MAGNETIC FLUCTUATIONS, EXCITATIONS, AND INDUCED MOMENTS IN A SYSTEM HAVING TWO SINGLET LEVELS

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MAGNETIC FLUCTUATIONS, EXCITATIONS, AND

INDUCED MOMENTS IN A SYSTEM HAVING TWO

SINGLET LEVELS

R. Lemmer, J. Lowther

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, suppldment au no

8,

Tome 39,

aotit 1978, page C6-803

MAGNETIC FLUCTUATIONS, EXCITATIONS, AND INDUCED MOMENTS

IN A SYSTEM

HAVING TWO SINGLET LEVELS

R.H. Lemer and J.E. Lowther

Department of Phgsics, University of the Witwatersrand, Johannesburg, South Africa

RQsum6.- L'influence des excitations et des fluctuations sur les moments induits dans un systsme 1 deux niveaux singulets est Btudide en utilisant la technique du diagramme de Vaks, Larkin et Pikin.

On applique ces resultats au calcul de l'aimantation du sous-rLseau du compost5 UN. On montre que la prise en compte des excitations magnetiques rend la courbe d'aimantation moins rigide que celle ob- tenue dans l'approximation du champ moyen, ce qui conduit 1 un resultat meilleur en accord avec l'ex- pgrience.

Abstract.- The influence of magnetic excitations and fluctuations on the induced moments in a system having two singlet levels is studied using the diagram technique of Vaks, Larkin and Pikin. Applica- tion of these results are made to a calculation of the sublattice magnetization of uranium mononitride. It is shown that the inclusion of magnetic excitations renders the magnetization curve less rigid than that obtained from mean field theory, leading to a better agreement with experiment.

I In a recent-comunication/l/ the competition where H =-

-

1 2 Maf3My6f ag(k) 'y~

(-k)

(2) between single ion crystal field splitting and ex-

is the additional interaction energy. The notation change interactions between ions in the ordered sta-

is as follows : f (r) =la,r><~,rl are basis ope- te of uranium monopnictides was analysed in a mean aB

-

field approximat.on (mA) The electronic configura- rators 141 at site

5

and f _aB(k) their Fourier trans-

_-

tion of the uranium ion in these materials has been form. V is the transform of the exchange interac- _k postulated to be either 5f2 /1,2/ or 5f3 131. The tion ~(z-r-') and Vk its maximum value. The basis ground state configuration 3 ~ , of 5f2 gives a satis- states (l> and (2y0 separated by A'

=

A

J

are

~

factory description of the magnetic properties of connected by matrix elements M l l = asin 28-<J >, uranium mononitride (UN) within a two-level appro- M12 =

-

acos 28 and M22 =

-

asin 28- <JZ>. ximation to the crystal field splitting of J = 4 in

a cubic field. This description is possible because the

I A

_I> ground state is only directly coupled to

the

IT^^>

member of the triplet

IT^>

by the total

spin operator J.' Both IA1> and / T ~ ~ > have a singlet character. The mean exchange field mixes these levels to induce equal and opposite moments (JZ)l = asin 28 = -(Jz)22 in the admixed states ) I > and )2> of the ion. Here a= < A ]J

IT

> =

1 z 'l

and.tan 28 = 2a(V /A)<JZ> = Am in'terms of the

!50

unperturbed crystal field splitting A, the mean field interaction Vk and the induced moment <J >=m.

-0

The preceeding discussion ignores all effects due to fluctuations in the induced moments or magnetic excitations of the crystal. In the following

The contribution of H to the free energy is

1

calculated by using the diagram technique of Vaks, Larkin and Pikin 151 to construct the correlation functions at temperature

B-'.

B

(k,iun)= dTei%T <f(fug(ky~)fi6(-g(ky0)> (3) Ka~;r6

-

.

B

The VLP diagram technique classifies diagrams according to inverse powers of the interaction radius relative to a given spin site. The low- est order contribution to (3) is obtained by sum- ming the set of all chain diagrams 151. Using the rules given in /4/ and 151, one finds the correlation functions

,S - we sunrmarize the calculation of such effects in a

associated with magnetic excitations of energy two-level system. The free energy of N ions subject

_{- 2 ~ f ~ ( ~ ~ + ~ / ~ ' ) D ~ d ~ ~ a n d}

to both crystal field and exchange interactions is

given by

C MololMBBKuC1 ;

BB

iun) = 6 (5)

sinh BA' a,B n,o

-@F =

-

L

NBV < J ~ > ~

+

Nin

{-l+

Ln{~r erp

.m2

D'

2 k, I-' '&+&a aa a

( - 6 ~ ~

)

1-

(1) associated with fluctuations in the induced moments.

(3)

Here D12 = (D -D ) is the thermal population diffe- 1 2

rence of

I

I> and 12>, while D; = (D,- .):D Special caseof (4) have been derived and discussed pre- viously by Fulde /6/ and especially by Cooper 171. The average interaction energy follows immediately from (4), (5) and (2). A coupling constant integra- tion then yields the change in free energy

sinh

@BE

C M ~ D*} B(F-F~) =

P

fin (sinh ₊

₇

_fin(l-Wk+k

,

k - - D

-

( 6 ) -BF0 being given by the first two terms of (1)

Equation (6) shows that the contributions to the free energy from magnetic excitations and fluctuations are additive, i.e. they are non-interacting in the present approximation. We now determine the magnetization m = <J > self-consistently by minimi- zing (6) with respect to <Jz>(and

&).

The resul- ting equation for m or A' =

-

A

is given appro- ximately by

1 1 1

A V

1

2a2(v /A1)(tanh $Af+

CEO~II$A'

-

--coth$~J )=l

L

_-

k €k

_-

-

(7) if we neglect the fluctuation contribution. Drop- ping the magnetic excitations entirely leads back to the MFA results /6,7/. Their retention leads to a numerical problem where the specific crystal struc- tureenters through the Vk. Figure I shows the magnetization for UN (a type

i

antiferromagnet

181)

with and without the inclusion of magnetic excitations.

Fig. I : The sublattice magnetizationa= m(T)/m(O) of UN as a function of temperature. Solid curve :

mean field approximation using parametersX= 0.42 and A 177 K that reproduce the experimental va- lues / 2 / of the low temperature moment, (0.75~~) and N&el temperature TN = 53 K. The broken curve shows the effect of including magnetic excitations for

X

unchanged but A increased to 237 K (to main- tain the same TN). Open circles denote the measure- ments of Curry

181.

ratures due to our method of approximation the L-sum. The main effect of these excitations is to render the magnetization curve less rigid because the crystal field gap

A'

is spread out into the band of ener- gies ck. One expects a similar renormalization from the fl~ctuation contribution.

Previous calculations /6,7/ for the two-level system using decoupling procedures give a disconti- nuity in m at higher temperatures indicative of a first order phase transition. It would be of consi- derable interest to see whether this particular fea- ture is retained in a more complete treatment of the problem, with fluctuations included in the manner described by (6).

References

/ l / Lemmer,R.H. and Lowther,J.E., J.Phys.C : Solid State Physics (to appear March, 1978)

/ 2 / Grunzweig-Genossar,J., Kuzneitz,M. and Friedman, F., Phys.Rev.

173

(1968) 562

/3/ Long,C. and Wang,Y., Phys.Rev.3 (1971)1656

/ 4 / Yang,D.H. and Wang,Y., Phys.Rev.E(l974) 471'4

Phys .Rev.a(1975) 1057

/5/ Vaks,V.G., Larkin,A.I. and Pikin,S.A., Sov.Phys. JETP E(1968) 188

/6/ Fulde,P. and Peschel,I., Ad. in Phys. Z(1972)l /7/ Cooper,B.R., in Magnetic Properties of Rare

Earth Metals, ed. R.J. Elliott (Plenum Press, London and New York, 1972) p. 17

/8/ Curry,N.A., Proc.Phys.Soc.~(1965)1193