CHARACTER OF THE CHARGE DENSITY DISTRIBUTION SURROUNDING IMPURITIES IN GOLD METAL

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CHARACTER OF THE CHARGE DENSITY

DISTRIBUTION SURROUNDING IMPURITIES IN

GOLD METAL

P. Huray, C. Tung, F. Obenshain, J. Thomson

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 12, Tome 37, Ddcembre 1976, page C6-371

CHARACTER OF THE CHARGE DENSITY DISTRIBUTION SURROUNDING

IMPURITIES IN GOLD METAL

P. G. HURAY (*), C. M. T U N G (**), F. E. OBENSHAIN, and J. 0. THOMSON University of Tennesse, Knoxville, Tennesse 37916, U. S. A.

and

Oak Ridge National Laboratory, Oak Ridge, Tennessee 38830 (***), U. S. A.

Rksum6. - Les mesures Mijssbauer du deplacement isomkrique de 197Au dans des solutions solides a 1 , 2, 3, 4 pour cent atomique de Ca, Sc, Ti, V, Cr, Mn, Fe, CO, Ni, Cu, Zn, Ga, Ge, Ag, Cd, In, Sn et Sb dans de l'or de grande purete ont et6 extrapolkes B 0 pour cent atomique pour deter- miner {'influence des impuretes isolkes sur la bande electronique de conduction de l'or. L'ecran forme par le nuage electronique present autour des impuretes modifie en valeur relative la densit 5 de charge Bectronique au noyau d'or voisin (distant de rj), de la quantite Apse(rj)/p~u(O). NOUS avons mesure cet effet grilce a la difference de deplacement isomerique dans les alliages cites, et donc

m

obtenu Ci A p ~ ~ ( r j ) / p ~ ~ ( O ) . Nous avons analyse le caractkre de cette distribution de charge a j = 1

I'aide du traitement de perturbations de Friedel, les paramMres etant des puits de potentiel carrks. De plus, nous avons mesure la rksistivitk Blectrique residuelle, la variation du parametre de reseau moyen, et avons pris les spins atomiques des impuretks Bgaux B (a) S = 0, (b) S = S 4 , 2 K, pour

obtenir un ensemble de quatre expressions simultanees faisant intervenir les dephasages. En resol- vant ces equations, nous avons obtenu un ensemble cohkrent de dkphasages en perturbation 61,,(KF) de profondeurs de puits, et les charges d'kcran associkes dans I'hypothkse ou 60, 61, SZf et

sont les termes prepond6rants.

Abstract. - Mossbauer isomer shift measurements of 1 9 7Au for nominally, 1 , 2 , 3 and 4 atomic

percent solid solutions of Ca, Sc, Ti, V, Cr, Mn, Fe, CO, Ni, Cu, Zn, Ga, Ge, Ag, Cd,In, Sn,'and Sb with high purity gold have been extrapolated to 0 atomlc percent to determine the Influence of single impurities upon the electron conduction band of gold. The screening cloud of electrons distri- buted about the impurities fractionally change the electron charge density at neighboring gold nuclei (rj away) by an amount A p ~ ~ ( r ~ ) / p ~ ~ ( o ) . We have measured this effect through the differential

m

isomer shift for the alloys listed and have thereby determined Ci Apsc(ri)/pi\u(O). The character

j = 1

of this charge distribution has been analyzed through the Friedel perturbation treatment and parameterized by square wcll potentials. We have additionally measured the residual electrical resistivity, the average lattice parameter change, and have taken the impurity atomic spins (a) S = 0 and (b) S = 5'4.2 K, to deduce a set of four simultaneous expressions in terms of the phase shifts. The

equations have been solved to obtain a consistent set of perturbation phase shifts 61,,,(kF), well depths, and their associated screening charge with the assumptions that 60, al, SZt, and 67,, are the dominent terms.

1. Introduction. - When a n impurity atom is dis- solved in a host metal, it can be viewed as presenting a perturbing potential t o an otherwise perfectly periodic potential of the host lattice. The influence of the perturbing potential upon the host conduction band may result in a redistribution of electron charge densities in the metal. The perturbing potential will attract o r repel electrons so as t o provide screening in

(*) Support partially with funds provided by a Research Corporation Grant.

(**) Work performed in partial fulfillment of the Ph. D. degree in Physics at Louisiana State University.

(***) Work supported by Union Carbide Corp. under contract with U. S. E. R. D. A.

the region of the impurity. For sufficiently attractive potentials, discrete localized bound states occur. For weaker perturbations the states are virtually bound with energies lying the conduction band. I n this case Friedel charge oscillations surround the impurity and provide the screening charge.

We have in this study attempted to construct a set of gold solid solution alloys which were dilute enough that the dominant effects on the electronic structure arise from single impurities. By extrapolation t o zero solute concentration we hope t o reveal those effects in the electron charge density distribution which would be produced by a single impurity (and its associated local distortion) in a one t o one exchange with a gold atom

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C6-372 P. G . HURAY, C. M. TUNG, F. E. OBENSHAIN AND J. 0. THOMSON

in pure gold. Account must also be taken of the change in average charge density in the alloy arising from volume effects [l] since a lattice expansion will in general decrease the amplitude of the electronic wave- functions over the whole alloy.

The lg7Au isomer shift depends on the charge density at the lg7Au nucleus. Thus, we picture the single impurity as producing a scattering center for the Au host conduction electrons which distribute themselves in a screening cloud about the scattering center and perhaps extend for many lattice spacings outward. This cloud of electrons is then an additional charge density perturbation to that of pure gold metal arising from a perturbation potential AV and increases or decreases the average electron charge density at the 12 near neighbouring gold nuclei, at the 6 second near neighbours, etc. The gold near neighbours to such an impurity thus have a different Mossbauer isomer shift than do pure metallic gold nuclei. Since all near neighbours are equally radially displaced from the impurity we assume an equal influence at all twelve but allow the six second near neighbours to have a yet different charge density and so on for the jth nearest neighbour shells of atoms. The measurement of Mossbauer isomer shift for the solid solution alloy is thus different from that of pure gold (after making volume corrections) by the influence of the average electron charge density scattered from the impurity to the various shells of gold neighbouring atoms. Each shell of neighbours will then have its own characteris- tic isomer shift. However, the Mossbauer linewidth for 1 9 7 A ~ is substantially greater than the typical isomer shift of the f t h shell of neighbours and, there-

fore, discrete satellites are not resolvable.

2. Mossbauer isomer-shift and other eIectrica1 measu- rements.

-

In pure gold we have previously found [2] the Mossbauer isomer shift to depend upon the average volume per atom 52 as

where

a,,

is the atomic volume per atom in normal uncompressed gold. The dependence was determined theoretically [2] through a relativistic Wigner-Seitz Hartree Slater-Latter wavefunction code. By comput- ing p(0) for several different atomic volumes near

Q,,,,, y was found to be approximately 0.86. Changes in p(0) due to variations in 52 were found to be predo- minantly due to changes in the 6s wavefunction at the origin. However, the 0.86 power (rather than 1.0) can be ascribed to the screening effects of 5d electrons in the core which also depend upon 52. The dependence (Eq. 1) was also previously [ l ] experimentally verified through Mossbauer isomer shift measurements with pure gold at high pressures (up to 70 kbar) which produced changes in Q up to 0.03

a,.

We assume that by adding to some of the gold

atomic sites in the compressed (or expanded) lattice a perturbation potential AV, we can simulate the dilute alloy of interest. A perturbation calculation [3] shows that the charge density at an average gold nucleus in solid solution alloy may be written as

Here

<

A psc(o)

>

is the average fractional change in PA~(O)

charge density at an alloy gold nucleus located at the origin brought about by scattering from foreign (solute) atoms in the alloy,

<

nj

>

is the average number of foreign atoms in the jth shell or neighbours to the gold atom at r = 0 and A psc(rj)/pAu(0) is the fractional scattered charge density (screening cloud) associated with the perturbation A V located at a point

r j away from r = 0. A choice of the form of A V leads to a theoretical value for this scattered charge density distribution.

For an alloy with no short range order (the random distribution of solute atoms on an f. c. c. lattice),

<

nj

>

= mcj there m is the atomic fraction of solute atoms in the alloy and cj is the total number of atoms in the jth shell of neighbours (c, = 12, c, = 6, ...).

We thus see that the differential Mossbauer isomer shift 6(AE) = k[pa,lOy(O) - pAU(O)]/~Au(O) is given

within the model approximations. We have previously determined the constant k to be

-

8 mm/s. By measur- ing p(AE) vs. m (nominally

+

0.01, 0.02, and 0.04) for the solid solution alloys of gold with Sc, Ti,'V, Cr, Mn, Fe, CO, Ni, Cu, Zn, Ga, Ge, Ag, Cd, In, Sn, and Sb, and correcting for volume (as in Eq. (4)) we may determine

(the total scattered charge density change brought about at gold nuclei due to surrounding impurities) for each of these foreign atoms in gold. Our sensitivity gives this value to better than 111 000 of the charge density of a 6s electron at pure gold nuclei, p,,(O).

We have seen that the differential isomer shift 6(AE)

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CHARGE DENSITY DISTRIBUTION SURROUNDING IMPURITIES IN GOLD METAL C6-373

upon the type of impurity introduced and we display this variation in figure 1. The error is given approximately by the size of the data point for all cases except Ti, Mn, and CO where it is approximately twice the size of the point shown.

0 4 2 3 4 5 6 7 6 9 NOMINAL A 2

FIG. 1 . ~ - Average scattered charge density change brought about at gold nuclei due to surrounding impurities in the gold

lattice.

3. Theory of charge density distribotions.

-

Our model for the screening cloud of electrons is based on the use of an impurity potential similar to those introduced by Mott [4] and by Friedel [5]. In the absence of detailed information about the perturbing potential for each impurity the results have been parameterized in terms of square well potentials. Using a partial wave analysis, we have calculated the average S, p and d charge density ch.anges at neighbouring gold nuclei in terms of phase shifts of incident gold wave functions of either spin due to various depths of potential wells.

From the point of view of scattering theory, where an impurity atom is substituted for an atom of host metal, a number of impurity electrons (corresponding to those shells whose atomic energies lie within the host

metal conduction band) will be delocalized and go into the host conduction band. Consequently, these will result an excess positive charge at the impurity site. Friedel [5] has pointed out, however, that as a result of scattering, charge will accumulate in the vicinity of the impurity and locally screen the impurity potential. The excess charge, Ne, must be screened in such a manner that the phase shifts obtained from the screened potential will satisfy the Friedel sum rule.

where the phase shifts 6,,, are to be evaluated at Fermi level k,. We have chosen only first few phase shifts 6,, 6, and 62 to satisfy eq. (5) because we do not expect the orbital quantum number 1 8 3 to significantly contri-

bute on the basis of free atom calculations for the impurities and host metals studied here.

For magnetic alloys, exchange energy will for some impurities favor spin alignment in the l = 2 shell. The impurity will then acquire a moment with 6, ,t (corresponding to spin up scattering) and 6,,1 being different. The spin of the impurity, S, is given in terms of the

phase shifts by

We will ignore the small exchange interactions that may exist among the s or p electrons.

The treatment of the impurity problem from the point of view of scattering theory also suggests that the impurities will affect the conduction electron mean free path and thereby lead to a residual resistivity

where m is the atomic fraction of impurities.

The fractional change in the electron charge density distribution [5] of a free electron gas which surrounds a perturbation ~otential A V will be given by

-

sin 2 G,,,(k) j,(kr) n,(kr)

)

k2 dk

+

<

Apsc(r)

'

bound, I , a] (8) ~ ~ " ( 0 1

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C6-3.74 P. G. HURAY, C. M. TUNG, F. E. OBENSHAIN AND J. 0. THOMSON

This asymptotic form follows from eq. (8) when the wavenumber dependence of the electron impurity scattering can be neglected. Geldart [6] has shown that. for a reasonably small potential eq. (9) may not be correct even in sign for r near the perturbation potential.

4. Application to the impurity screening doud. -

The Friedel sum rule, an expression for electrical residual resistivity and the impurity magnetic moment are given in terms of the phase shifts at the Fermi level for an arbitrary scattering potential, in eq. (5), (7),

and (6) respectively.

The fractional change in the electron charge density distribution is likewise given by eq. (8) and the quantity 4 nr2 ApI(r)/pAu(0) (in [ ] in eq. (8)) for 1 =0,1 and 2 is shown in figure 2. We have chosen a - 1 eV potential

7 l l 1 1 1 l 1 1 1 1 l '

SCATTERED CHARGE DENSITY FROM

A SQUARE WELL POTENTIAL I N Au - SCATTERING RADIUS =1.439 (81 _- WELL HEIGHT=-1.000 eV ( + I F REPULSIVE ) FERMI LEVEL = 5.503 eV - NEIGHBOR DISTANCE = R@ ( N = INTEGER) R= 2.878

(a)

- RADIUS ( 8 )

FIG. 2. - The fractional scattered charge density distribution 4 z r 2 A P ~ ( Y ) / P A ~ ( ~ ) for 1 = 0 , 1, and 2 for a

-

1 eV square well scattering potential of radius 1 439 A in gold as a function of radial distance in A from the scattering center. The vertical lines along the abscissa indicate the radial position of the neighbouring shells of atoms (12 at 2 878 A, 6 a t 4 070 8. etc.).

as an example. The integrated charge density distribution

1:

Ap,(r) 4 nr2 dr

may be used to check the Friedel sum rule (eq. (5)) for each value of CT and 1. The fractional average scatter-

ed charge density

<

Ap, >/pA,(0) per atomic percent is given as

<

'

_{per atomic}

_%

₌

~ ~ " ( 0 1

where r i is the iih neighbour distance from the origin and ci is the total number of atoms in the i~ shell of neighbours (c, = 12, c, = 6, etc). The fractional

average scattered charge density per atomic percent impurity in gold as a function of phase shifts is shown in figure 3, where we have averaged scattered charge

density over 64 shells of neighbours for various scattering potential depths in gold.

FIG. 3. -The calculated fractional scattered charge density change averaged over the 64 neighbouring gold nuclei shells and reported per atomic percent of impurity ingold as a function of Fermi level phase shift for a square well potential of radius

1 439 A for I = 0 , 1, and 2.

A semiempirical scheme has been developed for the construction of phase shifts from the investigation of alloy properties which depends on the electron redistribution in dilute alloys. The method utilizes para-

-4 1 I l I I I I l I I

I

0 4 2 3 4 5 6 7 8 9

NOMINAL A Z

FIG. 4. - Electron configuration corresponding to the total scattered electronic charge for various impurities in gold with impurity atomic spin S chosen = 0. The results are given relative to the electron configuration for the conduction band of pure gold. For comparison the electronic states expected

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CHARGE DENSITY DISTRIBUTION ]SURROUNDING IMPURITIES IN GOLD METAL C6-375

meterization through an 1 (and sometimes a) depend- .ent square-well potential around the solute atom. Residual electrical resistivity data [7], magnetic susceptibility data, Mossbauer isomer shift data, and the nominal valence difference corrected for local strain [l] provide the four independent pieces of information. By solving the set of simultaneous eq. (S), (6), (7) and (10) a consistent set of four phase shifts at the Fermi level can be obtained. We have chosen to neglect all 6,,, for l 2 3 and have assumed that = _and

6, ,? = 6,

,l.

These assumptions allow us to evaluate 6,(kF), 6, (k,), 6, ,t(kF), and 6,

,,

(k,) within our square well model calculations.

For magnetic alloys, we have evaluated phase shifts

at T = 0 K assuming a singlet ground state S = 0, and at T = 4.2 K using p,, = gpB JS(S

+

1) from susceptibility data. The resultant phase shifts at

T = 0 K for all the impurities studied are shown in

figure 4. For comparison we have included the electronic states expected for a free atom of the impurity in the figure. Since our isomer shift measurements are made relative to pure gold it should be remembered that the results in figure 4 are given relative to the electronic angular momentum character of the conduction band of pure gold. As indicated in the figure, the net S, p, and d types of charge are found roughly consistent with expected values for a free atom of the impurity.

References

[l] ROBERTS, L. D., BECKER, R. L., OBENSHAIN, F. E. and THOM- [3] HURAY, P. G., ROBERTS, L. D. and THOMSON, J. O., Phys. Rev.

SON, J. O., Phys. Rev. 137 (1965) A 895 : B 4 (1971) 2147.

ROBERTS, L. D., p ~

D. o.,

THOMiON, ~ J. ~0 . and ~ MOTT, ~ N.'F., ~ r o c . ~ Cambridge ~ h i l . ~ ,SOC. 32 (1936) 281. LEVEY, R. P. Phys. Rev. 179 (1969) 656. [5] FRIEDEL, J., NUOVO Cimento 7 (1958) 287.

[61 GELDART, D. J. W., Phys. Lett. 38A (1972) 25.