SINGLET-GROUND-STATE MAGNETS COUPLED WITH NUCLEAR SPINS. SYSTEMS WITH SINGLET-DOUBLET AND SINGLET-TRIPLET IONS

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SINGLET-GROUND-STATE MAGNETS COUPLED

WITH NUCLEAR SPINS. SYSTEMS WITH

SINGLET-DOUBLET AND SINGLET-TRIPLET IONS

N. Suzuki, Y. Fukuyama

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, SupplGment au no 12, Tome 49, dkcembre 1988

SINGLET-GROUND-STATE MAGNETS COUPLED WITH NUCLEAR SPINS.

SYSTEMS WITH SINGLET-DOUBLET AND SINGLET-TRIPLET IONS

- N. Suzuki and Y. fikuyama

Department of Material Physics, Faculty of Engineering Science, Osaka University, 1-1 Machikaneyama, Toy- onaka 560, Japan

Abstract. - Magnetic properties are studied of a system having two types of electronic singlet-ground-state ions, d-ion (singlet-doublet) and t-ion (singlet-triplet), by taking account of the hyperfine coupling. It is shown that the exchange interaction between the d-ion and the t-ion electronic spins affects crucially the magnetic properties. The results are discussed in connection with the experimental results of Pr metal.

Double-hexagonal-close-packed (DHCP) Pr metal has attracted much interest as an electronic singlet- ground-state (SGS) system. It contains two kinds of SGS ions, hexagonal-site and cubic-site ions. The first excited state is doublet a t the hexagonal site whereas it is triplet at the cubic site. From now on the hexagonal- site and the cubic-site ions will be called simply d-ion and t-ion, respectively.

From analysis of the magnetic excitations in Pr at T 5 K Houmann et al. [l] evaluated the energy separation between the ground and the first excited states: A d ~ 3 . 5 meV for the d-ion and At& meV for the t-ion. It is believed [l-31 that the magnitude of the exchange couplings between electronic moments is insufficient to induce spontaneous ordering and that the magnetic ordering at low temperatures in Pr [2] can occur with a help of the nuclear spins. Further- more, since At is much larger than Ad, it has been considered that the t-ions would hardly contribute to the magnetic ordering in Pr and hence little attention has been paid to the interaction between the d- and t- ions (inter-sublattice interaction). Fkcently, however, Kawarazaki et al. [4] observed evidence for nuclear- spin order at both the d-ion and the t-ion sites by neutron diffraction measurements, which indicates im- portance of the inter-sub-lattice coupling. The aim of the present report is to elucidate effects of the coupling between the two types of SGS ions on the magnetic properties.

We assume the following two-sublattice model Hamiltonian:

ii' jj' i j

where i (it) and j ( j ' ) run over the d-ion and the t-

ion sites, respectively, and I and 3 represent respec- tively the nuclear spin and the electronic total angular momentum which is called electron spin from now on (J = 4 in Pr). The first and the second terms denote the crystal-field Hamiltonian, the third and the fourth terms represent the isotropic hyperfine coupling and

the remaining terms are the exchange interaction between the electron spins. In real Pr ion I = 512, but for simplicity we assume I = 112.

The form of the Hamiltonian (1) is essentially equiv- alent t o that of a system with residual orbital angular momentum [5, 61. In the present case the hyperfine coupling A1.J takes the place of the spin orbit coupling XL

.

S. Therefore the treatment developed in ref- erences [5, 61 is applied to this system. The exchange interaction between the electron spins is treated by the molecular field approximation, and then the effective single ion Hamiltonian is diagonalized exactly as

a coupled system of the electron and the nuclear spins. Further, since we are concerned with magnetic properties at low temperatures, we take into account only the ground and the first excited states of

vd

and

vt.

The critical temperature T, of a transition from the paramagnetic state to an ordered state modulated by a wave vector q is determined from

which is obtained from a condition of divergence of the transverse susceptibility. Here (L = d, t) denotes the single-ion transverse susceptibility in the paramagnetic phase and K;' represents the Fourier transform of the exchange coupling. It is noted that q5e (L = d, t ) diverges in proportion to A'/ICBT as the temperature goes to 0 K. It is also noted that, if A is much smaller than Ad ( A t ) ,

4d

(4t) takes an almost constant value

c u 2 / a d ( - y 2 / ~ t ) in the temperature range A

<

ICBT

<

0.lAd (A

<

ICBT

<

O.lAt) where a and y represent the matrix elements of J+ linking the ground and the

first excited states (in P r a =

6

and y =

m).

If A = 0, (4t) takes the finite value a2/&

( r 2 / ~ , )

even at 0 K.

The transition temperature depends sensitively on

2 dd

the parameters, q d ~ l - 2a Kq /Ad,

v t ~

2

l

-

2 - y 2 ~ ; / ~ t and'X S mqt- (2ayK$)

/

(AdAt).

If qd<O ( ~ t < 0)

,

the d-ions (t-ions) can order without a help of the hyperfine coupling as well as the inter- sublattice coupling. In case of r]d> 0, qt> 0 and X

<

0

the inter-sublattice coupling enables the electron spins t o order by themselves without a help of the nuclear spins. Finally, if qd> 0, qt> 0 and X

>

0, spontaneous magnetic ordering is impossible without the hyperfine

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C8 - 1554 JOURNAL DE PHYSIQUE

interactions. Pr metal is considered to correspond to the last case.

Next we consider the ordered state. For simplicity we assume the ferromagnetic order. The molecular field acting on a d-ion (t-ion) is given by

where the moment direction is assumed to be along the *direction. Now ( J : ) and ( J : ) are determined simul- taneously from the usual selfconsistency condition and then ( I : ) is determined easily by using the resultant

effective single-ion Hamiltonian.

Since (J:) and ( J : ) show a rich variety of temperature variations depending on values of the parameters

m ,

qt, X and A, we foucus our attention to a case appropriate to Pr metal. Taking account of the parameters estimated for P r [l] (Adw3.5 mev,

2 dd

At-8 meV, A = 50 mK, 2 a

KO

/Ad=0.9 -1.0 and K:-Ktd) we carried out actual calculations by fix- ing the ratios of At, A,

~g~

and

KF

to Ad as fol-

2 dd

lows: At/Ad=2.5, A/Ad=0.005, 2a KO /Ad=0.975 and 2y2 ~ ? / ~ , = 0 . 6 5 .

In figure l a we show the temperature dependence of

( J : ) and (I:) (l = d, t ) calculated for 2ayK?/ad= 0.2. In this case the value of X is 0.001 and the transition temperature is TC=0.079bd/ks. It should be noted that the inter-sublattice coupling K? gives rise to finite values, though small, of ( J : ) and ( I : ) up to

Tc=0.08Ad/k~. If we neglect K?, the t-ion sublattice orders only below I O - ' A ~ / ~ B .

In figure l b we show the temperature dependences of Rj ( J : )

/

(J:) and RI E ( I : )

/

( I : ) . An in-

Fig. 1. - The temperature dependences of (a) ( J : ) ,

(J:) , (1:) and ( I : ) , and (b) RJ =

(4)

l

(J:) and RI = ( I : )

l

(I:). The used parameters are At/Ad=2.5, A/ad=0.005, 2 c ~ ~ ~ , d ~ / ~ , = 0 . 9 7 5 , 2 y 2 ~ h t / ~ d = 0 . 6 5 and

2c~y~$/A,=0.2.

teresting aspect is that in the temperature region 2A (= 0.01Ad) < ~ B T

<

IcBT, (= O.cl79Ad) RI equals

to RJ and their values are almost constant. These

facts can be understood as follows. In this temperature region the molecular field acting on a t-ion, H:=

2 ~( J : ) +2K? ( J : ) 2

,

is very small and therefore the moment of the t-ion can be approximately expressed as

(J:)

= 4tH;. Since

4t

takes a constant value y 2 / a t in this temperature region as noted previously, we get

Thus, RJ is temperature independent and its value

is determined by K? if the value of Kkt and At are fixed. As for the nuclear spins they are regarded as b e ing subjected t o an effective field A ( J : ) or A ( J : ) in this temperature range. Therefore (I:) atld ( I : } are

approximately determined from (I:} = QIA ( J : ) and

( I : ) = ~ I A (J:

)

where

41

= 2 1 ( I $. 1) / ~ B T denotes the nuclear Curier susceptibility. Hence it is clear that

RI = R j . Below a temperature T

-

2 A / k ~ the ratios

RJ and RI starts to deviate from the constant value

defined by equation (3), and Rj becomes smaller than

RI.

Real Pr metal is a much more complicated system than that described by the model Handtonian (1). For example, (a) the unit cell contains 4 a,toms (two d-ions and two t-ions), (b) the magnetic structure is not a ferromagnetic but a sinusoidal-incommeosurate structure specified by a wave vector close t o the point [2] and (c)'the exchange interaction is anisotropic [l]. How- ever, we believe that the results obtained in this paper could be applied qualitatively to real Pr. According t o the results of neutron diffraction measurements between 20 mK and 65 mK by Kawarazaki et al. [4], the observed and R;~" show temperature variations similar to those of RI and Rj below O.OlAd/ka in fig-

ure lb, i.e. RybS = = 0.04 at 65 mK and increases more steeply than below 65 mK. From this we can estimate the value of 2cryK$/h, to be

--

0.1.

Acknowledgments

This work is supported by the Kurata Foundation.

[l] Houmann, J. G., Rainford, B. D., Jensen, J. and Mackintosh, A. R., Phys.

Rew.

B 20 (1979) 1105. [2] McEwen, K. A. and Stirling, W. G., J. Phys.

C 14 (1981) 157.

[3] Murao, T., J. Phys. Soc. J p n 33 (1972) 33.

I41 Kawarazaki, S., Kunitomi, N., Arthur, J. R., Moon, R., Stirling, W. G. and IMcEwen, K. A.,

Phys.

Rev.

B 37 (1988) 5336.

[5] Suzuki, N., J. Phys. Soc. J p n 45 (1978) 1791. [6] Takeuchi, H., Suzuki, N. and hlotizuki, K., J.