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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
WW
Onnu 22569
Cont760935"MASTER

NOV 2 9.1966
12.C. 7.00, MN 50
PSEUDO-ATOM PHASE SHIFTS OF LIQUID METALS AND ALLOYS
(Short title: Pseudo-atom phase shifts)
Axel Meyer
Solid State Division, Oak Ridge National Laboratory
Oak Ridge, Tennessee, U.S.A.
C. W. Nestor, Jr.
Mathematics Division, Oak Ridge National Laboratory
Oak Ridge, Tennessee, U.S.A.
W. H. Young
Department of Physics, University of Sheffter
Sheffield, England

RELEASED FOR ANNOUNCEMENT
IN NUCLEAR SCIENCE ABSTRACTS
ABSTRACT
Ziman pseudo-atom phase shifts have been evaluated, as functions
of & and kg, for monovalent ions, using a pseudopotential technique. The
results of their application to a number of physical problems suggest
that our data are reasonably accurate.
S 1.
INTRODUCTION
Recent advances in the theory of the electronic structure of the
simpler metals have been reviewed by Ziman (1964), who drew particular
attention to the concept of a pseudo-atom. Such an entity is to be thought
of as the ion under consideration, together with a surrounding cloud of
neutralizing valence charge. The distribution of the latter is to be
computed as though the ion were completely alone in a free electron gas,
the density of which is that of the original averaged valence distribution.
Research sponsored by the U. S. Atomic Energy Commission under contract
with Union Carbide Corporation.
LEGAL NOTICE
This report was prepared as an account of Government sponsored work. Neither the United
States, aor the Commission, nor any person acting on behalf of the Commission:
A. Makes any warranty or reprosentation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information contained in this report, or that the use
of any information, apparatus, method, or process disclosed in this report may not infringe
privately owned righto; or
B. Assumes any liabilities with reapect to the use of, or for damages resulting from the
use of any information, apparatus, method, or process disclosed in this report.
As used in the above, "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 employee of such contractor prepares,
dioseminates, or provides access to, any Information pursuant to his employment or contract
with the Commission, or his employment with such contractor.
v
.
.
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.
.
The various partial wave phase shifts, no, for electrons of the Fermi level.
of the model system are of special importance, having immediate applica-
tion to a variety of physical problems.
To assess, fully, the merit of this method, it is necessary to kno:
the no, not merely at the pure metal Fermi level, but over a range of kps
so that, for example, applications to alloys can be made. In this paper
we present such data for the monovalent metals, explain how they are derived,
and briefly review a number of applications. The latter, on the whole, are
fairly successful, indicating that the pseudo-atom concept is a fruitful
one and that our results may be applied to it with reasonable accuracy.
.
§ 2.
THEORY
We consider a free electron gas with Fermi energy ** (atomic units
being used in general throughout) and which contains a single ion. The
Schrödinger quation for a free electron is then taken to be
n
imi
----
---
- { v204 + (Uy + Us + Up) - į km Of
. (1)
where V, is the Hartree-Fock-Slater potential of the bare ion, as tabulated
by Herman and Skillman (1963), Us is the screening potential, teken through-
-----
---
-.:.
out as
ins._--
;
-
Peo
(r < 1/80?
...
-
-
Us
-
(1/
(r ? 1/80).
... -
-
...
...
where e, is a constant which will be used to satisfy the Friedel (1952) sum
rule, and Up is the core-orthogonalization term which always arises in
pseudopotential theory.
....
:*...
-
.
-
..
-
We choose V, in the Austin (1962) form
Up okt -{ «Vc/* Uslove Vic
(3)
where summation is over all core states, We of the ion. If, now, a
Yom (0,0), and, as expected (Cohen and Heine 1961) cancellation of u against
Uy is good in the core, then in this region, as a first iterate, the radial
part of or can be taken to vary with r'. Hence, we have
Up $k = UGH $k = - { **!°H *us?em? Ok.
.
()
and thus obtain a semilocal potential (i.e., a simple potential function of
r for each l).
Conventional partial wave analysis may now be applied, the d-th.
partial wave equation reading
luego o
* U * Ug = Wine Cuix * ugdpret 2drs Winter 2 so
they
where ne and Rke are respectively the radial parts of vc and $*• The
screening parameter e, of Eq. (2) is now varied until the phase shifts ob-
tained from (5) satisfy the Friedel sum rule. The graphs thus obtained for
(24 + 1)n, versus ky for the monovalent metal ions are shown in fig. 1.
$ 3. LIMITS OF VALIDITY OF DATA
The results will become invalid at both high and low kp, near where
fundamental changes in the electronic configuration occur. As ke is lowered,
the formation of a bound state is signalled by a change in the sign of the
scattering length (see, for example, Messiah 1961.). In this way, by formally
replacing the right side of (1) by zero and solving for potentials calculated
self-consistently (in the sense described in s 2), we obtain the critical
data given in row (2) of table 1. It is to be expected that in the regions
so defined, the phase shifts are fluctuating rapidly and will be most diffi-
cult to compute with accuracy.
We have not investigated the corresponding effect at high kipy but pre-
sume that the doininance of a single phase shift in the Friedel sum (cf. fig. 1)
for Li, K and Cs is significant in this respect.
'In the cases ož Rb and Cs, the upper limit of usefulness of our data
is decided on mathematical grounds. By examination of our solutions, it is
possible to check the assumption that, to a first approximation, Rko a
inside those cores containing bound & states. In general, this approxima-
tion is very satisfactory but it works least well in the case l = 2. As an
example, we give, in fig. 2, the solutions appropriate to liquid Rb at its
melting point. It will be seen that the approximation is extremely good for
S- and p-waves, but breaks down for d-waves in the outer regions of the core.
This will not be important if the effect of this component is, in any case,
small, and our experience suggests that the data should be treated with
caution if ng exceeds about 0.15 to 0.2 (depending upon the application).
which bound d-states are absent, nor, indeed, to the noble metals where ng
is always very small.
§ 4. APPLICATIONS
It is not our present purpose to discuss applications in detail, but
merely to comment on such results as have been calculated on the basis of
- 5-
our phase shifts, and, thereby, to indicate the degree of accuracy of fig. 1.
The background theory, on which most of the following is based, has been
provided by Ziman et al. (1961, 1962, 1964, 1965) and, consequently, detailed
formulae will not, in general, be quoted. :
ti
§ 4.1.
Interatomio Foroes
2.
Por interatomic distances much in excess of one core diameter, the
interaction energy between two like ions should take its asymptotic form
(Meyer, Nestor and Young 1965)
V = - p(2Kp)(a/r3) cos(2kpr + $) ,
where a and $ are known functions of the no, and p(2k) is the 2k-th Fourier
component of the charge density (electronic and nuclear) on an ion.
In this way (table 1), Meyer, Young and Dickey (to be published) have
obtained correlations (though not of high precision) with the asymptotic
forces found by Johnson, Hutchinson and March (1964), on analyzing the ex-
perimental neutron diffraction data of Gingrich and Heaton (1961) or earlier
x-ray data on the liquid alkalis.
By fitting these same experimental data to a theoretical hard sphere
structure factor, Ashcroft and Lekner (1966) were able to define atomic diam-
eters. By relating (6) to the average thermal energy of the ions, we have
been able to define diameters in essential agreement with these. For present
purposes, whenever a structure factor is required below, we have used the
Ashcroft-Lekrer form.
"In a private communication by N. H. March, he has stated that X-ray data
were used in the case of sodium.
N
eue.
-6.
... ---
-
The interaction energy between two ions in a liquid metal is the
same (neglecting the small variation of kp) as that between two vacancies
in the corresponding solid (Harrison 1966). In the case of the noble
metals, the latter are known approximately and comparison with the values
calculated from (6) are shown in table 1.
§ 4.2, Electrical Resistivities and Thermopowers
of Pure Liquid Metals
Meyer, Young and Kilby (to be published) have obtained the results
shorm in table 1 for the resistivities, p, and the usual dimensionless
quantities & characterizing the thermopowers. The following points may be
noted.
.. (i)
The Slater average exchange approximation is known to be poor
for Li.
If, then, we use a Slater ls orbital and the associated Hartree
field in this case, we calculate a satisfactory resistivity of 21.6 uncm.
The actual differences in the phase shifts are not great (n, decreasing by
16 percent) but, as is well-known, the resistivity is extremely susceptible
to even small changes in the scattering ampíitude. In other less critical
applications, the phase shifts, derived for Li on the basis of a Hartree-
Pock-Slater core, appear to be adequate.
(ii) If, for Cs, we had used the experimental Gingrich-Heaton
structure factor, which lies rather low in the momentum transfer region
(0,2kg), we would have found a resistivity of 39.9 unem. It seems probable,
though, that the more likely explanation of our high result is that with
--
-
-
--
.
-7-
ng = 0.15, we are bordering on a situation (recall § 3) where one may expect
poorer results, especially for as sensitive a property as the resistivity.
(iii) The thermopower results contain the r-term of Ziman et al.
Without it, the correlation with experiment is considerably poorer. In-
volved is a calculation of the gradients of the n, across the Fermi level
for fixed Friedel-adjusted potentials. Thus, in this case, rather more
information is required than is embodied in fi
$ 4.3. Vacancy Resistivities in the Solids
In the case of the noble metals, where these are approximately
known from experiment, comparison with our calculated values (table 1) is
satisfactory. This would not appear to be a difficult criterion to meet,
however, as all the monovalent metals, with their rather different sets of
phase shifts, yield vacancy resistivities between 1.1 and 1.6 uncm/atomic %.
HUS
...
.!!..
T
§ 4.4. Impurity Resistivities
Dickey, Meyer and Young (1966 and to be published) have evaluated :
the impurity resistivity of every monovalent metal in every other monovalent
metal, both in the liquid and solid states. The results, in general, are .
gratifying when comparison with experiment is possible and when our data
satisfy the criteria of $ 3.
As an illustration, we give, in table 1, impurity resistivities in
liquid Na together with the experimental values of Freedman and Robertson
(1961). For, Li, Na and K, the calculations are successful; for Cs at the
Fermi level of pure Na, we have ng = 0.37, which is too large for complete
quantitative success. The discrepancies in the cases of Ag and Au should
be due to the proximity of a bound state (recall the discussion in s 3
and see table 1; for Na, kp = 0.474).
i
*1
-*.
*
.
To get things ?
1
1
:
Shame the modernistan
.
-8-
It is also worth mentioning, in passing, that for the case of K in
Rb, on melting, the calculated impurity resistivity changes sign, as is
experimentally observed (Kurnakow and Nikitinsky 1914).
in
a
§ 4.5. Pressure Dependence of the Resistivity
in Pure Liquid Metals
Using hard sphere diameters defined as in § 4.1 above, Dickey, Meyer
and Young (to be published) have evaluated as a function of kq as the
total volume V is varied under pressure. The general features of the Bridg-
man results (Lawson 1956) are obtained. The latter, admittedly, are for
solids, but it would seem that the dominant mechanism is the change in Fermi
level producing a variation of the pseudo-atom phase shifts.
At 1 atm., the values of [a in p/à in V]or are known experimentally
in the liquids, and comparison between these and our calculated results are
made in table 1. The negative value for Li is due to the exceptional in-
crease in efficiency as scatterers of individual ions of this metal as the
screening adjusts to an increase in Fermi level. .
.
.
.
-
§ 5. CONCLUSION
Above, we have presented phase shift data for use in the Ziman
theory. A feature of the latter is the generality of its approach, and we
have endeavored to match this on the numerical side. The uniform applica-
tion of a single computational procedure, starting from the Herman-Skillman
core states, themselves derived by a single systematic method, results in
the data of fig. 1.
The calculated phase shifts provide a rather coherent picture over
the range of physical phenomena discussed in f' 4. The reconciliation of

the Ashcroft-Lekner structure factor with (6) is particularly satisfactory.
For, this means that experiment need never be invoked at any intermediate
stage. All our computed quantities are completely theoretical, and, in
fact, with the one proviso discussed in connection with thermopowers, derive
entirely from fig. 1.
· REFERENCES
Ashcroft, N.W., and Lekner, J., 1966, Phys. Rev., 145, 83.
Austin, B.J., Heine, V., and Sham, L.J., 1962, Phys. Rev., 127, 276.
Bradley, C.C., Faber, T.E., Wilson, E.G., and Ziman, J.M., 1962, Phil. Mag.,
7, 865.
Cohen, M.H., and Heine, V., 1961, Phys. Rev., 122, 1821.
Dickey, J.M., Meyer, A., and Young, W.H., 1966, Phys. Rev. Letters, 16,
727.
..
Faber, T.E., and Ziman, J.M., 1965, Phil. Mag., 11, 153.
Freedman, J.F., and Robertson, W.D., 1961, J. Chem. Phys., 34, 769.
Friedel, J., 1952, Phil. Mag., 43, 153.
Gingrich, N.S., and Heaton, L., 1961, J. Chem. Phys., 34, 873.
Harrison, W.A., 1966, Pseudopotentials in the Theory of Metals (W.A. Benja-
min, Inc., New York).
Herman, F., and Skillman, S., 1963, Atomic Structure Calculations (Prentice-
Hall, Inc., New York).
Johnson, M.D., Hutchinson, P., and March, N.H., 1964, Proc. Roy. Soc.,
A, 282, 283.
Kurnakow, N.S., and Nikitinsky, A.J., 1914, Z. anorg. Chem., 88, 51.
Lawson, A.W., 1956, Progress in Metal Physics, 6 (Pergamon, London).
Messiah, A., 1961, buantum Mechanics (North-Holland Publishing Company,
Amsterdam), Vol. I, pp. 392, 408.
Meyer, A., Nestor, Jr., C.W., and Young, W.H., 1965, Physics Letters, 18, 10.
Ziman, J.M., 1961, Phil. Mag., 6, 1013; 1964, Advances in Physics, 13, 89.
-10.
FIGURE CAPTIONS
Figure 1
Pseudo-atom phase shifts for the monovalent metals; (a)
lithium, (b) sodium, (c) potassium, (d) rubidium, (e) cesium,
(f) copper, (g) silver, (h). gold.
Figure 2
Pseudopotential and partial wave solutions (at Fermi level)
for Rb for l = 0, 1, 2, 3, 4. Curve (1) shows the screened
core field, VH + Us, curve (2) the net pseudopotential
UH + Us + Uch, and (3) is the (arbitrarily normalized) par-
tial wave solution.
ORNL-DWG 66-8324
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TABLE I
boomwhaco.--
ORNL DWG 66-9428
Li
Na
Rb
Cs
Co
Ag
Ao
0.578
0.268
0.037 ((0.045)
0.49 0.36)
0.474
0.350
0.003 (0.007)
-0.30 (2,01)
.
0.382
0.280
0.005 (0.010)
0.66 (1.42)
0.357
0.286
0.010 (0.006)
0.63 (2.60)
0.699
0.50
0.332
0.239
0.047 (0.006)
0.32 (1.45)
0.611
1
0.49
0.614
0.53
(1) k, (pure liquid) (a.u.)
(2) k., (bouad state) (a.u.)
(3) kya?(a.u.)
(4) $(a.a.)
(5) Vac.-vac. b.e. (V)
(6) p (pure liquid) (un cm)
(7)
(8) p(vacancy) (1 cm/at. %)
(9) p(impurity in Na) (? cm/at. %)
(10) (a la pla la VI,
60.4
-5.31
(24.0).
(-8.8)
13.2.' (9.65)
0.39
0.39 (2.9)
(2.9)
14.6
1.54
1.54
(13.2)
(3.5)
(3.5)
-
.
25.4
0.44
(22.0)
(1.7)
56.7
-4.93
(36.0)
(-1.3)
0.7 (0.2)
15.9 (21.1)
-2.41 (-3.50
1.10 (1.5)
0.23
21.1
-1.61
1.43
0.88
(0.4)
(17.2)
(-1.9
(1.3)
(2.50)
0.35
25.0
-2.22
1.53
0.93
(0.1)
(31.2)
(-0.6)
(1.5)
(4.80)
0.34
-0.88
(0.28)
(-0.5)
1.11
(1.02)
(4.75)
2.01
1.15
7.16
(2.55)
(3.95)
(4.35)
(3.64)
(1.8)
Notes: The numbers in parentheses are experimental or, in the cases of (3) and (4), experimentally based, in that they are the results of theoretical work in which the input information is ex
perimental. He tabulate k a’ in (3) rather than a, because the former should roughly correlate (Reyer, Young and Dickey, to be published) with the resistivities of rom (6). The experimcatal aum-
bers in (10) were taken from Ziman (1961) and Bradley et. al. (1962).
AYTARTU
LES
titles
23
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.
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44
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DATE FILMED
1/ 31 / 67








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