Model calculation of the static magnetic susceptibility in light rare earth metallic systems

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Model calculation of the static magnetic susceptibility in light rare earth metallic systems

Y. Hammoud, J. Parlebas

To cite this version:

Y. Hammoud, J. Parlebas. Model calculation of the static magnetic susceptibility in light rare earth metallic systems. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.765-777. �10.1051/jp1:1991168�.

�jpa-00246369�

Classification

Physics

Abstracts

71.28 75.30M

Model calculation of the static magnetic susceptibility in fight

rare

earth metallic systems

Y. Hanunoud

(')

and J. C. Parlebas

f)

(') Physics

Department,

Syrian

A-E-C-, Damascus-S-A-R-,

Syria

(2) ^IPCMS

(*),

Universitk Louis Pasteur, 4 _rue Blaise Pascal, 67070

Strasbourg,

France

(Received

7

May

J990, reviYed

23January

J99J,

accepted

30January

J99J)

Rksumk. Nous utilisons le moddle d'Anderson I _une

impuretd

dans

l'approximation

des

grands Nr

^off

Nr

est la

ddgdndrescence

d'orbitale et de

spin

du niveau f et _nous calculons alors la

susceptibilitd paramagndtique statique (I tempdrature nulle)

dans les

systdmes mdtalliques

de terres rares

ldgdres.

Nous effectuons notre calcul _pour des valeurs de l'interaction de Coulomb Urr

grandes

_{par rapport} I

l'hybridation

V des

configurations

f _et f

avec les dtats de conduction

(c.-i~d.

la

configuration I°~i

_{nous ne} retenons que [es tenures [es

plus importants

dans _un

ddveloppement

_en

puissances

successives de

lIUrr

et V. Ensuite _nous discutons _nos rdsultats

numdriques

I

partir

d'une forrne

analytique simple

obtenue _pour la

susceptibilitd magndtique

en

fonction de la

position

du niveau f, de

l'hybridation

V, de la forrne et du

remplissage

de la bande de conduction et enfin des effets de Urr fiat. Finalement _nous

prdsentons

des courbes de

susceptibilitk

_{que nous avons} calculkes _en fonction de V et _en liaison _avec la transition

a y du cbrium, en utilisant les mdmes

paramdtres

_que ceux

qui

interviennent dans _un

prkcbdent

calcul de spectres

Lnj d'absorption

de _cwur. Ainsi,

kgalement

dans le _cas de la

susceptibilitk, l'hybridation

V

apparait-elle

_{ccmme un}

paramdtre important

_pour dkcrire le

changement

de

phase

_ay du cfirium.

Abstlract.

Using

the

impurity

Anderson model in the

large

Nr

approximation,

where

Nr

is the orbital and

spin degeneracy

of the f level, _we calculate the _zero temperature ^static

paramagnetic susceptibility

of

light

rare earth metallic systems. The calculation is

perforrned

for

large

values of the Coulomb Urr electron-electron interactions with respect of the V

hybridization

off' _{and f~}

configurations

vith the conduction states

(I.e.

I°

configuration)

_{: we}

only keep

the

leading

terms in _a

development

in successive powers of

I/Urr

and V. Our numerical results _on the

magnetic susceptibility

start from _a

simple analytic expression

and _are discussed in terms of the f level

position,

the

hybridization

V, the

shape

and

filling

of the conduction band and also the finite Urr effects.

Finally

_{we present} calculated _curves for the

susceptibility

_versus Vin connection with the _{a y} transition of cerium and

utilizing

the _same parameters as those used

previously

to obtain

core level

l~ii absorption

spectra : also in the case of the

susceptibility,

the

hybridization

_appears

to be _an

important

parameter to describe the

phase change

from _y to _a cerium.

(*)

UMR 46 CNRS.

1. Inboduction.

The valence

fluctuating

rare-earth systems ^show a Fermi

liquid

behaviour at low

tempera-

tures their

paramagnetic susceptibility

is of Pauli

type

and

approaches

_a finite

(although quite large)

value at _zero

temperature.

The

physics

of the finite

susceptibility

at low

temperatures

^is the renormalization of the local _rare earth moments

by

the conduction electrons

leading

to a

~paramagnetic) singlet ground

state of the system. ^Therefore no

magnetic ordering

has been observed

alihoujh

one of the valences of the considered

systems

has _a

magnetically

active

ground

state

(for example Ce~+

_as

compared

to

Ce~+).

From _a theoretical

point

of

view,

it has been demonstrated that the low temperature

properties

of the

impurity

Anderson model _can be described in _an

empirical

local Fermi

liquid theory ([1~4]

and Refs.

therein)

where the

impurity

is described in terms of what it does _to the conduction electron states. In the

present

_{paper we}

adopt

_an alternative way of

describing

_a

single

mixed-valent

impurity,

I.e. _we focus _on the

ground

state of the

impurity

Anderson

Hamiltonian where various atomic

configurations hybridize

with the conduction states so that

one has _a renormalized

configuration picture.

This

point

of view based _{on a}

I/Nr expansion (where Nr represents

^{both orbital and}

spin degeneracy

of the 4f

level)

has been

adopted

to calculate various spectra

~XPS, XAS, BIS...)

of rare-earth

compounds ([5-8]

and Refs.

therein)

_: the lowest-order version of this method

(in

terms of successive _powers of

I/Nr) provides

_an exact solution to the so-called filled band

(insulating)

model

[8]

where the

density

of states is _zero above the Fermi level _; in the _case of metallic systems

(where

the conduction states extend well above the fermi

energy)

it

provides

_an

asymptotically

correct solution in the limit

Nr-

co either for

U~-

_co

[5]

_or for finite

Urr [6, 7].

The

present

short paper is based _{on an} extension of _a

theory

of _core level spectra

by

Hammoud et al.

[9, 10]

in order to obtain _an

analytic

formulation of the _zero

temperature

static

magnetic susceptibility

for finite

(but large)

_or infinite Coulomb

U~

electron~electron

interactions. The outline of the paper is _as follows _: first

(Sect. 2)

we

briefly

recall how _to solve the

impurity

Anderson Hamiltonian

using

_an extended _space of states within which the

ground

state wave function is calculated

[6~8].

Also _a small external static

magnetic

field is switched _on in order _to define

analytically

the static

susceptibility [5] (Sect. 3.I).

Our

analytic

results _are then discussed

numerically

in the _case of infinite

(Sect. 3.2)

and finite

(Sect. 3.3).

U~. Especially

it is

quite interesting

to calculate _a

thermodynamic quantity

such _as the _zero-

temperature susceptibility utilizing

the _same parameters as those used to describe

spectro- scopic properties [9, 10].

^{This is done for the} ay transition of cerium

(Sect. 3.3).

2. Formulation and resolution of the Anderson model.

We consider _a metallic

light

_rare earth system

consisting

of _a conduction band

(CB)

treated _as

non-degenerate

and

hybridized

with _an

N~fold

orbital and

spin degenerate

f level of _energy

El.

The

paramagnetic instability

of the f level is

directly

reflected in the f static

susceptibility.

In order to calculate that

quantity,

it is _necessary to switch _{on a}

(small)

extemal static

magnetic

field h which _removes the

degeneracy

of the f level and

gives

rise to the Zeemann terms

corresponding

to the ionic

multiplet

of the _rare earth element.

Here,

for

simplicity,

_we

disregard

the field

dependence

of the CB levels _so that there will be _no CB electron

polarization

and _we write the

h-dependent split

f energy sub~levels _{s~ as} follows

s~=s/-gjp~mh; _{M=-J, -J+I;}

_+J

_(2,1)

where gj is ^the Land6

gyromagnetic

ratio of the considered rare-earth ion. The

magnetic

quantum

number, M,

represents ^both orbital number _m and

spin

direction _« of _an f state with

Nr

=

2J+1. From the h

dependence

of

equation (2.I)

_{we can} then write _an

implicit

h~

dependent

Hamiltonian _{as :}

H =

£

_s~

a£~

_a~~ +

£

_s~

ajj

_{a~ +}

U~ jj ^ajj

a~

a[,

_{a~, +}

£ V~~~ a[

_{a~~ +} h.c.

,

(2.2)

km M M»M' Mk«

where

a£~

^is an

operator creating

a conduction electron with _wave number

k, spin

_« and

energy s~ whereas

ajj

is that

creating

_an f electron in the M _state with energy

s~. The third term of

equation (2.2)

describes the Coulomb

repulsion _Urr

between two f electrons and the fourth term exhibits the

hybridization V~~~

between f and conduction states. It is well known that _a variational calculation of the

ground

state leads to the _same results _as first order

Brillouin-Wigner perturbation theory

when double

occupation

of the f level

(I.e. 4F configuration)

is

neglected [5, 9, 10].

^That

corresponds

to the

asymptotic

limit of

infinite Urr. ^More

generally,

the

(h-dependent) singlet

mixed valent

ground

state

#) of,£f, given by equation (2.2),

in which

hybridization

admixes

magnetic configurations,

_can be written _{as a} linear combination of basic states

corresponding

to 1°,

~

_{and f~}

configurations

and

preserving

the

singlet

character of the

ground

state [6~8]

~'~ ~~ ~

~

~~~" ~~

_{~~" ~}

l, ^{~ij ~~} ~~'~''~'

"' ~2"2

~~ ~ji, _ak,

«iak~

«~l

^tbb

) (2.3)

where

4~)

is the

singlet ground

state of the

41° configuration

with energy

E~,

I-e- the filled Fermi _sea of the CB electrons _up to the Fermi _{energy s~.}

< «

'ffib)

"

fl ~~« '°)

_i

l~b

"

I

_Sk

(~.4)

km km

In

equations (2.3)

and

(2.4),

the

inequality

< means that

only occupied

states are concemed

both in the summations and in the

product o)

is the _vacuum state. The second and third terms of

equation (2.3)

_are obtained

by transferring

_one and two electrons

(respectively)

from the CB to the f level which

corresponds

to the

4f'

_and

4f~ configurations. Now, using

the

(h-

dependent) Schr6dinger equation

Hip)

=

E(h) (~) (2.5)

in the limit of

large degeneracy

and at the lowest orders in

V~~~

and

I/U~ (for example, (V~~~)~/U~

and

higher

_power terms _are

neglected),

the energy variation

3E(h)

m

E(h) E~

of the considered

system

due to

hybridization

and extemal

magnetic

field

obeys

_a

characteristic self-consistent

equation

shown in

Appendix

A.

Then, using

the normalisation condition that

(~ ~

=

l,

_we obtain the

following

relation between the various

h-dependent

coefficients of the

ground

state

(2.3)

_:

1co(h)

_i~

=

ii

₊

ci(h)

₊

c~(h)1-

^'

(2.6)

Ci(h)

=

it lCMk«l~ (2.7)

~

«

and

=

" "

~~~~~ ~j I I '~MM'kj

« k _~ (~

~' _"' ~''

' ~ ~

(2.8)

Explicit equations

for

C~~~

and

C~~,

~~~~~~~~ are

given

in

Appendix

A. Also the

h-dependent

f electron number in the

ground

state is defined _as usual

by

:

~f"

I (~ (~~ _~M( ~) (~.~)

M

and

keeping

the

leading

terms in successive powers of

V~~~

and

I/U~

_we obtain _a

simple analytic expression

of nr, I.e.

n~=

~'(h)+2C2(h)

I ₊

Cl (h)

₊

C2(h) ^(2.10)

Of _{course we recover} the limit nr -

(I Co ~)

^when

U~

_- co

(see [9]

for

instance,

and also

Figs. la,

b and

cl.

~°(£> ^{(d> (e>}

~

n

o-s

.1.0 0.0 1-O

E/D

~

(i>

~

/

f-level

~

-2.O -I.O O-O I-O a-O

t-lmel

Fig.

I.-Behaviour of _nr and e, in the infinite Urr limit, with respect ^{to the f} level

position sllDj,

the

origin

of the

energies being

the Fermi level s~:

(a), (b)

and

(c)

_{represent nr} for

(V/Dj)

=

0, 0.04 and 0.08;

(d)

model DOS

represented by

two

semi-elliptic

curves vith

ej/Dj

=

0.50;

s~D~

=

0.25 and

DjDj

=

0.15

(e)

coefficient e vithin the _same model DOS _as

(d)

and

V/Dj

=

0.12 _; (o coefficient e within _a rectangular DOS of energy width 2Di (which allows _an analytic

investigation).

3. Calculation of the static

magnetic susceptibility.

In this

section,

^first _we ^formulate a new

analytic expression

of the static

paramagnetic

susceptibility

for finite but

large

values of U~r

(Sect. 3.I),

then _we look at the infinite

U~

limit and

present

some numerical results within this limit

(Sect. 3.2). Finally

we discuss _our results for finite

Urr (Sect. 3.3)

with _a

special emphasis

_on the

a y transition of cedum and

by using exactly

the _same

parameters

as in the

Lm

XAS

spectra

calculations

[9, 10].

3.I ANALWIC EXPRESSION. In the

preceding

section _we calculated how the total _energy of

a

polarizable singlet ground

state,

E(h) =E~+ 3E(h), depends

_{on an} extemal static

magnetic

field h. The

paramagnetic susceptibility

_can then be deduced

by differentiating

twice

3E(h)

with

respect

to the field and then

by taking

the limit of the

h~dependent

result _as h _- 0

[5]

_:

X = lim

°~~~)~~ ^(3.I)

h-o ah

Within the

preceding development

in successive _powers of

V~~~

and

I/U~

_{we can} cast X in the

following simple analytic

form

(see

^also

Appendix B)1

~~ ²

~

§ ^~~J

^~

^Bl~ ~~~

_~

l

r

(

_n

rj ^~~'~~

with

ici(li@11(o)1216 ~~~~liil~~~~~ ^~~'~~

In

equations (3.2)

and

(3.3),

r has the dimension of _an energy and is defined _{as :} r

= p

(s~) Nr ^V~

_p

(s)

= ~

£ V~~~

^~ 3

(s s~) (3.4)

fi~f ^~

Mk«

In

equations (3.4), p(s~)

represents the

hybridized density

of states

(D.O.S.)

at the Fermi energy e~. Before

hybridization

the

unperturbed

C.B.D.O.S. is

just

noted

p°(s)

_per

_spin

direction _:

p

°(E)

=

z

8

(E Sk) (3.5)

where N is the number of considered sites in the lattice. Let _us also remark that what _appears in

equations (3.4) is,

_as

usual,

_a constant effective

hybridization [4, 8]

:

(V~~)~

=

Nr V~ (3.6)

where

V~~

^is

independent

of

Nr

_so that V is of the order of

I/ @,

_{and for}

_example

_the

leading

term in successive powers of

(V/(Urr/

^~~ with

j

_< ³ is of the order

I/(Ni U~).

We notice

finally

from

equation (3.2)

that _X

depends

_very

sensitively

_{on ni,}

especially

it increases

(towards

_{« a}

magnetic

state

»)

when ni

approaches unity

_as is the _case for

(almost~)

trivalent Ce systems like

yce. However,

in addition _{to nr,} there appears also _a characteristic

coefficient e in _our

analytic

result

(3.2)

and _we will _see next what is the influence of e and _more

generally

what is the behaviour _{of X} first in the infinite

U~

limit

(Sect. 3.2)

and

then in the finite _one

(Sect. 3,3).

Our calculations _are all

performed

with

Nr

=

14,

I.e. _we

neglect

the

spin~orbit splitting,

3.2 CASE _oF INFINITE

U~.

In order to

study

the dimensionless coefficient e of

equations (3.2)

and

(3.3)

in _more

details,

_we take first the infinite

Urr limit,

in which _case

C~

_- ⁰ as well _as its derivative _versus

3E(h),

and _we

adopt

the

following simple

models for

the DOS _p

(s)

of the C.B.

(I)

_a

rectangular

DOS of width

Dj

centred at sj ;

then, everything

is

analytic and,

after

some

calculations,

the coefficient e reads _:

2[8E(0) s/J

_{+ sj}

_Dj

e

=

(3.7)

El

Di

(it)

Two

semi-elliptic

DOS _as in

YPd~ compounds

for

example ([9]

and Ref.

therein)

for which

p(s)

is

expressed

_as

(see Fig. ld)

_:

p

(E)

=

£ p;(E) p,(E)

=

p;I(D,)~ _(E E;)~l'°/D;, (3.8)

where s;,

D,

and _{p, are} the centre of _mass, the half-width and the maximum of the

i~th DOS

(I

=

1, 2).

The behaviour of e is shown in

figure

I with

respect

^{to the}

position e)

of the

level,

the

origin

of the energy axis

being

_{s~, curves}

(e)

and (f~

correspond

to the considered DOS models

(I)

and

(iii respectively. First,

the

particular shape

of the DOS has little influence _on the overall

function e of

El only

in the

limiting

_case of e-1. This

happens

for

deep

values of

El

_as

compared

to the

C.B.,

I.e.

typical

Kondo systems for which

Elm

_{s~ or more}

precisely

_:

Nr V~/(s~ El)

«

Dj

_nr < I

(Ce) (3.9)

Then it is

possible

to _recover a well-known result

[5,

6]

(although

less

general

than

equation (3.2)

which is valid for the whole range of

El values)

x =

j _{(gJp~)~ J(J}

+

1)

~

~) ^{(3. lo)}

where

kB TK

₌ nr ^r

(3.

I

I)

nr

defines _a Kondo temperature ^which represents the _energy

gain

due _to the stabilization of the

singlet (see

for _ex.

Eq. (4.12)

of Ref.

[6]).

In

equation (3. II)

the Kondo _energy is linear in

(I nr)

when V is

kept

constant,

leading

to very low

T~

and very

high (although finite)

_X

when nr tends to I like in

(almost)

trivalent Ce systems. ^{Gunnarsson and} Sch6nhammer

already pointed

out

numerically

the rather broad range of

validity

of

equation (3.10) (see Eq. (3.19)

and

Fig,

^of Ref.

[7]). However,

let _us

point

out

that,

for finite but

large

values of

U~

_or ^{for infinite}

U~,

_our

analytical

result

(3.2)

^is _more valid than

equation (3.10) especially

for

El

> s~ where e increases

linearly

with

El

and cannot be

replaced by

I. In the

vicinity

of s~, e is _a non-linear function of

El

via

8E(0).

Since _we will be

finally

interested in the behaviour of _X with respect to the

hybridization

V,

we also

plot

the e function _versus V

(Fig. 2)

for two values of the

El

level below and above the Fermi level _s~. the e function increases

sensitively

with

increasing

V in the two cases because

(8E(0) s/(,

which _expresses the e

dependence

_{upon V,} also increases with V whatever

El.

^{For the} curve of

figure

2

corresponding

to

El

above _s~, e _{can never} reach

I,

_even for

i~

~

o

CID #

5 ^f i

o.3

o.3

o-i ^' Q.z Q-a

M;.bfldization

Fig.

2. Coefficient e

(within

the model DOS of

figure

ld and Urr -

oJ)

with respect to

hybridization V/Dj

for two values of

e)/Dj

= 0.3 and 0.3.

vanishing

V.

Consequently

the

simple

form of _x

given by equation (3.10)

should differ

appreciably

from the _more

general equation (3.2), containing

e, unless the

typical

_case of Kondo systems ^is considered

according

to

equation (3.9).

When the

position El

is lowered from about the Fermi energy ^to below the CB

(Fig. 3a)

the

susceptibility

increases _more and _more whereas the situation is reversed for

SIN

_s~

(Fig. 3b).

From the various

hybridizations corresponding

to

figure

3a it is clear that

increasing hybridization

stabilizes the _«

non-magnetic

_{» state}

(low susceptibility)

while smaller

hybridi-

zation favours the

tendency

towards

magnetism,

I.e.

large (although finite)

values of the

susceptibility.

A similar

tendency

has been

already

encountered for

example

ⁱⁿ SmS

insulating compounds [12].

Instead of

varying hybridization

V, like in

figure 3a,

it is

possible

to obtain the _same

qualitative

behaviour _{of x} for _a

given

value of V

by varying

the

filling

of the CB

(Fig. 4a).

In the insert of

figure

4 _we checked the influence of

Urr

on x as is

explained

below.

3.3 CASE _{oF FINITE}

U~. Analytic equations (3.2)

and

(3.3)

allow _us to test the influence of

Urr

on the

susceptibility,

at least at the lowest order of _our

development

_versus

I/Urr,

_X

= X

(Urr-

co

) (I

+ O

(I/Uj)),

^I,e, the test will be _more correct for

large

values of Urr. ^{From the results of}

figure

4b

( _Urr

- co

)

and

figure

4c

( Urr

₌ 2 D

j

),

it appears that the

asymptotic

behaviour of _X is almost reached for

El

values in the

vicinity

of the bottom of the CB.

According

to the numerical results of Gunnarsson and Sch6nhammer

(see Fig.

I of Ref.

[7]),

the _X

dependence

_upon

Urr

^{which has been}

magnified by

the _use of _a Urr

logarithmic

scale is strong

only

for

relatively

small values of

Urr (typically Urr

_< bandwidth

la

V/D~« V/D~.

0.08

0.08 o, i o

o,16

d ^~'~~

p I

I

d

(

_(b)

~1

~

~n on m

0,14 ^M"e'

o, 16 2

(a>

o

-i,o -o~ o-o o~ i-o

f-level

Fig.

3.

-Susceptibility x' (in (Dj)~~ units)

_versus f level

position s)/Dj (within

the two

semi-elliptic

model DOS and in the infinite Urr

limit)

for various values of

hybridization

vith the

following

definition x'

= x

(gJHB)~

^~ (J +

1) (a)

the range of x' is [0, 12]

(b) magnified

view of x' in the range

[0,

3 0.5].

in the units of Ref.

[7]),

the details of the

dependence changing

with other

parameters.

We also obtain _a

significant Urr

^{variation of} _X in the low

Urr

range

(see Fig.

5a for

V = 0.96 eV and

El

= 2.5

eV) although

_{we are} not allowed to

explore

this

Urr regime

too

deeply

because of _our

asymptotic development

_versus

1/Urr,

therefore

V~/Urr

should still be

negligible.

Moreover the

Urr dependence

of _x is also found to be _very sensitive _upon the other parameters,

especially

the V

hybridization

and the

El position (Fig. 5).

Small effects of finite

Urr

^values on

susceptibility

_versus

hybridization

_are also shown in

figure

6a where the parameters chosen

correspond exactly

to those

adopted

in references

[9, 10]

to describe the _ay transition of cerium _: in order to mimic the 5d-6s band of

Ce,

_{we use}

here

only

_one

semi~elliptic

DOS of half-width

Di=4eV

and

(1/3)rd-filled

with

El

= 1.2 eV with

respect

to the Fermi level at 0. Then

again

the

susceptibility

obtained for

Urr=8eV (which

is _a reasonable value for

cerium)

is

quite

close to the result for

U~-

_co. In

figure

6b the

corresponding

_core

absorption spectra (observed experimentally and)

calculated in reference

[9] (and

Ref.

therein)

exhibit the

phase

transition from

yce

with V

< 0.20 eV _to ace with V

> 0.20 eV. These orders of

magnitudes

for

hybridi-

zation

yields

_a reduction factor of _at least

II10

when

going

from _X

( YCe

to X

( ace) (Fig. 6a).

Since the

present

^{model is}

quite crude,

it is hard _to make _{a more} detailed contact with

experimental susceptibilities;

in

particular

_as our calculations

only

include the f-level

2

1-S

~l~~l"

~

)

,~

(c) (b)

lli _j

)

o.50

o~

# I

~Q

(

o-w

-I.o -o.5 0.o 0.5

(

f-level

c9

0l5

'

o.25

",,

^(a)

0,oo

o

-i.o -o~ o-o o~ i-o

t-let,el

Fig.

4.

Susceptibility x' (in (Di)~

^' units same definition of x' as in

Fig.

3) versus the Llevel

position sllDi (within

the two

semi-elliptic

model

DOS)

_:

(a)

for various

fillings

of the band model, I.e. for

various

positions

of

ei/Di

(with s2/D~

=

ej/Dj

0.25,

V/Dj

=

0.12 and Urr _- oJ)_; (b) for

ej/Di

~ 0.50 _;

e2/D2

=

0.25

V/Di

~

0.08 and Urr _- oJ

(c)

for

ej/Dj

=

0.50 _; e~/D~ ₌ 0.25 _;

V/Dj

=

0.08 and Urr/Dj ₌ ^2.

contribution to X, the CB contribution has to be subtracted from any

comparison

to

experimental

data.

Moreover,

whereas the _zero temperature ^static

susceptibility

of ace is well

defined,

the

corresponding susceptibility

for

yce

is not, since _{we are} then

dealing

with _a

high temperature phase. Nevertheless, already

at room temperature,

X(ace)

measured under pressure ~p a 7.6 kbar

[13])

is

approximately

reduced

by

_a factor of

I/4

with respect to

X(YCe)

at lower

(or normal)

_pressure.

Finally

let _us also remark the

analogy

between

increasing

_pressure

(experiment)

and

increasing hybridization (Fig. 6a)

in order to make the transition _y

- a in cerium without

invoking

_a

large magnitude

of valence _v

change (see

for

ex.[14]):

_e.g. for V=0.16eV _we have _v= 3.06

(y phase),

whereas for

V

= 0.28eV _we obtain _v

= 3.38

(a phase).

Moreover _our

analysis

is very similar _to the

« Kondo Volume

Collapse

_» model

[15]

where _a ratio

J~/J~

_m ² has been used to describe the y - a

phase

transition in cerium. This

corresponds

also to _a ratio

V~/V~

₌

/

_{I.e. if}

V~

= 0.16 eV then

V~

₌ ^0.23

eV,

for which value the _a

phase begins

to appear in

figure

6b.

4~ Conclusion~

In this _{paper our} calculation of the

paramagnetic susceptibility

_x in

light

_rare earth metallic

systems

^{has been based} on a

I/Nr expansion.

We obtained _a rather

general

and _new

analytic

I'

(a)

e°

= 2.5 eV

f 4QQ

j

bQQ

]

~ ₌

~

j

n

~

a _?©Q

o.96

_ev

Q~

I

,tt

_eV

~ 5 IQ 16 2Q 25 JO

V ₌ (b)

e°

= o eV

4

o.96

_eV

? 1.92 eV

5 IQ 16

2i

_{25 JQ}

Uyy (eV)

Fig.

5.-

Susceptibility

x'

(in (eV)~'

units _same definition of x' as in

Fig.

3) versus Coulomb interaction

Urr(eV

)

(within

one

semi-elliptic

band of half-width

Dj

=

12 eV)

(a)

for

El

= 2.5 eV with V

=

0.96 eV and 1.44 eV (b) for

El

= e~ =

0 eV with V 0.96 eV and 1.92 eV.

loD

<a> (b)

ao

~ff 3

~'~ ~

JO

< 3.06

~ &o k.o ev

3 16

~

[

z _~Q YCe

0.16

~ ^~

u o;

i~ 40 8 ev

o-lo

30 _o-m

ace

~~

0.25

-8

lo Wievi

YCe _ace

0

o.16 o.2k 0.32 (eV)

Hybddizaticn

Fig.

6.

-(a) Susceptibility

x'

(in (eV)-'

units _{; same} definition of x' as in

Fig.

3) with respect to

hybridization

for various values of Urr = 4, 8 and

oJ(eV). (b)

Core

absorption

_spectra of Ce

(see

Refs. [9, 10]). The

following

model DOS for the CB has been used in

Fig.

6a _{: one}

semi-elliptic

_curve of half-width

Di

₌ 4 eV,

(1/3)rd

filled and with

El

=

-1.2 eV. The valence varies from 3.06

(yce

with V

= 0.16

ev~

to 3.38 (ace ^with V

=

0.28

ev~,

all parameters except

hybridization being

fixed.

expression

of _X, valid for the whole range of

possible

values of the f level

position El,

^and for finite

(but large)

_or infinite values of the Coulomb

repulsion

Urr. ^{In the} case of

typical

Kondo systems, ^I-e- ni= I for Ce systems, we recovered the standard result of _X at zero

temperature.

^Within _our

development

at the lowest order in V and

I/Urr,

the finite

Urr

^{result of} X for realistic Urr ^{values has been tested}

numerically

and shown to be close to the

infinite

U~

^result. For the determination of _X in the _y

- a transition of

cerium, exactly

the

same

parameters

^{have been used} as those

entering

the calculations of the

Lm-XAS

spectra,

demonstrating

_{once more} the

important

role of the

hybridization

effect

(in analogy

_to

pressure

effect)

for

describing

the _y

- a

phase

transition of Ce. More

precisely, starting

from

yce

with _a small

hybridization

and

increasing

the

latter,

_you

become,

_as it should

finally

in

ace,

less and less

magnetic ~x

smaller and

smaller)

the

important point

here is that _our

simple model,

in overall agreement ^with

experiments,

induces _a

change

in valence from _y to _a cerium which is much smaller than

expected

in older models

(e.g. promotional model).

Let _us

finally

mention that the concentration

dependent susceptibility

_X in

Ce(Pdj_~Rh~)~

_com-

pounds,

measured in reference

[16],

could be

successfully

calculated within _a similar theoretical

analysis [17]

as in the

present

paper. In that

example, CePd~

resembles

yce

and

by increasing «alloying

_pressure», I.e.

decreasing

_average lattice constant and

consequently increasing hybridization,

the

magnitude

of _x towards the _case of

CeRh~

is reduced

(to

^be

compared

to

ace).

Acknowledglnents.

We would like to thank Prof. F. Gautier for

stimulating

discussions at a first

stage

^{of this work}

as well _as Prof. A.

Kotani,

Drs. A.

Meyer,

J.

Besnus,

E.

Beaurepaire

and J. P.

Kappler.

One of the authors

~Y.H.)

is also

grateful

for

obtaining

^{the «Salam} Prize for Scientific

Advancement in

Syria

_{» as} well _{as a}

fellowship

^{from UNESCO} which enabled him to

spend

one month at ICTP in Trieste

during

the summer1990.

Appendix

A.

The characteristic self consistent

equation

for

8E(h)

is written _:

~~~~~

~~(h)

₊

~~~~~

_l~Mk«Mk«

^~~~~

with

TMk«M,k,

«, =

Z Z Z ( ("a)

Ml M2

~«j ~"2

ffib

~~«

~M

~~iij

_~~i2

_~k,

«j ~k~_«~ ^ffi

b) _(ffi

_b ~~2_«2

~~

«j ~M2 ~Mj

~~ji'

_~k'

«' ffi

b) 3E(h)

₊

s~~ ⁺ s~~

s~~ s~~ U~

(A2b) Equation (Al)

is _an extension to finite

(but large) U~

of the result obtained in reference

[9]

with

Urr

_- co. One of the effects of

Urr

and T

given by equation (A2)

is to renormalize the f level

positions s~)

Similarly

the

k-dependent

coefficients of

equation (2.3)

_can be

expressed respectively

as

follows _:

~'~~"

3E(h)

₊

~~~~~ _T~~~ ^~~~~

I I

<

_~M,

km ffi_b ~k~_«

~~

«,

~~i'

~M

~~M,

~k«

~ b)

~""'~

_"'~~

"~

~ ~~

2(8E(h)

_{+ s~~ + s~~ s~ s~,}

Urr)

~~~~

Appendix

B~

Here _{we use} another well~known method to derive _our

analytic expression

of the static

magnetic susceptibility

for finite but

large U~.

^We start

similarly

from the

appendix

B of reference

[6].

To the usual

impurity

Anderson Hamiltonian

Ho (with

finite

Urr)

^{such that}

Ho 1~)

~

Eol~) (Bl)

we add :

+J

H'

=

hsz

_;

Sz

=

£ Majj

_a~.

(82)

M= -J

Equation (Bl)

describes the

coupling

to _an extemal

magnetic

field h _;

Sz

is the Z component of the

spin

and

(Eo+AE)

is the

eigenvalue corresponding

to the total Hamiltonian

(Ho

₊

H').

In the present ^method we obtain the static

susceptibility

_{X as} the _z

= 0 limit of the

dynamical susceptibility x(z)

defined

by

_:

x

(z)

m

Gs(z)

₊

Gs(- z) (83)

with

G~(z)

=

~ Sz(z

+

Ho Eo)~ Sz ~ (84)

and then

x = rim _x

(z) (85)

Using

the _same basis states _as in

equation(2.3),

I.e.

[4b), aiiak«(tbb)

and

ajjajj,a~~a~,~, [4~)

for winch the

corresponding eigenvalues

of

Sz

_are

0,

M and

(M+ M') respectively,

_{we are} able to cast

G~(z)

as follows _:

~ ^~

"

ld~(CMk«'~

~~~~~

_~

l~

~

Z + hE

El

_{+ Sk}

_TMk«Mk«(Z)

<

(M

₊

M')~

_CMM~

k«k~

«'(

^~

~

~~~j

MM'~k'«'Z

₊ ~~ ~ _~

~~

_{~ ~~} ~ ~~'

where

T~~,~~,

has

already

been defined in

equation (A2). Finally using equations (83)

to

(85)

_{we recover}

equation (3.2)

for _x. See also reference

[17]

for _a similar

analytic

derivation

of _x.

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31.

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