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Hugoniot equation of state of the lanthanides
HUGONIOT
EQUATION OF STATE LANTHANIDES*
W. J. CARTER, J. N. Los Alamos
Scientific
FRITZ,
Laboratory
S. P.
of the University (Receired
and R. G. MCQL’EEN
MARSH
of California,
13 August
OF THE
NM 87.544. U.S.A.
Los Alamos,
1974)
Abstract-Shock wave studies of the lanthanide series show the existence of a very high pressure phase transition in all members of the series. The data show that this transformation is of necessity to a more incompressible phase and has been identified with melting. Thermodynamic considerations allow calculation of the solid-liquid phase boundary from these data; the results indicate that all the rare earths melt anomalously at sufficiently high pressures. This can be understood in the context of a “two-fluid” theory, in which the composition of the liquid along the phase boundary changes continuously with pressure due to the degree of pressure-induced electronic transition present in the liquid. Hence, at sufficiently high pressure, the density of the liquid becomes greater than the density of the contiguous solid and dP/dT becomes negative.
high speed smear
Briefly.
The
study
pressure recent
of lanthanides and temperature
years[ld], clues
elements
in the
program
to
the
metallic
techniques;
earths except Included
promethium
systems
(about
4 GPa)
energy
by standard
changes to the
the valence
adjacent
higher potential
0.7 GPa
in
well-known
cerium,
attributed
promotion[5],
if it occurs,
the periodic
the 4f and mixed atomic
number.
in conjunction
coupled
table,
velocity
at the interface
material
having
copper
experiments
[8].
The
melting
Rankine-Hugoniot
EH
together
at zero
such
direct
large
elsewhere[6,7]
and will
been
properties
of
transition
at
a 4f+5d
A similar
in these
equations
-
up j/u,
(1)
-
PO =
us4,
I vo
(2)
(3)
transition
transition
linear
and specific 0 refers
pres-
(E”)
energy
to the state of
representation
of the data is given by the
u, - u, relation
for u, = co + su,
gap between
appears
where
with
to occur
conditions
the Hugoniut
determined If rigidity
for
are
(4)
is linearly
of dynamic
Hugoniot
high pressure
adequately
of the
United
cur and the slope,
effects and possible the
low pressure
intercept
of an infinitesimal
should
co = [(aP/+),]‘12 related
bulk
to
the
modulus,
then reflects
phase changes
correspond
pressure
s, are
of least squares.
pulse,
to the
or the bulk
(at P = 0). Since the slope
pressure
(Sl,/aP),,
a nearly
linear
derivative a
of
linear
dependence
the
u, -u, of B,
on the pressure.
discussed in detail
intercept,
from the data by the method
neglected,
velocity
the auspices
(V,),
The subscript
generally
increases
shock
not be repeated
volume
then allow
quantities
material.
A convenient
electron
(u,, u,) points
as one proceeds
generally
under
the unshocked
here.
If the linear equations
*Work performed under Atomic Energy Commission.
of state.
standard
of the thermodynamic
along the Hugoniot.
which
series.
techniques
(16
with the experimental calculation
adiabatic have
and particle
and a standard
Eo = (PH - Po)( Vo - vH)/?
-
sure (PH), specific
Techniques The experimental
of Walsh [6],
equation
as the
conservation
PH
sound speed,
measurements
used
V”l V,, =
2. EXPERIMEVAL (a)
Hugoniot
was
the
u,, were
with
fee-fee
since the energy bands
method on pressure
of the sample
a known
Deoxidized
volume
bands
would be expected
The electronic
most of the lanthanide
allow
bands,
higher pressures
Sd-6s
with
conditions
velocities,
driven
the bulk to
is a case in point.
to occur at increasingly across
by the impedance-match
range extends
rearrangements
change
The
other rare earths,
(Z = 71).
or material
was used
u,, through
scandium
may well be expected
electronic
in turn profoundly material.
a
photography
velocities,
0.6. In view of the small
between
internal
using the continuity
shock
explosively
of the elements,
of the interatomic
produce
would
the
determined
are all the rare
high pressures,
A V/ V,,, of nearly
and
dynamic
Particle
(I GPa = IO kb) to nearly 200 GPa
These
differences
pressure
using
samples.
offer
of state of these elements
group III elements
the large compressibilities compressions.
various
camera
the shock-wave
these
undertaken
(Z = 39). The pressure
elements.
to determine
field in of
(Z = 61) and lutetium
from the lowest obtainable for some
have
in this study
also are the similar
(Z = 21) and yttrium
We
regime
included
of
such studies properties
state.
high pressure
conditions
an active
because
electronic
to study the equation
in the very
extreme
has become
primarily
valuable
wave
under
as a function
States
following 741
u,-u,
relation
holds, the Rankine-Hugoniot
can be used to express of
convenient
volume analytic
along
the pressure the
equations
and energy
Hugoniot (with
by
the
POand E,
742
W. J. CARl’FR
taken
to be zero):
t-t a/.
transverse
sound
individually
bulk sound velocities,
?y-V/V,,=I-$ we?t7
p,, =
TJS)?
(I E”
There
results
physical
properties
is a summary needed
Included
are both the directly
densities
and shock
this of
all
close-packing, neighbor
between
sequence
increased the
bee
only
and dipole-dipole
before
atom.
However,
average
The
values
Nuclear
part
from
The
the
made
samples
Research
Corporation
of
were
Table Faemen, z
Valence
2,
+3
hcp
with
Eu differs
from the other
elements
a significant
considerable
least well partly
speeds[9]
samples
are
on the samples obtained
Division
Longitudinal
with
of
and thermodynamic
Hugoniot
properties
E,Wtl-C.“ic Conflguratbn
3.196
because
such a highly
qualitatively
(kr?&
3.07
density
density
Ce,
change studied
variations
4.29
here,
between
the
inherent
in working
material. on
the
other
the rest of the rare
hand, earth
a K -L,
C. Ulka-K)
-Ia
5.60X10’
,2x,0-
,. 16 1.12
Y
39
+3
hcp
4d%’
4.576
4.38
2.52
3.26
3.09
10.6
La
57
+3
d!lcp
5d%s’
6.142
2.69
1.50
2.05
2.00
4.9
.3,
CF
56
+3(+-l)
ICC
4f’5d’6sa
6.729
2.33
I. 34
1.73
2.06
6.5
.37
pr
59
+3
dhcp
428s’
6.757
2.14
1.5,
2. I,
1.92
4.6
.34
tid
60
+3
dhcp
41’69’
6.963
2.64
L.60
2.16
2.09
6.7
Srn
62
+3
rhomb
4fsa
1.460
2.88
1.64
2.17
1.81
E”
63
+2
bee
4r?39=
5.290
2.34
l.64’
l.65
l
6.5 32
.45 *
differs
series.
at standard conditions
(kz;s)
in
has been used in the analysis)
of the difficulties reactive
in
structure
the data for Eu are the
of all the materials
of
from
ct ( km/e)
5.57
However,
of large
(an average
The
and
3d14s’
determined
because
and partly
for the
indicating
the
is either
at the transition.
of the 4f-band
that
this transition
shock
front.
Ce of a
is changing
indicate
in the
u$lm~~ SC
the data
volume
I. Physical
crystal Structure
small.
except
a characteristic
the conduction
Chemicals
America.
zero or very
associated
0.36 the slope
element
compressibility
cases
these
Eu and Ce.
of the tripositive
sound
of measurements
change
of
except
of a two wave
contains
and
The
whose
In most
This plane.
to describe
character
every
to a phase
rapidly.
(l-3).
that the data show the existence
at the expense
densities
used in this study. most
in view
state probably
pressure.
of
a
then
fits of the II,+,,
A V/ V. of approximately Hugoniot
to within
in the I(,+,,
general
transition volume
I are those of the
The
behind with care,
quantities
fit is required
by SO to 100 per cent,
to
at zero
II,-u,,
that,
least squares
increases
Er and Tm transform
state of most of these elements.
population.
the
l-5
is the same for all elements
At a compression of
more
free
of Sd-character
magnetic polymorphic
(densities,
to equations
in Figs.
data. (initial
in both the sample quantities
consistently,
according
adequately.
Hugoniots
is
valence admixture
second
data
all
neutral
band in the metallic
the
In addition,
listed in Table
errors
is presented
solid lines are linear
as pressure
melting
configurations
of
takes place in
temperature.
possibly
crystal
through
fee
these
Hugoniot
the derived
data; in all cases a two segment
in energy
+dhcp+
can hc measured
of one per cent:
information
coefficient,
quantities
has shown
The
modifications
forms
Experience
the
of the
and compressions
parameters
are
that
tenths
the
velocities
the
elements
[4] has noted
except
phase
pressure
these allotropic
wave).
of
measured
pressures,
which
Values
expansion
and the derived
velocities,
reflect
1,which
Eu.
hcp+rhomb
these elements
as one proceeds
for
at and near room
electronic
zero
electrostatic
Jayaraman
transitions
earths
few
zero-pressure
Except
these
differing
interactions. the
study.
wave
determined
from
calculated.
obtained from [IO]. 2 is a detailed listing
these velocities
in the
table. This is shown in Table
of appropriate
for
structures
similarity
of the rare
across the periodic
nearest
and discussion
is a striking
ch, were
heat, C,, and linear thermal
the shock (b) Experimental
method,
a, were Table
particle
2(1 - n.s)!.
c, and c,, were
specific
and the Cu standard)
Co2T2
=
velocities,
by the pulse-echo
.45 1.02
Gd
64
+3
hcp
4f’5d’6sa
7.877
2.95
1.69
2.21
2.99
6.4
.3,
Tb
65
+3
hCP
4f%8=
6.206
2.95
1.68
2.22
1.72
1.0
.60
Dy
66
+3
hcp
4f%’
6,398
3.07
1.76
2.26
1.74
6.6
.77
“0
67
+3
hCP
4?‘6e’
6.734
3.2,
1. 66
2.39
I. 64
9.5
.99
Er
66
+3
hcp
4f-36e2
9.039
3.13
1.83
2.31
1.61
9.2
.92
Tm
69
+3
hcp
4f%’
9.269
3.02
1.77
2.23
1.60
11.6
1.08
Yb
70
+2
fee
41”a=
7.036
1.94
1.12
1.45
1.45
25
1.09
Ce
Hugoniot
equation
Table 2. Hugoniot 00 nlms)
US (km/s)
2/i,
P ,GPaj
P (MC, ¶I?)
v/v0
u, v.3) (km/s)
143
of state of the lanthanides data
6% (~gjm')
U, (km/s)
6.756
2. II
6.764 6.762 6.75, 6.756 6.761 6.163 6.12, 6.751 6.761 6.728 6.134 6.765 6.75, 6.760 6.,30 6.756 6.764 6.762 6.154 6.164 6.764 6.759 6.759 6.755 6.769 6.765 6.763 6.755
2.39 2.45 2.65 2.82 2.89 3.03 3.06 3.13 3.28 3. 3, 3.73 3.96 4.2, 4.69 4.18 4.74 6.04 5.09 5.40 5.38 5.64 5.1, 5.14 6.06 6.15 6.32 6.24 6.35
6.983 6.9*2 6.984 6.983 6.965 6.965 6.982 6.983 6.921 6.976 6.983 6.980 6.983 6.994 6.978 6.984 6.991 6.084 6.990 6.988
2.16 2.48 2.55 2.72 2.14 2.91 2.90 3.02 2.99 3.1, 3.15 3.30 3.49 4.0, 4.62 5.18 5.44 5.56 5.76 5.83
7.460 7.468 1.464 7.469 7.464 1.463 1.463 7.460 7.470 7.463 7.463 7.465 1.454 7.465 1.460 7.459 7.459 7.454 7.460
2.1, 2.51 2.76 2.79 2.92 2.98 3.11 3.15 3.2, 3.28 3.40 3.71 4.31 4.50 4.55 5.32 5.4, 5.73 5.81
5.290 5.290 5.290 5.29" 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290
1.64 2.00 2.12 2.38 2.43 2.44 3.14 3.49 4.11 4.22 4.39 4.59 4.92 5.31 5.64
", (km/s)
IJ 0 (CI'a) (M&m',
vtvo
a (C") (km/w
Scandium 3.196 3. L96 3.196 3.193
3.192 3.191 3.19, 3.198
4.578 4.556 4.548 4.551 4.564 4.664 4.562 4.664 4.561 4.570 4.553 4.553 4.558 4.595 4.576 4.598 4.605 4.6R
4.29 5.2, 5.32 5.6, 6.02 6.69 1.04 7.63
3.26 3.73 3.75 3.86 3.93 4.0, 4.2, 4.31 4.3, 4.61 4.5, 4.62 4.91 5.35 5.76 6.39 6.59 7.27
0.15
13
0.95 1.40
16
1.93
25 3,
2.58
55
3.00 3.75
66 94
1.43 1.55
5 8 L1 13 L7 23 24 2, 30 32
I. 56 1.E6 2.19
33 42 54
2.52 3.03 3.10
i6 39 94 122
0.32 0.4, 0.64 0.73 0.91 1. 1, 1.21 1.34
3.58
3.20 3.73 3.89 4.24 4.70 6. I9 5.58 6.14
4.58 4.98 5.20 5.46 5.60 5.8, 6.29 6.35 6.5, 6.70 6.88 6.92 7.33 1.7, 8.14 8.73 8.10 9.21
1.000 0.857
4.74
0.822 0.752 0.679 0.615 0.6'13 0.521
4.94 6.42 5.98 6.6, 7.13 7.95
1.000 0.914 0.875 0.834 0.614 0.77, 0.725 0.118 0.694 0.682 0.662 0.656 0.622 0.592 0.562 0.52, 0.529 0.50,
4.29 4.45 4.63 4.73 4.92 5.21 5.26 5.39 5.51 6.62 5.66 6.98 6.3, 6.16 7.3, 7.4, 8.08
1.21 1.31
6 * 13 18 20 24 25 28
6.76 7.94 8.34 9.24 10.10 10.46 11.06 ,,.I2 11.64
1;45 1.66 1.76 2;01 2.19 2.21 2.29 2.45 2.49 2.62 2.64 2.83 2.83 2.88 3.02 3.10 3. 16 3.21 3.22
33 42 4, 58 69 71 73 84 86 95 96 108 IL0 112 124 12Y 135 136 138
11.19 12.16 12.2, 12.78 12.6* 12.54 13.09 13.15 L3.23 13.10 13.29 13.56 L3.28 L3.5, L3.45 13.60 13.56 13.93 13.69
0.35 0.46 0.11
0.93 1.02 1. 18
1.000
0.811 0.131 0.669 0.64, 0.61, 0.606 0.580 0.572 0.570 0.554 0.554 0.629 0.533 0.53, 0.516 0.514 0.511 0.516 0.509 0.499 0.509 0.498 0.502 0.49, 0.499 0.485 0.493
4.32 4.44 4.7, 4.96 5.05 5.24 5.2, 5.39 5.51 5.56 5.84 5.96 6.29 6.65 6.59 6.66 6.69 6.94 ,. 13 7.15 7.40 7.42 1.4, 7.68 7.18 7.88 7.92 ,.95
0.852
NCdpU”Ill
6.142 6.140 6.L35 6.130 6.138
2.05 2.40 2.54 2.70 2.70
6.133 6.136 6.132 6.L35 6.L51 6.13, 6.13, 6.13, 6.119 6.155 6.13, 6.082 6.136 6.140 6.166 6.153 6.154 6.149 6.156 6.134 6.142 6.110 6.15, 6.142
2.8, 2.82 2.81 3.02 3.14 3.32 3.50 3.76 4.28 4.16 4.64 5.01 5.18 5.33 5.39 5.66 5.91 5.9, 6.01 6.24 6.32 6.46 6.44 6.49
6.729 6.736 6.736 6.12, 6.132 6.130 6.726 6.134 6.719
1.73 1.5, 1.90 2.33 2.29 2.6, 2.64 2.85 3.09
6.734 6.726 6.773 6.778 6.730 6.759 6.750 6.732 6.770 6.739 6.764 6.731 6.772 6.731 6.771 6.756 6.750 6.732
3. 32 3.60 4.15 4.13 4.53 4.99 5.24 5.30 5.55 5.54 5.80 5.84 5.94 5.93 6.2, 6.26 6. 36 6.36
5 I
0.35 0.47 0.65 0.72
0.74 0.76 0.79 0.94 1.02 1.1* 1.31 1.44 1.7, 1.18 2.02 2.32 2.34 2.48 2.52 2.6E 2.8, 2.69 2.93 3.09 3.16 3.25
,I
12 13 13 I4 L, 20 24 28 33 46 46 58 7, 75 81 84 93 104 106 108 118 123 128
3.27
130
3.29
131
0.39 0.50
4 6
0.14
0.79
I2 12
0.94 0. 95
17 1,
1.02 1. I*
20 24
1.29 1.42 1.73 1.73 1.96 2.26 2.42 2.46 2.59 2.62 2.76 2.80 2.80 2.86 2.99 3.08 3.16
29 34 49 40 60 76 86 8R 9, 98 108 110 113 114 127
3.18
*30
136 137
6.14 7.20 7.54 8.09 8.29
1.000 0.852 0.814 0.758 0.740
4.31 4.44 4.63 4.11
8. 33 8. 38 8.51 8.92 9.12 9.53 9.19 9.96 10.44 10.7, 10.88 L1.34 11.20 Il.50 11.56 Il.69 11.9, 11.92 12.0, 12.14 12.30 12.28 12.52 12.48
0.736 0.132 0.720 0.688 0.675 0.644 0.62, 0.616 0.566 0.571 0.564 0.536 0.548 0.534 0.533 0.626 0.514 0.516 0.513 0.505 0.499 0.498 0.492 0.492
4.73 4.15 4.78 4.96 5.05 5.24 5.39 5.66 5.98 5.9* 6.29 6.66 6.12 6.89 6.94 7.15 7.40 7.42 7.4, 7.68 7.78 7.88 1.92 7.95
6.73 8.99 9.13 9.8, 10.32 10.40 10.64 10.52 IO.*5 11.03 II. I2 Il.63 11.68 Il.96 12.36 12.65 12.66 12.71 12.7, 12.93 12.93 12.84 13.00 12.93 13.31 13.42 13.51
1.000 0.749 0.736 0.6R2 0.652 0.64, 0.638 0.640 0.619 0.610 0.605 0.582 0.580 0.5,;3 0.54, 0.538 0.636 0.533 0.528 0.623 0.521 0.528 0.516 0.624 0.508 0.503 0.498
4.32 4.44 4.71 4.76 4.95 4.96 5.05 5.24 5.39 5.56 5.96 5.98 6.29 6.6, 6.89 6.94 7.13 7.15 7.36 1.40 7.42 ,:4, 7.68 7.78 7.86 7.91
1.13 1.15 L.28 1.42 I.64 2.21 2.64 2.79 2.88 2.9, 3.06
8 13 14 18 19 2, 21 25 25 30 35 52 ,, 96 106 112 120 124
8.12 8.50 9.41 9.4, LO. 12 10.22 10.43 10.41 10.83 11.02 11.42 11.7, 12.74 13.4" Ii25 14.38 14.12 14.37 14.69
1.000 0.860 0.822 0.742 0.138 0.688 0.683 0.670 0.666 0.644 0.634 0.611 0.593 0.549 0.521 0.490 0.486 0.484 0.486 0.476
4.32 4.44 4.71 4.73 4.95 4.96 6.05 6.05 5.21 6.24 5.39 5.56 6.10 6.59 7.15 ,.36 7.4, '1.61 7.71
0.34 0.69 0.11 0.89 0.9, 1.15 1.16 1.24 1.25 1.38 l.62 2.03 2.20 2.22 2.15 2.90 3.11 3. 14
6 14 15 19 22 2, 2, 30 31 35 4s 65 74 15 109 118 133 136
7.46 8.63 9.94 10.00 10.73 11.08 11.83 11.82 12.06 12.06 12.56 13.29 14.10 14.62 14.58 15.48 ,5.*ti 16.30 16.24
1.000 0.866 0.751 0.74, 0.695 0.674 0.631 0.63, 0.620 0.619 0.594 0.562 0.629 0.510 0.512 0.482 0.470 0.45, 0.459
4.32 4.72 4.74 4.95 5.04 5.25 5.2, 5.3, 5.38 5.54 5.85 6.39 6.61 6.63 1.36 1.54 7.82 7.8,
1.000 0.605 0.761 0.671 0.66, 0.57, 0.499 0.47, 0.446 0.420 0.422 0.405 0.406 0.403 0.370
4.31 4.43 4.71 4.,4 4.96 5.56 5.86 6.39 6.5, 6.69 6.92 7.16 7.4, 7.91
6.98 0.35
0.45 0.70 0. R 0.9, 0.92 1.00 1.00
6
Europlum 5.29 0.39 0.51 0.78 0.81 1.03 ,.5R
1.62 2.28 2.45
2;54 2.73 2.92 3.1, 3.65
4 6 10 IO 13
26 34 50 55 59
66 16 89 106
6.5, 6.95
7; 80 ,. 93 9.17
10.61 11.10 11.85 12.59 12154 *3.0* 13.04 13.12 14.26
W. .I. CARTER
144
et al.
Table 2 (Confinued)
PO (Mnlms)
” Om/s()
I. 877
2.2,
I. 876 7.877
2.49 2.6,
0. 32 0.44
7.875
2.80
0.87
P
“. (km/s)
VIVD
(GPa)
u. (CU) ocrn/S)
CNPalm’)
”
PC
f YKlrn’)
(km/s;
Il.
P
(km/e)
P
(GPP)
us (Cd (km/e)
VW.
(M./m’)
Gadolimum
0.69
6 9
7. a* 9.05 9.48
9.04
1.000 0.870
9.039
2.31
4.3,
9.058
2.6,
0.30
7
10.25
0.883
4.3,
4.44 4.72
a. 992
2.74 2.7,
0.40 0.4,
1” 10
10.54 10.68
0.853 0.847
4.43 4.44
2.88 2.86
0. 63 0.65
17 17
11.6, ,,.‘I4
0.780 0.77,
4.72 4.74
o;s2
22
,2;45
0.72,
4.95
1. 16 1.20
35
13.84 14.03
0.654
5.38
31
0.645
5.44
15
10.36
0.631 0.760
9.046 9.064
1.000
7,877
2.82
10.43
0.155
4.74
2.99 3.22
0;86
15
7.817 7.878
20
11.08
0.71,
9.056
L. 08
11.86
7.877
3.20
0.664 0.662
5.2,
9.054
3.0, 3.36
7.877
3.22
1.08 1.13
27 27
4.95 5.2, 5.27
7.876
3. 35
1.22
32
12.31
9.057 9.066
3.40 3.60
1.28
4,
14.3,
0.634
7.876
3. 36
12.64 L2.65 13.47
9.049
3.76
1.52
52
15.15
0.597
5.85
3.48 3.8,
34 37 48
0.623
7.815 7.881
1.27 1.35 1.58
a. 991 9.04,
4.13
1.85
69
Ifi. 3,
0.55,
6.29
7.818 7.877
4.26 4.6,
7.875 I. 864 I. 846
4.51 4.51 4.82
7.873
29
11.89 L2.16
0.646 0.636 0.613 0.565
5.38 5.44 5.54
1.92
65
L4.27
0.562
5.85 6.29
2; 15 2.18
76 76
14; 73 15.06
0.535 0.523
6.6, 6.63
2.21 2.37
79 90
15.19 ,5;45
0.518 0.508
6.66 6.89
4.88
2.40
15.48
7.860 7.860 7.843
5.15 6.26 5.39
2.53 2.72 2.76
92 102
0.508 0.509
6.94 7.13
0.483 0.488
7.36 7.42
I. 900
5. 34 5.62 6.8, 5.73 5.73 5.74
2.87
0.463 0.453 0.466 0.458 0.452 0.443
7.54 7. 62 7. 88 7.87 1.92 7. 95
7.870 7,666 I. 878 7.877 1.624
3.07 3. 10 3. 10 3. 14 3.20
a. 199
2.22 2.8, 2.6, 2.93 3.20 3.3,
0.66 0.68 0.79 1.07 1.2,
8.209 8.205 8.214 8.247 a. 181 a. 194 8.247 8.204 8.209
3.37 3.46 3.73 3.93 4.12 4.5, 5.23 5.61 5.70
1.25 1.33 1.56 1.65 1.85 2.,6 2.68 3.03 3.06
8.206 a. 194 a. 199 8.218
8.210
15.45
112 1613,
11,
12, ,36 14, 140 142 140
16.05 17. OH 17.3, I6.86 17.20 17.42 17.23
15 I6 19 28 33
10. 72 IO. 63 11.26 12: 33 I2.92
1.000 0.764 0.757 0.730 0.666 0.634
4.12 4.74 4.87 5.2, 5.38
35 38 *a 54 62 60 116 140
13.06 13.36 14. L3 14.24 14.87 IS. 74 16.90 17.85 17.14
0.629 0.613 0.581 0.579 0.560 0.62, 0.488 0.460 0.463
5.44 5.54 5.64 6.98 6.22 6.63 7.36 7.82 7.81
144
Dysprosium 8.398 6.41, 6.400 8.417 8.429 6.416 a.426 8.4LO 8.38, a. 409 8.420 8.4,, 8.422 a. 368 8.412 a. 408 8.422 6.419 8.410 a. 386 6.419
2.28 2.51 2.67 2.67 2.85 3.03 3.10 3.22 3.26 3.34 3. 39 3.40 3.52 4.03 4.53 4.5, 4.79 5. ,a 5.33 5.55 5.67
0.33 0.43
7 I”
0.65 0.67 0.64 0. 92 1.06 ,. I, 1. ,Y 1.19 1.23 1.3, 1.14 2.11 2.14 2.36 2.67 2.81 3.02 3.04
16 16 2, 24 29 30 33 34 35 39 59 80 a, 95 116 ,26 14, 145
6.40 9; 64 10.0, 10.90 11.03 11.65 11.96 12.50 L2.69 L3.04 12.91 13.20 13.44 14.70 15.73 16.02 16.63 17.37 17.76 18.40 18.16
L. 000 0.813 0.840 0.772 0.764 0.722 0.704 0.673 0.66, 0.645 0.649 0.637
4.32 4.44 4.72 4.74 4.95 5.04 5.2, 5.27 5.37 5.38 5.44
0.621 0.669 0.535 0.525 0.507 0.485 0.473 0.456 0.463
5.54 6. LO 6. 61 6. 63 6. 94 7. 36 7.54 7.62 7.87
9.046
4.43
2.0,
I6.54
0.547
6. 53
9.073
4.40
2.08
83
LT. 17
4.44 4.70 5.03
2.10 2.29 2.44
84 96 110
17.19 17.40 1,. 34
0.528 0.52,
6.6,
9.068 a. 927 a. 93,
0.513 0.615
9.055
5. 16
2.64
124
,a. 56
0.488
7. L3 7.40
8.926
5.29 5.5, 5.54
2.66
126
1,. 93
0.496
7.42
2.8, 2.92
137 145
18.09 l6.96
7.63 7. 78
5.6,
2.94
148
18.83
0.491 0.474 0.476
a. 987 a. 99, a. 946
5.76 5.65 5.69
2.9, 2.98 3.04
164 ,52 155
19.0, 19.23
0.485 0.473 0.465
7. a* 7.87 7. 95
9.289 9.318 9.311 9.271 9.260 9.269 9.310 9. 325 9.266 9.303 9.263 9.305 9.339 9,269 9.269 9.289
2.22 2.62 2.85 2.84 2.96 2.96 3.2, 3.23 3.44 3.72 4.06 4.35 4.35 4.80 5.33 5.63
0.63 0.65 0.62 0.83 1.04 ,. 07 L. 30 1.6, L. 84 2.05 2.06 2.47 2.76 3.17
1.000 0.882 0.78, 0.712 0. 725 0.122 0.675 0.669 0.622 0.595 0.546 0.528 0.525 0.486 0.479 0.436
4.32 4.7, 4.73 4.95 4.96 5.24 5.27 5.66 5.85 6.29 6.69 6.61 7.16 7. 6, a. 13
7.038 7.064 7.038 7.058 7.03, 7.050 7.026 7.028 7.028 7.037 1.028 7.021 7.049 7.049 7.068 7.070 7.016 7.039 7.019 7.036 7.040
1.45 1.78 1.77 1.85 1.86 2.0, 2.08 2.20 2.27 2.58 2.62 2.74 2.91 2.9, 3.04 3.46 3.90 4.2, 4.34 4.95 5.33
0.35 0.39 0.49 0.50 0.68 0. ‘I? 0.9, 0.94 1.19 1.23 1.35 1.43 1.44 1.53 1.8, 2.12 2.35 2.42 2.88 3.13
1.000 0.804 0.782 0.737 0.729 0.662 0.629 0.589 0.584 0.538 0.529 0.509 0.510 0.507 0.496 0.47, 0.457 0.44, 0.443 0.418 0.412
4.29 4.32 4.43 4.45 4.63 4.13 4.88 4.92 5.2, 5.26 5.39 5.50 5.5, 5.62 5.98 6.37 6. 67 6.76 7.37 7.11
8.878 a. 977 8.956
0.42 0.64 0.66 0. 63 1. I? 1.22 1.29 ,. 36 1.39 1.52 1.53 I. 61 1.93 1.99 2.08 2.12 2.67 2.97 3.02
10 16 17 22 35 36 40 42 44 50 5, 57 72 74 82 83 12, 145 147
a. 73 10.34 11.27 11.3, 12.00 13.37 13.55 13.80 L4.20 14.10 14.43 L4.58 15.31 15.80 16.44 36.26 16.70 18.04 18.61 19.21
1.000 0.842 0.777 0.774 0. 729 0.665 0.845 0.634 0.616 0.618 0.602 0.599 0.568 0.551 0.530 0.538 0.524 0.483 0.467 0.456
4.44 4.R 4.74 4.95 5.38 6.44 5.54 5.62 5.66 5.84 5.85 6.03 6. 39 6.45 6.6, 6.63 7.40 7. 82 7.87
0.7GPa.
18.63
form
at a rapidly manifested curve,
is
(the
the range
6. 63 6.89
7.82
as expected
decreasing decrease
at about
that the
to a form quite
in slope
48GPa.
This
so
whose rapidly,
of the
u,-u,
transition
is
transition.
point was determined sound
techni-
steep over the
more incompressible
rate. A transition not
tetravalent 15 per cent at
of explosive
indicating
a two segment measured
the
of Ce is quite
the fee solid-liquid
The transition
to
change of about
range.
by a slight
by imposing data
transition
of Ce is becoming
is observed
probably
7.04 8.78 9.00 9.58 9. 64 10.65 11.17 11.94 12.03 13.09 L3.28 13.82 13.83 L3.90 14.25 ,‘I. 83 15.35 15.95 15.84 16.82 17.09
4 5 6 7 10 I, I4 15 22 23 26 29 29 33 44 56 70 74 100 *,a
Hugoniot
increasing
compressibility
11.93 12.00 12.78 12.84 13.79 13.95 14.69 l5.64 I6.98 17.62 17.78 19.08 19.39 21.32
110 131 ,66
is below
experimental
tetravalent
17 17 23 23 31 32 4, 52 69 83 84
electronic
IA,-u,
9.29 10.56
a
with a volume
which
ques. The entire
0.3,
an
configuration 2.36 2.68 2.88 2.9, 3.06 3.39 3.43 3.53 3.55 3.63 3.62 3.8, 3.88 4.30 4.24 4.52 4.45 5. ,a 5.58 5.56
a,
Th”ll”lIl
undergoes a. 734 a. 709 a. 155 a. 753 a. 750 8.75, a. 749 a. 750 a. 745 a. 705 a. 684 8.129 a. 736 a. 105 a.715 a. 752 6.742 8.72, 8.723 a. 763
5.54
linear
in the II,+,
least squares
velocities
were
plane
fit on the not
used
Hugoniot
equation
of state of the lanthanides
a-
7 -
I
0
I
I
I
I
I
I
I
05
IO
I5
20
25
30
35
l
I
GADOLINIUM
tU,I
.
DYSPROSIUM
I”,*11
A
ERBIUM
I 0.5
2 0
40
I
n
IL!,*21
I 10
I
I
I5
20
J
UP I km/s
I
I 2.5
I 30
35
U,,(km/s)
Fig. I. Hugoniot II,-u, data for SC, Y and La. The Hugoniots of both SC and Y fail to extrapolate to the measured zero pressure sound speed, probably because of a sluggish low-pressure phase transformation and the presence of an elastic wave. Evidence of this transition has been found in free-surface velocity vs time measurements on Y. We have not attempted an analysis of the transition in SCand Yin this work. The behavior of La is typical of most of the remaining members of the rare earth sequence.
Fig. 3. Hugoniot u,-U, data for Gd, Dy and Er. Although the data are inconclusive, the u,-U, Hugoniot for Gd has been drawn with a break at high pressure and a subsequent decrease in slope. This is in agreement with the phase diagram for Gd presented in Fig. 16.
a
I
I
7 -
I
l
TERBl”M
.
HOLMIUM
IUs*l)
A
TH”Ll”M
iU,‘Zl
I
I
20
25
I
(“*I
6-
;; . E t
5-
; AL 4-
0 3-
28 0
05
IO
I5 U,
, 0
I
I
I
I
I
I
05
IO
I5
20
25
30
U,(
Fig.
2. Hugoniot
u,-u,
km/r
55
I
data
for Pr, Nd and Sm. These three slopes for the lower pressure phase. However, the Hugoniot fits do extrapolate well to the bulk sound speed, indicating that low-pressure transitions do not affect the shock wave data.
elements have anomalouslylow
because of the presence of low pressure phase changes) and minimizing the total deviation determined by leastsquares calculations. The resulting fits for most materials yielded standard deviations corresponding to no more than one per cent in zero-pressure intercept and one-half per cent in slope. The intersection of these straight lines then determined the velocities at the transition. If a small volume change is associated with the transition, this technique will give transition pressures that are slightly too high. However, the close spacing of the data points in
30
5
(km/r)
Fig. 4. Hugoniot U.-U, data for Tb, Ho and Tm. The elements all behave normally, with the transition being scarcely noticeable for Ho because of the large scatter in the data for the upper phase.
the transition region precludes this being more than a very small error. These Hugoniot u,-u, fits both below and above transition, the shock velocities at transition, and the corresponding transition pressure and compressions are listed in Table 3. Figure 6 shows the measured ultrasonic bulk sound speeds and the Hugoniot intercepts as a function of atomic number 2. The agreement in general is quite good, which is a further indication that the low pressure phase changes in the lanthanides are unimportant to interpretation of dynamic Hugoniot data. However, the agreement for the elements SC and Y is not good since the Hugoniot intercept lies well above the bulk sound speed. Such behavior is an indication either of a very large dynamic yield point or of a low pressure polymorphic phase
746 Table
ElUlX”t
7. Summary
Hugoniot l_owrr
data parameters
FL% upper
Pks*e B
co
of Hugoniot
co
8
VW,
“.
P
(km/s)
(km/s)
&m/s)
al Transition
F%asc
,c. Pa,
SC
4.76
.64
4.11
.99
5.94
.69
Y
3.31
.78
2.38
i. 35
4.71
.s3
37. cl
i-a
2.05
1.02
1.52
L.53
3.22
.64
22 .5
ce
.87
1.90
l.41
1.58
4.12
.56
46.0
PF
2.09
.79
0.94
I.68
3. 12
.59
27.0
Nd
2.16
.83
1.50
I.41
3.11
.63
25.0
sm
2.25
.76
1.56
1.35
3.13
.63
27.0
E”
1.62
.9s
1.12
1.29
2.43
.67
ICI.6
Cd
2.20
.92
1.77
1.29
3.40
.62
34.5
Tb
2.21
. 92
1.12
1.29
3,SO
.60
40.0
w
2.28
.8Q
1.91
1.22
3.27
.66
30.5
IlO
2.27
.96
I.91
1.22
3.62
.61
44.0
35.0
Er
2.31
.90
1.69
1.34
3.55
.fiI
44.0
*In
2.26
.Ql
1.90
1.19
3.38
.63
39. c!
Yb
1.46
.83
0.80
1.43
2.26
.57
16.0
‘7
5F’
“__.T..,
,
._-r,
r.
.
1
h
T
+
I
+ HUGONIOT
I _L-.._ I ~7s3S36oS 6263ekjl$5&zkz&70 LoCsR NdF7nBnEuGdTbDyHoErTmYb
Y
ATOMIC
I
I
co
I5
I
0
05
I 2.0
I
I
I
25
30
35
I 40
U,(km/sl
Fig. 5. Hugoniot u, -9, data for Ce, Eu and Yh. Although Yh behaves normally in this plane. hoth fe and Eu have anomalous r*,-U, Hugoniots. Eu is characterized by a large volume change at it< transformation. while Cc transform< immediatcl~ to the tetravalent fee form, which is quite incompressible. At higher pressures Ce again becomes somewhat softer and the U.-N, Hugoniol decreases slightly in slope.
change. shock
To check wave front
capacitor
this. the time-resolved through
technique.
measurement.
Figure
on all records,
account
for the discrepancy.
a phase change
this is assumed
indicating
that equilibrium
the
techniques. as well.
point
A similar
too small
has
not
transition
been
normally
however. Sm-type behind
of this sluggishness observed
undoubtedly
by occurs
Figure
8 shows the fitted values
of the IL-U,,
slopes for the two phases as a function
to
is pressure-dependent,
because
extrapolate 10 the \ound speed. Agreement for all the other elements is excellent. indicating that the known low-pressure close-packed ~olymorphs of the rare earths are quite similar Ihermody~micall~.
phases
in the wave
has not been attained
It is perhaps
Fig. 6. Bulk sound speeds and Hugoniot intercepts as a function of atomic number. The sound speeds were all obtained by the ultrasonic pulse-echo technique. except for the value for Eu which was obtained from compressibility measurementsI IO]. Ce. of course, transforms to rhe highly incompressible tetravalent form at pressures lower than shock pressures used here. and the two numbers are therefore not expected to agree. Similarly. hecause of hydrodynamic n~~nequilib~um behind the shock fronts at low pressure, the Hugoniots of Y and SC are not expected to
is
of about 0.4 GPa,
by a break
NUMBER
with
increasing
divalent
nearly
three
slopes
that
atomic
and
having
decrease
number. initial
times as great as the other lie on a smooth
curve
Hugoniot
of atomic
It will be noted that the slopes generally
at longitudi-
At high pressures,
transition
transition
of such a
to be to the rhombohedral
The
that
moving
l._-._i
of a
using the d.c.
but this is much
is indeed indicated
structure.
the shock front.
wave,
with an amplitude
observed
structure
7 shows the results
A small elastic
nal sound velocity
front;
Y was recorded
INTERCEPT 1
SC
IL
(SCHELBERG)
0 ULTRASONIC
.
4..
Eu
number. in both and
Yb.
compressibilities rare earths,
have
in the high pressure
regime determined by the elements normally characterized as trivalent. Ce has a very high initial slope, since it transforms
almost
immediately
lo the tetravalent
form.
The low values of the initial slopes for Pr. Nd, and Sm are
static
at present
in SC
a very
unexplained,
sluggish
equilibrium
even
although
solid-solid under
this may again be due to
phase change shock
conditions.
which
is not in
However,
the
747
Hugoniot equation of state of the lanthanides
Jayaraman[4,13] has measured the melting phase line in the P-T plane for most of the rare earths, in some cases to
TIME
(,,b)
Fig. 7. Free-surface velocity vs time for Y. The record was obtained by impacting a stationary Y target with a steel projectile and observing the motion of the free surface with the d.c. capacitor technique. Point I is the stress level where plastic flow first begins (the Hugoniot elastic limit). This low-pressure elastic wave propagates at essentially longitudinal sound velocity. The two-wave structure at point 2 is the nonequilibrium transition which leads to anomalously high values of the shock velocity using flash gap techniques. Here, the transition wave has been nearly overdriven.
a pressure of nearly 7GPa. Extrapolation of this data indicated that the melting curve and the Hugoniot do indeed intersect in the general neighborhood of the observed transition point in all cases. However, this is a considerable extrapolation and a more exact calculation of the phase line is desirable. To determine the location of phase lines the free energy function, G = H-TS, must be calculated, since a boundary between two phases is characterized by AG = 0. A convenient way to determine the free energy is first to calculate the temperature, T and entropy, S along the Hugoniot which can be combined with the known enthalpy to give the free energy along the Hugoniots of both phases. The results of these calculations can then be readily extended to give the free energy in the region of interest. This can be accomplished by use of the thermodynamic relations dE=TdS-PdV
(8)
T dS = (aE/aT), dT t T(aP/aT), dV
(9)
and the Rankine-Hugoniot relationships (l-3). By introduction of the Grtineisen parameter, y, defined i3S
. T-
20
‘.Liwq A UPPER
y =
V(aP/aE),
(10)
PHASE
equation (9) can be rewritten as
t
TdS=C,dT-C,(y/V)TdV
0, 21 SC
I 39 Y
I I I J L .1-l 57SSs96060 626364656667666670 lsCeRNdV4nhSmEuGdTbDyHoErTmYb ATOMIC
I
I
I
I
I
NUMBER
Fig. 8. Hugoniot u,-u, slopes as a function of atomic number. A definite trend across the periodic table is noted, with the slopes of both phases generally decreasing as atomic number is increased. It should be noted that, for a linear U.-U, relation, the slope is a direct measure of the rate of change of compressibility with pressure. The abnormally low values of the slope in the lower phase for the light rare earths Pr, Nd and Sm, and the abnormally high values of the slope in the upper phase for Pr, Er and Yb are difficult to explain, but presumably are also associated with electronic rearrangements. The initial slope of Ce, of course, is not to be compared with the other rare earths. The slopes of the upper phase are the experimental values, not those appropriate to the transformed metastable Hugoniots, although for most of these materials there is little difference between the two.
(11)
We shall assume that y is a function only of volume and, specifically, that the product my is constant. Experimental work on a number of materials[8], as well as theoretical considerations[l4], indicates this to be an adequate approximation. With this assumption, y(V) is given by y = (V/ VO)yO,where y0 is the thermodynamic value at standard conditions given by y. = 3aca2/C,. The assumption that y is a function only of volume implies that the specific heat C, is a function only of entropy. Since the characteristic Debye temperatures for all these elements are quite low, it is further assumed that C, is independent even of entropy (this assumption may not be good for elements such as Gd). Equations (3). (8) and (I 1) can then be combined to yield
dT,(v”VI rdP+[F-T(+‘)]dV
(12)
which can be used to determine the entropy through good agreement of the Hugoniot intercepts with the bulk sound speeds for these elements argues against this.
(13)
3. ANALYSIS
Of central interest in this study is the nature of the transitions which occur so persistently throughout the rare earth series. These transitions appear to be associated with melting, perhaps with an accompanying electronic rearrangement to the tetravalent state.
These have been integrated numerically. When this is done, a complete thermodynamic description along the Hugoniot for each of the two phases is obtained. Under the assumptions of constant (py) and C, thermodynamic quantities off the Hugoniot can be
W. J. CARTERet al.
748
computed through the relations (aP/U),
= pyC, = constant
P = Ph(V)+pyC,[T-
Th(V)].
(14)
The quantity Th( V) determined from equation (12), coupled with the known Ph( V) and equation (14), serves to determine V(P, T) implicitly. Equation (11) can then be used to determine the entropy, and the Mie-Griineisen equation E(P, V)=E~(V)+[P-P~(V)I/(yp)
(15)
serves to determine the internal energy at every point off the Hugoniot. To connect the two phases the difference in energy and entropy across the phase boundary at some point are needed. These differences may be computed from the Clapeyron equation if the slope of the phase line, (dP/dT)*, and the associated volume change is known at any point (P*, T*) on the phase boundary: AS = AV(dP/dT)* AE = AV(T*(dP/dT)*
- P*)
(16)
The appropriate values of P*, T* and (dP/dT)* were obtained from the P-T diagrams for melting presented by Jayaraman [4,13]. Before performing these thermodynamic calculations, it is necessary to center the upper-phase Hugoniot to a point (p(,, T,,, P,) appropriate to the transformed material. For convenience, we have chosen to extend the phase (in a metastable condition) down to room temperature and zero pressure. The method of recentering utilizes the thermodynamic calculations just described and by appropriate minimization procedures obtains a Hugoniot described by equation (4) centered at the desired point. This method has recently been described in detail [ 151.For this calculation, which results in a zero pressure density,
sound velocity and slope of the metastable Hugoniot, it is necessary to know or estimate the pressure, temperature and slope of the phase boundary where it intersects the Hugoniot. Since the intersection of the two linear u,-u, Hugoniots determines the pressure and temperature, only the slope of the phase line, dP/dT, is unknown. To determine this, an iteration procedure was used. The metastable Hugoniot was first calculated on the basis of a reasonable guess of dP/dT at the transition determined by an extrapolation of Jayaraman’s phase lines. This Hugoniot was then used to calculate the phase boundary as described above, and the slope, dP/dT, at its intersection with the lower phase Hugoniot determined from the results. This new value was then used to recenter the Hugoniot of the upper phase, and again the curve AG = 0 calculated. Two iterations of this type were generally sufficient to yield invariant values of P*, T*, and dP/dT* at the transition. Table 4 lists the parameters for the metastable Hugoniot used to establish the thermodynamic properties of the upper phase. The recentering procedure used to establish these metastable Hugoniots generally resuts in ill-defined minima determining the values of the parameters pX, c 8. and s *. While they appear to be well established, both ~‘6and pQ could be in error by as much as ten per cent; for the sake of consistancy and impartiality, we have always shown the values at the absolute minimum. It should be emphasized that nowhere in these calculations is the predicted value of the coordinate (P, T)* at melting on the Hugoniot required to agree with the experimental value; the Hugoniot data is used only to establish the thermodynamics of the two phases separately through equations (14) and (15). Furthermore, there has been no attempt to adjust any parameters in order to predict the transition points. In view of the abnormally low values of the Griineisen parameter, ye, for these metals and the importance of this parameter in the calculation of thermodynamic quantities, most of the possible errors in these calculations probably arise from the assumption of constant (py) and the lack of knowledge of y in the liquid region. Because of this
Table 4. Metastable Hugoniot parameters 0
Elenmnt
PO (Mglm3)
::c
C”
s*
kin/s)
1.75 Ucc) >. GO (liquid) Pr
6. 92
1.05
1. G!J
i-id
7.00
1.20
1.40
Srn
7.73
1. 64
1.37
Eu
4.82
1.03
1.27
Gd
8.24 9. 00
l.GG 2.10
1.33 1.25
Tb
9.03
2. on
1.27
w
8.31
1.89
1.22
Ho
7. 63
1.56
1.24
Er
7.68
1.36
1.31
Tm
G.G6
1.77
1.21
Yb
8.10
1.12
1.48
(bee) (liquid)
-
Hugoniot
uncertainty,
heat, uncertainties
questions
about
formulation
analysis
validity
immediately melting.
to the
However,
70
is in general
more
I
because
dense
I
HUGONlOl
30-
20,
slopes
Ce has a
/
._
’
/
10 -
1’
/
I.lO”lO
alfccl
state
upon
the
phase
I
_ CALCULATED .SOLIO- LIQUID P”ASE BOUNDARY
// /
:
-./‘,’ j/c
//
for all the
pressure
I
2
/’
/+ // //
0 ;;
it transforms
tetravalent
increasing
and
quite good.
Ce are positive;
presumably
CERIVY 40,
boundaries.
the calculated
Y and SC. The initial
except
with
I
phase
---
SOY-
a meaningful
phase diagrams
here except
initial dPldT*
of
Mie-Grtineisen
to perform
pressure
all the elements
the
between
show proposed
studied
negative
of
calculated
transition
(10-22)
elements
the
the agreement
experimental Figures
for
_.
in the static data, and even
itself, it is impossible
Nevertheless,
for
the
149
of state of the lanthanides
as well as that due to lack of knowledge
the specific
error
equation
1
I
0-YYCfl’=) 0
500
.1-I_ lxx)
IOCU
._..
2030 K)
Tl
-. 2500
3&u
35&-
Fig. I I. Proposed phase diagram for Ce. The extremum at 1.1 GPa has been attributed by Jayaraman to the completion of 4f-5d electron promotion in the liquid similar to the y-o transition in the solid. The metal then melts normally up to at least 5OGPa. The Hugoniot crosses the phase line at a point where dPldT is positive; this is manifested by a decrease in slope in the I,.-I,, plane. Because of the important low-pressure phase change. it was necessary to recenter the Hugoniots of both the upper and lower phases. This makes the calculation of temperature less certain. since small errors in the metastable density of the collapsed fee form can lead to large contributions to the PdV term of equation (I!)
30
---
Erblum
----
PRASEODYMIUM
20 0
IO
05
IS
2s
20
30
33
G
Us (km/s)
ii
Fig. 9. Hugoniot U.-U, data for porous erbium. The dashed lines are calculated, using (py) = constant, the Griineisen relation, and the crystal data indicated by the solid line. The agreement is seen to be satisfactory. indicating that the assumed volume dependence of the Grtineisen parameter is sufficiently accurate for our purposes.
z
/
I IO-
/’
I
I
’
,
.--_.-.,.
1
_A /’ ’ / /
,
/
/
CPILC”LATE0
,soLIO-LIO”I0 /
/
PHblK
I
1000
T(K)
BXlNMRY
bee
I ISCYJ
I
LlQUlO
IOOf
I500
Fig. 12. Proposed phase diagram for Pr. The phase line is initially nearly vertical. and the slope changes very little with pressure. It is possible. in view of the discrepancy between the measured and calculated transition pressures, that the fee-liquid transition is not observed on the Hugoniot and the electronic rearrangement producing the less compressible form doer not occur until the Hugoniot is extended well into the liquid phase. This is true for a number of the rare earths.
lines of all the other axis,
indicating
liquid relative
l_lO”lO
bee so0
I
I
/
/
OO-
500
-,A
dkQ
T(K)
\ \
I
,
..___.
LANTHANUM
HUGONIOT -,,
I
i-, __y I /
01 0 30 ,
CALCULATED .Sa.IO-LIQUID PHASE BOUNDARY;
/’
A0
Fig. IO. Proposed phase diagram for La. The low-pressure phase lines were obtained from Jayaraman’s work[41. The Hugoniot P-T locus and the solid-liquid phase line were calculated by the methods discussed in the text. The intersection of these two calculated curves is in excellent agreement with the experimental transition pressure, which is indicated by the symbol 0 in this and the following figures. An extremum in the phase line occurs at about 18 GPa and 1620 K. It is interesting that the region of bee phase stability increases progressively from La up to Eu. where presumably the only stable phase is bee. This is evident in the succeeding figures.
elements
bend toward
a continuously to the contiguous
indeed
regime
found
extremum
pressure
In an attempt tions,
by
available him.
to check
cold pressed
PAV
heating
and
the was
elements
these phase boundary of porous
from a fine meshed
powder
density.
obtained
Here.
the
Hugoniots
melting
line at pressures
calculaerbium to 92.87
use is made of the
by shocking
als; the porous phase
other
the Hugoniots
and 79 per cent of crystal large
the
up
lies well within
to Jayaraman.
For
of the
was too high for static techniques.
we have obtained
samples
density
solid as one proceeds
the phase line. For Eu. this extremum experimental
the pressure
increasing
porous
would be expected considerably
materi-
to reach the below
that
W. J. CARTER
ef al.
EOr---70 ,-
-
-
-1
GADOLlNlUM
/. ESTIMATED HUGONIOT
60.
/’ /
,
,,*’
HUGONIOT.
JO-
’
I’
,I-
1 --__ I 0-l
I
I
I-.
m
0
bee
i
/’
t
mw
T (K)
T(K)
Fig. 13. Proposed phase diagram for Nd. The fee-bee and bee-liquid phase lines have been extended to meet at a triple point, and a very steep fee-liquid dPldT assumed at this point. Hence, the calculated solid-liquid phase line for this element is in considerable doubt. even though the calculated and measured transition pressures are in good agreement.
30
/
ESTIHATED SOLIDLIQUID PHASE 6O”NWJW
I
20
,,
1.-.-
I
SAMPiRlUH
Fig. 16. Proposed phase diagram for Gd. .Although the region of bee phase stability has begun to decrease and will continue to do so until the element Yb. for Gd this region is still wide enough that the Hugoniot apparently intercepts the rhomh-bee phase line hefore the material melts. Gd is therefore the only rare earth without an anomalous melting curve. the electronic transition presumably taking place al or near the rhomb-hcc phase line. Temperature-pressure loci in the hcc phase are only estimated. as is the hcc-lrquid boundary.
5oI
4
I
‘-,---
’
40;. /
7 u
to-
IWASE
/
‘/
i ”
7
\
;
,’ ,
’ CALCULATED I. -SOLID~LlOUlD PHASE BOUNDARY
/ lI
/
I
,
2000 LIQUID ._I
0
2Ocu
phase diagram for Sm. The dhcp, bee. and liquid expected to meet at a triple point at several However, the dhcp-bee phase line was ignored calculations. so that the dhcp-liquid phase line is again in question.
30,
/ ’ / ;; CALCULATED ,,SOLIL-Llaulo PHASE BOUNDARY
,
I
/
20HUGONIOT-.
E
HUGONIOT
‘\
\
,I
\\
/
\
/
a
CALCULATED SOLID-LIOUID PHASE BOUNDARY
\
/
\
IO bee
Fig. IS. Proposed phase diagram for Eu. The extremum at 3 GPa was found by Jayardman et (I/. The agreement between calculated and experimental transition pressures is almost exact.
\
. I
_~._.. LIOUID
\
\
(rhombi / r-y--l
‘\
!
DYSPROSIUM
\
\
5;.
Fig. 17. Proposed phase diagram for Tb. The low-pressure hcp-rhomh transition is based on conjecture. although static reristivity measurements indicate an anomaly at 2.7GPa. and experience has shown that the rare earths transform in the sequence hcp-rhomb-dhcp-fee as pressure is increased [ 121. The calculated solid-liquid phase boundary exhibits a maximum at about 35GPa. The region of hcc phase stability has hegun IO decrease.
I
EUROPIUM
_ a.
3030
r
_
15.
2eQJ
T(K)
T(K)
Fig. 14. Proposed phases are again hundred kilobars. in performing the
-.
/
I
\
/
J /
20t \
IO-
HUGONIOT
301
0
\
0
/y
/
\
/
HUGONIOT
/
BOUNDARY
bee IS00
2000
T tK) Fig. lg. Proposed phase diagram for Dy. Again. the hcp-rhomb transition at SGPa is conjecture hased on resistivity measurements; since no transition has been discovered along the P = 0 axis, this phase boundary probably ends at a triple point with the hcp and bee phases. An extremum in the solid-liquid phase boundary is predicted to lie at about 5 CPA and 1950 K, which unfortunately is outside the range of present static techniques.
751
Hugoniot equation of state of the lanthanides 50
‘I’
t”’ HOLMIUM
40 ;; & lJ
SD-
THULIUM
J /’
/
!
CALCVLATED I/SOLID-LIOUID PHASE BOUNDARYI
7’
a.
HUGONIOT,/ / /
\
/ /
/ /
I
IOC bomb) : ,- --, 0 0
CALCULATED /
\
/
ZO-
‘1
I
/
HUGONlOT
40
kc
b I 1000
500
I I500 T(K)
/
,
I 2500
I 3000
0
Fig. 19. Proposed phase diagram for Ho. The calculated solid-liquidphase boundary exhibits an extremum at about 14GPa and 2340K. The agreement between calculated and experimental transition pressures is excellent.
I
50
I
I
I
1 /
30-
HUGONIOT
/
1500
2000
I
CALCULATED ,SOLlDLIOUID \ PHASE EOUNOARY
Y’
-
a c1 :
I
/’
HUGONIOT 10
/
\
/
/
)_’
,’ \i
IO-
0
SW
I 1000
I I500 T (Kl
/I 14 2ocQ
I 2500
/’
0
LIOUID
b
I
I
I
I
I
1’
II
3000
YTTERBIUM
ZD
\
/
0
I
2500
i
/‘!
/
ZO-
I
1000
LIOUID
Fig. 21. Proposed phase diagram from Tm. The extremum in the calculated solid-liquid phase boundary lies at about 6GPa and 2160K.
1
= p 0 0
I
500
\ / _-r
T(K)
EREIUM
40
0
I
I
\
I ir
LlOUlD
A,' 2000
0
500
IODD T(K)
/
/’ , 1500
LIOUID zoo0
I 3000
Fig. 20. Proposed phase diagram for Er. An extremum in the solid-liquid phase boundary is predicted at about 11GPa and 2200K. The field of bee-phase stability is now closed. required for the crystal density material. Unfortunately,
the transition in porous materials is masked at these lower presstires by the effects of void collapse and perhaps a wave front which is not in equilibrium. However, the higher-pressure data can alternatively be used to check the validity of the metastable equation of state derived earlier. These Hugoniot points do attain very high temperatures and almost certainly lie in the liquid phase regardless of the details of the phase boundary locus. Hence, if the parameters listed in Table 4 and used for the phase line calculation are truly representative of the liquid phase, then a simple recentering of this liquid Hugoniot to the initial porous density using equation (10) should reproduce the porous Hugoniot data (it should be noted that AH of melting is very small compared to the total energy on the Hugoniot). Except for the very low pressure end, the comparison between data and calculation, shown in Fig. 9, is quite satisfactory. 4.CONCLUSIONS
It would appear to be well established that all the rare earths except Ce and possibly Cd melt anomalously at sufficiently high pressures, and that Ce itself melts anomalously at atmospheric pressure but normally above about 4 GPa. The presence of extrema in melting phase lines is an interesting phenomena so far known to occur in only about fifteen materials (the present investigation
Fig. 22. Proposed phase diagram for Yb. The extremum of the solid-liquid phase boundary, predicted earlier by Jayaraman[l9] now appears to lie at about 11GPa and 1650K. The region of bee phase stability has again opened to include a large part of the P-T plane.
nearly doubles this list, of course). From the Clapeyron relation, assuming the entropy change of transition does not vary appreciably with pressure, such behavior can only mean that the density of the liquid continuously increases relative to the density of the solid as one proceeds up the phase line, which implies that eventually the average interatomic distance in the liquid becomes less than that in the solid. This can come about through one of only two mechanisms; either the coordination number of the liquid becomes greater than that of the solid, or else the ionic radius characteristic of the liquid is less than that of the solid. High coordination numbers are not unknown in liquids, but it seems unlikely that they could exceed the coordination of the close packed structures characteristic of most of the lanthanides, at least above the region of bee phase stability. Hence, the second possibility seems more likely. An abrupt decrease in ionic radius upon melting implies a corresponding change in the electronic structure of the metallic ion. Since the ionic radii of the lanthanides are known to decrease in the order divalent 7 trivalent > tetravalent, one conclusion is that an additional electron is forced into the valence band from the 4j shell by the application of sufficient pressure. (We invoke the Jamieson criterion [ 161 by saying that an electron arbitrarily ceases to be a 4j electron when the single-particle wave function describing it has less than 50 per cent 4j-type admixture.) The
152
W. .I. CARTERet al.
curvature of the phase line toward the pressure axis is then accounted for by an extension of the two-fluid model recently proposed by Klement[l7], in which an increasingly greater proportion of atoms in the liquid undergoes a 4f+5d electron promotion at the phase boundary as pressure is increased, leading to the necessary density relationship between solid and liquid. An alternative possibility is that there is a continuous electronic transformation occurring in the lower pressure solid region. The evidence for this is the very low slopes of the u,-u, Hugoniots in this region. It is known that the parameter s is a very strong measure of the type of repulsive potential existing in the material. With a slope that remains less than unity, a potential is implied that would allow infinite compression on the Hugoniot at finite pressure, an obviously impossible situation. Thus, if there were a continuous change in the amount of s- and dbonding with pressure, the small rate of increase in the incompressibility could be explained. The anomalous melting would have to be associated with the more rapid completion of the electronic transition in the liquid state adjacent to the solid material. Band structure calculations have shown that the 5d and 6s bands for the rare earths in the metallic state are broad and strongly mixed, while the 4f bands are quite narrow and localized and have little direct effect on the bulk properties of the materials. The studies of Royce[l8] and Al’tshuler et al. [2], have indicated that the presence of a significant population in the d-band results in a low compressibility, while s-bonded elements generally have much higher compressibilities. Probable electronic ground state configurations of the lanthanides in the free atom state as presented in Table 1, together with the normal valence of +3, suggest that there is probably significant d-bonding for all the lanthanides even at zero pressure. The increasing rate at which the compressibility decreases beyond the transition would indicate a further increase in d-band population at this point, most likely but not necessarily at the expense of the f-band occupation (in the cases of SC, Y, and possibly La the transition is certainly not due to a 4f-5d electron promotion). The exact nature of the electronic rearrangement probably will
not
be known until careful band structure calculations are carried out over large ranges of interatomic spacings. Acknowledgements-The authors would like to thank Dr. John Taylor for performing the free-surface velocity experiments on yttrium as well as providing much stimulation and encouragement, and Charles Shelberg for performing the zero-pressure sound speed measurements.
REFERENCES
4. 5. 6.
I.
8.
9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
19.
Stager R. A. and Drickamer H. G., Phys. Reo. 133, A830 (1964); Science 139, 1284 (1963). Al’tshuler I.. V., Bakanova A. A. and Dudoladov 1. P., JEPT Letters 3, 483 (1966); JETP 26, 1115 (1%8). Duff R. E., Ross M., Gust W. H., Royce E. B., Mitchell A. C., Keeler R. N. and Hoover W. G., Symposium HDP IUTAM, Paris. Gordon and Breach, New York (1968); Gust W. H. and Royce E. B., Phys. Rev. B 8, 3595 (1973). Jayaraman A., Phys. Rev. 139, A 690 (1%5). Lawson A. W. and Tang T. Y., Phys. Rev. 76, 301 (1959). Rice M. H., McQueen R. G. and Walsh J. M., Solid State Phys. (Edited by Seitz F. and Turnbull D.), Vol. 6. Academic Press, New York (1958). McQueen R. G., Metallurgical Society Conferences (Edited by Gschneider K. A., Hepworth M. T. and Parlee N. A. D.), Vol. 22. Gordon and Breach, New York (1964). Carter W. J., Marsh S. P., Fritz J. N. and McQueen R. G., Accurate Characterization of the High-Pressure Environment, (Edited by Lloyd E. C.). NBS Special Publication 326 (1971). Shelberg C. (private communication). Kremers H. E., Rare Metals Handbook (Edited by Hampel C. A.), 2nd Edition. Reinhold Publishing Corporation, London (1961). Bridgman P. W., Proc. Am. Acad. Arts Sci. 83, 3 (1954). Javaraman A. and Sherwood R. C., Phvs. Reu. 134, 691 (1964); Phys. Rev. Letters 11, 22 (1964). Jayaraman A., Phys. Rev. 137, A 179 (1%5). Walsh J. M., Rice M. H., McQueen R. G. and Yarger F. L., Phys. Rev. 108, 196 (19.57). McQueen R. G., Marsh S. P. and Fritz J. N., .I. Geophy. Res. 72, 4999 (1%7). Jamieson J. C., NONR-2121 (03) (l%O). Rapoport E., J. Chem. Phys. 46, 2891 (1967); J. Chem. Phys. 48, 1433 (1968). Royce E. B., UCRL-50152 (1%6); Phys. Rev. 164,929 (1967); Proceedings of the International School of Physics, Course XLVIII, (Edited by Caldirola P. and Knoeppel H.). Academic Press, New York (1971). Jayaraman A., Phys. Ren. 135, 1056 (1964).
EQUATION OF STATE LANTHANIDES*
W. J. CARTER, J. N. Los Alamos
Scientific
FRITZ,
Laboratory
S. P.
of the University (Receired
and R. G. MCQL’EEN
MARSH
of California,
13 August
OF THE
NM 87.544. U.S.A.
Los Alamos,
1974)
Abstract-Shock wave studies of the lanthanide series show the existence of a very high pressure phase transition in all members of the series. The data show that this transformation is of necessity to a more incompressible phase and has been identified with melting. Thermodynamic considerations allow calculation of the solid-liquid phase boundary from these data; the results indicate that all the rare earths melt anomalously at sufficiently high pressures. This can be understood in the context of a “two-fluid” theory, in which the composition of the liquid along the phase boundary changes continuously with pressure due to the degree of pressure-induced electronic transition present in the liquid. Hence, at sufficiently high pressure, the density of the liquid becomes greater than the density of the contiguous solid and dP/dT becomes negative.
high speed smear
Briefly.
The
study
pressure recent
of lanthanides and temperature
years[ld], clues
elements
in the
program
to
the
metallic
techniques;
earths except Included
promethium
systems
(about
4 GPa)
energy
by standard
changes to the
the valence
adjacent
higher potential
0.7 GPa
in
well-known
cerium,
attributed
promotion[5],
if it occurs,
the periodic
the 4f and mixed atomic
number.
in conjunction
coupled
table,
velocity
at the interface
material
having
copper
experiments
[8].
The
melting
Rankine-Hugoniot
EH
together
at zero
such
direct
large
elsewhere[6,7]
and will
been
properties
of
transition
at
a 4f+5d
A similar
in these
equations
-
up j/u,
(1)
-
PO =
us4,
I vo
(2)
(3)
transition
transition
linear
and specific 0 refers
pres-
(E”)
energy
to the state of
representation
of the data is given by the
u, - u, relation
for u, = co + su,
gap between
appears
where
with
to occur
conditions
the Hugoniut
determined If rigidity
for
are
(4)
is linearly
of dynamic
Hugoniot
high pressure
adequately
of the
United
cur and the slope,
effects and possible the
low pressure
intercept
of an infinitesimal
should
co = [(aP/+),]‘12 related
bulk
to
the
modulus,
then reflects
phase changes
correspond
pressure
s, are
of least squares.
pulse,
to the
or the bulk
(at P = 0). Since the slope
pressure
(Sl,/aP),,
a nearly
linear
derivative a
of
linear
dependence
the
u, -u, of B,
on the pressure.
discussed in detail
intercept,
from the data by the method
neglected,
velocity
the auspices
(V,),
The subscript
generally
increases
shock
not be repeated
volume
then allow
quantities
material.
A convenient
electron
(u,, u,) points
as one proceeds
generally
under
the unshocked
here.
If the linear equations
*Work performed under Atomic Energy Commission.
of state.
standard
of the thermodynamic
along the Hugoniot.
which
series.
techniques
(16
with the experimental calculation
adiabatic have
and particle
and a standard
Eo = (PH - Po)( Vo - vH)/?
-
sure (PH), specific
Techniques The experimental
of Walsh [6],
equation
as the
conservation
PH
sound speed,
measurements
used
V”l V,, =
2. EXPERIMEVAL (a)
Hugoniot
was
the
u,, were
with
fee-fee
since the energy bands
method on pressure
of the sample
a known
Deoxidized
volume
bands
would be expected
The electronic
most of the lanthanide
allow
bands,
higher pressures
Sd-6s
with
conditions
velocities,
driven
the bulk to
is a case in point.
to occur at increasingly across
by the impedance-match
range extends
rearrangements
change
The
other rare earths,
(Z = 71).
or material
was used
u,, through
scandium
may well be expected
electronic
in turn profoundly material.
a
photography
velocities,
0.6. In view of the small
between
internal
using the continuity
shock
explosively
of the elements,
of the interatomic
produce
would
the
determined
are all the rare
high pressures,
A V/ V,,, of nearly
and
dynamic
Particle
(I GPa = IO kb) to nearly 200 GPa
These
differences
pressure
using
samples.
offer
of state of these elements
group III elements
the large compressibilities compressions.
various
camera
the shock-wave
these
undertaken
(Z = 39). The pressure
elements.
to determine
field in of
(Z = 61) and lutetium
from the lowest obtainable for some
have
in this study
also are the similar
(Z = 21) and yttrium
We
regime
included
of
such studies properties
state.
high pressure
conditions
an active
because
electronic
to study the equation
in the very
extreme
has become
primarily
valuable
wave
under
as a function
States
following 741
u,-u,
relation
holds, the Rankine-Hugoniot
can be used to express of
convenient
volume analytic
along
the pressure the
equations
and energy
Hugoniot (with
by
the
POand E,
742
W. J. CARl’FR
taken
to be zero):
t-t a/.
transverse
sound
individually
bulk sound velocities,
?y-V/V,,=I-$ we?t7
p,, =
TJS)?
(I E”
There
results
physical
properties
is a summary needed
Included
are both the directly
densities
and shock
this of
all
close-packing, neighbor
between
sequence
increased the
bee
only
and dipole-dipole
before
atom.
However,
average
The
values
Nuclear
part
from
The
the
made
samples
Research
Corporation
of
were
Table Faemen, z
Valence
2,
+3
hcp
with
Eu differs
from the other
elements
a significant
considerable
least well partly
speeds[9]
samples
are
on the samples obtained
Division
Longitudinal
with
of
and thermodynamic
Hugoniot
properties
E,Wtl-C.“ic Conflguratbn
3.196
because
such a highly
qualitatively
(kr?&
3.07
density
density
Ce,
change studied
variations
4.29
here,
between
the
inherent
in working
material. on
the
other
the rest of the rare
hand, earth
a K -L,
C. Ulka-K)
-Ia
5.60X10’
,2x,0-
,. 16 1.12
Y
39
+3
hcp
4d%’
4.576
4.38
2.52
3.26
3.09
10.6
La
57
+3
d!lcp
5d%s’
6.142
2.69
1.50
2.05
2.00
4.9
.3,
CF
56
+3(+-l)
ICC
4f’5d’6sa
6.729
2.33
I. 34
1.73
2.06
6.5
.37
pr
59
+3
dhcp
428s’
6.757
2.14
1.5,
2. I,
1.92
4.6
.34
tid
60
+3
dhcp
41’69’
6.963
2.64
L.60
2.16
2.09
6.7
Srn
62
+3
rhomb
4fsa
1.460
2.88
1.64
2.17
1.81
E”
63
+2
bee
4r?39=
5.290
2.34
l.64’
l.65
l
6.5 32
.45 *
differs
series.
at standard conditions
(kz;s)
in
has been used in the analysis)
of the difficulties reactive
in
structure
the data for Eu are the
of all the materials
of
from
ct ( km/e)
5.57
However,
of large
(an average
The
and
3d14s’
determined
because
and partly
for the
indicating
the
is either
at the transition.
of the 4f-band
that
this transition
shock
front.
Ce of a
is changing
indicate
in the
u$lm~~ SC
the data
volume
I. Physical
crystal Structure
small.
except
a characteristic
the conduction
Chemicals
America.
zero or very
associated
0.36 the slope
element
compressibility
cases
these
Eu and Ce.
of the tripositive
sound
of measurements
change
of
except
of a two wave
contains
and
The
whose
In most
This plane.
to describe
character
every
to a phase
rapidly.
(l-3).
that the data show the existence
at the expense
densities
used in this study. most
in view
state probably
pressure.
of
a
then
fits of the II,+,,
A V/ V. of approximately Hugoniot
to within
in the I(,+,,
general
transition volume
I are those of the
The
behind with care,
quantities
fit is required
by SO to 100 per cent,
to
at zero
II,-u,,
that,
least squares
increases
Er and Tm transform
state of most of these elements.
population.
the
l-5
is the same for all elements
At a compression of
more
free
of Sd-character
magnetic polymorphic
(densities,
to equations
in Figs.
data. (initial
in both the sample quantities
consistently,
according
adequately.
Hugoniots
is
valence admixture
second
data
all
neutral
band in the metallic
the
In addition,
listed in Table
errors
is presented
solid lines are linear
as pressure
melting
configurations
of
takes place in
temperature.
possibly
crystal
through
fee
these
Hugoniot
the derived
data; in all cases a two segment
in energy
+dhcp+
can hc measured
of one per cent:
information
coefficient,
quantities
has shown
The
modifications
forms
Experience
the
of the
and compressions
parameters
are
that
tenths
the
velocities
the
elements
[4] has noted
except
phase
pressure
these allotropic
wave).
of
measured
pressures,
which
Values
expansion
and the derived
velocities,
reflect
1,which
Eu.
hcp+rhomb
these elements
as one proceeds
for
at and near room
electronic
zero
electrostatic
Jayaraman
transitions
earths
few
zero-pressure
Except
these
differing
interactions. the
study.
wave
determined
from
calculated.
obtained from [IO]. 2 is a detailed listing
these velocities
in the
table. This is shown in Table
of appropriate
for
structures
similarity
of the rare
across the periodic
nearest
and discussion
is a striking
ch, were
heat, C,, and linear thermal
the shock (b) Experimental
method,
a, were Table
particle
2(1 - n.s)!.
c, and c,, were
specific
and the Cu standard)
Co2T2
=
velocities,
by the pulse-echo
.45 1.02
Gd
64
+3
hcp
4f’5d’6sa
7.877
2.95
1.69
2.21
2.99
6.4
.3,
Tb
65
+3
hCP
4f%8=
6.206
2.95
1.68
2.22
1.72
1.0
.60
Dy
66
+3
hcp
4f%’
6,398
3.07
1.76
2.26
1.74
6.6
.77
“0
67
+3
hCP
4?‘6e’
6.734
3.2,
1. 66
2.39
I. 64
9.5
.99
Er
66
+3
hcp
4f-36e2
9.039
3.13
1.83
2.31
1.61
9.2
.92
Tm
69
+3
hcp
4f%’
9.269
3.02
1.77
2.23
1.60
11.6
1.08
Yb
70
+2
fee
41”a=
7.036
1.94
1.12
1.45
1.45
25
1.09
Ce
Hugoniot
equation
Table 2. Hugoniot 00 nlms)
US (km/s)
2/i,
P ,GPaj
P (MC, ¶I?)
v/v0
u, v.3) (km/s)
143
of state of the lanthanides data
6% (~gjm')
U, (km/s)
6.756
2. II
6.764 6.762 6.75, 6.756 6.761 6.163 6.12, 6.751 6.761 6.728 6.134 6.765 6.75, 6.760 6.,30 6.756 6.764 6.762 6.154 6.164 6.764 6.759 6.759 6.755 6.769 6.765 6.763 6.755
2.39 2.45 2.65 2.82 2.89 3.03 3.06 3.13 3.28 3. 3, 3.73 3.96 4.2, 4.69 4.18 4.74 6.04 5.09 5.40 5.38 5.64 5.1, 5.14 6.06 6.15 6.32 6.24 6.35
6.983 6.9*2 6.984 6.983 6.965 6.965 6.982 6.983 6.921 6.976 6.983 6.980 6.983 6.994 6.978 6.984 6.991 6.084 6.990 6.988
2.16 2.48 2.55 2.72 2.14 2.91 2.90 3.02 2.99 3.1, 3.15 3.30 3.49 4.0, 4.62 5.18 5.44 5.56 5.76 5.83
7.460 7.468 1.464 7.469 7.464 1.463 1.463 7.460 7.470 7.463 7.463 7.465 1.454 7.465 1.460 7.459 7.459 7.454 7.460
2.1, 2.51 2.76 2.79 2.92 2.98 3.11 3.15 3.2, 3.28 3.40 3.71 4.31 4.50 4.55 5.32 5.4, 5.73 5.81
5.290 5.290 5.290 5.29" 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290 5.290
1.64 2.00 2.12 2.38 2.43 2.44 3.14 3.49 4.11 4.22 4.39 4.59 4.92 5.31 5.64
", (km/s)
IJ 0 (CI'a) (M&m',
vtvo
a (C") (km/w
Scandium 3.196 3. L96 3.196 3.193
3.192 3.191 3.19, 3.198
4.578 4.556 4.548 4.551 4.564 4.664 4.562 4.664 4.561 4.570 4.553 4.553 4.558 4.595 4.576 4.598 4.605 4.6R
4.29 5.2, 5.32 5.6, 6.02 6.69 1.04 7.63
3.26 3.73 3.75 3.86 3.93 4.0, 4.2, 4.31 4.3, 4.61 4.5, 4.62 4.91 5.35 5.76 6.39 6.59 7.27
0.15
13
0.95 1.40
16
1.93
25 3,
2.58
55
3.00 3.75
66 94
1.43 1.55
5 8 L1 13 L7 23 24 2, 30 32
I. 56 1.E6 2.19
33 42 54
2.52 3.03 3.10
i6 39 94 122
0.32 0.4, 0.64 0.73 0.91 1. 1, 1.21 1.34
3.58
3.20 3.73 3.89 4.24 4.70 6. I9 5.58 6.14
4.58 4.98 5.20 5.46 5.60 5.8, 6.29 6.35 6.5, 6.70 6.88 6.92 7.33 1.7, 8.14 8.73 8.10 9.21
1.000 0.857
4.74
0.822 0.752 0.679 0.615 0.6'13 0.521
4.94 6.42 5.98 6.6, 7.13 7.95
1.000 0.914 0.875 0.834 0.614 0.77, 0.725 0.118 0.694 0.682 0.662 0.656 0.622 0.592 0.562 0.52, 0.529 0.50,
4.29 4.45 4.63 4.73 4.92 5.21 5.26 5.39 5.51 6.62 5.66 6.98 6.3, 6.16 7.3, 7.4, 8.08
1.21 1.31
6 * 13 18 20 24 25 28
6.76 7.94 8.34 9.24 10.10 10.46 11.06 ,,.I2 11.64
1;45 1.66 1.76 2;01 2.19 2.21 2.29 2.45 2.49 2.62 2.64 2.83 2.83 2.88 3.02 3.10 3. 16 3.21 3.22
33 42 4, 58 69 71 73 84 86 95 96 108 IL0 112 124 12Y 135 136 138
11.19 12.16 12.2, 12.78 12.6* 12.54 13.09 13.15 L3.23 13.10 13.29 13.56 L3.28 L3.5, L3.45 13.60 13.56 13.93 13.69
0.35 0.46 0.11
0.93 1.02 1. 18
1.000
0.811 0.131 0.669 0.64, 0.61, 0.606 0.580 0.572 0.570 0.554 0.554 0.629 0.533 0.53, 0.516 0.514 0.511 0.516 0.509 0.499 0.509 0.498 0.502 0.49, 0.499 0.485 0.493
4.32 4.44 4.7, 4.96 5.05 5.24 5.2, 5.39 5.51 5.56 5.84 5.96 6.29 6.65 6.59 6.66 6.69 6.94 ,. 13 7.15 7.40 7.42 1.4, 7.68 7.18 7.88 7.92 ,.95
0.852
NCdpU”Ill
6.142 6.140 6.L35 6.130 6.138
2.05 2.40 2.54 2.70 2.70
6.133 6.136 6.132 6.L35 6.L51 6.13, 6.13, 6.13, 6.119 6.155 6.13, 6.082 6.136 6.140 6.166 6.153 6.154 6.149 6.156 6.134 6.142 6.110 6.15, 6.142
2.8, 2.82 2.81 3.02 3.14 3.32 3.50 3.76 4.28 4.16 4.64 5.01 5.18 5.33 5.39 5.66 5.91 5.9, 6.01 6.24 6.32 6.46 6.44 6.49
6.729 6.736 6.736 6.12, 6.132 6.130 6.726 6.134 6.719
1.73 1.5, 1.90 2.33 2.29 2.6, 2.64 2.85 3.09
6.734 6.726 6.773 6.778 6.730 6.759 6.750 6.732 6.770 6.739 6.764 6.731 6.772 6.731 6.771 6.756 6.750 6.732
3. 32 3.60 4.15 4.13 4.53 4.99 5.24 5.30 5.55 5.54 5.80 5.84 5.94 5.93 6.2, 6.26 6. 36 6.36
5 I
0.35 0.47 0.65 0.72
0.74 0.76 0.79 0.94 1.02 1.1* 1.31 1.44 1.7, 1.18 2.02 2.32 2.34 2.48 2.52 2.6E 2.8, 2.69 2.93 3.09 3.16 3.25
,I
12 13 13 I4 L, 20 24 28 33 46 46 58 7, 75 81 84 93 104 106 108 118 123 128
3.27
130
3.29
131
0.39 0.50
4 6
0.14
0.79
I2 12
0.94 0. 95
17 1,
1.02 1. I*
20 24
1.29 1.42 1.73 1.73 1.96 2.26 2.42 2.46 2.59 2.62 2.76 2.80 2.80 2.86 2.99 3.08 3.16
29 34 49 40 60 76 86 8R 9, 98 108 110 113 114 127
3.18
*30
136 137
6.14 7.20 7.54 8.09 8.29
1.000 0.852 0.814 0.758 0.740
4.31 4.44 4.63 4.11
8. 33 8. 38 8.51 8.92 9.12 9.53 9.19 9.96 10.44 10.7, 10.88 L1.34 11.20 Il.50 11.56 Il.69 11.9, 11.92 12.0, 12.14 12.30 12.28 12.52 12.48
0.736 0.132 0.720 0.688 0.675 0.644 0.62, 0.616 0.566 0.571 0.564 0.536 0.548 0.534 0.533 0.626 0.514 0.516 0.513 0.505 0.499 0.498 0.492 0.492
4.73 4.15 4.78 4.96 5.05 5.24 5.39 5.66 5.98 5.9* 6.29 6.66 6.12 6.89 6.94 7.15 7.40 7.42 7.4, 7.68 7.78 7.88 1.92 7.95
6.73 8.99 9.13 9.8, 10.32 10.40 10.64 10.52 IO.*5 11.03 II. I2 Il.63 11.68 Il.96 12.36 12.65 12.66 12.71 12.7, 12.93 12.93 12.84 13.00 12.93 13.31 13.42 13.51
1.000 0.749 0.736 0.6R2 0.652 0.64, 0.638 0.640 0.619 0.610 0.605 0.582 0.580 0.5,;3 0.54, 0.538 0.636 0.533 0.528 0.623 0.521 0.528 0.516 0.624 0.508 0.503 0.498
4.32 4.44 4.71 4.76 4.95 4.96 5.05 5.24 5.39 5.56 5.96 5.98 6.29 6.6, 6.89 6.94 7.13 7.15 7.36 1.40 7.42 ,:4, 7.68 7.78 7.86 7.91
1.13 1.15 L.28 1.42 I.64 2.21 2.64 2.79 2.88 2.9, 3.06
8 13 14 18 19 2, 21 25 25 30 35 52 ,, 96 106 112 120 124
8.12 8.50 9.41 9.4, LO. 12 10.22 10.43 10.41 10.83 11.02 11.42 11.7, 12.74 13.4" Ii25 14.38 14.12 14.37 14.69
1.000 0.860 0.822 0.742 0.138 0.688 0.683 0.670 0.666 0.644 0.634 0.611 0.593 0.549 0.521 0.490 0.486 0.484 0.486 0.476
4.32 4.44 4.71 4.73 4.95 4.96 6.05 6.05 5.21 6.24 5.39 5.56 6.10 6.59 7.15 ,.36 7.4, '1.61 7.71
0.34 0.69 0.11 0.89 0.9, 1.15 1.16 1.24 1.25 1.38 l.62 2.03 2.20 2.22 2.15 2.90 3.11 3. 14
6 14 15 19 22 2, 2, 30 31 35 4s 65 74 15 109 118 133 136
7.46 8.63 9.94 10.00 10.73 11.08 11.83 11.82 12.06 12.06 12.56 13.29 14.10 14.62 14.58 15.48 ,5.*ti 16.30 16.24
1.000 0.866 0.751 0.74, 0.695 0.674 0.631 0.63, 0.620 0.619 0.594 0.562 0.629 0.510 0.512 0.482 0.470 0.45, 0.459
4.32 4.72 4.74 4.95 5.04 5.25 5.2, 5.3, 5.38 5.54 5.85 6.39 6.61 6.63 1.36 1.54 7.82 7.8,
1.000 0.605 0.761 0.671 0.66, 0.57, 0.499 0.47, 0.446 0.420 0.422 0.405 0.406 0.403 0.370
4.31 4.43 4.71 4.,4 4.96 5.56 5.86 6.39 6.5, 6.69 6.92 7.16 7.4, 7.91
6.98 0.35
0.45 0.70 0. R 0.9, 0.92 1.00 1.00
6
Europlum 5.29 0.39 0.51 0.78 0.81 1.03 ,.5R
1.62 2.28 2.45
2;54 2.73 2.92 3.1, 3.65
4 6 10 IO 13
26 34 50 55 59
66 16 89 106
6.5, 6.95
7; 80 ,. 93 9.17
10.61 11.10 11.85 12.59 12154 *3.0* 13.04 13.12 14.26
W. .I. CARTER
144
et al.
Table 2 (Confinued)
PO (Mnlms)
” Om/s()
I. 877
2.2,
I. 876 7.877
2.49 2.6,
0. 32 0.44
7.875
2.80
0.87
P
“. (km/s)
VIVD
(GPa)
u. (CU) ocrn/S)
CNPalm’)
”
PC
f YKlrn’)
(km/s;
Il.
P
(km/e)
P
(GPP)
us (Cd (km/e)
VW.
(M./m’)
Gadolimum
0.69
6 9
7. a* 9.05 9.48
9.04
1.000 0.870
9.039
2.31
4.3,
9.058
2.6,
0.30
7
10.25
0.883
4.3,
4.44 4.72
a. 992
2.74 2.7,
0.40 0.4,
1” 10
10.54 10.68
0.853 0.847
4.43 4.44
2.88 2.86
0. 63 0.65
17 17
11.6, ,,.‘I4
0.780 0.77,
4.72 4.74
o;s2
22
,2;45
0.72,
4.95
1. 16 1.20
35
13.84 14.03
0.654
5.38
31
0.645
5.44
15
10.36
0.631 0.760
9.046 9.064
1.000
7,877
2.82
10.43
0.155
4.74
2.99 3.22
0;86
15
7.817 7.878
20
11.08
0.71,
9.056
L. 08
11.86
7.877
3.20
0.664 0.662
5.2,
9.054
3.0, 3.36
7.877
3.22
1.08 1.13
27 27
4.95 5.2, 5.27
7.876
3. 35
1.22
32
12.31
9.057 9.066
3.40 3.60
1.28
4,
14.3,
0.634
7.876
3. 36
12.64 L2.65 13.47
9.049
3.76
1.52
52
15.15
0.597
5.85
3.48 3.8,
34 37 48
0.623
7.815 7.881
1.27 1.35 1.58
a. 991 9.04,
4.13
1.85
69
Ifi. 3,
0.55,
6.29
7.818 7.877
4.26 4.6,
7.875 I. 864 I. 846
4.51 4.51 4.82
7.873
29
11.89 L2.16
0.646 0.636 0.613 0.565
5.38 5.44 5.54
1.92
65
L4.27
0.562
5.85 6.29
2; 15 2.18
76 76
14; 73 15.06
0.535 0.523
6.6, 6.63
2.21 2.37
79 90
15.19 ,5;45
0.518 0.508
6.66 6.89
4.88
2.40
15.48
7.860 7.860 7.843
5.15 6.26 5.39
2.53 2.72 2.76
92 102
0.508 0.509
6.94 7.13
0.483 0.488
7.36 7.42
I. 900
5. 34 5.62 6.8, 5.73 5.73 5.74
2.87
0.463 0.453 0.466 0.458 0.452 0.443
7.54 7. 62 7. 88 7.87 1.92 7. 95
7.870 7,666 I. 878 7.877 1.624
3.07 3. 10 3. 10 3. 14 3.20
a. 199
2.22 2.8, 2.6, 2.93 3.20 3.3,
0.66 0.68 0.79 1.07 1.2,
8.209 8.205 8.214 8.247 a. 181 a. 194 8.247 8.204 8.209
3.37 3.46 3.73 3.93 4.12 4.5, 5.23 5.61 5.70
1.25 1.33 1.56 1.65 1.85 2.,6 2.68 3.03 3.06
8.206 a. 194 a. 199 8.218
8.210
15.45
112 1613,
11,
12, ,36 14, 140 142 140
16.05 17. OH 17.3, I6.86 17.20 17.42 17.23
15 I6 19 28 33
10. 72 IO. 63 11.26 12: 33 I2.92
1.000 0.764 0.757 0.730 0.666 0.634
4.12 4.74 4.87 5.2, 5.38
35 38 *a 54 62 60 116 140
13.06 13.36 14. L3 14.24 14.87 IS. 74 16.90 17.85 17.14
0.629 0.613 0.581 0.579 0.560 0.62, 0.488 0.460 0.463
5.44 5.54 5.64 6.98 6.22 6.63 7.36 7.82 7.81
144
Dysprosium 8.398 6.41, 6.400 8.417 8.429 6.416 a.426 8.4LO 8.38, a. 409 8.420 8.4,, 8.422 a. 368 8.412 a. 408 8.422 6.419 8.410 a. 386 6.419
2.28 2.51 2.67 2.67 2.85 3.03 3.10 3.22 3.26 3.34 3. 39 3.40 3.52 4.03 4.53 4.5, 4.79 5. ,a 5.33 5.55 5.67
0.33 0.43
7 I”
0.65 0.67 0.64 0. 92 1.06 ,. I, 1. ,Y 1.19 1.23 1.3, 1.14 2.11 2.14 2.36 2.67 2.81 3.02 3.04
16 16 2, 24 29 30 33 34 35 39 59 80 a, 95 116 ,26 14, 145
6.40 9; 64 10.0, 10.90 11.03 11.65 11.96 12.50 L2.69 L3.04 12.91 13.20 13.44 14.70 15.73 16.02 16.63 17.37 17.76 18.40 18.16
L. 000 0.813 0.840 0.772 0.764 0.722 0.704 0.673 0.66, 0.645 0.649 0.637
4.32 4.44 4.72 4.74 4.95 5.04 5.2, 5.27 5.37 5.38 5.44
0.621 0.669 0.535 0.525 0.507 0.485 0.473 0.456 0.463
5.54 6. LO 6. 61 6. 63 6. 94 7. 36 7.54 7.62 7.87
9.046
4.43
2.0,
I6.54
0.547
6. 53
9.073
4.40
2.08
83
LT. 17
4.44 4.70 5.03
2.10 2.29 2.44
84 96 110
17.19 17.40 1,. 34
0.528 0.52,
6.6,
9.068 a. 927 a. 93,
0.513 0.615
9.055
5. 16
2.64
124
,a. 56
0.488
7. L3 7.40
8.926
5.29 5.5, 5.54
2.66
126
1,. 93
0.496
7.42
2.8, 2.92
137 145
18.09 l6.96
7.63 7. 78
5.6,
2.94
148
18.83
0.491 0.474 0.476
a. 987 a. 99, a. 946
5.76 5.65 5.69
2.9, 2.98 3.04
164 ,52 155
19.0, 19.23
0.485 0.473 0.465
7. a* 7.87 7. 95
9.289 9.318 9.311 9.271 9.260 9.269 9.310 9. 325 9.266 9.303 9.263 9.305 9.339 9,269 9.269 9.289
2.22 2.62 2.85 2.84 2.96 2.96 3.2, 3.23 3.44 3.72 4.06 4.35 4.35 4.80 5.33 5.63
0.63 0.65 0.62 0.83 1.04 ,. 07 L. 30 1.6, L. 84 2.05 2.06 2.47 2.76 3.17
1.000 0.882 0.78, 0.712 0. 725 0.122 0.675 0.669 0.622 0.595 0.546 0.528 0.525 0.486 0.479 0.436
4.32 4.7, 4.73 4.95 4.96 5.24 5.27 5.66 5.85 6.29 6.69 6.61 7.16 7. 6, a. 13
7.038 7.064 7.038 7.058 7.03, 7.050 7.026 7.028 7.028 7.037 1.028 7.021 7.049 7.049 7.068 7.070 7.016 7.039 7.019 7.036 7.040
1.45 1.78 1.77 1.85 1.86 2.0, 2.08 2.20 2.27 2.58 2.62 2.74 2.91 2.9, 3.04 3.46 3.90 4.2, 4.34 4.95 5.33
0.35 0.39 0.49 0.50 0.68 0. ‘I? 0.9, 0.94 1.19 1.23 1.35 1.43 1.44 1.53 1.8, 2.12 2.35 2.42 2.88 3.13
1.000 0.804 0.782 0.737 0.729 0.662 0.629 0.589 0.584 0.538 0.529 0.509 0.510 0.507 0.496 0.47, 0.457 0.44, 0.443 0.418 0.412
4.29 4.32 4.43 4.45 4.63 4.13 4.88 4.92 5.2, 5.26 5.39 5.50 5.5, 5.62 5.98 6.37 6. 67 6.76 7.37 7.11
8.878 a. 977 8.956
0.42 0.64 0.66 0. 63 1. I? 1.22 1.29 ,. 36 1.39 1.52 1.53 I. 61 1.93 1.99 2.08 2.12 2.67 2.97 3.02
10 16 17 22 35 36 40 42 44 50 5, 57 72 74 82 83 12, 145 147
a. 73 10.34 11.27 11.3, 12.00 13.37 13.55 13.80 L4.20 14.10 14.43 L4.58 15.31 15.80 16.44 36.26 16.70 18.04 18.61 19.21
1.000 0.842 0.777 0.774 0. 729 0.665 0.845 0.634 0.616 0.618 0.602 0.599 0.568 0.551 0.530 0.538 0.524 0.483 0.467 0.456
4.44 4.R 4.74 4.95 5.38 6.44 5.54 5.62 5.66 5.84 5.85 6.03 6. 39 6.45 6.6, 6.63 7.40 7. 82 7.87
0.7GPa.
18.63
form
at a rapidly manifested curve,
is
(the
the range
6. 63 6.89
7.82
as expected
decreasing decrease
at about
that the
to a form quite
in slope
48GPa.
This
so
whose rapidly,
of the
u,-u,
transition
is
transition.
point was determined sound
techni-
steep over the
more incompressible
rate. A transition not
tetravalent 15 per cent at
of explosive
indicating
a two segment measured
the
of Ce is quite
the fee solid-liquid
The transition
to
change of about
range.
by a slight
by imposing data
transition
of Ce is becoming
is observed
probably
7.04 8.78 9.00 9.58 9. 64 10.65 11.17 11.94 12.03 13.09 L3.28 13.82 13.83 L3.90 14.25 ,‘I. 83 15.35 15.95 15.84 16.82 17.09
4 5 6 7 10 I, I4 15 22 23 26 29 29 33 44 56 70 74 100 *,a
Hugoniot
increasing
compressibility
11.93 12.00 12.78 12.84 13.79 13.95 14.69 l5.64 I6.98 17.62 17.78 19.08 19.39 21.32
110 131 ,66
is below
experimental
tetravalent
17 17 23 23 31 32 4, 52 69 83 84
electronic
IA,-u,
9.29 10.56
a
with a volume
which
ques. The entire
0.3,
an
configuration 2.36 2.68 2.88 2.9, 3.06 3.39 3.43 3.53 3.55 3.63 3.62 3.8, 3.88 4.30 4.24 4.52 4.45 5. ,a 5.58 5.56
a,
Th”ll”lIl
undergoes a. 734 a. 709 a. 155 a. 753 a. 750 8.75, a. 749 a. 750 a. 745 a. 705 a. 684 8.129 a. 736 a. 105 a.715 a. 752 6.742 8.72, 8.723 a. 763
5.54
linear
in the II,+,
least squares
velocities
were
plane
fit on the not
used
Hugoniot
equation
of state of the lanthanides
a-
7 -
I
0
I
I
I
I
I
I
I
05
IO
I5
20
25
30
35
l
I
GADOLINIUM
tU,I
.
DYSPROSIUM
I”,*11
A
ERBIUM
I 0.5
2 0
40
I
n
IL!,*21
I 10
I
I
I5
20
J
UP I km/s
I
I 2.5
I 30
35
U,,(km/s)
Fig. I. Hugoniot II,-u, data for SC, Y and La. The Hugoniots of both SC and Y fail to extrapolate to the measured zero pressure sound speed, probably because of a sluggish low-pressure phase transformation and the presence of an elastic wave. Evidence of this transition has been found in free-surface velocity vs time measurements on Y. We have not attempted an analysis of the transition in SCand Yin this work. The behavior of La is typical of most of the remaining members of the rare earth sequence.
Fig. 3. Hugoniot u,-U, data for Gd, Dy and Er. Although the data are inconclusive, the u,-U, Hugoniot for Gd has been drawn with a break at high pressure and a subsequent decrease in slope. This is in agreement with the phase diagram for Gd presented in Fig. 16.
a
I
I
7 -
I
l
TERBl”M
.
HOLMIUM
IUs*l)
A
TH”Ll”M
iU,‘Zl
I
I
20
25
I
(“*I
6-
;; . E t
5-
; AL 4-
0 3-
28 0
05
IO
I5 U,
, 0
I
I
I
I
I
I
05
IO
I5
20
25
30
U,(
Fig.
2. Hugoniot
u,-u,
km/r
55
I
data
for Pr, Nd and Sm. These three slopes for the lower pressure phase. However, the Hugoniot fits do extrapolate well to the bulk sound speed, indicating that low-pressure transitions do not affect the shock wave data.
elements have anomalouslylow
because of the presence of low pressure phase changes) and minimizing the total deviation determined by leastsquares calculations. The resulting fits for most materials yielded standard deviations corresponding to no more than one per cent in zero-pressure intercept and one-half per cent in slope. The intersection of these straight lines then determined the velocities at the transition. If a small volume change is associated with the transition, this technique will give transition pressures that are slightly too high. However, the close spacing of the data points in
30
5
(km/r)
Fig. 4. Hugoniot U.-U, data for Tb, Ho and Tm. The elements all behave normally, with the transition being scarcely noticeable for Ho because of the large scatter in the data for the upper phase.
the transition region precludes this being more than a very small error. These Hugoniot u,-u, fits both below and above transition, the shock velocities at transition, and the corresponding transition pressure and compressions are listed in Table 3. Figure 6 shows the measured ultrasonic bulk sound speeds and the Hugoniot intercepts as a function of atomic number 2. The agreement in general is quite good, which is a further indication that the low pressure phase changes in the lanthanides are unimportant to interpretation of dynamic Hugoniot data. However, the agreement for the elements SC and Y is not good since the Hugoniot intercept lies well above the bulk sound speed. Such behavior is an indication either of a very large dynamic yield point or of a low pressure polymorphic phase
746 Table
ElUlX”t
7. Summary
Hugoniot l_owrr
data parameters
FL% upper
Pks*e B
co
of Hugoniot
co
8
VW,
“.
P
(km/s)
(km/s)
&m/s)
al Transition
F%asc
,c. Pa,
SC
4.76
.64
4.11
.99
5.94
.69
Y
3.31
.78
2.38
i. 35
4.71
.s3
37. cl
i-a
2.05
1.02
1.52
L.53
3.22
.64
22 .5
ce
.87
1.90
l.41
1.58
4.12
.56
46.0
PF
2.09
.79
0.94
I.68
3. 12
.59
27.0
Nd
2.16
.83
1.50
I.41
3.11
.63
25.0
sm
2.25
.76
1.56
1.35
3.13
.63
27.0
E”
1.62
.9s
1.12
1.29
2.43
.67
ICI.6
Cd
2.20
.92
1.77
1.29
3.40
.62
34.5
Tb
2.21
. 92
1.12
1.29
3,SO
.60
40.0
w
2.28
.8Q
1.91
1.22
3.27
.66
30.5
IlO
2.27
.96
I.91
1.22
3.62
.61
44.0
35.0
Er
2.31
.90
1.69
1.34
3.55
.fiI
44.0
*In
2.26
.Ql
1.90
1.19
3.38
.63
39. c!
Yb
1.46
.83
0.80
1.43
2.26
.57
16.0
‘7
5F’
“__.T..,
,
._-r,
r.
.
1
h
T
+
I
+ HUGONIOT
I _L-.._ I ~7s3S36oS 6263ekjl$5&zkz&70 LoCsR NdF7nBnEuGdTbDyHoErTmYb
Y
ATOMIC
I
I
co
I5
I
0
05
I 2.0
I
I
I
25
30
35
I 40
U,(km/sl
Fig. 5. Hugoniot u, -9, data for Ce, Eu and Yh. Although Yh behaves normally in this plane. hoth fe and Eu have anomalous r*,-U, Hugoniots. Eu is characterized by a large volume change at it< transformation. while Cc transform< immediatcl~ to the tetravalent fee form, which is quite incompressible. At higher pressures Ce again becomes somewhat softer and the U.-N, Hugoniol decreases slightly in slope.
change. shock
To check wave front
capacitor
this. the time-resolved through
technique.
measurement.
Figure
on all records,
account
for the discrepancy.
a phase change
this is assumed
indicating
that equilibrium
the
techniques. as well.
point
A similar
too small
has
not
transition
been
normally
however. Sm-type behind
of this sluggishness observed
undoubtedly
by occurs
Figure
8 shows the fitted values
of the IL-U,,
slopes for the two phases as a function
to
is pressure-dependent,
because
extrapolate 10 the \ound speed. Agreement for all the other elements is excellent. indicating that the known low-pressure close-packed ~olymorphs of the rare earths are quite similar Ihermody~micall~.
phases
in the wave
has not been attained
It is perhaps
Fig. 6. Bulk sound speeds and Hugoniot intercepts as a function of atomic number. The sound speeds were all obtained by the ultrasonic pulse-echo technique. except for the value for Eu which was obtained from compressibility measurementsI IO]. Ce. of course, transforms to rhe highly incompressible tetravalent form at pressures lower than shock pressures used here. and the two numbers are therefore not expected to agree. Similarly. hecause of hydrodynamic n~~nequilib~um behind the shock fronts at low pressure, the Hugoniots of Y and SC are not expected to
is
of about 0.4 GPa,
by a break
NUMBER
with
increasing
divalent
nearly
three
slopes
that
atomic
and
having
decrease
number. initial
times as great as the other lie on a smooth
curve
Hugoniot
of atomic
It will be noted that the slopes generally
at longitudi-
At high pressures,
transition
transition
of such a
to be to the rhombohedral
The
that
moving
l._-._i
of a
using the d.c.
but this is much
is indeed indicated
structure.
the shock front.
wave,
with an amplitude
observed
structure
7 shows the results
A small elastic
nal sound velocity
front;
Y was recorded
INTERCEPT 1
SC
IL
(SCHELBERG)
0 ULTRASONIC
.
4..
Eu
number. in both and
Yb.
compressibilities rare earths,
have
in the high pressure
regime determined by the elements normally characterized as trivalent. Ce has a very high initial slope, since it transforms
almost
immediately
lo the tetravalent
form.
The low values of the initial slopes for Pr. Nd, and Sm are
static
at present
in SC
a very
unexplained,
sluggish
equilibrium
even
although
solid-solid under
this may again be due to
phase change shock
conditions.
which
is not in
However,
the
747
Hugoniot equation of state of the lanthanides
Jayaraman[4,13] has measured the melting phase line in the P-T plane for most of the rare earths, in some cases to
TIME
(,,b)
Fig. 7. Free-surface velocity vs time for Y. The record was obtained by impacting a stationary Y target with a steel projectile and observing the motion of the free surface with the d.c. capacitor technique. Point I is the stress level where plastic flow first begins (the Hugoniot elastic limit). This low-pressure elastic wave propagates at essentially longitudinal sound velocity. The two-wave structure at point 2 is the nonequilibrium transition which leads to anomalously high values of the shock velocity using flash gap techniques. Here, the transition wave has been nearly overdriven.
a pressure of nearly 7GPa. Extrapolation of this data indicated that the melting curve and the Hugoniot do indeed intersect in the general neighborhood of the observed transition point in all cases. However, this is a considerable extrapolation and a more exact calculation of the phase line is desirable. To determine the location of phase lines the free energy function, G = H-TS, must be calculated, since a boundary between two phases is characterized by AG = 0. A convenient way to determine the free energy is first to calculate the temperature, T and entropy, S along the Hugoniot which can be combined with the known enthalpy to give the free energy along the Hugoniots of both phases. The results of these calculations can then be readily extended to give the free energy in the region of interest. This can be accomplished by use of the thermodynamic relations dE=TdS-PdV
(8)
T dS = (aE/aT), dT t T(aP/aT), dV
(9)
and the Rankine-Hugoniot relationships (l-3). By introduction of the Grtineisen parameter, y, defined i3S
. T-
20
‘.Liwq A UPPER
y =
V(aP/aE),
(10)
PHASE
equation (9) can be rewritten as
t
TdS=C,dT-C,(y/V)TdV
0, 21 SC
I 39 Y
I I I J L .1-l 57SSs96060 626364656667666670 lsCeRNdV4nhSmEuGdTbDyHoErTmYb ATOMIC
I
I
I
I
I
NUMBER
Fig. 8. Hugoniot u,-u, slopes as a function of atomic number. A definite trend across the periodic table is noted, with the slopes of both phases generally decreasing as atomic number is increased. It should be noted that, for a linear U.-U, relation, the slope is a direct measure of the rate of change of compressibility with pressure. The abnormally low values of the slope in the lower phase for the light rare earths Pr, Nd and Sm, and the abnormally high values of the slope in the upper phase for Pr, Er and Yb are difficult to explain, but presumably are also associated with electronic rearrangements. The initial slope of Ce, of course, is not to be compared with the other rare earths. The slopes of the upper phase are the experimental values, not those appropriate to the transformed metastable Hugoniots, although for most of these materials there is little difference between the two.
(11)
We shall assume that y is a function only of volume and, specifically, that the product my is constant. Experimental work on a number of materials[8], as well as theoretical considerations[l4], indicates this to be an adequate approximation. With this assumption, y(V) is given by y = (V/ VO)yO,where y0 is the thermodynamic value at standard conditions given by y. = 3aca2/C,. The assumption that y is a function only of volume implies that the specific heat C, is a function only of entropy. Since the characteristic Debye temperatures for all these elements are quite low, it is further assumed that C, is independent even of entropy (this assumption may not be good for elements such as Gd). Equations (3). (8) and (I 1) can then be combined to yield
dT,(v”VI rdP+[F-T(+‘)]dV
(12)
which can be used to determine the entropy through good agreement of the Hugoniot intercepts with the bulk sound speeds for these elements argues against this.
(13)
3. ANALYSIS
Of central interest in this study is the nature of the transitions which occur so persistently throughout the rare earth series. These transitions appear to be associated with melting, perhaps with an accompanying electronic rearrangement to the tetravalent state.
These have been integrated numerically. When this is done, a complete thermodynamic description along the Hugoniot for each of the two phases is obtained. Under the assumptions of constant (py) and C, thermodynamic quantities off the Hugoniot can be
W. J. CARTERet al.
748
computed through the relations (aP/U),
= pyC, = constant
P = Ph(V)+pyC,[T-
Th(V)].
(14)
The quantity Th( V) determined from equation (12), coupled with the known Ph( V) and equation (14), serves to determine V(P, T) implicitly. Equation (11) can then be used to determine the entropy, and the Mie-Griineisen equation E(P, V)=E~(V)+[P-P~(V)I/(yp)
(15)
serves to determine the internal energy at every point off the Hugoniot. To connect the two phases the difference in energy and entropy across the phase boundary at some point are needed. These differences may be computed from the Clapeyron equation if the slope of the phase line, (dP/dT)*, and the associated volume change is known at any point (P*, T*) on the phase boundary: AS = AV(dP/dT)* AE = AV(T*(dP/dT)*
- P*)
(16)
The appropriate values of P*, T* and (dP/dT)* were obtained from the P-T diagrams for melting presented by Jayaraman [4,13]. Before performing these thermodynamic calculations, it is necessary to center the upper-phase Hugoniot to a point (p(,, T,,, P,) appropriate to the transformed material. For convenience, we have chosen to extend the phase (in a metastable condition) down to room temperature and zero pressure. The method of recentering utilizes the thermodynamic calculations just described and by appropriate minimization procedures obtains a Hugoniot described by equation (4) centered at the desired point. This method has recently been described in detail [ 151.For this calculation, which results in a zero pressure density,
sound velocity and slope of the metastable Hugoniot, it is necessary to know or estimate the pressure, temperature and slope of the phase boundary where it intersects the Hugoniot. Since the intersection of the two linear u,-u, Hugoniots determines the pressure and temperature, only the slope of the phase line, dP/dT, is unknown. To determine this, an iteration procedure was used. The metastable Hugoniot was first calculated on the basis of a reasonable guess of dP/dT at the transition determined by an extrapolation of Jayaraman’s phase lines. This Hugoniot was then used to calculate the phase boundary as described above, and the slope, dP/dT, at its intersection with the lower phase Hugoniot determined from the results. This new value was then used to recenter the Hugoniot of the upper phase, and again the curve AG = 0 calculated. Two iterations of this type were generally sufficient to yield invariant values of P*, T*, and dP/dT* at the transition. Table 4 lists the parameters for the metastable Hugoniot used to establish the thermodynamic properties of the upper phase. The recentering procedure used to establish these metastable Hugoniots generally resuts in ill-defined minima determining the values of the parameters pX, c 8. and s *. While they appear to be well established, both ~‘6and pQ could be in error by as much as ten per cent; for the sake of consistancy and impartiality, we have always shown the values at the absolute minimum. It should be emphasized that nowhere in these calculations is the predicted value of the coordinate (P, T)* at melting on the Hugoniot required to agree with the experimental value; the Hugoniot data is used only to establish the thermodynamics of the two phases separately through equations (14) and (15). Furthermore, there has been no attempt to adjust any parameters in order to predict the transition points. In view of the abnormally low values of the Griineisen parameter, ye, for these metals and the importance of this parameter in the calculation of thermodynamic quantities, most of the possible errors in these calculations probably arise from the assumption of constant (py) and the lack of knowledge of y in the liquid region. Because of this
Table 4. Metastable Hugoniot parameters 0
Elenmnt
PO (Mglm3)
::c
C”
s*
kin/s)
1.75 Ucc) >. GO (liquid) Pr
6. 92
1.05
1. G!J
i-id
7.00
1.20
1.40
Srn
7.73
1. 64
1.37
Eu
4.82
1.03
1.27
Gd
8.24 9. 00
l.GG 2.10
1.33 1.25
Tb
9.03
2. on
1.27
w
8.31
1.89
1.22
Ho
7. 63
1.56
1.24
Er
7.68
1.36
1.31
Tm
G.G6
1.77
1.21
Yb
8.10
1.12
1.48
(bee) (liquid)
-
Hugoniot
uncertainty,
heat, uncertainties
questions
about
formulation
analysis
validity
immediately melting.
to the
However,
70
is in general
more
I
because
dense
I
HUGONlOl
30-
20,
slopes
Ce has a
/
._
’
/
10 -
1’
/
I.lO”lO
alfccl
state
upon
the
phase
I
_ CALCULATED .SOLIO- LIQUID P”ASE BOUNDARY
// /
:
-./‘,’ j/c
//
for all the
pressure
I
2
/’
/+ // //
0 ;;
it transforms
tetravalent
increasing
and
quite good.
Ce are positive;
presumably
CERIVY 40,
boundaries.
the calculated
Y and SC. The initial
except
with
I
phase
---
SOY-
a meaningful
phase diagrams
here except
initial dPldT*
of
Mie-Grtineisen
to perform
pressure
all the elements
the
between
show proposed
studied
negative
of
calculated
transition
(10-22)
elements
the
the agreement
experimental Figures
for
_.
in the static data, and even
itself, it is impossible
Nevertheless,
for
the
149
of state of the lanthanides
as well as that due to lack of knowledge
the specific
error
equation
1
I
0-YYCfl’=) 0
500
.1-I_ lxx)
IOCU
._..
2030 K)
Tl
-. 2500
3&u
35&-
Fig. I I. Proposed phase diagram for Ce. The extremum at 1.1 GPa has been attributed by Jayaraman to the completion of 4f-5d electron promotion in the liquid similar to the y-o transition in the solid. The metal then melts normally up to at least 5OGPa. The Hugoniot crosses the phase line at a point where dPldT is positive; this is manifested by a decrease in slope in the I,.-I,, plane. Because of the important low-pressure phase change. it was necessary to recenter the Hugoniots of both the upper and lower phases. This makes the calculation of temperature less certain. since small errors in the metastable density of the collapsed fee form can lead to large contributions to the PdV term of equation (I!)
30
---
Erblum
----
PRASEODYMIUM
20 0
IO
05
IS
2s
20
30
33
G
Us (km/s)
ii
Fig. 9. Hugoniot U.-U, data for porous erbium. The dashed lines are calculated, using (py) = constant, the Griineisen relation, and the crystal data indicated by the solid line. The agreement is seen to be satisfactory. indicating that the assumed volume dependence of the Grtineisen parameter is sufficiently accurate for our purposes.
z
/
I IO-
/’
I
I
’
,
.--_.-.,.
1
_A /’ ’ / /
,
/
/
CPILC”LATE0
,soLIO-LIO”I0 /
/
PHblK
I
1000
T(K)
BXlNMRY
bee
I ISCYJ
I
LlQUlO
IOOf
I500
Fig. 12. Proposed phase diagram for Pr. The phase line is initially nearly vertical. and the slope changes very little with pressure. It is possible. in view of the discrepancy between the measured and calculated transition pressures, that the fee-liquid transition is not observed on the Hugoniot and the electronic rearrangement producing the less compressible form doer not occur until the Hugoniot is extended well into the liquid phase. This is true for a number of the rare earths.
lines of all the other axis,
indicating
liquid relative
l_lO”lO
bee so0
I
I
/
/
OO-
500
-,A
dkQ
T(K)
\ \
I
,
..___.
LANTHANUM
HUGONIOT -,,
I
i-, __y I /
01 0 30 ,
CALCULATED .Sa.IO-LIQUID PHASE BOUNDARY;
/’
A0
Fig. IO. Proposed phase diagram for La. The low-pressure phase lines were obtained from Jayaraman’s work[41. The Hugoniot P-T locus and the solid-liquid phase line were calculated by the methods discussed in the text. The intersection of these two calculated curves is in excellent agreement with the experimental transition pressure, which is indicated by the symbol 0 in this and the following figures. An extremum in the phase line occurs at about 18 GPa and 1620 K. It is interesting that the region of bee phase stability increases progressively from La up to Eu. where presumably the only stable phase is bee. This is evident in the succeeding figures.
elements
bend toward
a continuously to the contiguous
indeed
regime
found
extremum
pressure
In an attempt tions,
by
available him.
to check
cold pressed
PAV
heating
and
the was
elements
these phase boundary of porous
from a fine meshed
powder
density.
obtained
Here.
the
Hugoniots
melting
line at pressures
calculaerbium to 92.87
use is made of the
by shocking
als; the porous phase
other
the Hugoniots
and 79 per cent of crystal large
the
up
lies well within
to Jayaraman.
For
of the
was too high for static techniques.
we have obtained
samples
density
solid as one proceeds
the phase line. For Eu. this extremum experimental
the pressure
increasing
porous
would be expected considerably
materi-
to reach the below
that
W. J. CARTER
ef al.
EOr---70 ,-
-
-
-1
GADOLlNlUM
/. ESTIMATED HUGONIOT
60.
/’ /
,
,,*’
HUGONIOT.
JO-
’
I’
,I-
1 --__ I 0-l
I
I
I-.
m
0
bee
i
/’
t
mw
T (K)
T(K)
Fig. 13. Proposed phase diagram for Nd. The fee-bee and bee-liquid phase lines have been extended to meet at a triple point, and a very steep fee-liquid dPldT assumed at this point. Hence, the calculated solid-liquid phase line for this element is in considerable doubt. even though the calculated and measured transition pressures are in good agreement.
30
/
ESTIHATED SOLIDLIQUID PHASE 6O”NWJW
I
20
,,
1.-.-
I
SAMPiRlUH
Fig. 16. Proposed phase diagram for Gd. .Although the region of bee phase stability has begun to decrease and will continue to do so until the element Yb. for Gd this region is still wide enough that the Hugoniot apparently intercepts the rhomh-bee phase line hefore the material melts. Gd is therefore the only rare earth without an anomalous melting curve. the electronic transition presumably taking place al or near the rhomb-hcc phase line. Temperature-pressure loci in the hcc phase are only estimated. as is the hcc-lrquid boundary.
5oI
4
I
‘-,---
’
40;. /
7 u
to-
IWASE
/
‘/
i ”
7
\
;
,’ ,
’ CALCULATED I. -SOLID~LlOUlD PHASE BOUNDARY
/ lI
/
I
,
2000 LIQUID ._I
0
2Ocu
phase diagram for Sm. The dhcp, bee. and liquid expected to meet at a triple point at several However, the dhcp-bee phase line was ignored calculations. so that the dhcp-liquid phase line is again in question.
30,
/ ’ / ;; CALCULATED ,,SOLIL-Llaulo PHASE BOUNDARY
,
I
/
20HUGONIOT-.
E
HUGONIOT
‘\
\
,I
\\
/
\
/
a
CALCULATED SOLID-LIOUID PHASE BOUNDARY
\
/
\
IO bee
Fig. IS. Proposed phase diagram for Eu. The extremum at 3 GPa was found by Jayardman et (I/. The agreement between calculated and experimental transition pressures is almost exact.
\
. I
_~._.. LIOUID
\
\
(rhombi / r-y--l
‘\
!
DYSPROSIUM
\
\
5;.
Fig. 17. Proposed phase diagram for Tb. The low-pressure hcp-rhomh transition is based on conjecture. although static reristivity measurements indicate an anomaly at 2.7GPa. and experience has shown that the rare earths transform in the sequence hcp-rhomb-dhcp-fee as pressure is increased [ 121. The calculated solid-liquid phase boundary exhibits a maximum at about 35GPa. The region of hcc phase stability has hegun IO decrease.
I
EUROPIUM
_ a.
3030
r
_
15.
2eQJ
T(K)
T(K)
Fig. 14. Proposed phases are again hundred kilobars. in performing the
-.
/
I
\
/
J /
20t \
IO-
HUGONIOT
301
0
\
0
/y
/
\
/
HUGONIOT
/
BOUNDARY
bee IS00
2000
T tK) Fig. lg. Proposed phase diagram for Dy. Again. the hcp-rhomb transition at SGPa is conjecture hased on resistivity measurements; since no transition has been discovered along the P = 0 axis, this phase boundary probably ends at a triple point with the hcp and bee phases. An extremum in the solid-liquid phase boundary is predicted to lie at about 5 CPA and 1950 K, which unfortunately is outside the range of present static techniques.
751
Hugoniot equation of state of the lanthanides 50
‘I’
t”’ HOLMIUM
40 ;; & lJ
SD-
THULIUM
J /’
/
!
CALCVLATED I/SOLID-LIOUID PHASE BOUNDARYI
7’
a.
HUGONIOT,/ / /
\
/ /
/ /
I
IOC bomb) : ,- --, 0 0
CALCULATED /
\
/
ZO-
‘1
I
/
HUGONlOT
40
kc
b I 1000
500
I I500 T(K)
/
,
I 2500
I 3000
0
Fig. 19. Proposed phase diagram for Ho. The calculated solid-liquidphase boundary exhibits an extremum at about 14GPa and 2340K. The agreement between calculated and experimental transition pressures is excellent.
I
50
I
I
I
1 /
30-
HUGONIOT
/
1500
2000
I
CALCULATED ,SOLlDLIOUID \ PHASE EOUNOARY
Y’
-
a c1 :
I
/’
HUGONIOT 10
/
\
/
/
)_’
,’ \i
IO-
0
SW
I 1000
I I500 T (Kl
/I 14 2ocQ
I 2500
/’
0
LIOUID
b
I
I
I
I
I
1’
II
3000
YTTERBIUM
ZD
\
/
0
I
2500
i
/‘!
/
ZO-
I
1000
LIOUID
Fig. 21. Proposed phase diagram from Tm. The extremum in the calculated solid-liquid phase boundary lies at about 6GPa and 2160K.
1
= p 0 0
I
500
\ / _-r
T(K)
EREIUM
40
0
I
I
\
I ir
LlOUlD
A,' 2000
0
500
IODD T(K)
/
/’ , 1500
LIOUID zoo0
I 3000
Fig. 20. Proposed phase diagram for Er. An extremum in the solid-liquid phase boundary is predicted at about 11GPa and 2200K. The field of bee-phase stability is now closed. required for the crystal density material. Unfortunately,
the transition in porous materials is masked at these lower presstires by the effects of void collapse and perhaps a wave front which is not in equilibrium. However, the higher-pressure data can alternatively be used to check the validity of the metastable equation of state derived earlier. These Hugoniot points do attain very high temperatures and almost certainly lie in the liquid phase regardless of the details of the phase boundary locus. Hence, if the parameters listed in Table 4 and used for the phase line calculation are truly representative of the liquid phase, then a simple recentering of this liquid Hugoniot to the initial porous density using equation (10) should reproduce the porous Hugoniot data (it should be noted that AH of melting is very small compared to the total energy on the Hugoniot). Except for the very low pressure end, the comparison between data and calculation, shown in Fig. 9, is quite satisfactory. 4.CONCLUSIONS
It would appear to be well established that all the rare earths except Ce and possibly Cd melt anomalously at sufficiently high pressures, and that Ce itself melts anomalously at atmospheric pressure but normally above about 4 GPa. The presence of extrema in melting phase lines is an interesting phenomena so far known to occur in only about fifteen materials (the present investigation
Fig. 22. Proposed phase diagram for Yb. The extremum of the solid-liquid phase boundary, predicted earlier by Jayaraman[l9] now appears to lie at about 11GPa and 1650K. The region of bee phase stability has again opened to include a large part of the P-T plane.
nearly doubles this list, of course). From the Clapeyron relation, assuming the entropy change of transition does not vary appreciably with pressure, such behavior can only mean that the density of the liquid continuously increases relative to the density of the solid as one proceeds up the phase line, which implies that eventually the average interatomic distance in the liquid becomes less than that in the solid. This can come about through one of only two mechanisms; either the coordination number of the liquid becomes greater than that of the solid, or else the ionic radius characteristic of the liquid is less than that of the solid. High coordination numbers are not unknown in liquids, but it seems unlikely that they could exceed the coordination of the close packed structures characteristic of most of the lanthanides, at least above the region of bee phase stability. Hence, the second possibility seems more likely. An abrupt decrease in ionic radius upon melting implies a corresponding change in the electronic structure of the metallic ion. Since the ionic radii of the lanthanides are known to decrease in the order divalent 7 trivalent > tetravalent, one conclusion is that an additional electron is forced into the valence band from the 4j shell by the application of sufficient pressure. (We invoke the Jamieson criterion [ 161 by saying that an electron arbitrarily ceases to be a 4j electron when the single-particle wave function describing it has less than 50 per cent 4j-type admixture.) The
152
W. .I. CARTERet al.
curvature of the phase line toward the pressure axis is then accounted for by an extension of the two-fluid model recently proposed by Klement[l7], in which an increasingly greater proportion of atoms in the liquid undergoes a 4f+5d electron promotion at the phase boundary as pressure is increased, leading to the necessary density relationship between solid and liquid. An alternative possibility is that there is a continuous electronic transformation occurring in the lower pressure solid region. The evidence for this is the very low slopes of the u,-u, Hugoniots in this region. It is known that the parameter s is a very strong measure of the type of repulsive potential existing in the material. With a slope that remains less than unity, a potential is implied that would allow infinite compression on the Hugoniot at finite pressure, an obviously impossible situation. Thus, if there were a continuous change in the amount of s- and dbonding with pressure, the small rate of increase in the incompressibility could be explained. The anomalous melting would have to be associated with the more rapid completion of the electronic transition in the liquid state adjacent to the solid material. Band structure calculations have shown that the 5d and 6s bands for the rare earths in the metallic state are broad and strongly mixed, while the 4f bands are quite narrow and localized and have little direct effect on the bulk properties of the materials. The studies of Royce[l8] and Al’tshuler et al. [2], have indicated that the presence of a significant population in the d-band results in a low compressibility, while s-bonded elements generally have much higher compressibilities. Probable electronic ground state configurations of the lanthanides in the free atom state as presented in Table 1, together with the normal valence of +3, suggest that there is probably significant d-bonding for all the lanthanides even at zero pressure. The increasing rate at which the compressibility decreases beyond the transition would indicate a further increase in d-band population at this point, most likely but not necessarily at the expense of the f-band occupation (in the cases of SC, Y, and possibly La the transition is certainly not due to a 4f-5d electron promotion). The exact nature of the electronic rearrangement probably will
not
be known until careful band structure calculations are carried out over large ranges of interatomic spacings. Acknowledgements-The authors would like to thank Dr. John Taylor for performing the free-surface velocity experiments on yttrium as well as providing much stimulation and encouragement, and Charles Shelberg for performing the zero-pressure sound speed measurements.
REFERENCES
4. 5. 6.
I.
8.
9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
19.
Stager R. A. and Drickamer H. G., Phys. Reo. 133, A830 (1964); Science 139, 1284 (1963). Al’tshuler I.. V., Bakanova A. A. and Dudoladov 1. P., JEPT Letters 3, 483 (1966); JETP 26, 1115 (1%8). Duff R. E., Ross M., Gust W. H., Royce E. B., Mitchell A. C., Keeler R. N. and Hoover W. G., Symposium HDP IUTAM, Paris. Gordon and Breach, New York (1968); Gust W. H. and Royce E. B., Phys. Rev. B 8, 3595 (1973). Jayaraman A., Phys. Rev. 139, A 690 (1%5). Lawson A. W. and Tang T. Y., Phys. Rev. 76, 301 (1959). Rice M. H., McQueen R. G. and Walsh J. M., Solid State Phys. (Edited by Seitz F. and Turnbull D.), Vol. 6. Academic Press, New York (1958). McQueen R. G., Metallurgical Society Conferences (Edited by Gschneider K. A., Hepworth M. T. and Parlee N. A. D.), Vol. 22. Gordon and Breach, New York (1964). Carter W. J., Marsh S. P., Fritz J. N. and McQueen R. G., Accurate Characterization of the High-Pressure Environment, (Edited by Lloyd E. C.). NBS Special Publication 326 (1971). Shelberg C. (private communication). Kremers H. E., Rare Metals Handbook (Edited by Hampel C. A.), 2nd Edition. Reinhold Publishing Corporation, London (1961). Bridgman P. W., Proc. Am. Acad. Arts Sci. 83, 3 (1954). Javaraman A. and Sherwood R. C., Phvs. Reu. 134, 691 (1964); Phys. Rev. Letters 11, 22 (1964). Jayaraman A., Phys. Rev. 137, A 179 (1%5). Walsh J. M., Rice M. H., McQueen R. G. and Yarger F. L., Phys. Rev. 108, 196 (19.57). McQueen R. G., Marsh S. P. and Fritz J. N., .I. Geophy. Res. 72, 4999 (1%7). Jamieson J. C., NONR-2121 (03) (l%O). Rapoport E., J. Chem. Phys. 46, 2891 (1967); J. Chem. Phys. 48, 1433 (1968). Royce E. B., UCRL-50152 (1%6); Phys. Rev. 164,929 (1967); Proceedings of the International School of Physics, Course XLVIII, (Edited by Caldirola P. and Knoeppel H.). Academic Press, New York (1971). Jayaraman A., Phys. Ren. 135, 1056 (1964).