PHYSICAL
REVIE%'
B
VOLUME
48,
NUMBER
2
1 JULY
1993-II
Effect
of
pressure
on
atomic
volume
and
crystal
structure
of indium
to
67
GPa
Olaf Schulte and Wilfried
B.
Holzapfel
Fachbereich
Physik,
Uni
Uersitat
Paderborn,
D
33095
Paderborn,
Germany
(Received
10
December
1992;
revised
manuscript
received
16
February 1993)
The
crystal
structure of indium is
studied
by
energy-dispersive
x-ray diffraction
with diamond
anvil
cells
at
ambient
temperature
under
pressures
up
to
67 GPa. The
present
results are
compared
with
pre-
vious volumetric
data for the lower-pressure
region
as
well
as
with
data
from shock-wave
measurements
extending
to
higher
pressures.
A
comparison
of these data
with different
forms
representing equations
of
state
for solids
under
strong compression
shows that
indium
can be classified
as a
"simple"
solid.
No
structural
phase
transitiojis are observed in
the
present
experimental
range
up
to 67
GPa.
I.
INTRODUCTION
The
development
of diamond anvil cells for
the
genera-
tion of
pressures
in the
range
to
100
GPa,
'
together
with
suitable x-ray-diffraction
techniques,
allows for the
determination of structural
parameters including
equa-
tions of
state
(EOS)
on solids
under
strong
compression,
which were
previously
only
accessible
in
shock-wave
cornpressions
with
all
the
limitations
inherent in that
technique.
A
critical
analysis
of various EOS forms
commonly
used
for the
representation
of
experimental
data
extending
into the
region
of
strong
compression
has
shown that one
specific
series
expansion
allows not
only
for
simple
interpolations to
very
strong
compression but
results also in
a
classification of EOS forms
representing
"ideal,
"
"simple,
"
or more
complex
solids. In
addition,
a detailed
analysis
of the
crystal
structure
of
indium
to
pressures
above 50 GPa was stimulated
by
recent
results
of
Takemura,
which
gave
some
evidence for the begin-
ning
of
a
phase
transition
at
pressures
of about
44
GPa.
II. EXPERIMENTAL
TECHNIQUE
A diamond anvil
cell of
Syassen-Holzapfel
type
'
was
used
with
diamonds either in the
standard form
(600-pm
flat diameter) or
beveled with 300-pm
inner
flat
diameter.
Preindented
inconnel
gaskets
(X750)
with
sample
areas
of
180-
or
100-pm
diameter,
respectively,
were loaded
with
the indium
samples
of 99.
9%
purity,
with
a
ruby sphere
of
typically
5-pm
diameter and with either
liquid
nitro-
gen
or mineral oil
as
pressure
transmitting Quid. The
pressure
was determined with the
ruby
luminescence
technique
and
the use of the nonlinear
pressure
scale.
'
Energy-dispersive
x-ray
(EDX)-difFraction
patterns
were
taken either
with a conventionell
2-kW
tungsten
tube and
the conical
slit
system"
or
with
synchrotron
radiation
(SR)
at the
energy-dispersive
spectroscopy
(EDS)
station
in
HASYLAB,
DESY,
also
previously
described.
'
III. RESULTS
Typical
EDX
patterns
of indium
at
two
different
pres-
sures
taken
at HASYLAB are shown
in
Fig.
1. While
the
pattern
at
43.
3 GPa
was
measured in
1500
s,
a
counting
time
of 5000
s
was used
for
the
pattern
at
63.8 GPa.
Since
all the
peaks
can be
indexed
with
respect
to
either
K and
K&
fluorescence
of
indium,
or to
escape
peaks
(e)
from
the Ge
detector,
or diffraction
lines either
from the
gasket
(g)
or from
indium in
its body-centered-tetragonal
(bct)
structure with the
given
hkl
values,
no indication
can be
found
for the occurrence
of
any
contribution
from
any
other
phase
or
any
other
material
even at the
highest
pressure,
with no
support
of
the
previous
observations
about
a possible
phase
transition
of
indium
around
44
GPa.
Also a
careful
analysis
of the
linewidth
of
the
diffraction
peaks
shows
only
the
normal
slight
increase
2000
CO
cU
~o
1500
Q7
CL
V)
1 000
O
O
500
lO
6)
(a)
CD
10
20
p
=
43.4 GPa
O
O
Gt
O
CV
cn.
,
~
U)
O
D)
'~~M~~~~t
~~
30 40
50
60
energy
(
keV
)
3500
3000-
2500-
2000-
~o 1500
O
~
1000—
500-
O
O
O
p
=
63.8 GPa
10 20
30 40
energy
(
keV
)
50
60
FIT&. 1. EDX
diffraction
pattern
of indium at
43.4 and
63.
8
CJPa
with
indexing
for
the bct structure. The
lines labeled
with
g,
K,
and
e refer to
gasket,
fluorescence,
and
escape
peaks,
re-
spectively.
0163-1829/93/48(2)/767(7)/$06.
00 767
1993
The American
Physical
Society
768
OLAF
SCHULTE
AND
WILFRIED
B.
HOLZAPFEL
48
with
pressure
and
no anomalies.
A detailed
evaluation of
55
EDX
patterns,
including
the data of Ref.
13
with
47
patterns
taken
on
increasing
pressure
and 8
on
pressure
release,
results
in the
systematic
variation of
eight
different
lattice
spacings
di,
k&
for the
bct
structure of
indi-
um as illustrated
in
Fig.
2. Best fits
of
the lattice parame-
ters a and
c
to
these
data
are shown in
Fig.
3,
where the
size
of
the dots
represents
the
standard
deviations
in
the
fits of
typically
0.
1%
for
both a and
c.
Effects
from
slight
nonhydrostatic
stresses and
pressure gradients
between
the
ruby
pressure
sensor and the center of the indium
sample
may
thus account for most
of
the
scatter in these
data. No
systematic
differences
are observed
in
prepara-
tions with the
different
pressure
transmitting
media
used
in different experiments.
With the values
ao
=0.
325
20(6)
nm
and
co=0.
4947(l)
nm
from
the
literature'
for
the
lattice
parameters
at
am-
bient
pressure
and temperature
and with the
normalized
values
x,
=a/ap
or
x,
=c/co, respectively,
one can
use
the
analytic
form
of
an
EOS
for
strong
compression
pre-
viously
discussed for
isotropic
solids'
also
for the repre-
sentation of
the
pressure dependence
of
the
lattice
param-
eters a and c
by
the form
p=3Kox;
(1
—
x;)exp[Co(l
—
x,
)],
with
the
two
free
parameters
K,
o
and
C;0
for i
=a
or
c.
The least-squares
fitting
of
this form with minimization
of
the deviations in
x;
at the
given
values of
p
results
in
the
continuous lines
included
in
Fig.
3,
which
correspond
to the
parameter
values
K,
o=90(18)
GPa,
C,
O=25(5),
and
K,
o=
153(17)
GPa,
C,
0=18(5) whereby
the statistical
errors
in
parentheses
do
not include
the mutual
uncer-
tainty
due
to
the correlations
in these
parameters
in
the
fitting
procedure.
The ratio
K,
o/K,
0=1.
7(4))
1 shows
that the
compression
of
indium
at
low
pressures
is slight-
ly
anisotropic,
and
the
ratio
C,
o/C,
0=0.
4(2)
(
1 indicates
that this
anisotropy
decreases
under
pressure.
This
fact
is illustrated
more
clearly
in
Fig.
4,
where the
data
points
are evaluated
from the data
in
Fig.
3
with
V=a
c/2
for the
atomic
volume
of indium in its bct
structure
and
Vo=0.
02615(1)
nm
from the literature
0.
50
~l
0.
45-
~
~
E
0.
40-
E
c5
+~
0.
35-
CD
O
'0
=
030-
05
o~~
woo~o
——
0,25
0
10
30
40 50
pressure
(
GPa
)
60
70
FIG.
3. Effect
of
pressure
on the
lattice
parameters
of indium
derived
from the data
in
Fig.
2.
values.
'
As one
can
see
from
this
Fig.
4,
the
present
data correspond
to a
slight
increase of
c/a
in the
initial
range
of
weak compression
(
V/Vo
~0.
9)
with a
satura-
tion
of c/a
~1.
54
at
strong
compression
(V/Vo
~0.
7).
The few data
points
of
the
present
measurements
in the
range
0.8 (
V/Vo
(1
do
not allow us
to draw
any
further
conclusions
on
the
existence of a
maximum in
c/a
previ-
ously
observed
in
a
detailed
x-ray
study
on
indium
just
in
this
pressure
region,
'
however,
the
present
data as
well
as
the
recent data
also
covering
an
extended
pressure
range
(up
to
56
GPa)
(Ref.
6)
fit
better
to
a
smooth
in-
crease
in
c/a only.
For comparison
with a
possible
face-
centered-tetragonal
(fct)
indexing
of
this
body-centered-
tetragonal (bct)
structure,
the
corresponding
scale
for
(c/a)&„
is
given
in
Fig.
4
on
the right-hand
side.
This
scale
may
show
more
directly
that the
initial
slight
(8%)
distortion
with
respect
to a
more
symmetric
fcc
structure
increases
further
(slightly)
and
stabilizes
under
strong
compression
without
any
sign
to
approach
either the
ideal
value
for
a
fcc
or
bcc
structure.
A detailed comparison
of the
present
EOS data
with
previous
results
from
the literature
*
'
'
'
is
given
in
Fig.
5.
A close
agreement
can be
noticed
with
the recent
x-ray data
by
Takemura in the
range
to 56
GPa as
well
as
with
the data
from the
AIP
Handbook.
These
later
re-
sults
were
derived
from
shock-wave
data
by
correcting
0.25
-.
~
0.20
~
~
Qo
~
~
%~y
~
~
o
~
+,
,
~
o,
~
'-'
%
~
~
'=
ohio
~
~
o
Wo
o
~
~
o
oo
101
~
~
~
~
002
~
oQ
~
~
110
1.
6—
1.
5—
04
~
'
~
ye
—
1.1
fct
0.15
L
0.1
0
-'oog
10
~
A
~
~
oo
I
eooo
112
'
~
o
~
~
~
oy
~
103
o
200
o
oo
~
~
~
j
oj
j
j
j
211 /202
6020 30 40 50
pressure
(
GPa
)
70
1.
4
0.6
I
0.7
I
0.8
v/vp
I
0.
9
1.0
1.0
FICx.
2. Effect
of
pressure
on
lattice
spacings
dqkl
of
indium
with
indexing
for
the bct
structure
at ambient
temperature.
Data for
upstroke
and downstroke
experiments
are
represented
by
full
and
open
circles,
respectively.
FIG.
4. Variation of c/a
ratio
vs relative volume
V/V0
with
additional data
from
the
literature represented
by
dash-dotted
(Ref. 17)
and
dash-triple dotted (Ref.
6)
lines,
respectively.
The
continuous line
illustrates
the
present
best fit
by
the
analytic
form
given
in the
text.
48
EFFECT OF PRESSURE ON ATOMIC
VOLUME
AND CRYSTAL.
. . 769
for
the thermal
pressure
in
these
measurements
and,
thus,
the
close
agreement between the
present
static isothermal
measurements
with these
data shows that
the assump-
tions
made in these
previous data reductions
were indeed
very
reasonable.
Slight
deviations of
the
early
volumetric
data'
to lower
pressures
as well
as for the
first x-ray
data'
towards
higher pressures
can
be noticed. A
more
detailed
comparison
of
these
different
data can be
made,
however,
if one looks
at the
parameters
extracted
by
fitting specific analytic
EOS
forms to
these
data as shown
in
Table I. Since
the
specific
values
derived for
these
pa-
rameters
depend
also on
the
specific
EOS form
used
in
the
fitting
of the
data,
a
close look
at the effect
from
different
analytic
EOS forms
seems to be
necessary
at this
point.
IV.
DISCUSSION
Since
the
present
data
for
indium
cover a wide
range
in
compression (0.
6(
V/Vo
(1)
without
any
indication
of
anomalies or structural
phase
transitions,
Fig.
5,
these
1.
00
Q.
9Q
Y
0.
80—
Vp
0.
70—
0.
60
'
0
I
10
I
20
I I
30 40
pressure
(
Gpa
)
I
50
60 70
FIG. 5. Effect of
pressure
on
relative volume
V/Vo
at
am-
bient
temperature.
Present data
are scaled
with V0=0.02615
nm
for ambient
pressure
and
temperature
from
the
literature
(Ref.
14).
Previous EOS
data
from static measurements
are
represented
by
dotted (Ref.
16),
dash-dotted
(Ref.
17)
and
dash-
triple
dotted (Ref.
6)
lines and
results
from
shock-wave
mea-
surements {Ref.
3)
by
the dashed
line,
respectively.
TABLE I. Numerical
results
for
fits of different EOS
data
sets with different EOS forms. The names
for
the EOS
forms
are
explained
in the text. Isothermal values derived from ultrasonic measurements
are included for
comparison
in
the last section marked as US.
E
(GPa)
41(2)
38(2)
36(2)
37(2)
37(2)
39
39
39
39
39
4.
5(2)
5.
5(3)
6.
0(3)
5.
8(3)
5.
8(3)
4.
6(2)
5.
3(3)
5.
6(3)
5.
4(3)
5.
4(3)
p
(GPa)
67
67
67
67
67
67
67
67
67
67
a.
p-
(%)
0.790
0.
713
0.
686
0.
698
0.696
0.804
0.727
0.734
0.
728
0.727
0.976
0.983
0.
982
0.
982
0.
985
EOS
MU2
BE2
MV2
H02
H12
MU2
BE2
MV2
H02
H12
Ref.
Fit of
present
data
Fit of
present
data
with
K,
fixed
42(3)
52(2)
30(4)
39(1)
41(1)
37(2)
4.
0(9)
4.
5(5)
7(2)
5.
6{9)
5.
1(2)
5.
6(3)
10
26
11
5
56
90
0.
029
0.048
0.658
0.008
0.086
0.043
0.
979
0.975
0.
985
0.
977
0.
984
0.997
H12
16
17
18
19
6
3
41(3)
38(3)
37(3)
38(3)
38(3)
4.
4(3)
5.
3(5)
5.
8(5)
5.
6(5)
5.
6(6)
90
90
90
90
90
0.624
0.542
0.516
0.525
0.527
0.
936
0.954
0.949
0.949
0.955
MU2
BE2
MV2
H02
H12
Fit of
all
data
37(3)
39
5.
7{2)
5.
7
90
90
0.541
0.
759
H11
A11
data
39(1)
40.
3(8)
6.
2(1)
US
24
25
770
OLAF
SCHULTE
AND
WILFRIED
B.
HOLZAPFEL
48
data can
serve
very
reasonably
for
a
detailed
comparison
of
different
analytical
EOS
forms,
which
are either
com-
monly
used
or
just
recently
proposed
specifically
for
solids under
strong
compression.
'
The
commonly
used
EOS forms
are
denoted
here,
respectively,
by
=
3
2 2
MU2:
p
=
box
'(1
—
x
'),
with
C~
=3KO
2
BE2:
p
=
3ICox
(1
—
x
)[1+C2(x
—
1)],
with
Cz
=
3
(Eo
—
4),
MV2:
p
=3%ox
(1
—
x)exp[C2(1
—
x)],
with
C~
=
—,
'(ICO
—
1),
with x
=
(
V/
VD
)
'
.
Ko
represents
the
bulk
modulus
at
ambient
pressure,
and
the difFerent
parameters
C2
are
re-
lated to
the
pressure
derivative
Ko
by
different
well-
known relations also
given
here. These three EOS
forms
were derived
by
different
approaches,
however,
only
for
the representation
of EOS
data
of solids under moderate
compression
and therefore suffer from the fact that
they
diverge
(in
different directions) from
the
expected
behavior of
"regular"
solids under
very
strong
compres-
sion.
Thus,
we
arrive at the
question
of whether the
as-
sumptions
inherent in these different
forms
lead also to
some noticeable
discrepancies
in the evaluation of the
present
EOS data
for
indium,
which cover
a
reasonably
wide
range
in
compression.
On the other
hand,
more
re-
cently,
two other
(second-order)
EOS
forms
were
pro-
posed
'
with the correct
asymptotic
behavior
at
very
strong
compression.
While the form
H02:
p
=3Kox
(1
—
x)exp[Co&(1
—
x)],
With
C02
(+0
may
be
considered
just
as
a
small modification
with
respect
to
the form
MV2,
using,
however,
the
correct
asymptotic
exponent
5
for the term x
',
the
best
asymp-
totic fit is
obtained with the
form
H12:
p
=3Kox
'(1
—
x)exp[C, O(1
—
x)
+xC,
2(1
—
x)],
strong
compression,
since
the
scaling
by
the
Fermi
gas
pressure
pFG=p„G
/x
leads to
the
limiting
value
of
0
gH
~0
for x
—
+0
and
thus allows
for
a
simple
interpola-
tion
between
this value
for
ultimate
compression
and the
finite
value of
qH
=
in(3Ko/pFo
)
=
—
C&o
for x
=
1
corre-
sponding
to
zero
pressure.
A direct representation
of all
EOS data
for indium
in this
gH-x
linearization scheme
is
shown
in
Fig.
6.
In this
figure,
one can
see
more
clearly
differences
between the data
sets
than
in
Fig.
5. One
can
notice a
deviation
of the reduced
shock-wave
data at
low
compression
(small
dots),
which
may
be due to the
use
of
a
wrong
value for
Vo.
A new
fitting
of these
data
with
Vo
as
fit
parameter
and rescaling
with this new
Vo
results
in
the data
points displayed
as
small
crosses in
Fig.
6. The
value of this
new
Vo
is
slightly
larger
than the value
from
the literature.
'
All
further
calculations done
with this
data set
are
using
this new
VO=0.0264 nm .
Drastic
de-
viations are
shown
by
the data
of Refs.
17 and 18. These
data
sets
are omitted
in the
following
fits.
Figure 7(a)
il-
lustrates
in
similar form
the
present
data
together
with
various fitted
curves
in the experimental
region
0.85
~
x
~
1.
For a
more
detailed
discussion of the
different
EOS
forms an extrapolation
of
all the fitted
curves
into the
region
of
very
strong
compression
(x
~0)
is
given
in
Fig. 7(b),
where the
divergency
becomes
obvi-
ous.
Since
the scatter
of the
present
as
well
as
previous
data
at x
~1
becomes
very large,
the
limiting
value
gH
0
using
the isothermal
value for
Ko
from ultrasonic
mea-
surements
'
is also
included
in these
diagrams.
The
fitted
curves, corresponding
to
the data
given
in the first
block
of
Table
I,
seem to
deviate
from each
other
only
marginally
within
the experimental
region
0.85 x
~
1,
however,
due to
the
different
curvatures
inherent
in these
second-order
fits, slight
differences
for the
corresponding
values at
x
=
1 can be
noticed also
in
this
diagram
ac-
cording
to
the
differences in the
parameters given
in
Table I. Significant
differences
between
the curves
H02
and
H12 cannot be
noticed in this
limited
range
of x and
the
agreement
with the
ultrasonic values
appears
to be
very
good
for
any
of
these curves.
However,
if one looks
at
the
g~-x
plot
for the
extended
region
in 0&x
&1,
with
C,
o
=
—
ln(3It.
o/p„o
)
and
C,
p
=
—
(Ko
3)
C~o.
Thereby
p„o
=a„o(Z/Vo
)
with
a„o
=23.
37 MPa
nm
represents
the
pressure
of
a
free-electron
gas
(Fermi
gas)
with an electron
density
of Z electrons in the volume
Vo
for the solid
at
ambient conditions. It has been discussed
previously'
'
that EOS data
can
be
represented
very
conveniently
in different
'
linearization
schemes"
to
illus-
trate
specific
features
of these
data.
Thereby,
a
close
rela-
tion can
be
noticed
also
between
a
specific
EOS
form
and
the related
linearization
scheme.
'
The
specific
lineariza-
tion scheme
with the
"generalized
stress
parameter,
"
-3.
0
-35—
-40—
-4.
5
0.80
vvv
ref.
24,
25
ref.1 6
ref.
1 7
ref.
1 8
ref.1 9
ref.3
ref.
3,
Vo
=
0.0264
nm~
ref.
6
present
0.85
~
~
~
~ ~
~
~
0.90
X
I
0.95
@+pe.
~~.
.
*
'
'
~
~ ~
.
~
g~~
~
v4
1.
00
=ln
px
pFo
(1
—
x)
is
specially
suited
to
represent
data for solids under
FIG. 6.
Different data sets
(present
and from
the literature)
represented
in
qII-x
scaling.
The small dots
refer to
shock-wave
data
from
Ref.
3
and the
crosses
represent
the
same
data
after
rescaling
with
V0
=0.
0264
nm'.
48
EFFECT
OF PRESSURE
ON ATOMIC VOLUME
AND
CRYSTAL.
.
.
771
-3.
5
-4.
0
eE2
MV2
-4.
5
0.8
0.9
1.0
-2
-3
-4
-5
'
0.
0
i
0.
2
I
0.
4 0.6
I
Fpi.
7a
0.
8
1.0
drastic
divergences with
respect
to the
limiting
behavior
(ilH
~0)
can
be
noticed for all the
previously
used
forms
(MU2, BE2,
MV2).
It is also
seen
that
the different
cur-
vatures
inherent
in
these forms are
responsible
for
the
(slight)
differences
in the values for
ICO
derived
in the
fits
of
these forms
to the
present
data
(Table
I,
first
block,
and
Fig.
7).
The
deviation
between the
linear
(two-
parameter)
form
H02 and the
(more reasonable)
nonlinear
(two-parameter) form
H12
is
very
small
even
at
very
strong compression
and
suggests
that
already
the
one
pa-
rameter form
H11
with
C,
2=0
and the
corresponding
correlation
for %0=3
—
(
—,')ln(3Ko/p„z
)
related
to
the
slope
of
this
straight
line
represents
the
data
perfectly.
Thus,
the
one-parameter
erst-order
EOS
form H11
de-
rived from
H12
with
C,
2=0
and with
Eo
coupled
to
Ko/p„z
as
discussed before Ats all the
present
data per-
0
fectly
even if
just
the ultrasonic value
for
Eo
is used
without
any
free
parameter
left for
Atting,
while
all the
other
(second-order)
forms
(MU2, BE2,
MV2)
would
re-
quire
still
fitting
of their values for
Eo
even if
the value
for
Eo
is
used
from
ultrasonic
measurements.
One
may
now
wonder
how
it comes that
one
special
one-parameter
form
(Hl
1)
can fit
the
experimental
data
equally
well or
FIG.
7.
Representation of
the
present
EOS
data
with
diferent
Atted
EOS
forms
denoted
by
MU2, BE2,
MV2,
H02,
and H12
using
the
scales
stress
parameter
gH
and the
scaled
length
parameter
x
(for details
see
text). The
values for
gH
at
ambient
pressure
from
ultrasonic
data
(Table
I)
are
represented
by
the
diamond
(Ref.
24)
and
the
arrow
(Ref.
25),
respectively,
whereby
the
direction
of the
arrow
corresponds to the
ultrason-
ic value
for
Eo.
The
extended
diagram
for
extreme
compression
{O~x
~1)
includes
also
the
EOS for
"ideal
solids"
as
a thin
double line.
even better
(see
Table
I)
than
other
two-parameter forms
(MU2,
BE2,
MV2)
more
commonly
in
use.
Due
to
the
correct
limiting
behavior
for
x~O
incorporated
into
H11,
however,
more
physically
reasonable
assumptions
are incorporated
into H11.
Therefore,
H11 is
very
ap-
propriate
for
the
representation
of EOS data for
"simple"
solids.
Special
solids with
special
electronic
"transitions"
under
pressure
can
show,
however,
significant
deviations
from this
simple
behavior
(Hl
1),
as
discussed
just
recent-
ly
with
respect
to the
special
behavior
of the
regular
rare-earth
metals or with
respect
to
thorium under
strong
compression.
"
In
any
case,
it can
be
noticed
in
Fig.
7
that the
(best
fitting)
EOS for indium also
comes
rather close
to
the
double
line which
represents
the
general
behavior
of
"ideal"
solids
just
discussed in a recent
publication.
At
this
point,
the critical
comparison
of the
present
data
with
previous
results
represented
in Table
I needs some
further
discussion. Since the
comparison
of the
Ats with
different
EOS
forms for the
present data,
either with free
or fixed
(best
ultrasonic)
values for
Ko
in
the first and
second blocks
of Table
I,
shows
in
general
a
slight
su-
periority
in the
standard
deviations
crz
for H12
(or
at
least no
significant
deterioration
with
respect
to
any
oth-
er
form),
the
fit
of
the
previous
data is therefore
per-
formed
only
with
H12. If
one
compares
then
the
numeri-
cal values and statistical
errors for
the
parameters
Eo
and
E
o,
one can notice
that
some
of
the
values are in fact
in-
compatible
with each other within the
given
statistical
er-
rors,
and
these inconsistencies become even more evident
when
all the
experimental
data are
compared,
as shown
in the third
section of Table I.
However,
if one takes into
account that the values
of
the
fitted
parameters
Ko
and
Eo
are
strongly
(anti-)
correlated,
as shown
by
the
data
for
the
correlation coefticients
—
e
—
+1,
one can notice in
the representation
of
the
corresponding
error
ellipsoids
il-
lustrated
in
Fig.
8 that
only
the earliest
x-ray-diffraction
data'
are inconsistent with the other
volumetric,
shock-
wave,
and
x-ray results. The
value
for
Xo
from
the
high-
pressure
ultrasonic
study
may
be
affected
by
a
larger
er-
ror
than
given
in
the
original
work,
since this
ultrasonic
high-pressure
study
did
not use
any
pressure
transmitting
fluid to avoid
possible
systematic
errors due to
nonhydro-
static stresses.
Whereas
the former
discrepancy
could be
resolved
possibly
by
the use of
a
revised
pressure
scale in
the evaluation of the
early
x-ray
data,
'
a
difference
be-
tween
ultrasonic
and volumetric measurements could
be
expected
(after the standard correction from adiabatic
to
isothermal
conditions),
at
least
in
principle,
for noncubic
materials
such as indium in
polycrystalline
form,
where
different
elastic-plastic
boundary
conditions between
the
individual
grains
in the
compacted
samples
can lead to
a
distinction between two
limiting
cases
'
—
the
(stress-
free)
Reuss
case
for ideal
slipping
of the
grains
and the
(strain-free)
Voigt
case for ideal
sticking.
What is
most
striking
in
Fig.
8
(as
well as in
Fig.
9,
to be discussed
later)
is,
however, the
fact
that the correlation of
Ko
and
Eo
incorporated
in H11
by
the
assumption
of
C&2
=0
and
represented
in
Figs.
8 and
9
by
the
dotted
curve
(Hl1)
cuts
through
all
ellipsoids
(besides
the
erroneous
one
of
Ref.
17)
just
in
the
region
of
largest overlap.
Thus the
