Page 1

PHYSICAL REVIEW 8

VOLUME 52, NUMBER 3

15 JULY 1995-I

Theoretical zero-temperature

phase diagram

for neptunium

metal

Per Soderlind, Borje Johansson,

of Physics,

(Received 21 March 1995)

and Olle Eriksson

of Uppsala, P.O. Box 580, S 751 -21 Uppsala,

Condensed

Matter Theory Group, Department

Umversity

Siveden

Electronic structure

gradient

approximation

graphic

properties

such as crystal structure

ment. The calculated

in accordance with experiment.

two crystallographic

Mbar and second a P-Np—+ bcc phase transition

by small volume collapses of the order of 2—

theoretically

up to 10 Mbar. A canonical theory for the f-electron contribution

bcc structures

is presented.

calculations,

to the exchange

of neptunium

and atomic volume,

bulk modulus

based on the density-functional

and correlation

metal

under

compression.

are found to be in excellent

and the 6rst pressure

Our theory predicts that neptunium

phase transitions

upon compression:

theory

with the generalized

the crystallo-

ground-state

agreement

of the bulk modulus

at low temperature

first a transition

o.-Np

at 0.57 Mbar. These transitions

3%%uo. The high pressure

phase (bcc) is also investigated

energy,

are used to study

Calculated

properties,

with experi-

are also

undergoes

+ P-Np at 0.14

are accompanied

derivative

to the hcp, fcc, and

I. INTRODUCTION

TABLE I.

(Ref. 5) for o.-Np and P-Np.

Experimental

equilibrium

lattice

constants

o,-Np

P-Np

a (A.)

4.723

4.897

b/a

1.035

1.000

c/a

1.411

0.692

Neptunium

This series consists of radioactive

shell is progressively

series. For the elements Th-Pu the 5fstates have itiner-

ant (band) characteri

as part of a 5ftransition

ground-state

properties

analogous to the 4d and 5d tran-

sition metals.

The increasing

structures

&om Th to Pu is, however, in sharp contrast to the high-

symmetry

crystal structures

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

transforms

to a tetragonal

ble up to 900 K where the bcc structure

see Fig.

1. Details of the n-Np and P-Np structures

are given in Tables I and II . Recently

tablished

between the crystal structures

tinide metals and the d transition

it was shown that both a d and afmetal

simple

(high-symmetry

close-packed)

symmetry

open-packed)

crystal structures

their density.

This suggests that actinide metals should

generally

show a phase diagram

tures

become

increasingly

applied

pressure.

Unfortunately

belongs to the series of actinide

metals.

metals

where the 5f

filled as one proceeds through

the

sand these metals can be viewed

series with a behavior

of the

complexity

of the crystal

as the 5fband is being filled when proceeding

observed in the d transition-

neptunium

forms (P-Np), which is sta-

becomes stable,

a link was es-

of the light ac-

metals. In this work

can have

or complex

depending

(low-

on

where the crystal struc-

symmetric

as a function

very few high-pressure

of

0.03—

cu

~ 0.02-

0

0.01—

O'.-Np

P-Np

bcc()

Liquid

I

300

600

Temperature

(K)

900

FIG. 1. Experimental

tunium

metal (Refs. 20 and 21).

low-pressure

phase diagram of nep-

experiments

metals and so far this conjecture has not been confirmed

experimentally,

to our knowledge.

As will be seen, the present

port previous

findings

and are also in agreement

the theoretical

results

for Th, Pa, and U reported

%Pills and Eriksson

and the recently published

tions for Np.

sufEciently

high pressure the bcc crystal structure

become stable in Np. Soderlind

the bct phase (c/a=0.85) was close to being stable in the

vicinity of the a-Np ~ bcc transition.

atively poor agreement

with experiment

ground-state

data for these previous

lated equilibrium

volume

was about

experimental

data), the accuracy of the theoretical tran-

have been published

for the light actinide

calculations

for Np sup-

with

by

calcula-

' The latter calculations

'

showed that at

should

et al.

also noticed that

Due to the rel-

as regards

results

(the calcu-

15'%%uo smaller

the

than

0163-1829/95/52{3)/1631(9)/$06.00

52

1631

1995 The American Physical Society

Page 2

PER SODERLIND, BORJE JOHANSSON, AND OLLE ERIKSSON

TABLE II.

(Ref. 5) for o.-Np and P-Np.

Experimental

equilibrium

atomic

positions

Atom type

n-Np(I)

n-Np(II)

P-Np(I)

P-Np(II)

yx—

—0.208, zq —

Atomic positions

+(4,yi,zi)

+(4,—,' —yi, zi + —, ')

(0,0,0)

—0.036, yq—

—0.842, z2——0.319, u = 0.375.

sition pressures

therefore they were not published.

the ground-state

tions are performed

tion of the exchange and correlation

II),which also allows for a more accurate calculation of

the pressure

for the crystallographic

Hence we find it motivated to publish our improved cal-

culations for the various allotropes ofNp 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

an approximative

transformation

and bcc structure.

Furthermore,

results for the bcc Np equation-of-state

pressures.

In connection to the discussion

symmetry

close-packed

crystal structures

sures, a simple canonical band picture

used to evaluate

which particular

an itinerant fmetal will attain at very high pressures.

Since the present investigation

structure

for Np metal, we take the opportunity

some earlier theoretical

work on this material.

et al.

were the first to calculate the equilibrium

and bulk modulus

of Np (and the other light actinides);

they

which relied on the atomic sphere approximation

and the replacement

of the true structure

arrangement.

Subsequently,

spin-orbit

interaction

by solving the Dirac equation

the light actinides

including

and for a fcc structure.

Soderlind

ilar technique,

and &om the obtained

they were able to calculate the thermal expansion for Np

and the rest of the early actinide metals.

ports quoted above, the complicated

neptunium

was replaced. by the fcc structure.

electronic structure

calculation

low-temperature

crystal structure

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

The reported

results

are

structure

calculations

for neptunium

(n-Np ~ bcc) could be questionable

and

To try to improve on

of Np, the present

with a recently

developed

functional

properties

calcula-

formula-

(see Sec.

phase transitions.

in more detail; we study

path between the P-Np

we present

up to ultrahigh

theoretical

of the high-

at high pres-

is presented

close-packed structure

and

deals with the electronic

to quote

Skriver

volume

d.id this by means of scalar relativistic

calculations,

(ASA)

by a fcc atomic

incorporated

Brooks

the

for

Np, also within

et al.

the ASA

applied a sim-

equation-of-state

In all the re-

crystal structure

of

The first

the true

that considered

of Np was presented

DETAILS

Rom

in the seven crystal

obtained

electronic

structures

compression.

ified Schrodinger

tem (the Dirac equation

within

density-functional

are included in the Hamiltonian,

interaction

term,

recipe proposed

expanded

in linear muon-tin

lapping mufBn-tin spheres that surround

in the crystal.

We make use of a so-called double basis

set by allowing

two tails with di6'erent

for each numerical

basis function

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 5f,and

to the pseudocore

orbitals 68 and 6p, were defined.

side the muKn-tin

spheres, in the interstitial

wave functions are Hankel or Neumann

are represented

by a Fourier series using reciprocal lat-

tice vectors.

The same expansion

the charge

d.ensity

and the potential.

of the wave function,

charge density,

not adopt any geometrical

scribed type of computational

to as a full potential linear muffin-tin orbital method (FP-

LMTO).

The present

version of the method,

oped by Wills and co-workers,

successfully

been applied

for calculating

structure

for actinide materials

tions are therefore

a natural

publications.

In the density-functional

become common to make a local approximation

exchange

and correlation

interactions

trons.

Since the recently presented

approximation

has been shown to significantly

the accuracy of the results for actinide metals,

chosen to adopt this approximation

correlation

energy functional

In the calculation

of the one-electron

the special k-point method

sampling

densities of the k points.

ture, 16 k points in the irreducible

zone (IBZ) were used whereas for P-Np the corresponding

number

was 18. This might

numbers of k points in the IBZ for these structures

the IBZ is 1/2 and 1/4 of the full Brillouin

for n-Np and P-Np, respectively.

the number of k points to 32 (n-Np) and 40 (P-Np) only

lowered

the total energy

with about 0.1 mRy/atom

the theoretical equilibrium

volume.

ture for Np in the o.-U crystal structure

with

2 atoms/cell)

was obtained

the IBZ (1/8 of the FBZ). In the case of the more sym-

metric bcc, fcc, and bct structures,

bct unit cell was consistently

of 150 k points was used in the IBZ for those (1/16 of

the FBZ). For the hexagonal

to have an ideal c/a ratio,

IBZ (1/12 of the FBZ). To further

n-Np, P-Np, n-U, bct, bcc, fcc, and hcp under

The present

ab inito method solves a mod-

equation for the total energy of the sys-

is solved for the core electrons),

theory.

including

which

is considered

by Andersen.

The wave functions

orbitals inside the nonover-

All relativistic

efFects

the spin-orbit

according

to the

are

each atomic site

kinetic

energy

inside the mufBn-tin

Out-

region, the

which

functions,

is used to represent

This treatment

and potential

approximations

method is usually referred

does

and the de-

which

is devel-

has previously

the electronic

and the present calcula-

extension

of the previous

approach,

it has

to the

the elec-

gradient

improve

we have

between

generalized

for the exchange and.

in the present calculations.

band structures,

has been used with various

In the o.-Np struc-

wedge of the Brillouin

seem to be relatively

small

where

zone (FBZ)

However, an increase of

at

The electronic struc-

(orthorhombic

using 100 A: points in

the symmetry

of the

applied and a total number

lattice, which

we used 30 k points

investigate

we assumed

in the

the con-

Page 3

52

THEORETICAL ZERO-TEMPERATURE PHASE DIAGRAM FOR. . .

1633

vergence of the A:-point sampling

structures,

we put the crystallographic

5/a, and positional parameters)

so that the structure

ical equilibrium

volume,

that the total energy

was converged to about 0.3 mRy/atom.

for the various crystal

parameters

for o.-Np, P-Np, and bct

became bcc. Close to the theoret-

these test calculations

difference

between

(c/a,

showed

the structures

III. RESULTS

Experimentally

of data available

tunium.

to crystallize

8 atoms/unit

served in Np below 0.52 Mbar in the study by Dabos et

at."The axial ratios (c/a and b/a) showed,

small pressure dependence.

had decreased

by about

170 and the b/a value had in-

creased with a similar

amount

Other

high-pressure

x-ray diffraction

indicate,

however,

an increasing

of the axial ratios,

which

ble tetragonal

structure

by the authors.

pressures,

o.-Np is stable &om zero temperature

about 600 K, see Fig. 1. Above this temperature,

undergoes

a phase transition

quently,

around 900 K, thepphase (bcc) of neptunium

found.

No phase transitions

been published

for neptunium.

We have chosen to theoretically

served n, P,bct, and bcc phases of neptunium.

parison we also investigated

structures.

The o,-U structure

as a representative

for the open and complex crystal

structures

of the light actinides,

ture has been used to model the light actinide metals in

several early calculations.

in Fig. 2 where the total energies

structures

are plotted as a function of volume.

ternal crystallographic

parameters

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

The calculated total energy of the bct structure

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

(

energy.

As an example,

we display

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

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

for the fcc phase at this volume, a fact that also can be

seen in Fig. 2. The total energy calculations

furthermore

show that the o.-Np phase

there

as regards

conditions

in a simple

cell (o.-Np).

is only a rather

the phase diagram

Np is, however,

orthorhombic

No phase transition

limited

body

of nep-

found

with

was ob-

At ambient

structure

however,

a

At 0.5 Mbar, the c/a value

in their investigation.

studies

tendency

was interpreted

on o.-Np

unity

as a possi-

At low (zero)

toward

up to

Np

to the P phase and subse-

is

'

at higher pressures

have

investigate

the ob-

For com-

the o.-U, hcp, and fcc crystal

has sometimes

been used

whereas

the fcc struc-

'

Our main results are shown

for the seven crystal

The in-

for the n-Np, P-Np,

'

efforts.

was min-

16—20 A.s), a c/a value of 0.85 minimized

the total

in Fig. 3 the calcu-

distortion,

whereas

lower

(

21 mRy) than

in Fig. 2

has the lowest

0.12

0.10—

E

0

0.08

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

theoretical

equilibrium

temperature

experimental

in agreement

with experiment.

at the equilibrium

volume,

structures

is ordered according to their complexity

at both the

volume

as well as at the room

equilibrium

volume.

Figure 2 also shows that

the energy of the different

This is

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

total energy as a function of the c/a value in the bct structure,

for neptunium

at the experimental

A. ). The bcc and the fcc structures

values of 1 and ~2, respectively.

path, i.e.,the

equilibrium

are obtained

volume (19.2

for c/a

Volume (A

l

FIG. 2. Calculated

for the n-Np, P-Np, n-U, bct, bcc, hcp, and fcc crystals of

neptunium

metal. The points represent

the solid lines connecting them show the Murnaghan

as obtained

by a least-square

fit to the calculated

total energies as a function of volume

calculated

values and

functions

energies.

Page 4

1634

PER SODERLIND, BORJE JOHANSSON, AND OLLE ERIKSSON

52

openness;

and Gnally the more close-packed structures.

At a moderate

pressure of about 0.14 Mbar, see Table

III, neptunium

is calculated

sition to the P form of Np.

energy

difFerence between the studied crystal structures

(except hcp and fcc) as a function of volume.

structure

is here taken as a reference structure,

phase defines the zero energy

be seen in. this figure, the

rapidly increasing their energies relative to the structures

with higher crystal symmetry,

structure.

The calculated bct and bcc energies, however,

are very close to each other in the volume

17 A. , and this makes it difficult to accurately

late a transition

between

these two phases.

has been drawn to our attention

neptunium

has been seen experimentally

row pressure

range.

Theoretically

transition,

but at a larger compression,

transition

pressure of 0.57 Mbar, we Gnd that neptunium

adopts the bcc crystal structure,

being only marginally

higher in energy.

prediction

of a bcc phase in Np at ultrahigh

was already made earlier8'

experimentally.

Notice in Fig. 4 that the P-Np, bct, and bcc phases

are very close in energy at a volume of about 15As. As

mentioned

earlier these three structures

o.-Np structure

can be viewed as distortions

lattice. We illustrate

this for the P-Np structure

5 where a unit cell is shown.

6gure are of atom type I and the shaded atoms of type

II, as described

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.

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

below the center of the faces. Their locations are deter-

mined

by the crystal parameter

By choosing c/a to be I/~2 and u to be 0 5, we obtain

ci.-Np is lowest, then comes P-Np,

ci.-U, bct,

to undergo

In Fig.

a phase tran-

4, we show the

The bcc

and this

As can

level of the plot.

ci.-Np and P-Np structures

are

i.e.,the bct and the bcc

range 15—

calcu-

In fact, it

that the bct phase in

within a nar-

we do not And this

at a calculated

with the bct structure

The theoretical

pressures

and has now been con6rmed

as well as the

of the bcc

in Fig.

The open atoms in this

in Table II. The unit cell looks similar

For P-Np, the c/a ratio

slightly

above or

u, de6ned in Table II.

30—

20

E0

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

energy of the bcc structure,

as a function of volume.

relative to the total

the bcc crystal structure.

the c/a value,

can approximately

by translating

u parameter

is therefore

a function of the positional

show the total energy for Np as a function of the atomic

position

u at three atomic volumes;

In this 6gure, u = 0.5 corresponds

and u = 0.375 gives an efFective P-Np structure

is an approximation

to the true P-Np structure

is set equal to I/~2). Notice in Fig. 6 that for the vol-

ume 18 A.sthe bcc structure

Since for the P-Np structure

see Table I, is indeed close to I/~2, we

transform

P-Np to the bcc structure

the atoms of type II, i.e., by changing

from the observed

value of 0.375 to 0.5. It

interesting

to calculate the total energy

parameter

the

as

u. In Fig. 6 we

14, 15, and 18 A. .

to the bcc structure

(which

since c/a

shows a local energy inax-

TABLE III. Equation-of-state

shown in Fig. 2. Experimental

(Vp) and Murnaghan

modulus

and its pressure

Bo and Bo, respectively.

are labeled B. Pressure quantities

data obtained from Murnaghan

data are obtained at room temperature

fits to data from Dabos et al. (Ref. 17) (Bcand Bo). The calculated

derivative

evaluated

at the theoretical

The corresponding

calculated data evaluated

are given in Mbar and volumes

B

1.3

1.4

1.3

1.2

1.2

1.5

1'1

1.2

6.0

fits to the FP-LMTO calculations

from Donohue

(Ref. 22)

bulk

equilibrium

at the experimental

in A .

~amp

17.3

16.7

volume

are denoted

volume

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

Bo

5.5

5.3

4.9

5.0

4.9

4.1

5.6

pa-+p

0.14

0.14

pp-+bcc

0.57

0.57

gp-+bcc

14.9

14.5

Page 5

52

THEORETICAL ZERO-TEMPERATURE PHASE DIAGRAM FOR. . .

1635

FIG. 5. The unit cell of the P-Np crystal structure.

shaded atoms symbolize

the atoms of type II (see Table I).

The crystallographic

parameter

plicitly in the figure.

The

u (see Table I) is shown ex-

imum,

This is in agreement

at ambient

c/a = 0.69).

A),the previous

turns over into a local energy minimum

for u, which gives the global minimum

creased to about 0.40. The energy

whereas there is a global minimum

with the experimental

conditions,

where

At a somewhat

maximum

for u =0.36.

data

for Np

u = 0.375 + 0.015 (with

compressed

in energy for u =0.5 (bcc)

whereas the value

in energy, has in-

difference

volume

(15

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

position, u, for a crystal structure,

P-Np structure

imated to have the bcc value of 1/~2.

structure,

c/a = 0.69. Therefore

structure

is equivalent

the energy is normalized

calculations

done for Np close to the equilibrium

A. ). We also show results for compressed

(15 A.)and solid circles (14 A ). Lines connecting

points are polynomial

fits and serve as a guide to the eye.

total energy

as a function

which is equivalent

of atomic

to the

for u =0.375, except the c/a ratio is approx-

For the true P-Np

with u = 0.5, the present

to the bcc crystal structure

to zero. The solid diamonds

for which

refers to

volume (18

Np as open squares

the data

the bcc structure

also decreased

When neptunium

the bcc structure

even a local minimum

ture. The calculations

the transition

due to a soft phonon, corresponding

of the neptunium

tional parameter

a maximum

for the energy of the bcc phase at volumes

corresponding

to zero pressure, implies that the observed

high temperature

(HT), ambient

to be stabilized

by entropy

pressure HT bcc phase in Np requires a boot-strapping

mechanism

for stabilization

required for the pressure induced (zero-temperature)

phase. Therefore the HT bcc phase in Np seems to have

a diferent

physical origin. The observed

(Fig. 1) is in accordance with this conclusion,

HT bcc phase region forms a closed area in the (P, T)

plane.

Next we comment

upon the calculations

and fcc crystal structures

for neptunium.

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

compared to calculations

ture for the light actinides Pa (bct), U (n-U), and Np is

considered.

In the present

the calculated

equilibrium

volume

to the observed

volume.

In light of the present

for Np, this better agreement

atomic volume,

when assuming

be a fortuitous

result.

For the bcc phase we have extended

to even higher

densities,

and in Fig.

high-pressure

equation-of-state

phase. The pressure is obtained as the volume derivative

of the calculated

total energy.

total energy calculated

in the large volume range shown

in Fig. 7 is not appropriate.

volume

interval

into several pieces and numerically

ferentiated

the total energy to obtain the pressure.

instance at the smallest

investigated

obtained

a pressure

of about 10 Mbar.

accurately

calculate the total energy for neptunium

der more than a twofold compression,

treat the pseudocore

states (6s and 6p) as band states.

As an inset in Fig. 7 we also compare

pressures for o.-Np with the Murnaghan

mental data by Dabos et al.

of Np. Apart from the small discrepancy at low pressures,

our zero-temperature

calculations

room temperature

experimental

Murnaghan

fit was done for data in the pressure

0—0.52 Mbar and pressures

beyond that are therefore an

extrapolation.

We summarize

a number

and the effective P-Np structure

by more than

is compressed

further to about 14 As,

has the lowest energy and there is not

close to the effective P-Np struc-

presented

in Fig. 6 suggest that

from P-Np to bcc at large compressions

has

50'%%uo at this compression.

is

to the displacement

by the posi-

atom of type II defined

u. The fact that our calculations

show

pressure bcc phase has

eKects.

Therefore

the zero

of its phonons.

This is not

bcc

phase diagram

since the

for the hcp

Even though

volume is obtained

'

where the true crystal struc-

case of Np this means that

for fcc Np is closest

results

for the

seems to

with experiment

a fcc structure,

the calculations

7 we show the

in the bcc

for neptunium

A Murnaghan

fit to the

Instead, we have divided the

dif-

For

volume,

8 A.s, we

To be able to

un-

it was necessary to

our theoretical

fit to the experi-

for the low-pressure

regime

are very close to the

data. The experimental

range

of our calculated

quantities

Page 6

1636

PER SODERLIND, BORJE JOHANSSON, AND OLLE ERIKSSON

6

0

bc

—

~theory

expt

at room temperature,

temperature.

ume and the bulk modulus

cially neptunium

the unusually

This temperature

Debye-Gruneisen

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

tive to their zero temperature

small discrepancy

between our present theory and room

temperature

data is almost

consider temperature

efFects. Hence, where experimental

data are available, the calculations

o.-Np structure

give very good results.

whereas our theory applies to zero

The temperature

efFect on the atomic vol-

for the actinides,

and plutonium,

is considerable

large thermal expansion

efFect can be well accounted

theory

and a temperature

and espe-

due to

in these metals.

for by

correc-

'

by about

Thus the already

8'%%up, rela-

values.

entirely

removed

when

we

for neptunium

in the

10

Atomic Volume

12

14

16

(A )

FIG. 7. Theoretical

and atomic volume

neptunium.

data (n-Np) compared

data (Ref. 17) for neptunium

equation

of state (pressure

in Mbar

in A.)for the high-pressure

The inset shows, in the same units, theoretical

to a Murnaghan

at lower pressures.

bcc phase of

fit to experimental

for Np in Table III.

published

before.

data for neptunium

not available

experimental

couraging to notice that apart from the highly unstable

hcp and fcc structures,

Table III reveals that among the

other, bcc related, crystal structures

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 ofBo is very sensitive

to the choice of volume interval for which the total ener-

gies are 6tted to the Murnaghan

librium

volume, the agreement

(4%%up). We note that the calculated

shows a rather

large dependence

ture, and that only the calculation

again apart &om the excited hcp and fcc phases,

a value that is comparable

density. The fact that our calculated equilibrium

is slightly smaller than the measured

hances the computed

bulk modulus at this volume since

the bulk modulus

increases with density.

theoretical

bulk modulus

for this efFect we also show in

Table III the bulk modulus

equilibrium

volume;

we have chosen to denote this quan-

tity as B. With this volume correction

good agreement

with experiment

(8%%up). For Bpthe agreement

experiment

is also very encouraging.

All the experimental

data in Table III are measured

Some of these results

As mentioned,

at high pressure

to the authors.

low-pressure

data for neptunium.

have been

detailed experimental

are unfortunately

In Table III we present

It is en-

the best agreement

equation.

with experiment

equilibrium

on the crystal struc-

of the o.-Np phase,

For the equi-

is good

volume

gives

to the observed

equilibrium

volume

value indirectly

en-

To correct our

evaluated

at the measured

we obtain a very

for the bulk modulus

between theory (u-Np) and

IV. DISCUSSION

The present

erant f-electron

of high-symmetry

This is clearly

equilibrium

regarding

in energy) are ordered in increasing complexity/openness

(with n-Np being lowest and bcc being highest in energy),

whereas at sufFiciently

high compression

crystal structures

is exactly reversed (with bcc being low-

est and n-Np being highest

this behavior

have been discussed

portant parameter

in this connection is the1bandwidth.

Due to a Peierls distortion, a system with narrow bands

close to the Fermi level can gain energy by lowering the

crystal structure

symmetry.

calculations

at the equilibrium

u =0.5 (bcc) the calculated density of states at the Fermi

level, D(E~), is about 60 (states/Ry/atom),

the distorted

structure

Np) it is lowered to about 36 (states/Ry/atom).

substantial

lowering ofD(E~) due to this distortion

dicates that bands are shifted away from this region and

this results in a decrease ofthe sum ofthe one-particle

ergies. For a more compressed

calculations

for the two structures

u = 0.375, respectively)

give almost

of about 30 (states/Ry/atom),

energy gained by a Peierls distortion

duced in this case.

The arguments

described

dict, for instance, the general trend of crystal structures

of the early actinides

(with bonding

tion of compression.

Since the 5f bands are broadened

under

compression

they become more d-band-like,

the crystal structures

of the actinide metals

termined

by the same interactions

crystal

structures

of the d transition

ent conditions.

Namely,

the balance

electron

eigenvalue

sum and the electrostatic

calculations

metals

crystal structures

shown

in Figs.

volume

the crystal structure

the hcp and fcc phases

support

the idea that itin-

will show an increasing

degree

upon compression.

2 and. 4 where at the

energies

which

are very high

(dis-

the order of the

in energy).

The reasons for

earlier

and the im-

This is manifested

volume for Np, where for

in our

whereas for

(u= 0.375, approximatively

P-

The

in-

en-

volume (14 A.),however,

(with

the same D(E~)

which reflects that the

is considerably

u = 0.5 and

re-

above

can be used to pre-

5fstates) as a func-

and

will be de-

that determine

metals at ambi-

between

the

the one-

Coulomb

Page 7

THEORETICAL ZERO-TEMPERATURE PHASE DIAGRAM FOR. . .

1637

interaction.

contribution

one-electron

erty for the determination

a comparison

can therefore use a canonical band picture to investigate

which close-packed. structure

ticular f-electron metal.

In Fig. 8, the canonical f-band energy is evaluated

a function of f-band occupation.

to the bcc and hcp crystal structures

structure

energy are shown.

pations

for Th-Pu (at the equilibrium

shown in this figure.

However,

an appreciable

~fpromotion

neptunium,

LMTO-ASA calculations

of about one-half 5felectron as Np is compressed

to about

clude &om Fig. 8 that Th at extreme compressions

become either fcc or hcp depending

occupation

at the given compression.

metals U, Np, and Pu will become bcc, whereas

Fig. 8 suggests that the high-pressure

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

results are consistent

with the picture obtained

canonical band theory, as well as with the calculations

Skriver.

The high density phase of the 4f cerium metal,

is in some ways the analog to thorium

Since for close-packed

to the total energy is nearly the same, the

energy (band energy) is the important

of the crystal structure.

between

the close-packed

structures

the latter

prop-

In

structures,

one

one should expect for a par-

as

Energies corresponding

relative to the fcc

The approximate

volume)

the early actinides

5f occu-

are also

show

due to spd

'

'

increase of the 5foccupancy

with increasing

compression.

reveal an increase

For

down

60%%uz of its equilibrium

volume.2sHence, we con-

will

on the exact 5f-band

Furthermore

the

for Pa

phase is likely to

' to be stable. These

from the

of

o.-Ce,

and has about one

4f-band electron.

to the hcp or fcc crystal structure

and for this foccupation.

predicted

by means of FP-LMTO calculations

extreme pressures

o.-Ce, like Th, adopts the hcp crystal

structure.

The reason why the bcc canonical f-band energy shows

a minimum

Fig. 9. The canonical f density of states for the bcc

structure

(upper panel)

shows a characteristic

with two pronounced

peaks with a deep valley in between.

This valley, where the canonical fdensity ofstates is very

low for the bcc structure,

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

the other hand, the canonical fdensity 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

fbands are similar to the bcc canonical

for the nonmagnetic

d transition

Figure 8 suggests a phase transition

at high compression

In fact, it has recently been

that at

forfoccupations

around

6 is evident &om

behavior

is positioned at afoccupation

(see Fig. 8). On

d bands, which

series gives rise to that

1.0

0.5

bcc

cn 00

bQ

20

1.0

~ 0.5

0

0.0

hcp

1.0—

0.5

I

6

8

f occupation

10

12

10

FIG. 8. Canonical

units) as a function of f-band occupation.

energy defines the zero energy level of the plot. Equilibrium

f-band occupations

for the light actinides

shown.

f-band

energy

differences

The fcc canonical

(arbitrary

(Th-Pu) are also

f occupation

FIG. 9. Canonical fdensity of states (arbitrary

function of f-band occupation for the bcc (upper panel), fcc

(middle panel), and hcp (lower panel) crystal structures.

units) as a

Page 8

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

phase is continuously

increasing as a function of pressure.

The 5f occupation

will also increase with pressure

to the pressure

induced

Fig. 8 this means that as a function of pressure,

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

neptunium.

between the fcc and the bcc

due

In

spd ~felectron transfer.

Np be-

in the phase diagram of

V. CONCLUSIONS

remember

formed at room temperature

the electronic structure at zero temperature.

For large compressions

stable and we provide a simple model, based upon canoni-

cal bands, which can be used to understand

pressure structure

in Np turns out to be bcc. The canon-

ical bands

also give insight to the crystal structure

ambient conditions.

For Np it is obvious &om the canon-

icalfbands that the Peierls distortion,

the open and complex structures

derives from a bcc (and not an fcc or hcp) parent atomic

arrangement.

Furthermore,

can be applied to the high-pressure

erant f-electron metals.

For instance,

lations based on canonical fbands predicts

to behave similar to Np at high compression.

the present

results

will encourage

tal high-pressure

crystallographic

tinides.

that the experiments

on neptunium

and that our theory treats

are per-

we 6nd that the bcc phase is

why the high-

at

which stabilizes

for the light actinides,

this simple canonical model

phases of other itin-

the model calcu-

plutonium

Hopefully

experimen-

accurate

studies of the light ac-

To conclude,

that a f-electron metal adopts high-symmetry

upon compression,

is valid also for Np. Unpublished

perimental

results

reports that first the bct phase and

then the bcc structure

has been observed in high-pressure

measurements.

In the present

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

error of a &action of a mRy in the total energy for the bct

phase may lead to a phase transition

tirely exclude this possibility

accuracy of our computational

we have

shown

that

our hypothesis,

structures

ex-

work, we do not find that

range (see Fig. 4), a small

and we cannot en-

because this is close to the

method.

One should also

ACKNOWLEDGMENTS

We would

for taking part in most aspects of the present

Akella, J. Zhu,

A.K. McMahan,

thanked

for valuable

discussions.

Swedish

National

Supercomputer

where most of the calculations

also to G. Magnusson

for useful help regarding

lations on the National

and O.E.)are thankful

Swedish Natural

Science Research Council.

like to give special thanks

to J.M. Wills

work. J.

and U. Benedict are

We are grateful to the

Center

in I inkoping

were performed.

Thanks

the calcu-

Supercomputer.

for 6nancial

Two of us (B.J.

support

from the

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calcula-