CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
THE AB-INITIO STUDY OF STRUCTURAL STABILITY OF URANIUM
Andrey Kutepov, Svetlana Kutepova
Institute of Technical Physics, P.O.Box. 245,456770 Snezhinsk, Chelyabinsk reg., Russia
Abstract. The completely relativistic variant of a full-potential linear augmented plane wave method
supplemented by basis of local functions (FRLAPW+LO) was used to search for the stable structure of
uranium up to pressures of 5 Mbar. At each volume the optimization of all three free parameters of
alpha-uranium: b/a, c/a and internal parameter y was carried out. The candidates for stable structure
apart from alpha-uranium surveyed are face-centered cubic, body-centered cubic, and two variants
body-centered tetragonal structure: with c/a>l and with c/a<l. Comparison of enthalpies of different
structures computed at T=0 forecasts existence of phase transition alpha-uranium - bet (c/a<l) at
pressure approximately 1.7 Mbar.
0.8 Mbar. However, these calculations used a rough
approach in which the parameters of the crystalline
structure of alpha-uranium didn't depend on volume
in contraction to experimental data. The authors of
this theoretical work have performed further
advanced calculations, having optimized for each
value of the volume the ratio da - the most
variable parameter at compressions up to 1 Mbar.
This optimization has shown increased relative
stability of an alpha form up to a maximal pressure
1 Mbar, considered in work [6].
In the given work the search for stable structures
was extended to pressures of 5 Mbar. For structure
a-U apart from ratio da the other two free
parameters (b/a and internal parameter v ) were
optimized at each value of volume. The values of
these parameters at high pressures are essentially
different from the values at ambient conditions.
Apart from a-U , the face-centered cubic (fee),
body-centered cubic (bcc), and two variants of
body-centered tetragonal structure: bctl (c/a > 1)
and bct2 (c/a < 1) were considered as the candidates
for stable structure. Two types of bet-structure were
used because the total energy of uranium in the
given structure, as function of c/a, has two
minimums - corresponding to c/a > 1 and c/a < 1.
INTRODUCTION
Uranium, one of the heaviest elements discovered
in the nature, has attracted considerable attention to
itself per the last decades, mainly due to its usage as
reactor fuel and in nuclear weapons. The interesting
properties of this element include unusual
temperature dependence of elastic constants [1-3],
availability of waves of charge density at low
temperatures [4-5], anisotropic expansion at heating
up and, similarly to other light actinides, unique
crystalline structure [6]. However, as opposed to
other light actinides having one or more phase
changes at compressions up to 1 Mbar, uranium, in
agreement with the last experiments [6], remains in
the alpha form up to 1 Mbar. It is necessary to note,
that there were reports [7], indicating existence of a
phase change at 0.7 Mbar, due to the occurrence of
two new reflections in the X-ray spectrum.
However, a more recent study on a diamond-anvil
cell [8] has shown, that the reflections noted in the
earlier
study, originated from
anisotropic
compression of the crystal axes rather from a
crystallographic phase transition. The first
theoretical study of the given problem some years
later [9] has estimated a transition a-U -^bct at
245
Besides,
Besides, from
from the
the study
study of
of structural
structural stability
stability of
of
uranium
uranium atat zero
zero temperature,
temperature, the
the problem
problem on
on
thermal
thermal properties
properties of
of this
this element
element was
was investigated
investigated
ininthe
the given
given work.
work. While
While at
at low
low pressures
pressures only.
only. The
The
study
study of
of the
the given
given problem
problem was
was grounded
grounded on
on
representation
representation of
of aa Helmholtz
Helmholtz free
free energy of
of a
crystal
crystal as
as aa sum
sum of
of three
three terms
terms -- static
static energy
energy of
of a
lattice
lattice EE00,, free
free energy
energy of
of vibrations
vibrations of
of atoms
atoms FFvib
vib
and
and contribution
contribution from
from thermal
thermal excitations
excitations of
of
electrons
electrons FFel ::
F (V , T ) = E0 (V ) + Fvib (V , T ) + Fel (V , T )
harmonics inside spheres up to Lmax
10.
max equal to 10.
The
assumed to
to be
The radii of muffin-tin
muffin-tin spheres were assumed
identical for all structures and corresponding
corresponding to
almost touching spheres for structure α
a—U.
−U .
Density and potential in the interstitial were
represented
represented by the Fourier series consisting from
3175;
3175; 3151;
3151; 3217
3217 and 11385 plane waves for bctl
bct1
and bct2; bcc;
a—U
bcc; fee
fcc and α
− U structures accordingly.
The
was
The optimization of geometry in αa-U
− U was
conducted by a descent technique on coordinates.
For structures bct1
bctl and bct2
bct2 optimization
optimization of
of the
the
relation
out too.
too.
c / a was carried out
relation da
The
The pressure was computed as following:
(1)
(1)
The
The procedure
procedure of
of calculation
calculation of
of each
each of
of the
the above
above
terms,
terms, together
together with
with the
the obtained
obtained results,
results, is
is given
given
below.
below.
P=−
∂E0
,
∂V
(2)
where EE00 - total energy and V - volume.
The obtained results for pressure, as the function
The
of volume, have appeared lying between
of
from different
different works [12,13].
experimental results from
K was detected by
The phase transition at T =
The
= 0OK
equalization of enthalpies of different structures as
as
functions of pressure
functions
METHODS AND
AND RESULTS
RESULTS
METHODS
The calculation
calculation of
of total
total energy
energy of
of aa static
static lattice
lattice
The
wascarried
carriedout
outwithin
within the
the framework
framework of
of the
the densitydensitywas
functional theory
theory in
in generalized
generalized gradient
gradient
functional
approximation [10].
[10]. The
The full
full potential,
potential, fully
fully
approximation
relativistic (i.e.
(i.e. solving
solving the
the equation
equation of
of Dirac)
Dirac)
relativistic
variant of
of aa method
method of
of linearized
linearized augmented
augmented plane
plane
variant
waves supplemented
supplemented by
by basis
basis of
of local
local functions
functions
waves
(FRLAPW+LO) was
was used.
used. All
All calculations
calculations have
have
(FRLAPW+LO)
used 82
82 local
local functions
functions per
per atom
atom differing
differing by
by orbital
orbital
used
quantum numbers
numbers and
and values
values of
of energies
energies at which
quantum
these functions
functions were
were determined.
determined. The
The number
number of
of
these
augmented plane
plane waves
waves was
was determined
determined by
by the
the
augmented
quantity |kk++G|
G (k
( k -- point
point of
of aa Brillouin
Brillouin zone; G
G quantity
the
vector
of
reciprocal
lattice)
and,
thus,
was
varied
the vector of reciprocal lattice) and, thus, was varied
bothfor
fordifferent
different points
points of
of aa Brillouin
Brillouin zone,
zone, and
and for
for
both
different structures.
structures. As
As aa whole,
whole, the
the maximal
maximal
different
number of
of basis
basis functions
functions (including
(including augmented
augmented
number
plane waves
waves and
and local
local functions)
functions) was:
was: for
for structures
structures
plane
bct1 and
andbct2
bct2 -- 490;
490; for
for bcc
bcc -- 484;
484; for
for fee
fcc -- 450
450 and
and
bctl
α − U -- 399
399 functions
functions per
per atom.
atom.
forstructure
structure a-U
for
The integration
integration over
over Brillouin
Brillouin zone
zone was
was carried
carried
The
out
by
the
improved
tetrahedron
method
[11].
We
out by the improved tetrahedron method [11]. We
used 126
126 irreducible
irreducible points
points for
for bctl
bct1 and
and bct2;
bct2; 84
used
84 for
for
bcc; 98
98 for
for fee
fcc and
and 125
125 for
for a-U
α − U .. Electronic
Electronic
bcc;
density and
and potential
potential inside
inside muffin-tin
muffin-tin spheres
spheres were
were
density
expanded
on
spherical
harmonics
up
to
equal
L
max equal
expanded on spherical harmonics up to Lmax
Basis functions
functions were
were expanded
expanded on
on spherical
spherical
toto 6.6. Basis
H (P) =
=E
PV .
E0Q++PV.
(3)
(3)
In figure
figure 1 the enthalpies of the studied structures
In
relative to the enthalpy of bcc structure are given.
relative
^^~~"
>. 0.06
o
8,0.04
>^
« °'°3
2
is °-°
>, 0.01
Q_
«
•s
LU -0.01
fee
X'""
-jy 0.05
-
/
./
^/
/
/
-/
;
0
bct(c/a>1)__
„...-.-"••-•••--•— --—--'-""
^•~"~~~~~"
bcc
,„,--"•—""""*
_„,__——""""
:
-Wa-Tj"""'
0
500
1000
~~brt(5i<n~1500
2000 2500 3000
3500 4000 4500
Pressure, kbar
FIGURE 1.
1. Enthalpies
Enthalpies of
FIGURE
of different
different structures
structures of
of U.
U.
246
As
As shown
shown in
in the
the figure
figure the
the given
given work
work forecasts
forecasts
the
at aa
the existence
existence of
of phase
phase change
change a-U
α − U -^>bct2
→ bct 2 at
pressure
of
1.7
Mbar
approximately.
pressure of 1.7 Mbar approximately.
Apart
T=
Apart from
from the
the study
study of
of properties
properties at
at T
= 0OK
K ,,
the
the investigation
investigation of
of thermal
thermal properties
properties of
of uranium
uranium at
at
low
low pressures
pressures was
was carried
carried out.
out. The
The calculation
calculation of
of aa
part
part of
of free
free energy
energy connected
connected with
with aa vibrating
vibrating lattice
lattice
was
Debye
was conducted
conducted in
in quasiharmonic
quasiharmonic
approximation.
approximation. The
The Debye
Debye temperature
temperature 0
ΘDD was
determined
at
each
value
of
volume
from
a
relation
determined at each value
from
 rB 
Θ D = 41.63  
M 
">-H£
1
CV =
wiε i f i ( f i − 1) *
i
Fermi-Dirac, µju -- chemical
chemical
where ff.i - function of Fermi-Dirac,
potential of electrons, defined
defined from
from relation
relation
∑w f
i i
2
=N,
(7)
(7)
i
(4)
of electrons,
electrons, wwii -- weight,
weight,
where N - full number of
connected with aa geometrical
geometrical location
location of
of the
the point
point in
in
summation in
in formulas
formulas (6)
(6) and
and
Brillouin zone. The summation
out over
over one-particle
one-particle states.
states.
(7) is carried out
internal energy,
energy, entropy
entropy and
and free
free
3) Calculation of internal
with use
use of
of standard
standard
energy of Helmholtz with
thermodynamic relations.
Density of alpha-uranium
alpha-uranium at
at zero
zero pressure
pressure for
for
different temperatures,
temperatures, obtained
obtained from
from minimums
minimums of
of
free energy, is represented
represented on
on figure
figure 22 in
in
comparison with experimental
experimental data
data [15,16].
[15,16]. Density
Density
obtained while disregarding
disregarding thermal
thermal excitation
excitation of
of
electrons is also given in the figure.
9
NkB Θ D + 3Nk BT *
8
3 ΘD

,
T 3
Θ
x d x
ln  1 − e− D T  −  T 
 

 
ΘnD  ∫0 e x − 1 
 e


∑
 ∑ w j f j ( f j − 1)(ε j − µ )
 , (6)
(6)
 j

−
ε
+
µ


i
∑j w j f j ( f j − 1)




where
/M where rr -- Wigner-Seitz
Wigner-Seitz radius
radius in a.u., M
nuclear
nuclear mass,
mass, BB -- bulk
bulk modulus
modulus in
in kbar.
kbar. Formula
(4)
(4) was
was used
used with
with success
success in
in work [14] for
calculation
calculation of
of thermal
thermal properties
properties of a series of
metals.
metals. The
The knowledge
knowledge of
of Debye
Debye temperature
temperature as a
function
function of
of volume
volume allows
allows the
the computation of the
contribution
contribution to
to free
free energy
energy
Fvib (T ) =
1
k BT 2
(5)
where NN -- number
number of
of atoms
atoms in
in unit
unit cell,
cell, kBB where
Boltzmann’s
constant.
Boltzmann's constant.
The contribution
contribution of
of thermal
thermal excitation of
The
electrons was
was determined
determined on
on the
the basis
basis of Fermielectrons
Dirac statistics
statistics and
and the
the one-particle
one-particle spectrum
spectrum of the
Dirac
T
=
0
K
.
The
crystal
computed
for
crystal computed for T = OK The sequence of
calculations for
for each
each value
value of
of volume was:
calculations
19.2
19.1,
19
„
18.9
.0 18.8
D)
*-exp. data [16]
o - exp. data [15]
1
1) Carrying
Carrying out
out the
the self-consistent
self-consistent calculation
calculation of
1)
crystal band
band structure.
structure. It yields one-particle
crystal
energies £.
with /i indicating
indicating both
both the
the point
point of
of
ε i with
energies
a
Brillouin
zone
and
band
number.
a Brillouin zone and band
18.5
18.4
dot - calc. without e" exc.
solid - calc. with e" exc.
18.3
18.2
400
600
Temperature, K
2) Calculation
Calculation of
of heat
heat capacity
capacity for
for all
all interesting
interesting
2)
values of
of temperatures
temperatures with
with the
the formula
formula
values
FIGURE 2.
FIGURE
2. Density
Density of
of alpha-U
alpha-U versus
versus temperature.
temperature.
As it
As
it is
is seen
seen from
from aa figure,
figure, the
the thermal
thermal excitation
excitation
of
electrons
is
essential
at
temperatures
of electrons is essential at temperatures higher
higher than
than
200?
200K.. Used
Used model
model does
does not
not describe
describe an
an anisotropy
anisotropy
247
of expansion of uranium at heating up. Probably, it
is one of the reasons for the not quite precise
reproduction of temperature dependence on density
of uranium. Other reason may be following:
following: we
didn’t take into account the anharmonicity of
didn't
vibrations of a lattice.
In figure 3 the comparison of the computed and
experimental [17-20] heat capacities Cpp at zero
confirms
pressure is given. This figure also confirms
importance of taking into account the electronic
excitations for more low temperatures.
describing properties of uranium, but we continue to
work in the given direction.
REFERENCES
1.
2.
3.
4.
5.
50
45
6.
40
35
^
JO) 30
7.
o
8.
E25
O
9.
*-exp. data [17,18,19]
o - exp. data [20]
dot - calc. without e" exc.
solid - calc. with e" exc.
400
10.
11.
600
Temperature, K
12.
FIGURE 3. Heat of capacity of alpha-U versus temperature.
13.
CONCLUSIONS
14.
The theoretical investigation of properties of
uranium at zero temperature, carried out in the given
forecasts existence of structural phase
work, forecasts
−U →
bct at 1.7 Mbar. A rather simple
transition α
a-U
—> bet
simple
model of the description of thermal properties of
this element at zero pressure has yielded quite a
good exposition of temperature dependence on
density of this element and its heat capacities.
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range equation of state of the given metal, the given
insufficient. First of all, more
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functions is required. In particular it
thermodynamic functions
concerns the structures which are not stable at low
temperatures (bcc, bct).
bet). Apparently, it is necessary
effects also.
to take into account the anharmonic effects
Usage of molecular dynamic methods can become a
natural way of the solution of the given problem.
However we are unaware of potentials adequately
15.
16.
17.
18.
19.
20.
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