of SrZ + in NaCl.

The enthalpy of association, Ah, = 0.57 eV, for the Ca2' -cation vacancy complex obtained in these diffusion experiments agrees exactly with the value reported by BCniCre et al. [16] by conductivity measu- rements. Such exact agreement between two indepen- dent laboratories using different experimental methods is rare in alkali halide transport studies and very satisfying. Both laboratories agree with the high value reported by Slifkin and Brebec [13]. There is no comparable agreement in the reported values of Ah, for the Sr2+-cation vacancy complex. The value 0.67 eV found in these experiments lies with the range reported by Allnatt et al. [4].

The plots of Aga us. T for both Ca2+ and Sr2* (Fig. 4) are linear, indicating that the impurity system used to describe the simultaneous diffusion was not perturbed by unsuspected impurity interactions.

Conclusions.

-

The work reported here is a portion of

a

rather tedious effort to measure diffusion and association parameters for spherical divalent ions in pure alkali halides. Drawing general conclusions about the behavior of divalent cations must await its completion, but some interesting observations can be made for the four ions that have been studied in NaCl [I].

An examination of the values of

D o

shows no simple functional relationship with r n - ' I 2 which holds for isotope effects [17]. This is, of course, unsurprising because unlike isotopes of the same element the size and chemical binding of the various impurities differ. However a purely empirical relationship between Do and mass does exist and is

where M is the mass number of the isotope. This

empirical relation has a correlation factor of 0.998 8 with the four values of

Do.

We do not wish to imply any physical significance to this expression. In fact several others exist that correlate almost as well with the data.

A portion of the migration energy arises from repul- sive forces and U should show some correlation with an expression of a Born-Meyer type. One approach is to assume a jump path and seek a correlation between the distortion of the lattice and the migration energy. If we assume a fixed lattice, i. e. neglect any lattice

relaxation around the impurity, in which the impurity follows a path with the least displacement of the host lattice ions. The displacement of the lattice when the impurity passes through the trigonal C1- position is given as Ar. For our purpose the actual path is not important ; any expression of the form of a constant minus the impurity radius would suffice. Here Tosi-Fumi [18] radii were used for the alkali and halide ions. The choice of radii for impurity ions is somewhat arbitrary as no convincing experimental evidence favoring a specific set is known to these authors. Both Goldschmidt [19] and Pauling [20] radii were used for the impurities in least squares fitting of the migration energies to an empirical Born-Meyer equation of the form

where A and B are constants. When Goldschmidt radii were used as impurity radii, A = 0.553 4 eV and B = 2.226 A. The correlation, R, was R = 0.966 5. However, Pauling radii gave a correlation of R = 0.994 1 with A = 0.725 9 eV and B = 2.001

A.

The constants A and B do not correspond with b and p usually associated with Born-Meyer equations [21].

The association parameters do not exhibit simple easily discernible features. The less ionic [22] impurities cadmium and lead have more negative association enthalpies and entropies than do the more ionic calcium and strontium.

References

[l] KRAUSE, J. L. and FREDERICKS, W. J., J. Phys. & Chem.

Solids 32 (1971) 2673.

[2J KRAUSE, J. L. and FREDERICKS, W. J., J. Physique Colloq.

34 (1973) C 9.

[3] SCHMIDT, E., Fopple Festshrift (Springer Verlag) 1924. 179.

(cited in : CRANK, J., The Mathematics of Diffusion.

(Oxford University, London) 1970, 187.)

[4] ALNATT, A. R., PANTELIS, P. and S m , S. J., Proc. Phys.

Soc. London (Solid State Phys.) 4 (1971) 1778.

[5] FREDERICKS, W. J., SCHEURMAN, L. W. and LEWIS, L. C.,

Final Reports on Air Force Contracts AF- AFOSR-217-73 and AF-AFOSR-217-66 (1966).

[6] LEDERER, C. M., HOLLANDER, J. M. and PERLMAN, I.,

Table of Isotopes. 6th ed. (Wiley, New York) 1967, 594.

[7] RANE, A. T. and BHATKI, K. S., Anal. Chem. 38 (1966)

1598.

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SIMULTANEOUS DIFFUSION OF CALCIUM AND STRONTIUM IN NaCl SINGLE CRYSTALS C7-391 [9] VORTISCH, E., Neues Jahrb. Mineral Geol. 38 (1914) 220.

(Cited in : LEVIN, E. M., ROBBINS, C. R. and MCMUR- DIE, H. F., Phase Diagrams for Ceramists. (American Ceramic Society, Columbus, Ohio) 1964.)

[lo] SANDONNINI, C., Gaz. Chim. Ztal. 441 (1914) 335.

(Cited in : LEVIN, E. M., ROBBINS, C. R. and McMm-

DIE, H. F., Phase Diagrams for Ceramists. (American

Ceramic Society, Columbus, Ohio) 1964.)

[ l l ] WANG, C. H. and WILLIS, D. L., Radiotracer Methodology

in Biological Science. (Prentice-Hall, Englewood

Cliffs, N. J.) 1965, 382.

[12] SHOEMAKER, D. P. and GARLAND, C. W., Experiments in

Physical Chemistry. (McGraw-Hill, New York) 1967.

1131 SLIFKM, L. and BREBEc, G., Rep. CEA-DM (1968) 1750. [14] ALLNAW, A. R. and PANTELIS, P., Trans. Farada.~ Soc. 64

(1968) 2100.

[15] B~NI~RE, F., Thesis, University of Paris, Orsay (1970). [16] BBNI~RE, F., BENIBRE, M. and CHELMA, M., C. R. Hebd.

Sean. Acad. Sci. 268 (1969) 1461.

1171 FRANKLIN, W. M., J. Chem. Phys. 57 (1972) 2659. [18] TOSI, M. P. and FUMI, F. G., J. Phys. & Chem. Solids 25

(1964) 45.

[19] GOLDSCHMIDT, V. M., Geochem. Vert. Elemente 8 (1926)

69 ; Ber. 60 (1927) 1263.

1201 PAULING, L., The Nature of the Chemical Bond. 3rd ed. (Cornell University, Ithaca) 1960.

[21] BOSWARVA, I. M., PYOC. Phys. Soc. London (Solid State

Phys.) 5 (1972) L5.

[22] PAULING, L., The Nature of the Chemical Bond. 1st Ed. (1939).

DISCUSSION L. SLIFKIN.

-

YOU find that the association ener-

gies of Cd2+ and pb2+ are approximately the same, and are larger than that for the alkaline earths. In AgCl and AgBr, we also find that all alkaline earths have about the same binding energy, while the d-shell ions have a different value - but in the silver halides, the binding energies for the alkaline earths is larger than that by the d-shell ions. Thus, in both systems our results agree in showing an effect of d-shells, but the silver halides and alkali halides apparently differ in which type of ion has the largest association energy.