Pressure shifts of F-center hyperfine interaction parameters in alkali chlorides

I. &I

Chcm. Solids. 1978.Vol. 39. pp. 445448

Pcqmon Press. Printed in Gruc lhimin.

PRESSURE SHIFTS OF F-CENTER HYPERFINE INTERACTION PARAMETERS IN ALKALI CHLORIDES R. D. ZWICKER Department

of Physics,

(Received

University

29 March

of the Witwatersrand,

1971; accepted

Johannesburg,

in revised form

26 August

South

Africa

1977)

Abstract-We report here calculated values of the pressure shift in the shell I hyperfine interaction constants for F-centers in KCI, NaCl and LiCI. The calculation is based on a modified version of the pseudopotential approach of Bartram et al. with electronic polarization included by means of an r-dependent polarization potential. The results agree reasonably well with the available experimental data, and the strong dependence of the pressure shift on ion size ratio is shown to be mainly due to changes in the indirect-overlap contributions to the spin density. These terms result from the requirement of mutual orthogonality of core states on neighbouring ions, and their dependence on the lattice spacing is stronger than that of other contributions to the spin density. Our calculations further indicate that although the predicted value of the pressure shift is sensitive to errors in the pseudopotential and polarization parameters, the relative importance of the indirect overlap contributions is evident in spite of such

errors. I. lNTRORUCl’tON High-pressure electron-nuclear double resonance (ENDOR) experiments have provided a considerable amount of information about the F-center ground state. In particular, compression-induced shifts in the F-center hypeAte interaction (hfi) parameters can be related directly to the corresponding changes in the P-electron spin density with varying interionic distance. In a recent study [ I] on KC1 an attempt was made to determine the main contributing factors governing these pressure shifts. It was shown that the indirect overlap (i-o) contributions[2] to the F-electron wave function play an important role in the interpretation of the pressure shift data, and in fact changes in the i-o terms can dominate the pressure shift even when the i-o contribution to the spin density is itself relatively small. These terms originate in the requirement of orthogonality of ion core functions at a given site to those on neighbouring sites, and at shells I and II the i-o contributions act in such a way as to decrease the spin density. In Ref. [I] it was suggested that the relative importance of the i-o terms could explain the experimentally observed[3,4] dependence of the shell I pressure shift on the ion size ratio R-/R+. In the present work we verify this suggestion with calculations on KCI, NaCl and LiCI. The calculations are based on a modified and expanded version of the pseudopotential model of Bartram er al. [S], and the results are expressed in terms of the dimensionless quantity e=

a dpr=Bdor 36 da

a,

dP

where D is the interionic spacing, p, is the shell I spin density, P is pressure, er is the contact hti parameter, and the effective local modulus I? is estimated as half the bulk modulus for these materials[6]. We obtain reasonable agreement with the available experimental data, and the decrease in 0 with increasing R-/R+ is JF’CS Vd. 39. No. 5-A

44s

interpreted in the light of the increasing importance of the indirect overlap contributions via shell II. Although the values of 0 are rather dependent on the manner in which the pseudopotential and polarization parameters are varied, the change in B with ion-size ratio is relatively insensitive to this.

The pseudopotential model used in the calculation of the pressure shifts has been described in detail elsewhere]71 and will only be outlined here. It is similar to the model of Bartram et ai.[S] (BSG) but is derived directly from the Phillips-Kleinman[8] pseudopotential as this approach has been shown to be compatable with a variational solution.[9] Also, the expansion of the Fcenter pseudowavefunction in ion core regions is carried to first order so as to include contributions from ionic p-states. This leads to an energy functional of the form

to be minimized with respect to the trial function 4. Here V,, is the point-ion potential, W, is the Madelung potential energy of an electron at ion site y due to all the other ions (in a perfect crystal), r, is the radial distance from the F-center to the yth site, and the sums are over all the ions in the crystal. The parameters A, and B, are effectively the same as the corresponding BSG parameters and are characteristic of the ion type. The J, and K, result from the inclusion of the first order term in the expansion of I$, and by symmetry these incorporate the ionic p-state contributions to the pseudopotential.

R. D. ZWXKER

446

Table t. Dielectric constant Y. anion polarization polarizability a_. and parameters p* appropriate to interionic spacing a (in a.u.)

Electronic polarization effects were taken into account by means of an t-dependent polarization potential suggested by Fowler[lO] and used with modfications by bpik and WoodIll] (8W). This has the form

K

a

U=

(

i-i

>

{-S(p.+p,)+f[l-;(e-p~~+e-ph’)]}

KC1

where K is the high-frequency dielectric constant and p,, is determined from the Mott-LittletonI 121 polarization energy U,,_ associated with the removal of a negative ion:

a.

P*

5.93 5.87 NaCl 5.31 5.25

2.130 2.150 2.250 2.274

19.405 18.924 19.365 18.934

0.2109 0.2139 0.2032 0.2059

LiCl

2.750 2.782

19.629 19.119

0.2107 0.2134

4.83 4.77

From these the true wavefunctions JI were calculated at lattice points r, from the formula Following dW, we have put p. = p,, in our calculations. Lattice distortion has been neglected throughout. The trial function # was taken as

where N is a normalization constant and the qJA are Slater orbital! q5,A= [(2Afa)2’+‘/4n(21f!]“2r~-‘e-*““.

The parameters 6, rl and C were varied independently to minimize the energy. We chose m = I and allowed n to take.on integer values from 2 to 6. 3. CALCULAIIONS

The pseudopotenti~ parameters used in these calculations were obtained from the free cation and crystal anion functions of Paschalis and Weiss(l3] The values for KCI and NaCl are given in Ref. [7], while for Li” we have A =45.11, B = 12.16, J = - 1.83 and K = 0. No Clcore functions appropriate to LiCl were listed by PW, so we have used in this case the values of A,B, J and K appropriate to NaCI. The error introduced in the pseudopotential by this is expected to be small compared to those which result from other approximations made, such as the neglect of indirect overlap terms. The variation in the anion pseudopotential parameters with lattice spacing during compression was estimated in all cases by linear extrapolation using the values appropriate to KC1 and NaCI. The polarization parameters ph were obtained from a first-order Mott-Littleton calculation for each value of the interionic spacing a, using the ionic polarizabilities of Tessman et aI.[l4]. The variation of the dielectric constant I( with lattice spacing was deduced for KC1 and NaCI from the results of Lowndes et ai. [IS]. For LiCl we have assumed that the value of - (V/c) (&/aV), is the same as that for reported in Ref. (151for KC1 and NaCl (0.31). The variation in anion polarizability was calculated from the Classius-Mossotti relation, and the resulting values used to calculate p,, for different u. The values of the polarization parameters are given in Table 1 for the lattice spacings chosen for these calculations. Minimization of the energy functional yielded the energies and pseudowavefunctions Q, listed in Table 2.

Here t&Tis the ith core state at lattice site y, &,(O) is evaluated at the symmetry center of the core function, and the S,, are overlap integrals between neighbouring ion core states. The second and third terms on the right of eqn (3) are called the direct and indirect overlap terms, respectively[2]. We have calculated the wave function in the zeroeth-order approximation, with all S,., = 0, and in the first-order in which the nearest neighbour (Inn) cation-anion and 2nn anion-anion overlaps SirVBwere included. In the j-sum of eqn (3) only the highest-energy core states of each symmetry type were taken into account. The results of these calculations are displayed in Table 3 along with the corresponding experimental results[I6-181. In the evaluation of the normalization constant N only the first two shells surTable 2. Pseudowavefunction

KC1 NaCl LiCl

parameters Ryd)

and energies (in

a

n

I

I)

C

E

5.93 5.87 5.31 5.25 4.83 4.77

6 6 6 6 6 6

2.037 2.027 1.869 1863 1.946 1.957

II.705 11.746 10.537 IO.571 9.866 9.873

0.39520 0.39124 0.35009 0.34575 0.46153 0.46679

-0.3959 -0.3975 - 0.4535 - 0.4558 - 0.4957 -0.4976

Table 3. Values of and first-order wave experimental values others

KCI NaCl LiCl

5.93 5.87 5.31 S.25 4.83 4.77

pseudowavefunction (b, zerofunction # and corresponding for shell 1. (4 in IO-* a.,.-In, in 10-l a.u.?‘)

1.2703 1.2920 1.6894 1.7181 1.9910 2.0225

3.4933 3.5752 2.63% 2.7052 1.9148 1.9584

3.0047 3.0367 2.1880 2.2056 1.3892 1.3849

3.M 2.30$ 1.05%

tKerstco R., Phys.StatusSo&di29.575(1968). SSeidclH., Z. Phys.165,218(l%l).

OWolbarst A. B., Phys. Status Midi (19721. , ,

(b) 49, 511

Pressure shifts in alkali

rounding the F-center were included, and ion-ion overlap was included in the first-order calculation. Although the use of NaCl anion core functions is not expected to introduce a large error in the LiCl pseudopotential, their use in the evaluation of the hfi parameters (particularly the pressure shifts) must be considered more carefully. Specifically, the NaCI anion functions tend to overestimate the i-o contributions to the spin density: We have therefore approximated the effects of the crystal environment in LiCl by reducing all 3p overlap integrals in the evaluation of # by 7% (based on linear extrapolation of the parameter K). Integrals occuringin the normalization factor N which involved two 2nn 3p functions were reduced by 15%. The pressure shift parameters 8 evaluated from the wavefunctions of Table 3 are listed in Table 4 along with the corresponding experimental parameters131 for KC1 and LiCI. To ascertain the relative importance of the various contributions to 6 we write the first-order wave function 4 in the form

where fd and fi are the direct and indirect overlap contributions from eqn (3). Then the parameter 8 can be written

where a = - 2a/3Aa, and f. and N, are average values. From Table 4 it is seen that the i-o contribution 8, is of the same order as 0, even when the i-o contribution to the wave function is relatively small (- IS% for KCl). Further, we note that the increasing magnitude of the negative t&accounts for the decrease in 8 with increasing ion size ratio, and the negative pressure shift in LiCl results from the dominance of ei over 8, for this crystal. This disproportionate i-o contribution to the pressure shift results from the strong dependence of the .&,, on lattice spacing. We note that the calculated value of the shell I pressure shift parameter 0 for KC1 is somewhat larger than the corresponding experimental value. This model further predictsa value of 0 for shell II for KCI which is too large, and is in fact positive in disagreement with the negative experimental value correctly predicted by the calculations of Ref. [ 11. The KC1 shell III results are also made worse in the present treatment. As the major difference in the calculations reported here and those of

441

chlorides

Ref. (I] is the use of the r-dependent polarization potential (which yields a somewhat more diffuse wave function[7] than that reported in Ref. [l]), we feel that the main source of error in our present results lies in the variation of this potential with the lattice spacing. Firstly, as mentioned by Wood and 6pik, the polarization potential is not known to have the correct rdependence at small r. Added to this is the fact that the variations of the dielectric constant K is known only approximately[lS] for Kcl and NaCI, and had to be assumed for LiCI. Further errors in these calculations stem from the method of variation of the pseudopotential parameters. Although the error introduced in the pseudopotential by neglecting indirect overlap terms is not likely to be large, the correspondiig error in the variation of the pseudopotential with interionic spacing is expected to be somewhat more significant. Our first-order calculations of the normalization constants N lead us to believe that changes in the i-o contributions would probably not dominate the pseudopotential variation (even for LiCI), but that inclusion of the i-o terms would nevertheless signiticantly reduce the effective change in the anion parameters A, B, I and K which we obtained by linear extrapolation. To test the sensitivity of our results to the method of variation of parameters we have recalculated the pressure shift parameters according to two further prescrip tions. In the tirst of these the polarization parameters were allowed to vary with lattice spacing while the pseudopotential parameters were held constant, and in the second both the pseudopotential and polarization parameters were held cqnstant while u was varied. The results of these and theprevious calculations are displayed in Fig. 1, which shows the values obtained for the shell I pressure shit parameter 8 by the different methods of parameter variation. These I.0

1

-ITS-

A A

KC1 L

-‘.O

NaCI

1

I.0

I

3.0

KC1

NaCl LiCl

I.36 :::

P-/R,

Fig. 1. Shell I pressure shift parameters t?J&iidined by d&rent methods of variation potential parameters with lattice spacing: 0 all parameters varied; A, pseudopotential parameters held constant; A, pseudopotential and polarization parameters held constant; 0, experimental values131 (effective local modulus B for LiCl taken as half the bulk modulus, as for KCl[6]).

Table 4. Contributions to shell I pressure shift parameter B. Experimental values from Wolbarst A. B., 1. Phys. Chm. SoUdsX3.2013(1972)

RJR,

LiCl

1

2.0

6,

4

4

-0.052 0.086 -0.137

1.326 1.122 Ml

-0.886 -1.042 -1.sOO

@N 0.307 0.308 0.160

8 0.69 0.47 -0.17

L 0.54 -0.06

448 are

from

R. D. ZWICKER

plotted against the appropriate ion-size ratio R-/R+ the ionic radii of Pauling. We note from Fig. 1 that

the calculated value of the pressure shift is rather sensitive to the manner in which the potential parameters

are varied. We further note however that the strong decrease in 8 with increasing R-/R+ is evident in ai cases. Our calculations thus indicate that the relative importance of the indirect overlap contributions 0, of Table 4 is not a model-dependent feature and is apparently insensitive to errors in the potential. If we regard our values of 0 for NaCl as upper and lower limits then our calculations predict an average value of 8 -0.3, or daJdP - 160kHz/kbar, assuming the effective local modulus B to be half the bulk modulus. [6] Linear interpolation of the experimental results of KC1 and LiCl gives about the same result.

3. Wolbarst A. B., 1. Phys. Chem. Solids 33, 2013 (1972). A smaller experimental value of 0 = 0.48 was reported for KC1 in Ref. [l]. 4. Mamola K. and Wu R., .I. Phys. Chem. Solids 36, 1323(1975). 5. Bartram R. H., Stoneham A. M. and Gash P., Phys. Rev. 176, 1014(1968). 6. Jacobs I. s., Phys. Rev. 93,993 (1954). 7. Zwicker R. D., Phys. Reu. to be published. 8. Phillips J. C. and Kleinman L., Phys. Rev. 116,287 (1959). 9. Zwicker R. D., Phys. Reu. B12.5502(1975). IO. Fowler W. B.. Phvs. Rev. 151,657(1966). 11. bpik U. and wodd R. F., Phys. Rev. 179, 772 (1%9); Wood R. F. and opik U., Phys. Rev.179,783(1%9). 12. Mott N. F. and Littleton M. J., Trans. Faraday Sot. 34, 485 (1938). 13. Paschalis E. and Weiss A., Theoref. Chim. Acro (Berl.) 13,

381(1%9). 14. Tessman J. R., Kahn A. H. and Shockley W., Phys. Rev. 92, 890 (1953). 15. Lowndes R. P. and Martin D. H., Proc. R. Sot. London A

316,351(1970). -Cm 1. Wolbarst A. B. and Zwicker R. D., Phys. Reo. Letters 37, 1487(1976). 2. Wood R. F., Phys. Status Solidi, 42,849 (1970).

16. Kersten R., Phys. Status Solidi 29, 575 (1%8). 17. Seidel H., Z. Phys. 165,218(l%l). 18. Wolbarst A. B., Phys. Status Soiidi (b) 49, 511 (1972).