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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS -1963
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2194
CEST DOCES
JUN 27956
CONF-660529-2
H.C. $ 7.00: MN_50
66-64
Votr:
This is a draft of a paper which will be presented at i con:rence
on The Calculation of the Properties of Vacancies and Intercitiais,
May 1-5, 1966, Proceedings of the conference will be published.
Contents of this draft should not be quoted or referred to without
permission of the authors.

MASTER
A Calculation of Force-Constant Changes for the U Center
R. F. Wood and R. L. Gilbert
LEGAL NOTICE
RELEASED FOR ANNOUNCEMENT
This report was prepared as an account of Government sponsored work. Neither the United
States, por the Commission, nor any person acting on behalf of the Commission:
A. Makes any warranty or representation, exprossed or impllod, with respoct to the accu-
racy, completanoSI, nr wefulness of the information contained in talo roport, or that the use
of any information, apparatus, method, or process disclosed in this report may not Infringe
printoly owned tghts; or
B. Asrmos any labutos with zospect to the use of, or for damage resulting from the
use of any information, apparatus, method, or process disclosed in this reports
As used in the abovo, "person acting on beball of the Commission" includes way om-
ployoo or contractor of the Commission, or employer of such contractor, to the axtent that

IN NUCLEAR SCIENCE ABSTRACIN
dienominator, or provides acconto, way Information purnuant to his employmeat or contruct
with the Commission, or his employment with such contractor.
Solid State Division
Oak Ridge National Laboratory
operated by
Union Carbide Corporation
for the
U.S. Atomic Energy Commission
Oak Ridge, Tennessee
w
April 1966
.
A Calculation of Force-Constant Changes for the U Center
R. F. WOOD and R. L. GILBERT
Solid State Division, Oak Ridge National Laboratory
Oak Ridge, Tennessee
. It was established some time ago that a hydride ion can replace a .
halide ion in alkali halide crystals. This defect, now known as the U
center, is responsible for a characteristic absorption in the ultraviolet
which has been intensively and extensively investigated. Through this
work many of the properties of crystals containing U centers have become
understood. More recently, attention has shifted to theoretical and ex-
perimental research on the infrared absorption of the defect, Shaefer's"
original experimental work on the subject coincides with an increasing
interest in the effects of a defect atom on the vibrational properties of
the host crystals, particularly in localized and quasi-localized modes.
The li center is an interesting example of the former case, in which a light
impurity atom produces a localized mode well above the limit of the band
of transverse optical vibrational frequencies (2.67 x 1013 sec-in kci)
of the perfect crystal. The problem of the localized mode connected with
the U center and of its interaction with the "in band" modes has been dis-
cussed by many people. It was claimed earlier that one could understand
the experimental results simply by considering the mass difference between
F
.
the Hº ion and the negative ion it replaced.
It is now generally recog-
nized that it is also necessary to take into account the changes in force
2
Research sponsored by the U. S. Atomic Energy Commission under contract
with Union Carbide Corporation.
Summer participant from Illinois Institute of Technology, Chicago, Illinois.
n
tirp
constants which must surely occur. In a recent paper, Fieschi et al.' go
EREX ----
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into this question rather thoroughly.
Last summer we carried out extensive calculations on the electronic
structure of the L center, and recently these have been taken up again.
Our original aim was simply to see how well methods which had been employed
---
.
for similar calculations on the F center could give the absorption energy
of the U center. Since lattice distortions around the defect are allowed
in our model, it occurred to us that we could actually calculate the poten-
.
.
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:
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tial energy curve for both the nearest neighbor relaxation and the movement
in a <100> direction of the H ion. We were thus in a position to calculati.
the force constant of the Hº ion from a fairly rigorous quantum niechanicai
iuc:nulation. It is the purpose of this paper to sketch the calculütions
of both the optical and infrared properties of U centers in kci, Kir, alü
KI. Unfortunately, the calculations are not yet finished, and this pape:
must really be considered as in the nature of a preliminary report.
We follow here very closely the work in reference 4. The wave func-
tion of a crystal containing a single U center is written as
Y(1,2,...n) = AX,(1,2)4c(3,4,...n)
(1)
in which \,(1,2) and c(3,4,...n) are appropriately antisymmetrized group
functions describing the H™ ion embedded in the crystal and the rest of the
crystal respectively. A is tien an antisyminetrizing operator which inter-
changes the electrons among the two groups i and C and appropriately nor-
malizes the resulting function. So-called "strong orthogonality" is
assumed, e.g., that
Swy (1,2)*c (3...k-1,1,k+1,...n)dt, = 0,
anů vy, and we are considered to be separatcły normalized. The Hamiltonian
for the problem can be written as
H = Hy + Her + Hint ,
where
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Want (1,2,3...)
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21.
Hor(3...n) = | ſ
Hoi vij
15w, Pul
ifw, i-Ev, jí
The expecta-
r , is the coordinate of the į electron on the voth ion, etc.
tion value of H with respect to V, 4c is
(VallclH VUVC) = (Vyl Hul Wy) + (*c Herluc) + (vulc Hintl Molle).
(7)
Antisymmetrization merely introduces exchange between the groups U and C.
We treat the first and third terms quantum mechanically and the second teru
by classical ionic crystal theory. The form of y, (1,2) is taken as
Vy (1,2) = N, [V,(194(2) + V(2)(1)] ,
where
Vo - No loa - I calv.1),j]
زوا
and
and
An exactly similar form is taken for Yp. This type of function, first used
for the helium atom, allows for some in-out correlation as well as giving
.
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-
.-
.
a fairiy adequate approximation to the Hartree-Fock doubly occupied orbi-
tal.
· From Eqs. (1)-(8) an expression for the energy can be derived which
we shall write, for purposes of discussion, as
.
-
-
.
.-
-
-
- -
-
--
Ein = Tu + Emad, u * Emad, u + Epent + Eex + Eov + Ecr.
(11)
The first three terns are respectively the kinetic energy, the Madelung
energy, and a correction to the Madelung energy due to the fact that some
of the charge density may be outside the various rings of neighboring ions;
together these three terms give the point ion approximation as discussed by
Gourary.' Eat is the coulomb energy due to the penetration of the defect
NL
..
electrons into the cores of the neighboring ions, Eey is the exchange energy
with these core electrons, and Ey is the energy introduced by the overlap
terms (alv,j) in Fq. (9). Er is the energy of the rest of the crystal.
All of the terms in Eq. (11), except Ty, depend directiy on the positions of
the neighboring ions and on the displacement of the H ion. Ali of the icr ils
except Ecr depend on the variation parameters a and b appearing iný, and Pig
The calculation consists in first minimizing En as a function of the varia-
tion parameters and the positions of the inn ions in a fi displacement.
Thus, we get the distortion of the crystal in the vicinity of the defect.
Holding the ions in their new positions, we then displace the # ion in a
<100> direction in order to map out Er as a function of this displacement.
According to the Born-Oppenheimer theorem, we can obtain in this way thc ef-
fective potential energy curve for the motion of the hº ion.
Some of the results of our calculations thus far are given in Tables
I and II. The optical transition energies in Table I have been obtained by
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using very simple approximations to the Hartree-Fock 3s and 3p k* core
orbitals constructed according to the Slater prescription. They are
$35 = Ng9x28*2.5837 and 43p = Ngpa?-2.583? cos o
where Ngs and Nzp are normalizing factors. Calculations employing accurate
Hartree-Fock orbitals for all core electrons give a ground state energy
level somewhat higher but, since the excited state energy level should be
raised by about the same amount, we do not expect the transition energies
to change greatly when the more accurate calculation is completed. Follow-
ing Gourary's earlier point ion calculations, we have lowered the ground
state energy by 0.389 ev to include the remaining correlation energy un-
accounted for by the form of Eq. (8). It is assumed that the correlation
energy is negligible in the excited state.
The results of the force constant calculations given in Table II show,
as expected, that the point ion approximation is totally inadequate to ac-
count for the observed local mode frequency. It is somewhat surprising that
the results using the Slater approximation to the 3s and 3p Hartree-rock
orbitals should be as close as they are to the experimental values. The in-
.
clusion of the il polarization discussed below would make this agreement
even better, but we are forced to assume that it is fortuitous. The results
using accurate Hartree-Fock orbitals for all core electrons appear to be
.
very inaccurate, but this calculation is not yet finished. We have still to
add the corrections due to orthogonalization to the core electrons in the
calculation of the expectation value of 1./12., and we have not reminimized
the energy for each displacement of the Hº ion.
The first of these steps
should decrease K significantly; the second will probably have only a small
.
1
.,
-6
effect.
In all of the calculations of K reported in Table II, the poten-
tial curves are very nearly harinonic.
We now wish to include in the calculations the possibility that the
Hº ion can be polarized during its vibrations. By far the easiest way of
doing this, within the framework of our model, is to assume that the entire
electronic shell of the H ion can be displaced relative to the proton. We
can then calculate the effective force constant between the shell and the
proton. In practice, we displace the hº ion a given distance assuming no
polarization, hold the proton at that position and allow the electronic shell
to displace until equilibrium is established. Thus, we can calculate an ef-
fective potential curve with and without polarization.
.
S
Somewhat to our surprise, the polarization of H calculated in this
way was negligible. The reason for this soon became apparent. The two
functions da and dy involved in (1,2) have exponential parameters a and b
whose magnitudes are about 1 and 0.5 respectively. ¢ is very nearly a free
hydrogen ls function, wh: le « gives a fairly adequate representation of the
F center ground state; compared to be a is rather compact. llence, when
the Hº ion is polarized, we expect most of the distortion from spherical sym
metry to be attributable to the Ph function. In our model this means that
Pb is shifted relative to the proton much more than is da: In fac., it
appears to be a good approximation to assume that is not displaced at all.'
Even this method of calculating the polarization may not be adequate. A very
simple calculation on the free hydrogen atom by the method of rigid shell
displacement gives a polarizability too small by a factor of six.
Our own
calculations give a polarizability of Hº somewhat larger (less than a factor
of 2) than the value of 1.9 ÅS estimated by Calder et al.' The calculations
**
-7-
-..--*
.-
..
-
...
indicate that the effect of the HⓇ polarization will be to lower the force
constants by 10 to 20 percent.
Finally, we would like to mention two problems which have arisen in
this calculation. First, there is the one of accuracy. The change in the
energy of the H™ ion as it is displaced slightly from equilibrium (say 2
percent of the nearest neighbor distance) is of the order of 10" eV. One
must calculate all quantities involved to very high accuracy. The second
problem really stems from the first. In calculations of this type, accuracy
is very expensive in terms of computer time.
This problem has been somewhat
alleviated for us recently by the installation of an IBM 360-75 at Oak Ridge,
and we hope to have these calculations completed scmctime in the near future.
We are hopeful of eventually obtaining the force constants to within 5 to 10
percent of the experimental values, but this may depend on the importance of
the polarization of the k* ions.
.
1
1
V
S
Table I. Optical absorption energies (in eV). The calculated
values were obtained with Slater orbitals on the k*
ions.
Crystal
Calculated
Experiment
Percent Error
KCI
5.86
5.79
1.2
KBT
5.68
5.44
5.32
5.08
Table II. Force constants, K, in various approximations. Column 1 shows
the equilibrium displacement of the inn ions in the three crys-
tals in % of Inn distance of the perfect crystal. The second
column contains the results for the force constants in the point
ion approximation, the third the results with the Slater approxi-
mation to the 3s and 3p K* llartree-Fock functions, the fourth the
results with accurate Hartree-Fock orbitals for all core func-
tions, and the fifth the experimental values.
10-K (in dynes/cm)
Slater Hartree-Fock
Cubic
Distortion
Crystal
Point Ion
Experimental
КСІ
2.78
19.80
26.10
14.43
KBT
2.5
2.31
14.95
18.63
11.44
KI
3.0
1.83
10.10
11.57
8.94
-10-
References
1.
For background information on the U center, see J.H. Schulman and
W. Dale Compton, Color Centers in Solids (Pergamon Press, Inc., New
York, 1962).
2. G. Shaefer, Phys. Chem. Solids 12, 233 (1960).
3. From a rather lengthy list we mention only the recent work or R. Ficschi,
G.P. Nardelli, and N. Terzi, Phys. Rev. 138, A203 (1965), whici seems
to be the most closely connected with our work.
4. 8. F. Wood and H. W. Joy, Phys. Rev. 136, A451 (1964).
5. B. S. Gourary, Phys. Rev. 112, 337 (1958).
6. We wish to thank 1. Öpik for pointing this out to us.
7. R. S. Calder, W. Cochran, D. Griffiths, and R. D. Lowde, Phys. Chem.
Solids 23, 621 (1962).
--
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