SOME IMPLICATIONS OF BLOCH ELECTRONS IN THE NUCLEAR QUADRUPOLE INTERACTION IN METALS

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SOME IMPLICATIONS OF BLOCH ELECTRONS IN THE NUCLEAR QUADRUPOLE INTERACTION IN

METALS

Y. Fukai

To cite this version:

Y. Fukai. SOME IMPLICATIONS OF BLOCH ELECTRONS IN THE NUCLEAR QUADRUPOLE INTERACTION IN METALS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-235-C3-238.

�10.1051/jphyscol:1972334�. �jpa-00215068�

JOURNAL DE PHYSIQUE

Collogue C3, supplement au n° 5-6, Tome 33, Mai-Juin 1972, page C3-235

SOME IMPLICATIONS OF BLOCH ELECTRONS

IN THE NUCLEAR QUADRUPOLE INTERACTION IN METALS

Y. FUKAI

Department of Physics, Chuo University, Kasuga, Bunkyo-ku, Tokyo, Japan

Resume. — Le calcul du facteur de renforcement et des gradients de champ electrique dans deux exemples particuliers : In dans In et In dans Pb, montre clairement Pimportance du role des electrons de Bloch dans l'interaction quadrupolaire dans les metaux et alliages.

Abstract. — Calculations of core-enhancement effects and electric field gradients are performed fort two specific examples : In in In and In in Pb. Results clearly demonstrate the importance of Bloch character of conduction electrons in the nuclear quadrupole interaction in metals and alloys.

I. Introduction. •— The nuclear quadrupole inter- action in metals and alloys is one of those properties of which the physics is well established [1], [2], [3], but quantitative calculations have never been suc- cessful. The purpose of this paper is to explore one aspect of the calculation, the anisotropy of so-called core-enhancement effect, and illustrate its signifi- cance in two specific cases.

All that is needed here is the electric field gradient (EFG) acting on a nucleus:

(1) and its calculation is straightforward once the charge distribution p(R) is known. What is of interest is that, contrary to free-electron expectations, conduc- tion electrons are known to be a major contributor to EFG both in pure metals and in alloys. Bare ion potentials do not contribute in any case because they are substantially screened out by conduction elec- trons. In pure metals, it is obvious that only deviations from free-electron character, i. e., Bloch character, can give rise to EFG in non-cubic lattice. In the case of alloys, recent site-by-site measurements of EFG in the vicinity of impurity atoms [4]-[9] provide a rare possibility and necessity to investigate the screen- ing charge distribution in detail. For example, it is now established that, in many Al and Cu alloys, EFG's on the nearest-neighbor of impurity atoms show a large deviation from axial symmetry, and in some cases, the principal axis is even directed perpen- dicular to the line joining the nearest-neighbor site to the impurity atom [9]. These results strongly indi- cate the large anisotropy of screening charge distribu- tion, which should be a manifestation of Bloch charac- ter of screening electrons. Thus, we have to face the problem of screening by Bloch electrons, and thereby constructing self-consistent potentials. These problems

have not been studied adequately to be applied to real systems.

In the present paper, we focus our attention on another aspect of EFG, the core-enhancement effect [2], [3]. Because of the factor 1/R

in eq. (1), EFG is extremely sensitive to details of wave functions close to the nucleus. The core-enhancement effect, which arise from the orthogonality requirement between conduction and core electrons, is exactly of this nature, and requires careful investigation. Calculation of this effect, with emphasis on its anisotropy, is descri- bed in the next section for two specific examples ; In in In, and In in Pb. Some implications of results are discussed in the last section.

II. Core-enhancement factor. — The EFG produced by conduction electrons at distance r from a scattering potential is approximately given by

eq(r) = (8 ne/3) adn(r) , (2) where 5n(r) is a smooth part of screening charge

density without taking account of a rapidly varying part arising from interactions with core electrons.

The effect of the latter is approximately factored out as a, the cOre-enhancement factor, defined by,

(3)

i/f

is a wave function of a conduction electron on a Fermi surface, and ty\ is the one approximated by a plane wave. y(R) is an anti-shielding function [1]

at distance R from the nucleus in question. Since y(R) is nearly zero inside the ion core, a is essentially determined by \]/

close to the nucleus. It should be noted that, owing to the Bloch character of i^

, a is a function of the direction of k.

16 Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972334

C3-236 Y. FUKAI In the present calculation [IO], we construct wave functions from the pseudopotential theory :

where w,(q) is a pseudopotential on the I-th lattice site, and the sum over q excludes q

0. In the case of pure metals, or in the evaluation of a for host atoms in dilute alloys, the second term may be written as

Cf ^w(G)/(Ek - OP Wk+c, in which G

0 is c

omitted from the sum over reciprocal lattice vectors.

For the calculation of a for impurity atoms, on the other hand, it is convenient to write eq. (4) as

in which wi(q) and w,(q) are pseudopotentials of the impurity and the host atom, respectively, and OPW's should be orthogonalized to core wave func- tions of the impurity atom.

Pseudopotentials are here approximated by Ash- croft's form [ll], using an RPA dielectric function without exchange correction, and core radii R, deter- mined from Fermi surface data. Values of R, adopted are : 0.586 A for In [12], and 0.538 A [13] and 0.57 A [6]

for Pb. The function y(R) is approximated by y,(l - exp - (RJp)'), where

is the antishielding factor for free ions, and p is taken to be Pauling's ionic radius. A value of

for In is 15.3 [14]. Core wave functions calculated by Herman and Skillman [I 51 are used in constructing OPW's.

A) In

In

- Since indium has a face- centered-tetragonal structure, the core-enhancement factor is calculated for k along five principal directions :

< l o o > , <OOp>, < 1 1 0 > , < l o p > , and

< ^{l l p} >. Results are fitted to an analytic form, 4 8 , @)

a, + a, cos 2 8 + a, cos 4 8

+ a, (1 - cos 2 8) cos 4

+

+ ^a,(l

cos 4 0) cos 4 8 , where angles refer to crystal axes, with c-axis as a polar axis. The result, shown in figure 1, differs stri- kingly from single-OPW calculations, which yield a - 58 irrespective of directions of k.

B) In

Pb

- We describe next calculations of a for an In atom embedded as a probe in Pb matrix. Wave functions are approximated by eq. ( 5 ) , in which the impurity potential wi(q) is constructed by assuming the same R, as determined in pure In. Values of a calculated for three directions of k, < ¹⁰⁰ >, < ¹¹⁰ > ^and < ¹¹¹ >, are fitted to lowest three cubic harmonics. Results obtained for two different values of R, for Pb are shown in figure 2.

The anisotropy of a is seen to be really pronounced.

FIG. 1.

Variation of core-enhancement factor

with direction of k in In.

denotes c-axis.

FIG.

-Variation of core-enhancement factor

with direc- ion of k for In in Pb. Two slightly different pseudopotentials are used for Pb, i. e. Ashcroft's potentials with core radii

0.538 A (solid line) and 0.57 A (dashed line).

It is also instructive to look into the breakdown of a.

Let the three terms in eq. (5) be labelled I, I1 and 111.

Neglecting terms of second order with respect to pseudopotentials, a is made up of three terms arising from I x I, I x I1 and I x 111. The first one gives a single-OPW contribution, the second is characterized by impurity species, the third gives rise to anisotropy.

While the first one is always positive, the second is positive or negative according as the valence of the impurity is higher or lower than that of the host atom.

111. Discussion. - Some implication of the aniso- tropic core-enhancement factor a found in the prece- ding section will be discussed here in connection to experiments. Since a is not a measurable quantity, comparison with experiments can only be made in terms of EFG.

Very approximate calculations of EFG are made for

In metal based on a pseudoatom concept. EFG is

SOME IMPLICATIONS

BLOCH ELECTRONS

then written as a sum over constituent atoms : eq

C eqi P, (cos ei), where

i

eqi

(8 ne/3) a ( % , @J 6n(ri) .

Implicit in this is an approximation of nearly spherical Fermi surface, which allows to identify the direction of k with the direction of r, a vector connecting a scattering center to the probe nucleus. The screening charge density is approximated by

6n(r)

A cos (2 kF r + 4)/r3 ,

with A and 4 treated as parameters. Results of calcu- lations of eq as a function of temperature are shown in figure 3, with experimental results. Indium is in

t

^I

Calc. B I

Calc 92

I

0 100 2 0 0 300 400

Temperature I

FIG. 3. - Electric field gradient

in In. Curves A 1 and A 2 are obtained without and with including thermaI averaging of lattice potentials, assuming for the screening charge distribution,

0.02 and 4

1.3. Curves B

and B

are corresponding results for

0.19 and 4

0.99. Curve C is obtained from an array

point charges of

Experimental results are

denoted by Exp.

Colc. C -

fact a peculiar metal in which the temperature depen- dence of EFG is very large [16]. Curves labelled A 1 and A 2 correspond to A

0.02 and 4

1.3, while those marked B 1 and B 2 to A

0.19 and 4

0.99.

Differences of curves A 1 and A 2, or B 1 and B 2, are caused by the effect of thermal averaging of w(G)'s on the factor a. The thermal average is cal- culated by Kasowski's method [17], and the phonon distribution function required is approximated by a2 f derived from tunneling experiments [18]. The effect proved to be small

a changes by a few percent.

The sensitivity of EFG to this small change in

is, however, worthy of notice. Although a good agree- ment of the curve B 2 with observation could hardly be meaningful in view of crudeness of the model, it does suggest that the large temperature dependence could be explained in terms of redistribution of conduc- tion electrons only without calling for any novel mechanisms.

The second example, In atom embedded in Pb, will now be considered [lo]. Dilute alloys of In in Pb is also a peculiar system, in which In nuclei are subject to anomalously weak EFG [19]. Judging from the value of residual resistivity, 1.0 pa. cm/at. % In, it is very unlikely that a scattering potential is weak. The fact that a is nearly zero for k along < ¹⁰⁰ > ^{is of}

particular interest in this regard : Scattered waves from an In atom will produce only a very small EFG on another In nucleus in the second nearest-neighbor position, which is in the < 100 > direction. It was shown in our previous paper [lo] that the first nearest- neighbor comes close to one of the nodes of screening charge distribution, and therefore feels very small EFG. Although the discussion here is rather quali- tative in nature, it demonstrates unambiguously the importance of the anisotropy of a in this case. It should be added in amplifying the statement above that analysis of quadrupole interactions studied by means of probe nuclei requires careful investigation of their local environment which is no longer the same as in the absence of probe nuclei.

Present calculations are all rather crude. For more realistic calculations of EFG, we have to base our scheme on full-scale band calculations on pure metals, and treat the screening by Bloch electrons in alloys.

But, whatever formalism we may use, there is one rule to be observed : A requirement of orthogonality between conduction and core electrons should be well satisfied because it is exactly the origin of core- enhancement effects.

Numerical calculations have been performed on HITAC 5020 E at the Computer Center of the Uni- versity of Tokyo.

Acknowledgment. - The author expresses his gratitude to Dr P. Averbuch, of Laboratoire de Spec- tromktrie Physique, Grenoble, for inviting him for the summer of 1971, which made his attendance to this conference possible. Hospitality of other members of the laboratory is also gratefully acknowledged.

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