The coupling of phonons to magnetic excitations in a system with two singlet levels

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The coupling of phonons to magnetic excitations in a system with two singlet levels

R. Lemmer

To cite this version:

R. Lemmer. The coupling of phonons to magnetic excitations in a system with two singlet levels.

Journal de Physique Colloques, 1979, 40 (C4), pp.C4-220-C4-221. �10.1051/jphyscol:1979468�. �jpa-

00218864�

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JOURNAL DE PHYSIQUE Colloque C4, supplément au n" 4, Tome 40, avril 1979, page C4-220

The coupling of phonons to magnetic excitations in a system with two singlet levels

R. H. Lemmer

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

Résumé. — On montre que les excitations magnétiques d'un système ayant deux niveaux singulets donnent naissance à des changements de vitesse des phonons longitudinaux à la fois dans les phases ordonnées et paramagnétiques du système. Dans ce dernier régime, l'effet s'étend sur une gamme de température de l'ordre de la séparation des niveaux. Le changement de vitesse des phonons transverses est limité uniquement à la région ordonnée, où il est strictement proportionnel à m quand on l'exprime par un changement de la constante élastique C<* (m est la constante magnétique du réseau ou du sous-réseau).

Abstract. — It is shown that the magnetic excitations of a system having two singlet levels give rise to velocity shifts of longitudinal phonons in both the ordered and paramagnetic phase of the system. The effect in the latter regime extends over a temperature range of order of the separation between the levels. The velocity shift of transverse phonons is confined to the ordered region only, where it is strictly proportional to m2 when expressed as a change in the elastic constant C« (m is the lattice or sublattice magnetization).

The coupling of phonons to magnetic excitations in a crystal limits both the lifetime of these excita- tions and renormalizes the phonon propagator. In this communication we report an investigation of such renormalization effects on phonons in a lattice of ions whose ground states have been strongly split by the crystal field. A case in point is uranium mononitride (UN) where it is known that strong crystal field effects determine the magnetic state of the uranium ion [1]. In this instance, when the crystal field splitting is strong, one finds that the two lowest states | A, > and | T

lz

) of the J — A multiplet in the cubic field give a good description of the magne- tic properties of the ion [2]. Both | A,) and | T

l z

> are singlet states. Our model therefore considers pho- nons in a lattice of ions having access to two singlet states only. The excitation spectrum of such ions in a frozen lattice consists of magnetic excitons [3]

rather than true spin waves, that are spread out into a band e, around the gap A of an individual ion ; / is the wave number of the excitation.

Lattice vibrations in the system cause a strain modulation of the exchange field between ions and this induces a phonon-magnetic exciton coupling.

Calling the coupling matrix kW, one calculates the self-energy for longitudinal phonons (k, &>) at tempe- rature / 3

¹

as

X (k, <o) = k

2

A'

2

(2M]

2

tanh^pA) x

x 2 | w ; r « '

^{{ N ( 5}

.

^g

^ "

^{N ( a}

' W

^}

(i)

"7 '

^L

' w + to + se, - s' e,

⁺ⁱ

where N(e,) = [exp(/3e,) - l]"

1

. We have used the diagram technique of Vaks [4], as extended by Jones

and Cottam [5], to obtain this result. M

l 2

= (/

2

)

12

is the transition matrix element of J

2

between the crystal field states that are separated by

A' = A V l + A

²

m

²

in the mean exchange field of strength A, and m = (J

z

) is the average magnetization [2].

Notice that 2 does not vanish in the paramagnetic regime (m = 0 ) . This contrasts strongly with the behaviour of phonons interacting with normal spin waves where the self-energy goes like m

2

and thus vanishes at the transition temperature [5, 6, 7]. The

102 t ^ \ ^.A = 177K

— °\

° " l 0 0 - * ^ ° ° o o

1 1 I

0 1-0 2-0 T / TN

Fig. 1. — A comparison of the acoustic data [8] (open circles) and neutron scattering measurements [9] (closed circles) of the square of the sublattice magnetization of UN, illustrating how well the relation (2) is obeyed in this case. The solid curve is a calculation of m2 using the two-level model of reference [2] with a gap A = 177 K. The measured and calculated values of m 2 have been arbitrarily adjusted near T = 0 to agree with the measured excess in C*, of 2.3 % above its unrenormalized value [8] of 0.77 x 10l! erg/cc just above the transition temperature.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979468

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THE COUPLING O F PHONONS TO MAGNETIC EXCITATIONS C4-221

poles in the exciton propagators on the right hand side of (I) can give rise to temperature anomalies in the phonon velocity that are not necessarily asso- ciated with the phase transition [7]. Apart from such detailed effects one anticipates from (1) that the velocity renormalization should extend over a tem- perature range of order A /

k , on the paramagnetic

side, behaving very roughly like PA sech2! PA. This is reminiscent of the Schottky factor in the specific heat of a two-level system [3], but this suggestion needs to be substantiated by a careful evaluation of (1). However the long paramagnetic tail seen in the C , , data of reference [S] does suggest an effect of this sort.

Equation (1) refers to longitudinal phonons. For

those transverse phonons in cubic crystals that pro- be the C , elastic constant, selection rules [?] show that W; vanishes. The renormalization of such pho- nons, as expressed in terms of a shift in C,, is given instead by

both in the Heisenberg model [6, 71 and in the present two-level model, except that m

= (

J,

)

is the thermal average of the induced moment in the latter case. The relation (2) is obeyed empirically in UN [S]. Figure 1 compares the acoustic [S] and magnetic [9] data with the two-singlet model calcula- tion of m [7].

References

[I] GRUNZWEIG-GENOSSAR, J., KUZNEITZ, M., FRIEDMAN, F., Phys. Rev. 173 (1968) 562.

[2] LEMMER, R. H., LOWTHER, J. E., J. Phys. C : Solid State Phys. (1978).

[3] COOPER, B. R., in Magnetic Properties of Rare Earth Metals, ed. R. J. Elliott (Plenum Press, London and New York)

1972, p. 17.

[4] VAKS, V. G., LARKIN, A. I., PIKIN, S. A., SOU. Phys. JETP 26 (1968) 188.

JONES, M. J . , COTTAM, M. G., Phys. Status Solidi (b) 66 (1974) 651.

PYTTE, E., Ann. Phys. (N. Y . ) 32 (1965) 377.

LEMMER, R. H., VIUOEN, J. de P., J. Magn. Magn. Mater. 5 (1977) 161.

[8] DUPLESSIS, P. de V., VAN DOORN, F., J. Magn. Magn. Mater.

5 (1977) 164.

[9] CURRY, N. A., Proc. Phys. Soc. 86 (1965) 1193.