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CONF-651-9
Bli 51964
A CORRELATION OF THE MOSS BAUER ISOMER SHIFT AND THE RESIDUAL
ELECTRICAL RESISTIVITY FOR 197A3 ALLOYS*

Louis D. Roberts, Richard L. Becker, F. E. Obenshein,
Oak Ridge National Laboratory
Oak Ridge, Tennessee
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J. 0. Thomson
University of Tennessee, Knoxville, Tennessee
I.
INTRODUCTION
In the theoretical discussion of metals 1t 18 often convenient
to describe the metallic sample by using a single potential well for the
conduction electrons.
The Bloch wave functions which extend throughout
the entire sample give an approximate solution to the corresponding
quantum mechanical problem. When an impurity is dissolved in the metal,
the electron wave functions again will, in general, extend throughout
the sample. One may then expect a correlation between different physical
phenomena associated with the impurity which may each depend dominantly
on the different regions of the wave function. For a suitable host and
impurity one should observe a correlation between the charge density at
the impurity nucleus pro) and the transport cross section per impurity
atom or which the impurity atoms present to the host conduction electrons.
For impurity atoms whose nuclei have resonant gamma ray transitions suitable
for Mössbauer effect studies", a measurement of the isomer shift will
give information about plo). Por suitable dilute alloys, 'the measurement
of the residual electrical resistivity per atomic per cent AR/C, will
give or Thus one may expect a correlation between the 1somer shift ve
"Research sponsored by the U. S. Atomic Energy Commission under contract
with the Union Carbide Corporation.
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of the resonance gamma ray energy E, of the nucleus of an impurity in a
dilute alloy with AR c due to that impurity in the alloy.
The 1sotope 19 Au has a suitable gamma ray for recoilless radiation
(Mössbauer) Investigations' and, since gold 18 a noble metal, 1t 18
particularly suitable for a study of the correlation of v, with AR/c.
Thus, if we assume that only the 68 shell of gold 18 appreciably modified
when gold 10 dissolved in a host metal, measurements of V, and of AR/C
may be used in conjunction to give information about the s-band wave
function in the gold alloy system. In the following we present measurements
of V, and of A R/C for gold alloyed in a variety of host metals and show
a correlation between these measured quantities using a simple theoretical
model which 18 based on the use of an impurity potential. Information
18 obtained about the impurity potential and about the s-band filling.
The 18omer shift v, aribes as a result of the Coulomb (electric
monopole) interaction between the nucleus and the atomic s-electrons which
penetrate it. In first order perturbation theory the Coulombic energy
shift of a nuclear state * produced by a single Dirac s-electron 18
D14.(0)|2 <x120%*). Here D 18 a calculable constant, 14.(0)/2
18 the nonrelativisitic electron density at the origin and r = (1 - 0??)
where z 18 the nuclear charge and a 18 the fine structure constant. If
the matrix elements <x/r2° differ for the nuclear ground and excited
states, E, will be modified by D p.(0)(<:20) - (20), where e and
8 refer to the excited and ground nuclear states and where
2
.
/
(0)
16 the total nonrelativisitic charge density at the
+
2
=
--!
"ima
.
12.
29
2
.
.
origin due to all of the s-electrons. Under the circumstance that p. (0)
1s different for the y ray source and for the absorber in the Mössbauer
source gamma ray energy to bring about resonance with the nuclei in the
absorber, namely
A Babe (fo,abelo) - Pejsource(o)] <220). (2207)).
(1)
This energy-increment may be supplied to B, as a Doppler energy. The
required Doppler velocity will be vabe = c AE D/E, where c 16 the velocity
of light. It 18 convenient to refer these energy shifts and the equivalent
Doppler velocities to pure gold as the absorber. The 18omer shift for a
gold alloy as absorber referred to pure metallic gold as the absorber and
expressed as a Doppler velocity V, 18 then
Yi (Valloy - "Au?
= 81 pówro) - Paulo))
where Paulo) 18 the charge density at a gold nucleus in metallic gold,
Pilo) 18 that charge density at a gold nucleus for a gold atom in an
alloy, and g 18 a constant. In general, primes will be used to designate
quant ties which describe alloy properties.
II. EXPERIMENTAL RESULTS
• Experimental values for v, in mm/sec have been determined at 4.2°K
as a function of gold concentration in the range from near zero to one
hundred atomic per cent of Au in binary alloys of Au with Cu, Ag, Pa, Pt,
or N1. Table I gives values for these alloys of v,(0), the 1somer shift
ty
FO:
extrapolated to zero gold concentration. The experimental techniques used
for these isomer shift measurements have been previously described.'
Bxperimental values for a R/c in microhn centimeters per atomic per
cent were measured at 4.2º, 770 and 300°K for gold concentrations of 0.50,
1.00, and 2.00 atomic per cent of gold in each of the above metals as host.
The AR/c values were found to be independent of concentration over the
range investigated and of temperature for Au in Cu, Ag, Pd, and Pt. For
nickel A R/C was also independent of concentration for the threr con-
centrations measured, but was found to rise nearly linearly and quite
strongly with temperature from 0.38 w cm/at. % at 4.2°K to 0.7 m 2 cm/at. %
at 300°K. Th.18 temperature dependence of a R/C must be related to the
ferromagnetism of Ni and Au-N1.' Table I gives the values of A R/C measured
at 4.2°k for the above alloy systems. Because of the ferromagnetism of Ni
and Au-Ni it may not be po681ble to interpret A R/c for the nickel gold
system in the same context with the alloys of Au with Cu, Aš, Pd, and Pt.
III. THEORETICAL DISCUSSION OF V (0) AND OF A R/c
Basically, the gamma ray 18omer shift of an impurity in an alloy
may be viewed ag an aspect of the screening of the impurity. In the case
where the gold impurity 16 d18solved in another noble metal the screening
will presunably be predominantly by the conduction s-band of the host. In
the case where the host 18 a transition metal, however, the situation may
be more complex. We shall take the usual view that the eigenstates of the
outer electrons in a transition metal host may be described by two bands,
one of dominantly s and the other of dominantly d-character. For dilute
**
'
,...,
-5.
gold alloys with the transition metals we further assure (a) that the
states of gold other than the 68 state, in particular the 5d states, are
not sufficiently modified, relative to pure gold, to significantly alter
their screening of the gold a shells, (b) that only the s band contributes
to the 1somer shift, e.g., we neglect any hybridization of gold s end P7/12
states into the a band, (c) that in a gold cell, the hybridization of gold
p and a functions into the s band 18 nearly the same as in pure gold, and
(a) that the residual resistivity 18 attributable entirely to the s band,
because of the high effective mass of the d band holes.
The interaction of the impurity atom with the electrons of the s
band of the host metal will be described by associating a perturbing pseudo-
potential with the impurity atom. The pseudo-potential will give rise to
a transport cross section Otor for the conduction electrons and will thus
produce a residual electrical resistivity per atomic per cent, AR/c. From
& measurement of AR/C, er for electrons at the Fermi level may be obtained.
Then from Otr and the requirement that the impurity atom must be screened
in an electrically conducting alloy as described through the Friedel sum
rule (see Eq. (11)) the effective potential for the l = 0 component of
Bloch waves in the 8 band may be estimated. Using this well, one may obtain
a calculated value womelo), and thus of the 18ɔmer shift to
within a constant which depends on the nuclear size change.
We consider the substitution of an impurity atom B for an atom A
of the host metal at the origin of the coordinate system. The wave function
i for the s-partial wave of an o-band electron in the alloy with energy
Es will be written as
.
$x(vr) 06(r)U(r),
(42)
where
minni
*
W
wiwittentive turistene
(40)
i smam...awn, as it
ugla), PER;
U(r) =
Ulr), r > R,
18 assured to be 11.dependent of k. Here v (r) or v (r) 18 taken to be the
wave function at the bottom of the s-band and within the Wigner-Seltz cell
of the pure host metal A or of the pure solute metal B. The radius R wiu
be taken as the radius of the Wigner-Seitz celi of the pure metal B. It
will be assumed that Un and V, have the same amplitude at the surface of
their respective Wigner-Seitz spheres, 1.e., U(R) = 0,(R). The charge
density Pn (0) at the origin, 1.e., at the Impurity nucleus B in the alloy,
arising from the s-partial wave of a given k state 18 then
PENCO) - lug(0)12.18660312
where k 18 the electron momentum vector. For pure gold
Paklo) - \U,(0)|2
and thus the isomer shift arising from a given k state vrk may be written
PIK - 8/09(0)12 (143812 - 1)
(7)
To obtain the 18omer shift, Vipe must be multiplied by the density of states
n(k) and Integrated over filled k states. Assuming n(k) to be proportional
to k', and where kr 18 the s-band electron momentum at the Fermi surface
of the host metal we have
V
د مرسده
(6)
Yg + 62 | 0360)12 saix (1460)/2 - 1) x2
(8)
+811440)12 ( - 1).
(80)
1.
.
.
.
.
.
.
-7
r ariwis.
riscont
b are
t i
.**--'-i
i
where 8, 18 a constant.
It 18 now necessary to obtain an estimate of 102(0)/. In the
asymptotic region
k(r) = sin (kr + Sc(k))/kr.
Here So(k) 18 the phase shift of the s-partial wave arising from the effect
of the impurity atom on a conduction 8-band electron state of energy Eck
It will be assumed that at the Fermi surface only so(k) and Sq(Kp)
(where S18 the phase shift of the p-partial wave) are appreziably
different from zero. Then an estimate of So(k) and S,(k) may be obtained
using the measured A R/c and the Friedel bum rue.
For infinitely dilute alloys Huang' has given the result
..
..
.
.
..
..
AR/c = (g/0.703 mm'3) Şle + 1) sinº [fe(key) - letz(ka) (10)
which describes A R/C in terms of the phase shifts -S,(kg) of the l-th
partial wave produced at the Fermi surface by the impurity. Here r. 18
the Wagner-Seitz cell radius 1n atomic units and YA 18 the number of b-band
electrons per atom for pure metal A. The Friedel sum rule,
2 Ş (2 + 1) Se(key)
(11)
where 2 16 the difference of charge between impurity and host atoms, gives
a second condition on the phase shifts. With the assumption mentioned,
that only do and S, are appreciable, and given the value for 2, the
simultaneous solution of Eqs. (10) and (11) will give an estimate of (k).
The difference of lonic charge z between impurity and host, Eq. (11),
16 assumed to have the form
2-178-Ya!-10A
(12)
Tale -
-8.
Here you like which was defined earlier, 16 the number of s-band
electrons per atom in the pure metal B, and (
2 2 ) gives the ionic
charge on the impurity to be screened by the s-band of the host metal A.
The second term on the right in Eq. (12) gives the contribution to 2 which
arises from the dilation of the host lattice A by the impurity B and 18 of
the form used by Blatt. Here vw 16 Poisson's ratio and Sa/a 18 the
change of lattice parameter in per cent of metal A per atomic per cent of
metal B dissolved, Table I.
Then to obtain an estimate of 104(0)2 we assume that folkp) 18
produced by a square well of radius R and of a suitably adjusted depth k?/2.
Using this well we may then obtain the result
7-1
10.60)2 - 1 -
(13)
[ (x2 + xəyfə 61m2 (VIII
a form given previously by Daniel' in obtaining an estimate of the Knight
shift in dilute alloys. The use here of Eq. (13) amounts to assuming only
that, in its effect on 8-waves over a limited range of energy, the pseudo-
potential may be replaced by a roughly equivalent square well. By sub-
stituting Ey. (13) into Eq. (82) and performing the indicated integration
an estimate of v, may be obtained from sR/c.
In obtaining Sc(k) and f (kg) from R/c and the sun rule, there
are in general two sets of solutions because Eq. (10) involves the squares
of the sine functions. We give below the results of the calculations for
both sets of solutions. For the cases we have treated one value of f(ke)
18 postive and the other negative. If the potential is weak, a positive
-9-
Solkype) corresponds to an attractive potential for s-waves, and a negative
S (K) arises from a repulsive potential. Consequently the two solutions
give strikingly different results for the isomer shift.
IV. COMPARISON OF THEORY WITH EXPERIMENT
As shown in Eq. (80) the 1somer shift V, 18 proportional to (p - 1).
This (Ž - 1) 18 estimated from A R/C by the calculation outlined in
Section III where, in the calculation, there are two free parameters, the
5-band filling in the host metal Ya, and the choice between the two
possible solutions of Eq8. (10), (11) and (12) from which the impurity
potential 18 obtained.
Because the experimental isomer shifts are all of the same sign and
% 18 less than or equal to me, for the alloys discussed here, one must
have the same kind of potential, attractive or repulsive, associated with
the s-partial waves for all of the alloys. In F18. 1 the calculated (P - 1)
18 presented for Au alloyed in Cu, A8, Pa, Pt and Ni as a function of s-band
filling for the case where the impurity potential is attractive and in Fig.
2 for the case where this potential 16 repulsive. The proportionality
constant Q = 87/.o)l2 of Eqs. (8), which relates V, and († - 1), 18
obtained by normalizing the theoretical value of Ž - 1 at na = 1 for Au
In Cu to the experimental value va = 4.4 1 0.2 mm/sec. This must be done
separately for the attractive well solution, yielding a constant G = 8.0 mm/sec,
and for the repulsive barrier, giving a constant GR = - 31 mm/sec. The
quantities (B-1) are also given as functions of Ya in Fig. 1, and the
quantities Q (- 1) are given similarly in M8. 2. For Cu and Ag, na
-
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-
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-10-
1s taken to be unity to the experimental 18omer shifts are plotted as points
at Y. -1 with the experimental errors indicated. For the three transition
metal alloys the value of 18 lese certain, so that the experimental
18omer shifts are plotted as crosshatched bands with heights equal to the
experimental errors.
-.-.-
An examination of Fig. 1 shows a good measure of agreement between
theory and experiment for the attractive potential case.
In detail, ex-
.-.-.-.-.-.-.-.-.
periment and theory agree for Au in Pd with a Po 8-band filling in the
-.
-.-.-.-.
range 0.56 to 0.60. This is in quite reasonable agreement with the usual
interpretation of the results of magnetic measurements on Pa. For the case
of Pt as host, the s-band filling at which experiment and calculation
agree, Fig. 1, 18 in the range 0.35 to 0.38. There have been several
previous papers on the subject of the 8-band filling in Pt. Some years
ago, Wohlfarthº gave an interpretation of measurements of the magnetic
susceptibility and of the specific heat of pure pt in which it was suggested
that the number of holes in the d-band was in the vicinity of 0.2 - 0.3.
The number of electrons in the s-band would then be expected to lie in a
similar range. A recent treatment of the Knight shift in Pt by Clogston
et al. also indicates an 8 band filling in this range. Thus there is
reasonable agreement between these results for the s-band filling in Pt and
the result obtained here. A much higher value of 0.58 for this g-band filling
has been suggested by Budworth, Hoare, and Preston'from a rigid band model
Interpretation of their specific heat measurements on platinum-gold alloys.
Perhaps estimates of the band filling of pure Pt may be derived more easily
1.
AN
"
:
-110
frca Investigations of the pure or nearly pure metal as in the work of
Wohlfarth, Clogston et al. or from the results reported here, than from a
rigid band model interpretation of the properties of concentrated alloys.
for the case of Au in Ag, Mg. 1, our experimental isomer shirt
lies about 30% below the theoretical curva. For the case of Au la Ni at
any -0.6 the experimental result lies about 60% higher than the
theoretical curve. The agreement between theory and experiment for the
case of Au io Ag could be made to appear better by changing the normalization
constant to a somewhat lower value near , * 7.0 m/sec. The experimental
results for both Au la Ag and Au in Cu would then lie almost within their
experimental errors of the theoretical curves and the results for the 8-band
rilling ia Ad and Pt would be very little changed. On the other hand the
disagreement between theory and experiment for Au in Ni would be somewhat
worse.
In our calculation of the isomer shift from the electrical re-
818 tance, ferromagnetism of the host was not taken into account in any way.
We find a reasonable agreement between theory and experiment for the para-
magnetic or nonagnetic cases but not where ferromagnetien occurs. As was
observed earlier, AR/C for Au in NI 16 strongly temperature dependent."
While this temperature dependence must be due mainly to magnetic scattering
1t 18 not absolutely clear that 1 R/C at T - Oºk gives a useful measurement
of the transport cross section associated with the Au impurity in the
context of our model.
Turaing to fig. 2, we may now lavestigate the comparison between
experiment and the theoretical model for the case of a repulsive potential.
PA
1.
-12-
With theory and experiment again normalized for Au in Cu we find . 31
/060. Here, however, for Au In 18, Pd, and Pt the theoretica nalwes
for vel are all greater than for Au la Cu while the experimental values
are all less. There is no agreement for these cases between theory and
experiment for any value of the s-band filling y. In the case Au in
Ni, theory and experiment still do not agree at n - 0.6 although the
disagreement in this case so less severe than for the attractive potential
Pig. l. Considering all of these cases, the use of a repulsive well in
the above calculation is not consistent with experimental results.
The results presented in Fig. 1 thus indicate that in Au alloys
or compounds where v, 1o greater than zero, the electron charge density
near the gold nucleus 1. greater than the density in pure gold,
12:,(0) · P., (0)) > 0. This result is in agreement with the conclusion
of Mott? that gold when alloyed in silver bould present an attractive
potential to valence band electrons. Prom a study of the correlation
between the 1somer shift of gold in a variety of alloys with the electro-
negativity difference between gold and the host metal, Shirley et al.14
have also concluded that ve > ņ corresponds to (Paulo) - Pau (0)) > 0. We
note from Eq. (1) in agreement with Shirley et al., that with
Paulo) - Pau (0)) > 0 one has ( 29) - (120) >0. This result
18 in agreement as to sign with the description of 197Au given by zeldes 25
using the shell model.
In the use of the theoretical model given above to describe the
1somer shift, the range of the pseudo-potential was taken to be the radius
.
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of a gold aton. Strictly, this assumption to not required in the above
theoretical model. We have thus repeated the above calculation for an
R of 0.25 rg, 0.50 rg, 0.75 rg and 1.25 rg. The best correlation between
1somer shift and residual resistance was found for the case presented in
detail, namely R = rp
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REFERENCES
1. R. L. Mössbauer. 2. Phy., 151, 124 (1958).
2. 0. C. Kistner and A. W. Sunyar, Phys. Rev. Lett., 4, 412 (1960).
3. L. D. Roberts and J. 0. lhomson, Phys. Rev., 129, 664 (1963).
L. D. Roberts, A. Pomerance, J. 0. Thomson, and C. p. Dan, Bull. Am.
Phyns. Soc., 1, 565 (1962).
4. L. D. Roberto, R. L. Becker and J. 0. Thonson, Bull. Am. Phys. Soc.
8, 42 (1963); R. L. Becker, L. D. Roberts and J. 0. Thomson, Bull.
Am. Phys. Soc., 8, 558 (1953). L. D. Roberts, R. L. Becker,
F. E. Obenshain, and J. 0. Thomson (to be published).
5. L. D. Roberts, F. E. Obehshain, R. L. Becker, and J. 0. Thomson,
Bull. Am. Phys. Soc., 2, 398 (1964).
J. Priedel, Phil. Mag. 43, 153 (1952); Adv. in Phys., 3, 446 (1954).
P. de Faget de Casteljau and J. Friedel, J. Phys. Radium, 27, 27 (1956).
J. Priedel, Il Nuovo Cimento Ser. X, 1, 287 (1958).
7. K. Auang, Proc. Phys. Soc., 60, 161 (1948).
8. F. J. Blatt, Phys. Rev., 108, 285 (1957); Phys. Rev., 108, 1204 (1957).
E. Daniel, J. Phys. Chem. Solid8, 10, 174 (1959).
10. E. P. Wohlfarth, Proc. of the Leeds Philosophical and Lit. Society,
Sci. Sect., 1948, p. 89-101.
11. A. M. Clogston, V. Jaccarino and Y. Yafet, Phys. Rev. 134A, 650 (1964).
12. D. W. Budworth, P. E. Hoare, and J. Preston, Proc. Roy. Soc., A257,
250 (1960).
ó
IT
MAMA
.
-15-
13. N. 7. Mott, Proc. Camb. Phil. 800., 22, 281 (1936).
14. D. A. Shirley, Rev. Mod. Phys., 26, 339 (1964); P. 8. Barrett,
R. W. Orant, M. Kaplan, and D. A. Shirley, J. Chem. Phys., 29,
1035 (1963).
15. N. Zldes, Muc. Phys., 3, 1 (1956/57).
TABLE I
Parameters uscd in the correlation of the isomer shirt with the residual resistance for Au. The phase shirts
S and S, are given for the case of a potential attractive for s-waves and for thc valencen, also given
below, whith results in the best correlation, Fig. 7. The quantity k/2 18 the corresponding well depth
in ev.
Host
mexperimental
residual
resistivity
Experimental
Isomer shirt
Poisson's
Host
valence
used
Phase
shirts
ratio
Well depth
k2/2 (ev)
Aplechts
(www/sec)
0.52 + 0.02
0.364 0.02
0.38 0.02
0.700.02
1.550.02
4.4 0.2
2.1 + 0.2
5.4 + 0.2
2.4 t 0.2
1.4 1 0.2
0.157 0.364
-0.00735 0.37
0.209 0.31
0.05280,0.398
0.040 0.39€
1.337
0.984
1.397
1.120
1.091
1.0
1.0
0.6
0.58
0.37
0.0562 -0.1951
0.2155 -0.06343
0.2872 -0.01113
0.423 +0.0425
0.624 +0.102
0.305
1.07
1.407
1.935
2.597
a) L. Vegard and A. Kloster, Zeit. Krist. 89, 560 (1934); C. S. Smith, Min. and Met. 9, 158 (1928).
b) Q. Sachs and J. Weerts, Zeit. Phys. 60, 481 (1930).
c) 3. C. Elwood and K. Q. Bagley, J. Inst. Metals, London, 80, 617 (1951/2).
a) V. G. Kuznecov, Izv. Sek. Platiny Akad. Nauk. SSSR, 20, 5 (1946), see Structure Reports, 10, 54 (1945/6).
e) 1. . Darling, R. A. Mintern and J. C. Chastoa, J. Inst. Metals, london, 81, 125 (1952/3).
8) from the A.I.P. Handbook, McGraw Hill, 1957.
8) Assumed the value for Pt.
b) y = 3(1-0)/(1+0)
1
FIGURE CAPTIONS
Fig. 1. Comparison between calculated and experimental 1soner shifts
for an attractive potential. The calculated values have been
normalized to give agreement for Au in Cu, with Q = 8.0 mm/sec,
Eq. (86).
M8. 2. Comparison between calculated and experimental isomer shifts
for a repulsive potential. The calculated values have been
normalized to give agreement for Au in Cu, with a = - 31.0 m/sec,
Bq. (80).
UNCLASSIFIED
ORNL-DWG 64-2846

-
-
kriminimai ir 11"
(P-1) FOR ATTRACTIVE POTENTIAL
W
V1, ISOMER SHIFT (mm/sec)
N
s
S
*
A
Pay
Ever..
in.
-
0.3
0.4
9.0
9.1
0.5 0.6 0.7 0.8 0.9
mu, S-BAND FILLING OF HOST
Fig. 1.
-
--
--
*** -
- -
IN
UNCLASSIFIED
ORNL-DWG 64-2847

Ag
NI MILIONIZIMIT
(P-1) FOR REPULSIVE POTENTIAL
Tvil, ISOMER SHIFT (mm/sec)
MIPA
0.4
0.5
0.6 0.7 0.8 0.9
na , S-BAND FILLING OF HOST
9.0
9.1
Fig. 2.
DATE FILMED
11/ 125 / 164






.
"
7
do
LEGAL NOTICE --

This report was prepared as an account of Government sponsored work. Neither the United
States, nor the Commission, nor any person acting on behalf of tho Commission:
A. Makes any warranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information containod in this report, or that the use
of any information, apparatus, method, or process disclosed in this report may not infringe
privately owned rights; or
B. Assumos any liabilities with respect to the use of, or for damages resulting from the
Use of any Information, apparatus, method, or process disclosed in this report.
As usod in the abovo, "person acting on behalf of the Commission” includes any em-
ployee or contractor of the Commission, or employee of such contractor, to the extent that
such employee or contractor of the Commission, or employce of such contractor prepares,
disseminates, or provides access to, any information pursuant to his employment or contract
with the Commission, or his employment with such contractor,

END


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