J. Phys. Chem. Solids.
Pergamon Press 1966. Vol. 27, pp. 1659-1665.
VACANCIES
Printed in Great Britain.
IN SOLID ARGON
H. R. GLYDE
University
of Sussex
School of Mathematical and Physical Sciences
Falmer, Brighton, England
(Received 11 February 1966; in revised form 7 April 1966)
Abstract-The
enthalpy and entropy of vacancy creation in solid argon are re-evaluated using the
two-body force approximation. At the triple point, the resulting vacancy free energy is g, = 1900
-4.0
RTcaljmole
vat. This does not agree well with previous calculations of g,, but agrees well
with experimental values of g, obtained from the argon specific heat”’ C, (but not with the questionable values obtained from CJ and with direct measurements (r3) of (n/N) = exp( -gJRT).
If the
present value of g, is correct, the agreement with experiment suggests that many-body effects
contribute little to the binding of argon atoms around a vacancy.
1. INTRODUCTXON
SOLID argon has captured the interest of physicists
for many years(r) largely because the many-body
interactions in the crystal may be well approximated by a sum of two-body interactions between
pairs of atoms. With this approximation
it has
been possible to carry through first principle
calculations of the properties of solid argon. The
properties of vacancies are of interest chiefly because of the information
they provide on the
validity of this two-body force approximation.
Previous calculations at O°K(2*3) and 80”K(4*5)
of the enthalpy h, and entropy s, of vacancy creation in argon using the two-body force approximation did not agree well with the experimental
values. Both h, and s, (though largely h,) were in
error suggesting that many-body
effects make a
large contribution’“)
(-25%)
to the binding
energy of the atoms around the vacancy. This large
contribution
appeared necessary to obtain relaxation around the vacancy so that the potential
energy increase of the crystal on vacancy creation
is not too large (h, M SV). Subsequently,
threebody calculations(**s) were attempted and enough
relaxation was found to provide agreement with
the experimental
h,. However, severe approximations were necessary in these difficult calculations and the results of JANSEN@) in particular,
seem to require a binding energy per atom in the
perfect crystal which is not consistent with the
sublimation energy Lo. For this reason and because
the original values c4u5)of h, and s, seem so large it
appears worthwhile to re-evaluate h, and s, using
the two-body force approximation
in an effort to
get better direct agreement with experiment.
In this paper we evaluate h, and s, using the
two-body force approximation
first assuming no
(or negligible)
relaxation
around the vacancy
(Section 3). The extent of relaxation possible with
two-body forces is then determined and its effect
on h, and s, included (Section 4). In the Discussion the results are compared
with previous
calculations,
the experimental
situation
is reviewed and the results compared with the reliable
experimental values to estimate the magnitude of
many-body effects (Section 5).
2. BASIC RELATIONSHIPS
The equilibrium
state of a crystal of N argon
atoms at constant pressure is that state for which its
Gibbs free energy, G = U+ Pu - TS, is a minimum. Should the otherwise perfect crystal contain 1zvacancies so that the N atoms occupy N+n
sites, its Gibbs free energy is changed by
6G = n(~i’IJ,+P6v,)-T6S
(1)
Here (SU,+P8u,) m h, is the enthalpy change of
the crystal per vacancy introduced
and SS is
1659
1660
H. R. GLYDE
the entropy
change
“mixing” part
which
has
two
parts;
N!
a
(2)
a’~= ’ log(N_+j-
and a part per vacancy s, due to loss of order
around each vacancy (SS = SS,,, + ns,).
Minimizing
G with respect to n to find the
equilibrium
number n(T) of such non-interacting
vacancies at a temperature T gives directly
= exp(s,/k)
- exp( - h,/k T)
h, = 6U,
= exp(-g,IkT)
as the equilibrium
fraction of vacant sites. With
as the enthalpy and entropy
changes of the crystal on introducing one vacancy
we now evaluate them first assuming negligible
lattice relaxation around the vacancy.
3. hhuAND su ASSUMING NEGLIGIBLE
RELAXATION
3.1 The enthalpy, h,
To evaluate h,, we note first that we may neglect
P&J, as SV~NV,_,the volume of one unit cell, and
Pv, N 2 Cal/mole vat. when P = 1 atm. h, thus
reduces to h, = SU, = 6V,+SE,,,,
the potential
and vibrational
energy changes of the lattice.
Secondly, there is a significant number of vacancies
to be of interest at high temperatures
only. We
may thus choose a lattice model suited to high
temperatures
such as the Einstein model. We
further restrict this to a harmonic Einstein model
as we shall find SE,,, < SVL so that we need not
be too precise in evaluating SE,,,. The perfect
lattice energy in the Einstein approximation using
the high temperature.
Thirring
expansion
is
then(lO) (for T > 50°K)
2
=
(4)
where h and k are Plant’s and Boltzmann’s
constants respectively and V~ is the Einstein frequency.
On creating a vacancy, the atoms in the neighbourhood of the vacancy relax to vibrate around
new equilibrium positions. To find the vibrational
energy change of these atoms, we assume that
Einstein model is still valid for each atom but that
= BVL+4$
[i
(v3(1))2 - NQ]
I
(3)
h, and s, so defined
U
the Einstein frequency of an atom near the vacancy
changes. Its new frequency will now depend on
how far away it is from the vacancy, i.e. on how
much its force constant is changed by the disappearance of the one atom. Although obviously
wrong in detail, this model should be satisfactory
for the approximate
high temperature
thermal
averages we require. Labelling the new frequency
of an Zth neighbour atom to the vacancy as vEu),
from (4), h, is
+gTiqYEy2
= 6V,
1
where SV, is the lattice potential change and
8~~~~)is the frequency change of the lth neighbour atoms when the vacancy is introduced.
3.1 .l The potential energy change 6 V,. The
potential energy of a perfect lattice in the two
body force picture is
V,=N.
[
&$l
1
1=NVol
(6)
where #(or)
is the binding energy of a given
atom due to the presence of its Zth neighbour a
distance rl away. On creating a vacancy, there are
still N atoms bound, but the binding energy of
each neighbour to the vacancy is reduced as there
is a term missing in the sum 4 C,& for these atoms.
On summing this binding change to each Zth
neighbour to the vacancy, we reproduce the sum in
(6) so that the binding energy change of the whole
lattice SV, is just V, the binding energy per atom
in the prefect lattice.
In Table 1 we have tabulated V, at different
temperatures
using a Lennard-Jones
(6,12) potential (with parameters from HORTON and LEECH)
which gives
V,(R)
= ;(c,,
(912-2G
($}
E = 119.Ok
rs = 3.818 A
C, = 14.45392
C,,
The variation
(7)
= 12.11388
of SV, = V. with T was found by
VACANCIES
IN SOLID
noting the change of R, the experimental
value
of the interatom spacing,(r3’ with T and evaluating
Vb(R) as a function of R.
Table 1. Enthalpy
of vacancy creation h, =
SV,+SE,,,
for no ~e~~fft~on arowd the vacancy
using efpatio~s (7) and (8)
P
_zzz=-.
(caljmole vacancies)
0
20
40
60
70
80
T.P.
2024
2022
2012
1984
1966
1943
1933
-23.2
-19.9
-17.4
-16.7
1961
1946
1926
1916
3.1.2 Vibrational
energy change S&&,.
The
Einstein frequency Q. of an atom can be related
to the restoring force cc set up by all neighbours k
of the atom (a = 7”’ = 3 x sV’(+ fl@)) by vE2 =
(2/~n)a. On creating a vacancy the frequency of the
neighbours to the vacancy is altered as there is
again a term missing in XU~, the change being
S(P~@)~ = -(2/m)&’ for aA Zth neighbour to the
vacancy. Summing
over all neighbours
to the
vacancy, the sum in (5) becomes I: ~(P$))~ =
I
-I: a” L-=-vE2. The vibrational
energy change
I
of the whole lattice is then
&g,,,=
_h2
2=
4KTYE
1 f&,2
-----=
4
T
1400
--Yj-(g)
where we have taken 0, = 53°K as obtained from
fits to experimental properties rather than relating
Ye to a particular potential. The resulting values of
SE,,,, are given in Table 1.
3.2 Eniropy of vacaplcycreata’an
The entropy of the perfect crystal (at zero or
small pressure) is
1661
ARGON
AYE
@*
kT
T
x=:---=-
and d#dT = -.x+/T which is obtained by neglecting any temperature dependence of vE’ We see that
the entropy also divides into a potential and vibrational part and we consider the changes in these
parts separately when a vacancy is introduced to
i&d s,.
3.2.1 Potential part of sv: swtPj. The change
SV, in potential energy of the lattice on introduction of a vacancy was SV, = V,, where - Vo is
the binding energy per atom in the perfect crystal.
Thus
which we evaluate by calculating ~V*(R~~~R using
the Lennard-Jones
(6,lZ)potentiaf and snbst~~ting
the experimental values of ~R~~T~13~ (see Table 2).
3.2.2 Vi~ationa~p~t of sv; stitv). sDtv) can be
calculated using the Einstein model from (9) and
the values obtained are nearly temperature
independent
between 60°K and the triple point.
We prefer, however, to use the more accurate
value s,o’) = 2.08 R obtained by NARDELLXand
TEFCZI(~)using a full lattice wave model at 80°K
and assume this is approximately correct down to
60°K (Table 2).
2. ~~~Y~~~
of vacamp creatim s,=syff3
+svCpjfor no ~e~a~a~~~aromd the vaca~
using
the Lemard-Jones (6,lZ)potentiak
Tabk
T"K
S"(V)
= av,jaT
S"(P)
60
O*77R
(2.08
R)
(2.9 R)*
70
1.05 R
(Z-08 R)
(3.1 R)
80
1*39R
2.08 R
3.5 R
T.P.
1.54R
2.08R
3-6 R
..__.__I_
x
[V~~3~~T~+3~~~ln(l
-e-z)]p
(91
&
.~
* Vakxes in brackets are estimated values using
s,(Y) = 2.08 R at 80°K for all temperatures.
1662
H. R.
4. RELAXATION
As it is not possible to take complete account of
an infinite lattice, some assumptions must be made
in evaluating the relaxation around a vacancy. The
basic assumption made here is that at large distances from the vacancy the perfect periodicity of
the lattice is retained. For practical purposes, this
distance was chosen as 6 neighbour atom distance
(~‘(6) R) an d ou t wards. The remaining 5 neighbour atom shells (the 12 first, 6 second, 24 third,
12 fourth and 24 fifth neighbour atoms to the
vacancy) were then allowed to relax either radially
inward or outward with the vacant site as centre.
At each position of these first 5 neighbour atoms
(of
T=SO’K
GLYDE
It is interesting to note that the atoms relax
outward rather than inward. This happens at
high temperatures because the inter-atom spacing
R is greater than the separation distance rO at
which two argon atoms have minimum potential
energy. Thus, an atom in the lattice is attracted to
all of its neighbours including its nearest neighbours. In a perfect lattice, this attraction is equal
on both sides of a given atom, but will be reduced
on one side if there is a vacant site on that side of
the atom. The atom is then attracted
away
from the vacant site by the unbalanced attraction
giving the observed outward relaxation. At low
temperatures where R < ro, an atom in the lattice
R=3%#38
(bf
T=70°K
R=3.8329
i~OO8C
T
1~0060
T
1~0000
(1)
L
FIG. 1. The
Neighbaur
distance
to vocmcy
AV a-13 Cal/moleYBC
L
(2)
Neighbour
(3)
I
_I.
(5)
to vacancy
R that the 1st~5th neighbours to the vacancy relax away or towards
units of the interatom spacing I?).
moving in or out, the complete lattice potential
was evaluated on the I.C.T. Atlas Computer. To
calculate the lattice potential each atom in the
crystal was connected to its first 5 neighbour atoms
through the 6-12 Lennard-Jones
potential;
the
interaction beyond the fifth neighbour shell being
neglected. In this way, the configuration
which
gave the minimum potential energy was found with
this con~guration considered as the relaxed lattice
configuration
and the corresponding
potential as
the relaxed lattice potential. The potential energies
obtained with the lattice at T = 70°K and SOOK,
expressed as a fraction of the original unrelaxed
lattice potential, were 0.9934 t O+OOOland 0.9912
F O*OOOlrespectively. The relaxed positions of the
atoms and the potential change in cal/mole vat.
are shown in Fig. 1.
(4)
the vacant site (in
is repelled by its nearest neighbours and attracted
by all other atoms. As the nearest neighbout interaction is dominant creation of a vacancy will result
in the atom being repelled by its neighbours toward the hole. This inward relaxation has been
found in previous 0°K calculations(2s3) and in a
test computer calculation by us.
4.1 The e#ect of relaxation on h, and s,
The enthalpy h, is reduced by the same amount
as AV. The entropy s, is increased by a contribution .rvcR)due to the temperature
dependence of
the decrease AV (through the (&3V/aT), term in
equation (13)). There will also be an increase in
entropy due to loss of order around the vacancy
as the atoms relax. This is difficult to evaluate and
has not been included so that the listed entropy
VACANCIES
IN
below will be on the low side due to omission of
this contribution.
The final results using the Lennard_Jones(6,12)
potential and including the effects of relaxation are
shown in Table 3.”
Table 3. Enthai~y h, and eniropy s, of vacancy
creation using the Lennard-Jones (6,12)potentiuland
including effects of relaxation
60
f984-(lo)-23
70
1966-13-20
= 1933
1*0.5+0*23+2.1
= (3.4)
80
1943-17-17
= 1909
1.39-t-0.27+2.08
= 3.8
1~54+0~30+2~08
(units of R)
= 4.0
T.P.
(
= 1951
1933-20-17
= 1896
(Cal/mole vat.)
)-estimated
0~77+0.2+2~1
= (3-l)
values.
HALL(~)and KASZAKI’~)evaluate the potential
energy change 6 V, of the perfect lattice when it has
a vacancy at 0°K. In relation to the present picture,
they determine the potential part of h, (or of
g, as h, = g, at T = 0°K). The extent of relaxation is evaluated anal~ically and found to be small
(wl%)givingh,
M 2000 calfmole vat. (h, w 8 V,).
Nardelli and Repanai Chiarotti evaluate g, from
0°K to 80°K. Their calcuIation of g, differs from
the present one in that (a) a higher numerical
value of 6YL for the unrelaxed lattice is quoted,
(b) the method of Kanzaki is used to evaluate the
relaxation and (c) thermal free energies which tend
to be large and difficult to determine precisely
rather than thermal energies are evaluated. h, and
s, a 80°K are then found from g, by defining
s, = dg,/dT at 80”K, considering su constant and
extrapolating
the straight line back to T = 0°K
* The potential parts of h, and sv were also evaluated,
for both relaxed and unrelaxed lattice, using the 6parameter Guggenheim-McGlashan
potential.(ia) In all
cases the results were qualitatively the same with h,
having a slightly higher value suggesting that the present
values are insensitive to reasonable potential changes.
SOLID
1663
ARGON
to
get h,. The intersection of this straight line with
the ordinate at T = 0°K defines h, in g, = h,, -s,T
givingg, = 2.540-6 RT. Although a definition of
h,, h, so defined is not the same as either that
defined here in Section 2 or that obtainable from
experiment.
It leads to high values of h, which
could not agree with experimental values, although
g, certainly could. In a following paper NARDELLI
and TERZF evaluates, separately finding s, = 8.1
R. This is larger than the present s, because a
much larger local expansion around the vacancy
is found. With h, defined in the same way the
larger s, leads to g, = 2700-8.1
RT at 80°K.
It seems to the author important to aim at
maximum precision of h, for, with h, M 6V,, it is
comparison between calculated and experimental
h, which decide whether the two body force picture
is adequate. As with the Hall and Kanzaki calculations, the three-body
force calculations(8*9)
evaluate only SV, at 0°K.
5.2 Summary of experimental results
With the recent excellent
X-ray measurements(13) of the lattice constant a0 right up to the
triple point some direct de~rmination
of the
vacancy content is now possible. These measurements provide the ideal, no vacancy argon density
pX = 4M/Lao3 which can be compared
with
measurement&Q
of the actual density p to obtain
the vacancy content through (n/N) = (p-&/p.
At the triple point, the density differences giveo3)
n
0
X
= (0.03 rt: 0*15)%
T.P.
or
n
0
z
< 0.27;
T.P.
with most of the error occurring in the actual
density measurement.
The X-ray measurements
also provide a new accurate value of /3, the expansivity, which is significantly larger than the older
values.
Historically, the first experimental values of h,
and s, were obtained from the contribution
M
the vacancies make to the specific heat of solid
argon.(14*6*7) These values were obtained in two
ways.
1664
H.
R. GLYDE
In the first method, the experimental value of
C, at low temperatures, where vacancies make no
contribution, was’extrapolated to high temperature
to obtain CPo for a hypothetical lattice containing
no vacancies. The difference between the observed
CP and CPo gives the vacancy contribution
AC,,(AC, = nhV2/kT2) which lead to the values:
g, = 1900-5.4
RT’“’
= o*lS~o
;
(
1 T.P.
and
g, = 1820-5*5RT’r5’
+
(
= 0.37%
1 T.P.
where (n~~)~,~, is the fraction of sites vacant at the
triple point predicted by each g,.
In the second method, G, is first converted to
C, using C, = C, -f12VTftc, Cv” is then calculated
theoretically and AC, found. The values of g, so
obtained are :
n
& = 1580 - 5 RT@)
g, =
1720 -6.9
= 0.96%
( N ) T.P.
= 2.7%
RT’@
0N
g, =
1280-3.4RT@’
g, =
1600-5.1
RT’7’
T.P.
n
0N
We conclude, therefore, that with the present
theoretical g, and the recent experimental
data,
there is no need to include many-body effects to
obtain agreement between theory and experiment.
Keeping in mind that (nl~)~.p. is fairly insensitive
to h,, it is possible that there is some many-body
contribution,
but this is unlikely to reduce the
binding energy of the atoms around the vacancy by
more than 5%. Finally, as many-body
contributions to binding near a vacancy will be much
greater (due to asymmetric
atomic distortion
around a vacancy (16)) than in a perfect lattice, it is
unlikely that the two-body force approximation
in a perfect lattice is in error by more than 4 to 1%.
= 1.3%
5
(
from C, which is also within the X-ray limits.
A more complete evaluation of the two-body
relaxation and full account of it included in su
should reduce g,, slightly in the direction of better
agreement with the value obtained from C,. The
previous theoretical values (gv = 2560-6 RT)(4’
and (go = 2700 -8.1 RT)(5) still do not agree well,
however, with the value obtained from C, as was
concluded by Foreman and Lidiard. If the present
theoretical value is correct, there seems to be no
need to invoke many-body binding corrections.
1 T.P.
= OG3%
T.P.
The values of g, obtained from C, are, however,
questionable
for two reasons. First, they were
calculated when only the earlier inaccurate values
of /3 were available; using the new value of /I, AC,
due to vacancies is only about one fifth’i”) of that
obtained previously. Second, these values of g,
predict (n/N),+,. above the limit (n/N),.,+ 6 0.2%
set by the direct measurements. This leaves, then,
only the values of g, obtained from C, available
for comparison with theory.
5.3 Comparison with experiment
We see that the present theoretical value g, =
1900 -4.0 RT which predicts (n/N),.,,
= 0.055%
is within the limits set by recent X-ray measurements (~~~)~.~. 5 0.2% and compares favourably
with the one value g, = 1900-5.4 RT obtained
Acknowledgments-The
author is indebted to Dr. M.
DE LEENER, Universitg Libre de Bruxelles, who set up
the computer programme and provided many helpful
suggestions along the way. He also wishes to acknowledge with thanks the C.I.B.A.
FeIIowship Trust who
provided a grant during 1964-65
and Professor
I.
PRICOCINE for extending to the author the facilities
of his department.
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VACANCIES
IN
10. PEIERLS R. E., Quantum
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ARGON
166.5
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15. MORRISONJ. A., see p. 10, Ref. 6.
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