Article

Theoretical zero-temperature phase diagram for neptunium metal

Physical review. B, Condensed matter (Impact Factor: 3.66). 08/1995; 52(3):1631-1639. DOI: 10.1103/PhysRevB.52.1631
Source: PubMed

ABSTRACT

Electronic structure calculations, based on the density-functional theory with the generalized gradient approximation to the exchange and correlation energy, are used to study the crystallographic properties of neptunium metal under compression. Calculated ground-state properties, such as crystal structure and atomic volume, are found to be in excellent agreement with experiment. The calculated bulk modulus and the first pressure derivative of the bulk modulus are also in accordance with experiment. Our theory predicts that neptunium at low temperature undergoes two crystallographic phase transitions upon compression. First a transition alpha-Np-->beta-Np at 0.14 Mbar and second a beta-Np-->bcc phase transition at 0.57 Mbar. These transitions are accompanied by small volume collapses of the order of 2-3 %. The high-pressure phase (bcc) is also investigated theoretically up to 10 Mbar. A canonical theory for the f-electron contribution to the hcp, fcc, and bcc structures is presented.

Full-text

Available from: Per Söderlind, May 24, 2014
PHYSICAL
REVIEW 8
VOLUME
52,
NUMBER
3
15
JULY
1995-I
Theoretical
zero-temperature
phase
diagram
for
neptunium
metal
Per
Soderlind,
Borje
Johansson,
and Olle
Eriksson
Condensed Matter
Theory
Group,
Department
of
Physics,
Umversity
of
Uppsala,
P.
O.
Box
580,
S
751
-21
Uppsala,
Siveden
(Received
21 March
1995)
Electronic
structure
calculations, based on the density-functional
theory
with the
generalized
gradient
approximation
to
the
exchange
and
correlation
energy,
are used to
study
the crystallo-
graphic
properties
of
neptunium
metal
under
compression.
Calculated ground-state
properties,
such as
crystal
structure
and atomic
volume,
are found to be in excellent
agreement
with
experi-
ment. The calculated
bulk modulus
and the 6rst
pressure
derivative of the bulk modulus are
also
in accordance
with
experiment.
Our
theory predicts
that
neptunium
at low
temperature undergoes
two
crystallographic
phase
transitions
upon
compression: first a transition o.-Np +
P-Np
at
0.
14
Mbar
and
second
a
P-Np
+ bcc
phase
transition
at 0.
57
Mbar. These transitions are
accompanied
by
small
volume
collapses
of the order of 2
3%%uo.
The
high
pressure phase
(bcc)
is also
investigated
theoretically
up
to
10
Mbar.
A
canonical
theory
for the f-electron
contribution
to
the
hcp,
fcc,
and
bcc structures
is
presented.
I. INTRODUCTION
TABLE I.
Experimental
equilibrium
lattice constants
(Ref.
5)
for o.
-Np and
P-Np.
o,-Np
P-Np
a
(A.
)
4.723
4.897
b/a
1.
035
1.000
c/a
1.411
0.692
Neptunium
belongs
to
the series of actinide metals.
This series consists
of radioactive metals where
the
5f
shell is
progressively
filled
as one
proceeds through
the
series.
For the elements
Th-Pu
the
5
f
states
have
itiner-
ant
(band)
characteri
s
and these metals can be viewed
as
part
of a 5
f
transition series
with a behavior of the
ground-state
properties
analogous
to the 4d and 5d
tran-
sition metals.
The
increasing
complexity
of
the
crystal
structures
as
the
5
f
band is
being
filled
when
proceeding
&om Th to Pu
is,
however,
in
sharp
contrast
to the high-
symmetry
crystal
structures observed
in the d
transition-
metal series.
In
neptunium
for
instance,
an orthorhombic
structures
(a-Np)
is found to
be stable
from
helium
tem-
peratures
up
to about
600 K and thereafter
neptunium
transforms to
a
tetragonal
forms
(P-Np),
which is
sta-
ble
up
to 900
K
where the
bcc structure becomes
stable,
see
Fig.
1. Details of the n-Np
and
P-Np
structures
are
given
in Tables I and II .
Recently
a link was
es-
tablished between the
crystal
structures
of the
light
ac-
tinide
metals
and the d transition
metals. In this work
it
was
shown
that both a d
and
a
f
metal can
have
simple
(high-symmetry
close-packed) or
complex
(low-
symmetry open-packed) crystal
structures
depending
on
their
density.
This
suggests
that
actinide metals should
generally
show
a
phase
diagram
where the
crystal
struc-
tures become
increasingly
symmetric
as a function of
applied
pressure. Unfortunately
very
few
high-pressure
0.
03—
cu
~
0.
02-
0
0.
01—
O'.
-Np
P-Np
bcc
()
Liquid
I
300 600
Temperature
(
K
)
900
FIG. 1. Experimental
low-pressure
phase
diagram
of nep-
tunium metal
(Refs.
20 and
21).
experiments
have been
published
for
the
light
actinide
metals
and so far this
conjecture
has not
been confirmed
experimentally,
to our
knowledge.
As will be
seen,
the
present
calculations
for
Np
sup-
port
previous findings
and are
also in
agreement with
the theoretical results
for
Th, Pa,
and
U
reported
by
%Pills
and Eriksson
and the
recently published
calcula-
tions for
Np.
'
The latter calculations
'
showed
that at
sufEciently
high
pressure
the bcc
crystal
structure should
become stable
in
Np.
Soderlind et al.
also noticed that
the
bct
phase
(c/a
=
0.
85)
was
close to
being
stable
in the
vicinity
of the a-Np
~
bcc transition.
Due
to the
rel-
atively
poor
agreement
with
experiment
as
regards
the
ground-state
data for these
previous
results
(the
calcu-
lated
equilibrium
volume
was about
15'%%uo
smaller
than
experimental
data),
the
accuracy
of the
theoretical
tran-
0163-1829/95/52{3)/1631(9)/$06.
00
52
1631
1995
The
American
Physical
Society
Page 1
PER
SODERLIND,
BORJE
JOHANSSON,
AND
OLLE ERIKSSON
TABLE
II. Experimental
equilibrium
atomic
positions
(Ref.
5)
for o.
-Np
and
P-Np.
Atom
type
n-Np(I)
n-Np(II)
P-Np(I)
P-Np(II)
Atomic positions
+(4,
yi,
zi)
+(4,
—,
'
yi,
zi
+
—,
')
(0,
0,
0)
yx
0.
208,
zq
0.
036,
yq
0.
842, z2
——
0.
319,
u
=
0.375.
sition
pressures
(n-Np
~
bcc)
could be
questionable
and
therefore
they
were not
published.
To
try
to
improve
on
the
ground-state
properties
of
Np,
the
present
calcula-
tions are
performed
with a
recently
developed
formula-
tion
of the exchange
and
correlation
functional
(see
Sec.
II),
which
also allows
for a more accurate
calculation
of
the pressure
for the
crystallographic phase
transitions.
Hence we find it
motivated to
publish
our
improved
cal-
culations
for the various
allotropes
of
Np
as a
function
of
compression
and
we
also
present
the calculated
pressures
at
which the
transitions are
predicted
to
occur. One of
these
transitions
is investigated
in more
detail;
we
study
an
approximative
transformation
path
between
the
P-Np
and bcc structure.
Furthermore,
we
present
theoretical
results
for
the
bcc
Np
equation-of-state
up
to
ultrahigh
pressures.
In connection to the
discussion
of
the
high-
symmetry
close-packed
crystal
structures at
high
pres-
sures,
a
simple
canonical
band
picture
is
presented
and
used to evaluate which
particular
close-packed structure
an itinerant
f
metal will attain at
very
high
pressures.
Since the
present
investigation
deals
with
the
electronic
structure for
Np
metal,
we take the
opportunity
to
quote
some earlier theoretical work on
this material. Skriver
et
al. were the first to
calculate
the
equilibrium
volume
and
bulk modulus
of
Np
(and
the other
light
actinides);
they
d.
id
this
by
means of
scalar relativistic
calculations,
which
relied
on
the
atomic
sphere approximation
(ASA)
and
the
replacement
of
the
true structure
by
a fcc atomic
arrangement.
Subsequently,
Brooks
incorporated
the
spin-orbit interaction
by
solving
the Dirac
equation
for
the
light
actinides
including
Np,
also within the
ASA
and for a
fcc
structure. Soderlind
et
al.
applied
a
sim-
ilar
technique,
and &om
the obtained
equation-of-state
they
were
able
to calculate
the thermal
expansion
for
Np
and the
rest of the
early
actinide metals.
In all
the
re-
ports
quoted above,
the complicated crystal
structure of
neptunium
was
replaced.
by
the
fcc structure.
The first
electronic
structure
calculation that
considered
the
true
low-temperature
crystal
structure of
Np
was
presented
by
Boring
et
al. who
reported.
electron
density
of
states
for
a-Np,
P-Np,
and bcc
Np.
In the
following
sections
we describe our computational
method
(Sec.
II)
and our
obtained
results
(Sec. III).
In
Sec.
IV,
we
discuss these
results and in the final
section
(Sec.
V)
we
present
our conclusions.
II. COMPUTATIONAL
DETAILS
The
reported
results
are obtained Rom
electronic
structure
calculations for neptunium
in
the seven
crystal
structures
n-Np,
P-Np,
n-U,
bct,
bcc,
fcc,
and
hcp
under
compression.
The
present
ab inito method
solves
a
mod-
ified
Schrodinger
equation
for the total
energy
of the sys-
tem
(the
Dirac
equation
is solved
for the
core
electrons),
within density-functional
theory.
All relativistic efFects
are included in the
Hamiltonian,
including
the spin-orbit
interaction
term,
which is considered
according
to the
recipe
proposed
by
Andersen.
The wave functions
are
expanded
in linear
muon-tin
orbitals
inside
the
nonover-
lapping
mufBn-tin
spheres
that surround each
atomic site
in the
crystal.
We
make use of
a
so-called
double basis
set
by
allowing
two tails with
di6'erent
kinetic
energy
for
each numerical
basis function inside
the
mufBn-tin
spheres.
The calculations were
done for
one, fully
hy-
bridizing,
energy
panel
in which
energy
values,
E„,
asso-
ciated with the valence
orbitals
7s,
7p,
6d,
and
5
f,
and
to the
pseudocore
orbitals
68 and
6p,
were
defined.
Out-
side
the
muKn-tin
spheres,
in the
interstitial
region,
the
wave
functions are Hankel or
Neumann
functions,
which
are
represented
by
a
Fourier series
using
reciprocal
lat-
tice vectors. The same
expansion
is used to
represent
the
charge
d.
ensity
and the
potential.
This treatment
of the wave
function,
charge
density,
and
potential
does
not
adopt
any
geometrical
approximations
and the
de-
scribed
type
of
computational
method
is
usually
referred
to as
a
full
potential
linear
muffin-tin
orbital method
(FP-
LMTO).
The
present
version of the
method,
which
is
devel-
oped
by
Wills
and co-workers,
has
previously
successfully
been
applied
for
calculating
the electronic
structure
for actinide materials
and the
present
calcula-
tions are therefore a
natural
extension
of
the
previous
publications. In the density-functional
approach,
it
has
become common to make
a local
approximation
to the
exchange
and correlation
interactions
between the
elec-
trons. Since the
recently
presented generalized
gradient
approximation
has
been
shown
to
significantly improve
the
accuracy
of the results
for actinide
metals,
we have
chosen to
adopt
this
approximation
for the
exchange
and.
correlation
energy
functional
in the
present
calculations.
In the
calculation of the
one-electron
band
structures,
the
special
k-point method has been used with various
sampling
densities
of
the k
points.
In the
o.
-Np
struc-
ture,
16 k
points
in
the irreducible
wedge
of
the Brillouin
zone
(IBZ)
were used whereas for
P-Np
the
corresponding
number was
18. This
might
seem to be
relatively
small
numbers
of k
points
in the IBZ
for
these structures where
the IBZ is
1/2
and
1/4
of
the
full
Brillouin
zone
(FBZ)
for
n-Np and
P-Np,
respectively. However,
an increase
of
the
number
of k
points
to 32
(n-Np)
and 40
(P-Np)
only
lowered the total
energy
with about 0.1
mRy/atom
at
the
theoretical
equilibrium
volume. The
electronic
struc-
ture
for
Np
in
the
o.
-U
crystal
structure
(orthorhombic
with 2
atoms/cell)
was obtained
using
100 A:
points
in
the IBZ
(1/8
of the
FBZ).
In the case of the
more sym-
metric
bcc, fcc,
and
bct
structures,
the
symmetry
of
the
bct
unit
cell
was
consistently
applied
and a
total number
of 150 k
points
was used
in
the
IBZ for
those
(1/16
of
the
FBZ).
For
the
hexagonal lattice,
which we assumed
to
have
an
ideal
c/a
ratio,
we used
30
k points
in the
IBZ
(1/12
of the
FBZ).
To
further
investigate
the
con-
Page 2
52
THEORETICAL
ZERO-TEMPERATURE
PHASE DIAGRAM
FOR. .
.
1633
vergence
of the
A:-point
sampling
for the
various
crystal
structures,
we
put
the
crystallographic
parameters
(c/a,
5/a,
and
positional
parameters)
for o.
-Np,
P-Np,
and bct
so that the
structure
became bcc.
Close to the
theoret-
ical
equilibrium volume,
these test
calculations showed
that the
total
energy
difference
between the structures
was
converged
to
about
0.
3
mRy/atom.
III.
RESULTS
Experimentally
there is
only
a
rather
limited
body
of
data available as
regards
the
phase diagram
of
nep-
tunium.
At ambient
conditions
Np
is,
however,
found
to
crystallize
in
a
simple
orthorhombic structure
with
8 atoms/unit
cell
(o.
-Np).
No
phase
transition was
ob-
served
in
Np
below 0.
52
Mbar
in
the
study
by
Dabos
et
at.
"
The
axial ratios
(c/a
and
b/a)
showed,
however,
a
small
pressure
dependence. At 0.5
Mbar,
the
c/a
value
had
decreased
by
about
170
and
the
b/a
value had
in-
creased with a
similar
amount in
their
investigation.
Other
high-pressure
x-ray diffraction studies on
o.
-Np
indicate, however,
an
increasing tendency
toward
unity
of
the
axial
ratios,
which
was interpreted as
a possi-
ble
tetragonal
structure
by
the
authors.
At low
(zero)
pressures,
o.
-Np
is stable &om zero
temperature
up
to
about
600
K,
see
Fig.
1. Above this
temperature,
Np
undergoes
a
phase
transition to
the
P
phase
and
subse-
quently,
around
900
K,
the
p
phase
(bcc)
of
neptunium
is
found.
'
No
phase
transitions
at
higher
pressures
have
been
published
for
neptunium.
We
have
chosen to
theoretically
investigate
the
ob-
served
n,
P,
bct,
and
bcc
phases
of
neptunium.
For
com-
parison
we
also
investigated
the
o.-U,
hcp,
and fcc
crystal
structures. The o,
-U
structure has
sometimes been used
as a representative
for
the
open
and
complex crystal
structures
of the
light
actinides,
whereas the fcc
struc-
ture has been used to model the
light
actinide metals in
several
early
calculations.
'
Our main
results
are
shown
in
Fig.
2 where the total
energies
for the seven
crystal
structures are
plotted
as a
function
of
volume. The
in-
ternal
crystallographic
parameters
for the
n-Np,
P-Np,
and
n-U
structures
were
kept
constant at the experimen-
tally
determined values
'
(see
Tables
I
and
II for
n-Np
and
P-Np)
in order to
reduce
the
computational
efforts.
The
calculated
total
energy
of the bct structure was
min-
imized with
respect
to
the value
of
the
axial
ratio,
c/a,
at
several
volumes,
and we
found that for all volumes
where
the bct structure
is lower in
energy
than the bcc structure
(
16
20
A.
s),
a
c/a
value
of
0.
85 minimized the
total
energy.
As an
example,
we
display
in
Fig.
3
the
calcu-
lated Bain transformation
path,
at
the
room
temperature
experimental equilibrium
volume
(19.
2
A.
).
This
figure
shows
that both
bcc
(c/a
=
1)
and fcc
(c/a
=
i/2)
are
locally
unstable towards a
tetragonal distortion,
whereas
the
bct
structure,
with
a
c/a
value
of
about
0.
85,
min-
imizes the total
energy.
Notice
also
that the
energy
for
the
bcc
phase
is considerably
lower
(
21
mRy)
than
for
the fcc
phase
at this
volume,
a fact that
also
can be
seen
in
Fig.
2. The total
energy
calculations
in
Fig.
2
furthermore
show that the o.-Np
phase
has
the
lowest
0.12
0.
10—
E
0.
08
0
0.06
bQ
0.
04
0
0.
02
0.
00
CC
ct
u-U
-Np
-Np
l
14
I
15
I
16
17 18
I
19
20
energy
among
the studied
crystal
structures
at both the
theoretical
equilibrium
volume
as
well
as
at
the room
temperature experimental
equilibrium
volume. This is
in
agreement
with
experiment.
Figure
2 also shows that
at the
equilibrium
volume,
the
energy
of the different
structures is ordered
according
to their
complexity
and
26—
22
~
~
~
~
18
bQ
10—
bCC
fcc
~
~
~~0
I
0.8
~
e+o
~
I
1.0
I
1.
2
I
1.
4
c/a ratio
1.6
I
1.
8
~
~
~
~
2.0
FIG. 3.
The
so-called
Bain transformation
path,
i.e.
,
the
total
energy
as
a
function
of the
c/a
value
in
the bct structure,
for
neptunium
at
the
experimental
equilibrium
volume
(19.
2
A.
).
The bcc and
the
fcc structures
are obtained
for
c/a
values
of
1
and
~2,
respectively.
Volume
(
A
l
FIG.
2. Calculated
total
energies
as a function
of volume
for
the
n-Np,
P-Np,
n-U,
bct, bcc,
hcp,
and fcc
crystals
of
neptunium
metal.
The
points represent
calculated
values and
the
solid
lines
connecting
them show
the
Murnaghan
functions
as obtained
by
a
least-square fit
to the calculated
energies.
Page 3
1634
PER
SODERLIND,
BORJE
JOHANSSON,
AND OLLE
ERIKSSON
52
openness;
ci.-Np is
lowest,
then
comes
P-Np,
ci.
-U,
bct,
and
Gnally
the more close-packed
structures.
At a
moderate
pressure
of about 0.14
Mbar,
see
Table
III,
neptunium
is calculated to
undergo
a
phase
tran-
sition to the
P
form of
Np.
In
Fig.
4,
we
show the
energy
difFerence
between
the
studied
crystal
structures
(except hcp
and
fcc)
as a
function of volume. The bcc
structure is here
taken
as a
reference
structure,
and
this
phase
defines
the
zero
energy
level of the
plot.
As can
be
seen in. this
figure,
the
ci.-Np and
P-Np
structures are
rapidly
increasing
their
energies
relative
to the structures
with
higher
crystal
symmetry,
i.e.
,
the bct and the bcc
structure. The
calculated
bct and
bcc
energies,
however,
are
very
close
to each other
in the volume
range
15—
17
A.
,
and this
makes
it difficult to
accurately
calcu-
late
a
transition between these two
phases.
In
fact,
it
has been drawn to our attention that
the bct
phase
in
neptunium
has been seen
experimentally
within
a
nar-
row
pressure
range.
Theoretically
we do
not
And this
transition,
but
at a
larger
compression,
at
a
calculated
transition
pressure
of 0.
57
Mbar,
we
Gnd
that
neptunium
adopts
the
bcc
crystal
structure,
with the bct structure
being
only
marginally
higher
in
energy.
The theoretical
prediction
of a bcc
phase
in
Np
at
ultrahigh pressures
was
already
made
earlier8'
and has now been con6rmed
experimentally.
Notice
in
Fig.
4 that
the
P-Np,
bct,
and
bcc
phases
are
very
close in
energy
at a
volume of
about
15
As.
As
mentioned
earlier these
three
structures
as well as the
o.-Np
structure can
be
viewed as
distortions
of the
bcc
lattice. We
illustrate
this
for
the
P-Np
structure in
Fig.
5
where a
unit cell is
shown. The
open
atoms
in this
6gure
are of
atom
type
I
and the
shaded
atoms of
type
II,
as
described in
Table
II. The
unit cell looks
similar
to
the fcc unit cell with
atoms in the
corners and on
the
faces
of the
cell. If
the
c/a
ratio of
I9-Np
is
equal
to
one and if
u is
equal
to 0.
5,
one
obtains the
atomic
ar-
rangement
of the
fcc
structure. For
P-Np,
the
c/a
ratio
is close
to
I/~2,
and
the
P-Np
unit
cell
is
in fact
very
close to
the bcc unit cell.
Notice that
the shaded
atoms
(atom
type
II)
are
located at positions
slightly
above
or
below the center
of the
faces.
Their locations
are
deter-
mined
by
the
crystal
parameter
u,
de6ned in
Table II.
By
choosing
c/a
to be
I/~2
and u
to be
0
5,
we obtain
30—
20
E
0
10—
V
C
0
bQ
&D
C
10—
CC
ct
-U
-Np
-20
-Np
I
I
14 15
17
I
18
I I
19 20
Volume
(
A
)
FIG. 4.
Calculated
total
energies
for the
n-Np,
P-Np,
n-U,
and bct
crystal
structures of
neptunium
relative to the total
energy
of the
bcc
structure,
as a
function of volume.
the
bcc
crystal
structure.
Since
for the
P-Np
structure
the
c/a
value,
see Table
I,
is indeed close to
I/~2,
we
can
approximately
transform
P-Np
to the bcc structure
by
translating
the
atoms of
type
II,
i.e.
,
by
changing
the
u
parameter
from
the
observed
value of 0.375 to 0.5. It
is therefore
interesting
to calculate
the
total
energy
as
a function of the
positional
parameter
u.
In
Fig.
6
we
show
the
total
energy
for
Np
as
a function
of the atomic
position
u at three atomic
volumes;
14,
15,
and 18 A.
.
In this
6gure,
u
=
0.
5
corresponds
to the bcc structure
and u
=
0.
375
gives
an efFective
P-Np
structure
(which
is an
approximation
to
the true
P-Np
structure since
c/a
is set
equal
to
I/~2).
Notice
in
Fig.
6 that for the
vol-
ume 18
A.
s
the bcc structure shows a local
energy
inax-
TABLE III.
Equation-of-state data
obtained
from
Murnaghan
fits to
the
FP-LMTO
calculations
shown
in
Fig.
2.
Experimental
data
are
obtained at
room
temperature
from Donohue
(Ref.
22)
(Vp)
and
Murnaghan
fits
to data
from Dabos et al.
(Ref.
17)
(Bc
and
Bo).
The
calculated
bulk
modulus and its
pressure
derivative evaluated at the theoretical equilibrium
volume are denoted
Bo
and
Bo,
respectively.
The
corresponding
calculated data
evaluated at the
experimental
volume
are
labeled
B.
Pressure quantities
are
given
in Mbar and volumes
in
A
.
o,-Np
P-Np
(x-U
bct
bcc
fcc
hcp
expt
Vp
18.4
17.7
17.7
17.6
17.4
18.7
18.
0
19.2
B()
1.
7
2.0
2.0
1.
9
2.0
1.6
1.
6
1.
2
B
1.3
1.4
1.
3
1.2
1.2
1.
5
1
'
1
1.
2
Bo
5.
5
5.
3
4.9
5.0
4.9
4.1
5.
6
6.0
pa-+p
0.14
0.14
~amp
17.3
16.7
pp-+bcc
0.
57
0.
57
g
p
-+bcc
14.9
14.
5
Page 4
52
THEORETICAL
ZERO-TEMPERATURE PHASE
DIAGRAM FOR.
. . 1635
FIG.
5.
The unit cell of the
P-Np
crystal
structure.
The
shaded atoms
symbolize
the atoms
of
type
II
(see
Table
I).
The
crystallographic
parameter
u
(see
Table
I)
is
shown
ex-
plicitly
in
the
figure.
imum,
whereas
there is a
global
minimum for u
=
0.36.
This is in
agreement
with
the experimental
data for
Np
at
ambient
conditions,
where u
=
0.
375 +
0.015
(with
c/a
=
0.
69).
At a somewhat
compressed
volume
(15
A
),
the previous
maximum
in
energy
for u
=
0.5
(bcc)
turns over into a
local
energy
minimum whereas the
value
for
u,
which
gives
the
global
minimum in
energy,
has
in-
creased to
about 0.40. The
energy
difference
between
12—
0
crt
4
CC
bQ
-4-
I
I
0.35
0.4
0.
45 0.5
0.55
Atomic displacement,
u
FIG.
6. Calculated
total
energy
as a
function of atomic
position,
u,
for a
crystal
structure,
which is
equivalent
to the
P-Np
structure for
u
=
0.
375,
except
the
c/a
ratio is
approx-
imated to have the bcc value
of
1/~2.
For the
true
P-Np
structure,
c/a
=
0.69. Therefore with u
=
0.
5,
the
present
structure is
equivalent
to the bcc
crystal
structure for which
the
energy
is normalized to zero.
The solid diamonds refers to
calculations
done for
Np
close
to
the
equilibrium
volume
(18
A.
).
We also show results for
compressed
Np
as
open
squares
(15
A.
)
and
solid circles
(14
A
).
Lines
connecting
the
data
points
are
polynomial
fits
and serve as a
guide
to the
eye.
the bcc
structure
and the
effective
P-Np
structure
has
also
decreased
by
more than
50'%%uo
at this
compression.
When
neptunium
is compressed
further
to about 14
As,
the bcc
structure has the lowest
energy
and
there is not
even
a
local
minimum close to the effective
P-Np
struc-
ture.
The calculations
presented
in
Fig.
6
suggest
that
the transition
from
P-Np
to
bcc at
large
compressions
is
due
to a soft
phonon,
corresponding
to the
displacement
of the
neptunium
atom of
type
II
defined
by
the
posi-
tional
parameter
u.
The fact that our calculations
show
a maximum for
the
energy
of
the bcc
phase
at volumes
corresponding
to
zero
pressure,
implies
that the observed
high
temperature
(HT),
ambient pressure bcc
phase
has
to be stabilized
by
entropy
eKects.
Therefore the zero
pressure
HT
bcc
phase
in
Np
requires
a
boot-strapping
mechanism for
stabilization of its
phonons.
This is not
required
for the
pressure
induced
(zero-temperature)
bcc
phase.
Therefore the HT
bcc
phase
in
Np
seems to
have
a
diferent
physical
origin.
The
observed
phase
diagram
(Fig.
1)
is in
accordance
with this
conclusion, since
the
HT bcc
phase
region
forms a
closed
area in
the
(P,
T)
plane.
Next
we
comment
upon
the
calculations for
the
hcp
and fcc
crystal
structures
for
neptunium.
Even
though
these
phases
are never close
to be stable
for
Np,
their
cal-
culated
equilibrium
volumes
are rather
close to the equi-
librium
volume for
n-Np
(see Fig.
2).
In
fact it has
been
found that in
calculations
where the fcc
crystal
struc-
ture is
assumed,
a
larger
equilibrium
volume is obtained
compared
to
calculations
'
where
the true
crystal
struc-
ture for
the
light
actinides
Pa
(bct),
U
(n-U),
and
Np
is
considered. In the
present
case of
Np
this means
that
the calculated equilibrium
volume for fcc
Np
is closest
to the
observed volume. In
light
of the
present
results
for
Np,
this
better
agreement
with
experiment
for the
atomic
volume,
when
assuming
a fcc structure,
seems
to
be a
fortuitous
result.
For
the bcc
phase
we have
extended the
calculations
to
even
higher
densities,
and
in
Fig.
7
we show
the
high-pressure
equation-of-state
for
neptunium
in the
bcc
phase.
The
pressure
is obtained as the
volume
derivative
of the
calculated
total
energy.
A
Murnaghan
fit
to the
total
energy
calculated
in the
large
volume
range
shown
in
Fig.
7
is not
appropriate.
Instead,
we have
divided the
volume
interval into
several
pieces
and numerically
dif-
ferentiated
the total
energy
to obtain
the
pressure.
For
instance at
the
smallest investigated volume, 8
A.
s,
we
obtained
a
pressure
of
about
10
Mbar.
To be
able to
accurately
calculate
the total
energy
for neptunium
un-
der more than
a
twofold
compression,
it
was necessary
to
treat the pseudocore
states
(6s
and
6p)
as
band states.
As an
inset in
Fig.
7
we also
compare
our
theoretical
pressures
for
o.
-Np
with
the
Murnaghan
fit
to the
experi-
mental
data
by
Dabos
et al. for the
low-pressure
regime
of
Np.
Apart
from the
small
discrepancy
at
low
pressures,
our
zero-temperature
calculations are
very
close to
the
room temperature
experimental
data. The
experimental
Murnaghan
fit
was
done for data in
the
pressure
range
0
0.52
Mbar and pressures
beyond
that
are
therefore an
extrapolation.
We
summarize a
number of our
calculated quantities
Page 5
1636
PER
SODERLIND,
BORJE
JOHANSSON,
AND OLLE
ERIKSSON
6
0
bc
expt
~
theory
at
room
temperature,
whereas our
theory
applies
to zero
temperature.
The
temperature
efFect on
the atomic
vol-
ume
and the
bulk modulus for the
actinides, and espe-
cially
neptunium
and
plutonium,
is
considerable
due to
the
unusually
large
thermal
expansion
in these
metals.
This
temperature
efFect can be well
accounted
for
by
Debye-Gruneisen
theory
'
and
a
temperature
correc-
tion
to our
present
results will further
improve
the agree-
ment with
experiment.
For
instance,
our
room tempera-
ture
equilibrium
volume
for o.-Np will increase
by
about
2%%up
and
the bulk modulus decrease
by
about
8'%%up,
rela-
tive
to their zero
temperature
values.
Thus
the
already
small
discrepancy
between our
present
theory
and room
temperature
data is
almost
entirely
removed
when
we
consider
temperature
efFects.
Hence,
where
experimental
data are
available,
the calculations
for
neptunium in
the
o.
-Np structure
give very
good
results.
10 12 14 16
Atomic Volume
(
A
)
FIG.
7.
Theoretical
equation
of state
(pressure
in
Mbar
and
atomic volume
in
A.
)
for
the
high-pressure
bcc
phase
of
neptunium.
The inset
shows,
in
the same
units,
theoretical
data
(n-Np)
compared
to
a
Murnaghan
fit to
experimental
data
(Ref.
17)
for neptunium at
lower
pressures.
for
Np
in Table
III. Some of these results have been
published
before. As
mentioned,
detailed
experimental
data for
neptunium
at
high
pressure
are
unfortunately
not available to
the authors.
In
Table
III
we
present
experimental
low-pressure data
for
neptunium.
It is
en-
couraging
to notice that
apart
from the
highly
unstable
hcp
and fcc
structures,
Table III
reveals that
among
the
other,
bcc
related, crystal
structures the
best
agreement
with
experiment
for
Vo,
Bo,
and
Bo
(the
pressure
deriva-
tive
of
the
bulk
modulus)
is obtained
for
n-Np. It should
however be noted
that the value of
Bo
is
very
sensitive
to the choice
of volume interval for which
the total
ener-
gies
are 6tted
to the
Murnaghan
equation. For the equi-
librium
volume,
the
agreement
with
experiment
is
good
(4%%up).
We note
that
the
calculated
equilibrium
volume
shows
a
rather
large
dependence
on the
crystal
struc-
ture,
and that
only
the calculation
of
the
o.
-Np
phase,
again apart
&om the
excited
hcp
and
fcc
phases,
gives
a value that
is
comparable
to the observed
equilibrium
density.
The
fact
that our
calculated
equilibrium
volume
is
slightly
smaller than the
measured value
indirectly
en-
hances
the
computed
bulk
modulus at this
volume since
the bulk modulus increases with
density.
To correct our
theoretical
bulk
modulus for this efFect we
also show
in
Table III the bulk modulus evaluated at
the measured
equilibrium
volume;
we
have
chosen to denote
this
quan-
tity
as B. With this
volume correction we obtain a
very
good
agreement
with
experiment
for the bulk
modulus
(8%%up).
For
Bp
the
agreement
between
theory
(u-Np)
and
experiment is also
very
encouraging.
All the
experimental
data in Table III
are measured
IV.
DISCUSSION
The
present
calculations
support
the
idea
that
itin-
erant
f-electron metals
will show
an
increasing
degree
of
high-symmetry crystal
structures
upon
compression.
This is
clearly
shown in
Figs.
2 and. 4
where
at the
equilibrium volume the
crystal
structure
energies (dis-
regarding
the
hcp
and fcc
phases
which
are
very
high
in
energy)
are
ordered
in
increasing
complexity/openness
(with
n-Np
being
lowest
and bcc
being highest
in
energy),
whereas at
sufFiciently
high
compression
the
order of
the
crystal
structures
is
exactly
reversed
(with
bcc
being
low-
est
and n-Np
being highest
in
energy).
The
reasons
for
this
behavior
have been
discussed
earlier
and
the
im-
portant
parameter
in
this
connection
is the
1
bandwidth.
Due to
a Peierls
distortion, a
system
with
narrow
bands
close to
the
Fermi level
can
gain
energy
by
lowering
the
crystal
structure
symmetry.
This
is manifested
in
our
calculations
at the
equilibrium volume
for
Np,
where
for
u
=
0.
5
(bcc)
the
calculated
density
of
states
at
the Fermi
level,
D(E~),
is about
60
(states/Ry/atom),
whereas
for
the
distorted
structure
(u
=
0.
375,
approximatively
P-
Np)
it is lowered
to about
36
(states/Ry/atom).
The
substantial
lowering
of
D(E~)
due
to this distortion
in-
dicates that
bands are shifted
away
from
this
region
and
this results
in a decrease of
the sum of
the one-particle
en-
ergies.
For a
more
compressed volume
(14
A.
),
however,
calculations for
the two
structures
(with
u
=
0.
5
and
u
=
0.
375,
respectively)
give
almost the
same
D(E~)
of about 30
(states/Ry/atom),
which reflects that the
energy
gained
by
a Peierls
distortion is
considerably
re-
duced in this case.
The
arguments
described above
can be used to
pre-
dict,
for
instance,
the
general
trend of
crystal
structures
of the
early
actinides
(with
bonding 5
f
states)
as
a
func-
tion of
compression.
Since the
5f
bands are broadened
under
compression
they
become
more
d-band-like,
and
the
crystal
structures
of
the
actinide
metals
will
be
de-
termined
by
the same
interactions
that
determine the
crystal
structures
of
the
d transition metals at
ambi-
ent
conditions.
Namely,
the balance between the
one-
electron
eigenvalue
sum
and
the electrostatic Coulomb
Page 6
THEORETICAL
ZERO-TEMPERATURE
PHASE DIAGRAM FOR. .
.
1637
interaction.
Since for close-packed structures the latter
contribution
to the total
energy
is
nearly
the
same,
the
one-electron
energy (band
energy)
is the
important prop-
erty
for the
determination
of the
crystal
structure. In
a
comparison
between
the close-packed
structures,
one
can
therefore use a canonical band
picture
to
investigate
which
close-packed. structure one should
expect
for a
par-
ticular f-electron
metal.
In
Fig.
8,
the canonical f-band
energy
is
evaluated
as
a function of
f-band
occupation.
Energies
corresponding
to the bcc and
hcp
crystal
structures relative to the fcc
structure
energy
are shown. The
approximate
5f
occu-
pations
for
Th-Pu
(at
the
equilibrium
volume)
are also
shown in this
figure.
However,
the
early
actinides show
an
appreciable
increase of the 5
f
occupancy
due
to
spd
~
f
promotion
with
increasing
compression.
'
'
For
neptunium,
LMTO-ASA
calculations
reveal
an
increase
of about
one-half
5
f
electron as
Np
is
compressed
down
to about
60%%uz
of
its
equilibrium
volume.
2s
Hence,
we
con-
clude &om
Fig.
8
that Th at extreme compressions
will
become either fcc or
hcp
depending
on
the exact
5
f-band
occupation
at the
given
compression.
Furthermore the
metals
U,
Np,
and Pu will
become
bcc,
whereas
for Pa
Fig.
8
suggests
that the
high-pressure
phase
is
likely
to
be
hcp
or
fcc.
For
Th,
accurate
calculations
(FP-LMTO)
(Ref.
15)
confirm
the results
of
Fig.
8;
for
U,
and
Np,
the
bcc
structure has been
determined
'
to be stable.
These
results are
consistent with
the
picture
obtained from
the
canonical band
theory,
as well
as with the
calculations of
Skriver.
The
high
density phase
of the 4f
cerium
metal,
o.-Ce,
is in some
ways
the
analog
to thorium and
has about one
4f-band electron.
Figure
8
suggests
a
phase
transition
to the
hcp
or fcc
crystal
structure at
high
compression
and for this
f
occupation. In
fact,
it
has
recently
been
predicted
by
means
of
FP-LMTO
calculations
that at
extreme
pressures
o.
-Ce, like
Th,
adopts
the
hcp
crystal
structure.
The reason
why
the bcc canonical f-band
energy
shows
a minimum for
f
occupations
around 6 is evident
&om
Fig.
9.
The
canonical
f
density
of states for the bcc
structure
(upper
panel)
shows
a
characteristic behavior
with two
pronounced
peaks
with a
deep
valley
in between.
This
valley,
where
the
canonical
f
density
of
states is
very
low
for the bcc
structure,
is
positioned
at a
f
occupation
of about six
electrons,
i.
e.
,
it corresponds
to the low
min-
imum
in
the
canonical f-band
energy
that
bcc
displays
compared
to the fcc and
hcp
structures
(see
Fig.
8).
On
the
other
hand,
the
canonical
f
density
of states
for fcc
and
hcp
looks
relatively similar,
due
to their
equal
num-
ber of nearest
neighbors.
This is reQected
in
their similar
canonical
f-band
energies as a
function
of
f-band
filling
in
Fig.
8. The double
peak
features of the bcc canonical
f
bands
are similar to the bcc
canonical d
bands,
which
for the
nonmagnetic
d transition series
gives
rise to
that
1.0
0.5
bcc
cn
00
bQ
2
0
1.
0
~
0.
5
0
0.0
hcp
1.
0—
0.5
I
6 8
foccupation
10
12
10
FIG. 8. Canonical
f-band
energy
differences (arbitrary
units)
as
a
function
of f-band occupation.
The fcc
canonical
energy
defines the zero
energy
level of the
plot.
Equilibrium
f-band
occupations
for
the
light
actinides
(Th-Pu)
are also
shown.
f
occupation
FIG. 9.
Canonical
f
density
of
states
(arbitrary
units)
as a
function of f-band
occupation
for the bcc
(upper
panel),
fcc
(middle
panel),
and
hcp
(lower
panel)
crystal
structures.
Page 7
PER
SODERLIND,
BORJE
JOHANSSON,
AND OLLE ERIKSSON
certain metals
(e.
g.
,
W
and
Ta)
are
very
stable
in
the bcc
structure. In
this
respect
Np
and
Pu could be interpreted
as
their actinide
counterparts.
To
further
analyze
our
Np
results,
we notice &om
Fig.
2 that
the
energy
diH'erence
between the fcc and the bcc
phase
is continuously increasing as a
function
of
pressure.
The
5f
occupation
will also increase with
pressure
due
to the
pressure
induced
spd
~
f
electron transfer.
In
Fig.
8
this
means that as
a
function
of
pressure,
Np
be-
comes more like Pu
and
in fact it
is
clear &om
this
6gure
that
the
bcc-fcc
energy
diBerence indeed should increase
in
agreement
with the accurate
(FP-LMTO)
calculations
shown
in
Fig.
2.
The fact
that
Np
has a
5f
occupa-
tion
which,
according
to the canonical
energies
in
Fig.
8,
gives
a
much lower bcc
energy
compared
to the
fcc
en-
ergy,
explains
the occurrence
of
bcc related
(as
opposed
to
fcc
related)
crystal
structures
in the
phase
diagram
of
neptunium.
V. CONCLUSIONS
remember that the
experiments
on
neptunium
are
per-
formed at room
temperature
and that
our
theory
treats
the electronic
structure
at zero
temperature.
For
large
compressions we 6nd that the bcc
phase
is
stable
and we
provide a
simple model,
based
upon
canoni-
cal
bands,
which
can be used
to understand
why
the
high-
pressure
structure in
Np
turns out to be
bcc.
The
canon-
ical bands also
give
insight
to the
crystal
structure
at
ambient
conditions.
For
Np
it
is obvious
&om
the
canon-
ical
f
bands
that
the Peierls
distortion,
which
stabilizes
the
open
and
complex
structures for
the
light
actinides,
derives
from
a
bcc
(and
not
an fcc
or
hcp)
parent
atomic
arrangement.
Furthermore,
this
simple
canonical
model
can
be
applied
to
the
high-pressure
phases
of other
itin-
erant
f-electron
metals. For
instance, the
model
calcu-
lations
based on
canonical
f
bands
predicts
plutonium
to behave
similar to
Np
at
high
compression.
Hopefully
the
present
results
will
encourage
accurate
experimen-
tal
high-pressure
crystallographic
studies of
the
light
ac-
tinides.
To
conclude,
we
have
shown that our
hypothesis,
that a f-electron
metal adopts
high-symmetry
structures
upon
compression,
is valid also for
Np.
Unpublished
ex-
perimental
results
reports
that first
the bct
phase
and
then the bcc structure has been observed
in
high-pressure
measurements.
In the
present
work,
we do not find that
the
bct
phase
is stable at
any
pressure,
but since the bct
(c/a
=
0.
85)
phase
and the bcc
phase
have
very
similar
energies
over a
large
volume
range
(see
Fig.
4),
a small
error of a &action of a
mRy
in
the total
energy
for the bct
phase
may
lead to
a
phase
transition and we cannot
en-
tirely
exclude this
possibility
because this
is
close to
the
accuracy
of our
computational
method. One should also
ACKNOWLEDGMENTS
We would like to
give
special
thanks
to J.M. Wills
for
taking
part
in most
aspects
of the
present
work. J.
Akella,
J.
Zhu,
A.
K.
McMahan,
and
U.
Benedict
are
thanked
for valuable
discussions.
We are
grateful
to
the
Swedish National
Supercomputer
Center in
I
inkoping
where most
of the
calculations
were
performed.
Thanks
also
to G.
Magnusson
for
useful
help regarding
the
calcu-
lations
on
the
National
Supercomputer.
Two
of us
(B.
J.
and O.E.
)
are
thankful for
6nancial
support
from
the
Swedish
Natural
Science
Research
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