Change of the transition probabilities for f-electron J-multiplets due to the dynamical exchange interaction

Solid State Communications, Vol. 16, pp. 839—842, 1975.

Perganion Press.

Printed in Great Britain

CHANGE OF THE TRANSITION PROBABIUTIES FOR f-ELECTRON J-MULTIPLETS DUE TO THE DYNAMICAL EXCHANGE INTERACTION A. Furrer Delegation für Ausbildung und Hochschulforschung am Eidgenossischen Institut für Reaktorforschung, CH-5303 Wurenlingen, Switzerland (Received 23 August 1974; in revised form 2 December 1974 by E.F. Bertaut)

The shortcomings of the elementaxy crystal field theory have recently been demonstrated by neutron inelastic scattering experiments in the neodymium monochaicogenides.1 A major improvement of the crystal field theory is obtained by including the dynamical exchange interaction which causes considerable changes of the transition probabilities for f-electron.1.multiplets.

I. INTRODUCTION

2. CRYSTAL FIELD THEORY

THE POWER of the neutron inelastic scattering technique in the exploration of crystal field effects in metallic rare earth systems is now well established. Whereas measurements of the bulk properties cannot reflect details of the crystal field because of its integral nature, neutron spectroscopy yields detailed information on the crystal field energy levels through the peak positions in the observed energy spectra, on the crystal field wavefunctions through the peak intensities, thestrucpeak widths.and Upon torelaxation the presentphenomena the crystal through field level

In elementary crystal field theory the electrostatic potential at the rare earth ion position arises from the electric charge distribution on the surrounding ions. If overlap with the 4f electrons is neglected, the electrostatic potential is a solution of Laplace’s equation = 0. (1) For f electrons, cubic symmetry, and the polar axis along the cube edge, the solution of equation (1) can be written as2 V = A 4)x 6)~ 4(r 4(O~+ 5O~)+A6(r 6(O°6 2lO~), (2)

tures have been determined in a large number of rare

—

earth systems, and qualitative information on relaxation effects is available. However, no special effort has beenfield made in orderpeak to examine the observed crystal transition intensities.

1) is the the n-thenergy where A~is a coefficient which determines moment of the distribution of the(r’ magnetic electrons, scale of the crystal field splitting,

Recent neutron inelastic scattering experiments in the neodymium monochalcogenides1 have demonstrated the shortcomings of the elementary crystal field theory. In particular, the analysis of the measured energy spectra revealed discrepancies between the ob-

Xn is a reduced matrix element, and the O~’are the

Stevens operator equivalents.8 Following Lea et al.4 (hereafter referred to as LLW) it is customary to rewrite equation (2) as

v=

WI!._(00 4 + 5O~)+ I

served intensities of the crystal field transition peaks and those calculated on the basis of the elementary crystal field theory. It is the purpose of this note to show that inclusion of the dynamical exchange interaction will give a major improvement of the elementary crystal field theory.

with

IF4

—

lxi (O°6 21O~)) (3) —

F6 4F4= W~,

4)~ A4(r

(4)

6)~ A6(r 6F6= W(1 lxi), (5) where —1 ~ x ~ 1 and the F,,, are numerical factors —

839

840

TRANSITION PROBABILITIES FOR f-ELECTRON J-MULTIPLETS

tabulated by LLW. By diagonalizing the crystal field Hamiltonian in the form of equation (3) the eigenvalues and eigenfunctions for each value of x are obtained. This has been done by LLW for the whole lanthanide group. For neodymium the tenfold degenerate ground state multiplet ~I9,2 is split into a doublet 1~’~ and ~ The eigenfunctions rof6 the andlevels two quartets are lineari combinations of the five Kramers degenerate wave-functions, mm>, m = 9/2, 1/2. . . . ,

3. NEUTRON INELASTIC SCATTERING

Vol. 16, No. 7

•‘

•

8

‘

•

I

•

~‘-“

_____

——.

I ~ .5

[

~

rr2’

0

5

r”~r

10 15 energy trsnsfer (meV)

20

In the analysis of measured energy spectra it iS very often found that a unique identification of the observed crystal field transition peaks is not possible on the basis of an energy level diagram alone. In addition one has to consider the relative intensities of the various transitions. For a system ofN noninteracting ions the thermal neutron cross-section for crystal field transitions in the small Q approximation is given by5

FIG. 1. Energy spectrum of neutrons scattered from polycrystalline NdS at room temperature. The broken line is the result of a least squares fitting procedure as described in reference 1.

d2a d~TZdw

quasielastic peak corresponding to transitions within the doublet and the two quartets. Its width is generally be distinguished from the elastic line. The inelastic larger than the resolution width, so that it can easily peaks are due to transitions between the different

n, m

=

exp

N1l.9 2 2 Z t~2mc2gj) F2(Q) j~exp kB T

{ 2W~

1< m I iii n >j2 6(E~ Em —

(6) —

(A))~

Z is the partition function, F(Q) is the form factor, k 0 and k1 are the wavenumbers of the incoming and outgoing neutrons, exp {— 2W } is the Debye—Waller factor, E,, and In> are the eigenvalues and eigenfunc. tions of the crystal field Hamiltonian given by equation (3), and Jj. is the component of the total angular momentum operator perpendicular to the scattering vector Q. The remaining symbols have their usual meaning. (mi by J InFurrer >12 for 3~in a The cubictransition field haveprobabilities been calculated Nd 6 Because of line broadening due to relaxation et al. lattice distortions, impurities, nonstoichiometry, effects, and instrumental resolution effects the 6-function ~ the cross-section formula (6) is generally replaced by a line-width ‘parameter Ynm~ Experimental results for NdS’ at room tempera. ture are shown in Fig. 1. NdS shows f.c.c. type ~ antiferromagnetic ordering7 below TN = 8 K. The scattering vector Q was held constant at 2.06 A-’, and the experiment has been carried out in the energy gain configuration. The energy spectrum can be qualitatively characterized as follows. There is an intense peak at

zero energy transfer corresponding to nuclear incoherent scattering. Its peak width is identical to the instrumental resolution for zero energy transfer. There is a second peak at zero energy transfer, namely the

crystal field levels. Since all transitions are allowed, we expect three crystal field transition peaks which are indicated by arrows in Fig. 1. A measurement of the “non-magnetic” sample LnS showed that quasielastic nuclear scattering as well as phonon contributions in the considered energy transfer range are negligible for the chosen scattering vector Q. The analysis of the energy spectrum of Fig. 1 gave the following crystal field level sequence: r~2>
=

—0.35 ±0.01 meV, 0.69 ±0.01.

There is a large discrepancy between the observed and calculated crystal field transition peak intensities listed in Table 1. Since the intensities are essentially given by the matrix elements of J between the different crystal field states, we conclude that the theoretical wave functions obtained from the crystal field Hamiltonian (3) are not adequate to describe the rare earth system under consideration.

Vol. 16, No.7

TRANSITION PROBABILITIES FORf-ELECTRON J-MULTIPLETS

Table 1. Observed and calculated intensities of the crystal field transition peaks in NdS. The intensities are normalized to the value 10 for the r~’~—ii2~ transition Transition

~~~~—10

—9 —E —_ 7 — 6

________ —

>

Intensities Observed

841

~ ~-1

Calculated with 1C~~ +

Calculated with ~ cf

dyn 3-8

r

6 ~ r~’~5.1 ±1.6 r6 ~ 10± 3 ~ 10± 2 quasielastic

180

2.2 19 10 100

±40

3.9 13 10 110

— / 2-2 1-1 ///29

~

—

—

C

)1 I

— —

—

—

—6-6 4-4 8-8

\~/1-10 \4-7

4. DYNAMICAL EXCHANGE INTERACTION

-~

The importance of including the dynamical exchange interaction term in the Hamiltonian of a rare earth8system been demonstrated by Furrertheory and Based has on results of general fluctuation Heer. they derived the following probability distribution of the dynamic exchange field HdYfl in the molecular field approximation: H~ 7,, (7) W(HdYfl)~—exp~~2kTxj

I

1

X is the molecular field parameter. The thermal average of an observable P is then given by (F>

=

j

W(Hdyfl) (F(HdY~)>dHdYfl/J W(HdYfl) dHdyfl

(8)

with Tr

(F(HdYfl)>

p(HdYfl) is the density matrix: p(Hdy~) =

where

I

exp

(9)

EPp(HdYfl)].

rI

~C(Hdyn kBT

~C(Hdy~) = lC~f gMBHd~fl

,

~j,

(10) (11)

is the single ion Hamiltonian a rare earth system in the paramagnetic state. lCcf isofthe crystal field potential given by equation (3). Symmetry considerations and equation (7) limit the integration procedure in equation (8) to the following argument range: 0 ~ HdYfl < ‘~/2ln 2k~rx. In order to demonstrate qualitatively the effect of the dynamical exchange interaction, Fig. 2 shows the eigenvalues and the matrix elements (mIJ~ln)for

~so

_____

I

~

100

150200 Hefl (kOe I

______

~

4-6 \9-9 7-7 10-10

250

FIG. 2. Energy levels E~and matrix elements ‘~m L1~in> forWa=Nd3~ ionmeV,x in a cubic crystal field (LLW parameters —0.16 = 0.76) as a function of an effective field Heyy. The numbers on the right hand side denote the energy level labels n. Nd~in a cubic field as a function of an effective field Heø. (The matrix elements (m iJ~in> and (m I.!, In> exhibit a similar behaviour). Some of the 10 energy levels and some of the 18 non-zero matrix elements are strongly dependent on the effective field Hett. In a dynamical sense, i.e. by taking into account the dynamical exchange field H dyn’ the rare earth system changes its energy level scheme as well as its matrix elements instantaneously. This means that the dynamical exchange field H ,,, can modify all the magnetic properties which are dependent either on the energy levels E~or on the matrix elements (mI4In>(cs=x,y,z). transition 2~ probFigure 3 shows thelevels r~”—r’~ ability and the energy of Nd3~in a cubic crystal field as a function of the LLW parameter x. Similar curves have been obtained for the other five crystal field transitions. Since the dynamical exchange interaction tends to admix different crystal field states, its influence is biggest in regions where the crystal field levels cross or interact with each other.

842

TRANSITION PROBABILITIES FORf-EL~CTRONJ-MULTIPLETS ~/ 10

~

In Table I the intensities of the crystal field

______

~

______

______

_______

______

______

~

~

.4~/

transition peaks for NdS are calculated including the

7-.,g

0

dynamical exchange interaction. For the paramagnetic molecular field parameter determining the dynamical field Hdyfl we took the value X = —43 x 1023 0e2/erg, which has been derived from susceptibility data.9 The agreement between the observed and calculated intensities turns out to be satisfactory (except for the quasielastic transition) which demonstrates the importance of the dynamical exchange

‘Ts/

/

7~—..!i~ /

______

-10

~~----~

/

_______

-10 _______

_______

r

~ —

f/~ / \j

/ //

~

1O~

,/J

______

_______

r(’Lr(2) ~\

/ 1U

/

_______

interaction.

\~, I

,7\

~ —-‘

-~

Vol. 16, No.7

0

0.5

—H

FIG. 3. r~’~—r~2~ transition probability and eigenvalues of Nd3~in a cubic crystal field for the LLW parameter W = 0.25 meV. The full line represents the crystal field only calculation. The values obtained with inclusion of the dynamical exchange interaction (room temperature, paramagnetic molecular field parameter X = 0.5 x 10~Oe2/meV)are given by the broken line.

5. CONCLUSION The dynamical exchange interaction can be an important term in the Hamiltonian of a rare earth system. In particular, it is a necessary improvement in the calculation of transition probabilities for f-electron J-multiplets. Consequently inclusion of the dynamical exchange interaction in the analysis of measured crystal field spectra turns out to be a sensitive method for the evaluation of exchange parameters. This is especially valuable in rare earth systems which do not order magnetically.

REFERENCES 1.

FURRER A. and WARMING E., J. Ploys. C: Solid StatePloys. 7, 3365 (1974).

2. 3.

HUTCHINGS M.T., Solid State Physics, Vol. 16, p. 227, Academic Press, New York (1964). STEVENS K.W.H.,1~vc.Phys. Soc. A65, 209 (1952).

4. 5.

LEA K.R., LEASK M.J.M. and WOLF W.P.,J. Ploys. Chem. Solids 23, 1381 (1962). DE GENNES P.G.,Magnetism, Vol. 3, p. 115. Academic Press, New York and London (1963).

6. 7. 8.

FURRER A., KJEMS J. and VOGT O.,J. Phys. C: Solid State Ploys. 5,2246 (1972). SCHOB1NGER-PAPAMAJ”JTELLOS P., NIGGLI A., FISCHER P., KALDLS E. and HILDEBRANDT V., J. Ploys. C: Solid State Phys. 7, 2023 (1974). FURRER A. and HEER H., Ploys. Rev. Lett. 31, 1350 (1973).

9.

ADAMYAN V.E. and LOGINOV G.M., Soy. Phys. JETP 24,696 (1967).