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

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Available from: Per Söderlind, May 24, 2014PHYSICAL

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

Council.

E.

A. Kmetko

and H.

H.

Hill,

in Plutonium 2970 and Other

Actinides,

edited

by

W.N. Miner

(AIME,

New

York,

1970);

A.

J. Freeman and D.D.

Koelling,

in The Actinides:

Elec-

tronic Structure and Belated

Properties,

edited

by

A.

J.

Freeman

and

J.

E.

Darby

(Academic

Press,

New

York,

1974),

Vol l.

B. 3ohansson and

A.

Rosengren, Phys.

Rev.

B

11,

2836

(1975).

A.

3.

Freeman

and D.D.

Koelling,

Phys.

Rev.

B

22,

2695

(1980).

H.

L.

Skriver,

B.

Johansson,

and O.K.

Andersen,

Phys.

Rev.

Lett

41,

42

(.

1978); 44,

1230

(1980);

M.

S.

S.

Brooks,

J.

Magn.

Mater.

29,

257

(1982);

J.

Phys.

F

13,

103

(1983);

M.S.

S

Brooks,

B.

3ohansson,

and H.L.

Skriver,

in

Hand-

book

on

the

Physics

and

Chemistry

of

the

Actinides,

edited

by

A.

J.

Freeman

and G.H. Lander

(North-Holland,

Ams-

terdam,

1984).

W.

H.

Zachariasen,

Acta

Crystallogr.

5,

660

(1952);

5,

664

(1952).

P.

Soderlind,

O.

Eriksson,

B.

Johansson,

J.M.

Wills,

and

A.

M.

Boring,

Nature

374,

524

(1995).

J.M.

Wills

and O.

Eriksson,

Phys.

Rev. B

45,

13879

(1992).

O.

Eriksson, J.M.

Wills,

P.

Soderlind, J.

Melsen,

R.

Ahuja,

A.

M.

Boring,

and

B.

Johansson, J.

Alloys Compounds

213/214,

268

(1994).

P.

Soderlind,

J.

M.

Wills,

A.

M.

Boring,

B.

3ohansson, and

O.

Eriksson, J.

Phys.

Condens. Matter

6,

6573

(1994).

P.

Soderlind, L.

Nordstrom, L.

Yongming,

and

B.

Johans-

son, Phys.

Rev. B

42,

4544

(1990).

A.

M.

Boring,

G.M.

Schadler, R.C.

Albers,

and P.

Wein-

berger,

J.

Less Common Met.

144,

71

(1988).

O.

K.

Andersen,

Phys.

Rev.

B

12,

3060

(1975).

J.

M. Wills

(unpublished);

J.M.

Wills and B.

R.

Cooper,

Phys.

Rev. B

36,

3809

(1987);

D.

L.

Price and B.

R.

Cooper,

ibid.

39,

4945

(1989).

P.

Soderlind,

O.

Eriksson, B.

Johansson, and

J.

M.

Wills,

Phys.

Rev. B

50,

7291

(1994).

P.

Soderlind,

O.

Eriksson, B.

3ohansson, and 3.M. Wills

(unpublished).

J.

P.

Perdew,

J.A.

Chevary,

S.

H.

Vosko,

K.

A.

Jackson, M.R.

Pederson, and

D.J.

Singh,

Phys.

Rev. B

46,

6671

(1992).

S.

Dabos,

C.

Dufour,

U.

Benedict, and M.

Pages,

J.

Magn.

Magn.

Mater.

63

0

64,

661

(1987).

S.

Dabos,

C.

Dufour,

U.

Benedict, and M.

Pages

(private

communication).

J.

Akella,

G.S.

Smith,

R.

Grover,

Y.

Wu,

and S.

Martin,

Page 8

52

THEORETICAL

ZERO-TEMPERATURE PHASE DIAGRAM

FOR.

. .

1639

High

Pressure Res.

2,

295

(1990).

D.A.

Y'oung,

Phase

Diagrams

of

the

Elements

(University

of

California

Press, Berkeley,

1991.

)

J. Akella

(unpublished).

J.

Donohue,

The

Structure

of

the

Elements

(Wiley,

New

York,

1974.

)

K.

A.

Gschneidner,

Jr.

,

Solid State

Physics:

Advances

in

Research and

Applications,

edited

by

F.

Seitz

and

D.

Turn-

bull

(Academic,

New

York,

1964),

Vol.

16,

p.

275.

V.

L.

Moruzzi,

J.F.

Janak,

and K.

Schwartz,

Phys.

Rev. B

87,

790

(1988).

O.

Eriksson,

P.

Soderlind,

and J.M.

Wills,

Phys.

Rev. B

45,

12 588

(1992).

A.K.

McMahan,

Scalar

relativistic

LMTO-ASA calcula-

tions

including

combined

correction terms

to

the ASA

(pri-

vate

communication).

H.

L.

Skriver,

Phys.

Rev. B

31,

1909

(1985).

Page 9

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

- [Show abstract] [Hide abstract]
**ABSTRACT:**The high-pressure crystal structures of Ce and Th are studied theoretically by means of a full potential linear muffin-tin orbital technique, where the generalized gradient correction to density-functional theory has been implemented. A crystallographic phase transition from fcc to bct is found for alpha-Ce at about 10 GPa, in good agreement with experiment. The calculated pressure dependence of the crystallographic c/a ratio (of the bct structure) is in good agreement with experiment. For alpha-Ce a subsequent transition to the hcp crystal structure is predicted at about 1600 GPa. For Th a continuous transformation from fcc to bct at 45 GPa is found, confirming recent experimental findings. Also for this material the calculated pressure dependence of the crystallographic c/a ratio (of the bct structure) is in good agreement with experiment. For Th we also predict a transformation to the hcp crystal structure at even higher pressures. -
##### Article: Cohesive properties of UO2

[Show abstract] [Hide abstract]**ABSTRACT:**The cohesive properties of some actinide metals (thorium, protoactinium and uranium), and of their oxides ThO2 and UO2, were studied by means of the linear muffin-tin orbital method in the atomic sphere approximation. This method gives accurate predictions of the stability of the close-packed structures, but it is found to be less accurate for the most open structure: the α-U structure. More specifically, the experimental cohesive properties of UO2 and ThO2 are particularly well reproduced. This leads us to propose, from ab-initio calculations, an estimation of the formation energy of oxygen vacancies in UO2. - [Show abstract] [Hide abstract]
**ABSTRACT:**The electronic structure, density of states, and x-ray photoelectron spectroscopy (XPS) intensities of the α, β, and γ phases of Np have been theoretically determined from self-consistent, fully relativistic, linear muffin-tin-orbital band-structure calculations. The calculated XPS intensities show a large f peak that narrows (due to the larger volumes per atom) and moves closer to the Fermi energy as one progresses through the α, β, and γ phases. Experimental XPS intensities of the α and β phases are also presented and compared to the theoretical calculations. Experimentally, the α and β intensities are found to be almost identical, and do not show any evidence for the theoretically predicted f-band narrowing. In addition, there is a large additional experimental intensity at higher binding energy that is not present in the calculations. This extra intensity may be evidence for quasilocalized behavior of the 5f electronic states that is induced by the XPS process. For example, the low-energy tail may be related to either multiplet or satellite structure due to final-state localized atomiclike correlation effects. © 1996 The American Physical Society.