Structure of liquid transition and rare earth metals

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Structure of liquid transition and rare earth metals

S.N. Khanna, F. Cyrot-Lackmann

To cite this version:

S.N. Khanna, F. Cyrot-Lackmann. Structure of liquid transition and rare earth metals. Journal de

Physique Lettres, Edp sciences, 1979, 40 (3), pp.45-48. �10.1051/jphyslet:0197900400304500�.

�jpa-00231566�

Structure

of

_liquid

transition and

rare

earth

metals

S. N. Khanna and F.

_{Cyrot-Lackmann}

Groupe des Transitions de Phases, C.N.R.S., B.P. 166, 38042 Grenoble _Cedex, France

(Re~u le 15 novembre _{1978, accepte} le 15 decembre ₁₉₇₈₎

Résumé. 2014 On montre

que les résultats

expérimentaux

des facteurs de structure des métaux de transition et terres rares

_liquides

sont raisonnablement _{décrits par} un modèle de

_plasma

(OCP). Ce modèle est utilisé aussi pour calculer le coefficient de température de la résistivité des métaux de transition

_liquides.

Les résultats sont en

bon accord avec

_{l’expérience.}

Abstract. 2014 It is shown that the observed structure factors of transition and

rare earth _liquid metals can be

reaso-nably

simulated _by a one _component

_plasma

_(OCP) model. The simulated OCP model has been used to calculate the _temperature coefficient of

_resistivity

in

_liquid

transition metals and _{good agreement} with

_experiment

has been obtained.

Classification Physics Abstracts 61.25M

During

recent _years, there has been considerable interest

_regarding

the structure of

_liquid

metals.

Large

numbers of

_experimental

measurements have been

_reported,

on the structure factors of

simple

and

_{non-simple liquid}

metals

_[1-6].

The well known

success of

_hard-sphere

model in _rare-gas

_liquids

encouraged

its use in

_liquid

metals and

_attempts

have been made to simulate the structure factors of

_liquid

metals in terms of

_hard-sphere

models.

_Except

for

some

_simple

and

_non-simple

metals,

at the end of the 3d transition metal

_series,

such

_attempts

have failed to

describe the

height

and

_positions

of the

_peaks

in the

structure

_factor,

_S(q).

The failure of

_hard-sphere

models can be understood

_by

the fact that the ionic

radii of metals are smaller than the observed mean

semi-interparticle spacing

in the

_{liquid phase}

_[1-8].

This indicates that the

_potential

seen

by

the ions is

different from the hard core

_{repulsive potential.}

The

difference is

_{particularly large}

for

V, Ti,

and rare earth metals which are

_precisely

the metals where the

hard-sphere

model is found to be most

_{unsatisfactory.}

The aim of this _paper is to consider the

_problem

of structure of

_liquid

metals from another

_point

of

view, namely by starting

from a soft interionic

potential.

All

_liquid

_metals,

to a first

_{approximation,}

can be considered as a set of ions immersed in a

responding

electron _{gas. The} effective interionic

potential

for such a _system shows Friedel oscillations : its actual calculation involves a

complicated

Fourier

transform of screened Coulomb

_potentials.

The work of Weeks et al.

_[9]

shows that the structure of a

liquid

is

_mainly

determined

_by

the

_repulsive

forces i.e.

a reference _system made _up of

_repulsive

forces alone leads to a structure

_fairly

close to the actual _system.

In view of this and the

_preceding

discussion,

we _expect a _{system, composed} of ions

_interacting

via Coulomb

repulsion

and immersed in a uniform

_{non-responding}

background

of

_opposite

_charge,

to

_fairly

simulate the structure factors of

_liquid

metals. Such a _system for

the case of

_point

ions is

_usually

referred to as one

component

plasma

(OCP)

and has been studied

numerically

by

Galam and Hansen

_[10].

For

_simple

metals,

Minoo et al.

_[11]

have shown that OCP model

fairly

well simulates their structure. In this

work,

we show that the OCP model can also describe

through

S(q)

the structure of

_non-simple

metals and its

_temperature

variation. As an

application,

we

have used simulated OCP models to calculate the

temperature coefficient of

_resistivity

in various

tran-sition metals.

As

_pointed

out

above,

the OCP model consists of a

classical fluid of

_{charged particles}

of

_charge

Ze

interacting through

a bare

_{Coulomb-pair-wise}

inter-action and

_placed

in a uniform

_background

of

_opposite

charge.

The interaction

_potential,

_u(r),

between two

ions

_{separated by}

a

_distance,

r, is :

where :

L-46 JOURNAL DE _{PHYSIQUE -} LETTRES

with :

Here k8

is the Boltzmann constant, n is the number

density

of ions and Tis the _temperature. The

_properties

of the model

_depend

on the _parameter f and Galam and Hansen

_[10]

have made Monte Carlo simulations

to calculate

_S(q)

and other

_quantities

of OCP model for various r values.

To simulate a

_liquid

metal

_by

OCP

model,

one

requires

a

_knowledge

of effective

_{charge, Zeff,}

on the

ions which

_{physically corresponds}

to the effective

charge

carried

_along

_by

ions. For

_non-simple

metals,

the calculation of

_Zeff

is difficult due to the _presence of d or f states. In view of

_this,

in this

_work,

we have

regarded

Zeff

as an

adjustable

_parameter

obtained

by

choosing

the F value so as to fit the first

_peak

of expe-rimental

_S(q).

In table

I,

we have

_given

the r values for various

_metals,

obtained as

_above,

_together

with

the

_{peak positions}

in

_S(q)

based on the

corresponding

simulated model and

_experiment

and

_{fairly good}

agreement

is

found.

The

_{corresponding}

_Zeff

have been found to have values between 1.2 and 1.5 for transition metals and between 1.0 and 1.6 for rare

earth metals. As

_already

stated,

their

_{physical meaning}

is not obvious. This

_point

shall be considered in the

future

_by

_{incorporating screening}

effects. In

_figure

1,

we have

_compared

the simulated

OCP,

S(q),

with

experimental S(q)

for Eu and Ce. For

Ce,

the simulated

Fig. 1. - Comparison of

experimental S(q) curves _{(solid curves)} for

Eu, and Ce with OCP model (0, A) based on _{corresponding} r

values (Table II). Dashed curve _{(- - - -) represents} the HS model S(q) for Ce.

HS

model,

S(q)

[4]

is also shown for

_comparison

with

corresponding

OCP,

S(q).

Let us now consider the

_temperature

variation of

S(q).

Measurements of Waseda and Tamaki

_[4]

on

_La,

Ce, Pr,

Eu and

_Yb,

show that as the

_temperature

is

increased the

_{peak heights}

in

_S(q)

decrease. In OCP

model,

an increase in _temperature decreases the

parameter

T

_{(eq. (2))}

due to its direct

_dependence

on

temperature

and due to the decrease in the number

density

of ions. The Monte Carlo structure factors

_[10]

show that a decrease in r results in a decrease of

_peak

Table I. -

Fig. 2. -

Temperature variation of the structure factor of La ;

-1 expt. S(q) at 1243 K; 0, corresponding OCP (F = 80);

...., expt. S(q) at 1343 K; A, corresponding OCP (r = 74).

heights

in

_qualitative

_agreement with

_experiment

(Fig.

2).

We shall now describe the calculation of the

temperature

coefficient of

_resistivity,

a,

given

by :

where _p is the

_resistivity

at

_temperature

T

_{and p}

is the

pressure.

Using

the relation between temperature

and volume

_dependence,

labelled

_respectively

_by

T

and V,

a can be

_expressed

as :

In this

_work,

we have

_separately

calculated the

terms of _eq.

₍₅₎

_using

a t-matrix formulation

[12],

by which,

where 0 is the atomic

volume,

VF the Fermi

velocity

and

_kF

the free electron Fermi wave number.

_S(q)

is the structure factor and

t(EF,

q)

is the t-matrix

describing

the

_probability

of an electron in a state

I

k F >

being

scattered to a

_state

_{kp )}

with

For one _energy shell

scattering,

it is

given

by

[12] :

where r~j

are the

_phase

shifts at _energy

_EF.

We have used the Fermi _{energy and the}

_phase

shifts at the Fermi _energy tabulated

_by

Hirata et al.

_[13]

for noble and transition metals. To calculate the

volume

coefficient -

(2013)

the resistivities of the

p

B~/F

noble and transition metals were calculated at r = 110

and r = 120

(table II)

using

OCP model

S(q)

values

[10].

The volume coefficient was obtained

by

calculat-ing

the

_change

in _temperature at constant volume

corresponding

to the above F

_change

and its values

are

_given

in table II.

For

2013

ap

an exact calculation of this

_quantity

’

p

r

q

Y

requires

a construction of the muffin-tin

potential

at two volumes and the calculation of

_respective

Fermi _energy and

_phase

shifts.

_However,

if the volume

change

is

small,

one can

_reasonably

well assume the

Fermi _energy and the

_{corresponding phase}

shifts

to remain

constant. ;

(2013~

T

is then evaluated

_by

p

av T

Y

changing

the volume

_{by 5 %}

and

_using

the structure

factor at the

_changed

r value

(table II).

Finally

a calculation of a

using

requires

a

_knowledge

of

_expansion

coefficient

Table II. - Various

L-48 JOURNAL DE _{PHYSIQUE -} LETTRES

For Ti and

V,

the

_expansion

coefficients were not

available and so it was not

_possible

to calculate a. For other

metals,

we have calculated the

_expansion

coefficients

_using

the tabulated densities

_[14].

The calculated (X values are

_given

in table II and

show

_good

_agreement

with

_experimental

a values when known

_[15].

To

conclude,

let us

_point

out that in view of the

present

work,

we

_expect

OCP to serve as a better reference _system than a

_hard-sphere

_model,

in the

calculations of

_{thermodynamic}

_properties.

Such a

conclusion is reinforced

_by

the work of Galam and

Hansen

_[10]

who have

_shown,

_using

a variational

method,

that the free

_energies

determined

_using

OCP reference _system are lower than those obtained

using

hard-sphere

reference _system and since both these

sets are _upper bounds to the exact free _{energy, the} OCP free

_energies

are closer to the actual value. The actual calculations are in _{progress and} shall be

reported

in a

_coming

_paper.

Acknowledgments.

- One of us

(S.

N.

Khanna)

is

grateful

to Professor J. P. Hansen for useful discus-sions.

References

[1] GREENFIELD, A. _{J., WELLENDORF,} J. and _{WISER, N.,} Phys.

Rev. A 4 (1971) 1607.

[2] HUIJBEN, M. J. and LUGT, V., VAN DER, W., J. _Phys. F 6

(1976) L 225.

[3] BREUIL, M. and TouRAND, G., Phys. Lett. 29A (1969) 506. [4] WASEDA, Y. and TAMAKI, S., Philos. _Mag. 32 ₍₁₉₇₅₎ 273 ;

Philos. _Mag.

_{36 (1977)}

1.

[5] WASEDA, Y. and MILLER, W. A., Philos. _Mag. 38B ₍₁₉₇₈₎ 21. [6] ENDERBY, J. E. and NGUYEN, V. T., J. Phys. C 8 (1975) L 112. [7] KITTEL, C., Solid State Physics (New York : John Wiley and

Sons) 1976.

[8] TAYLOR, K. N. R. and DARBY, M. I., Physics of Rare-Earth Solids (London : Chapman and Hall) 1972.

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[14] WEAST, R. C., Handbook of Chemistry and Physics (U.S.A. :

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