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Submitted on 1 Jan 1988
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NUCLEAR RELAXATION AND TRANSPORT
PROPERTY NEAR THE NUCLEAR ORDERING
TRANSITION IN INTERMETALLIC COMPOUNDS
H. Ishii
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988
NUCLEAR RELAXATION AND TRANSPORT PROPERTY NEAR THE NUCLEAR
ORDERING TRANSITION IN INTERMETALEIC COMPOUNDS
H. Ishii
Department of Physics, Osaka City University, Sumiyoshiku, Osaka 558, Japan
Abstract. - Nuclear relaxation rate Tcl and electrical resistivity are studied theoretically in substances with a singlet ground state. It is shown that Ti1 deviates from the Korringa law with decreasing temperature and vanishes at Tc.
The resistivity due to nuclear spins is discussed just like that due to localized electronic spins.
Nuclear ordering occurs in some intermetallic compounds of Pr [I]. In these substances the 3 ~ f- 4 electronic state of P r is split by the crystalline field and a singlet ground state is realized. When the indirect coupling between f-electrons on differrent sites is not strong enough to yield the ordering of electronic spins, the ordering of nuclear spins takes place by this in- direct interaction via hyperfine coupling enhanced by the van Vleck susceptibility 12-41.
Since it is conduction electrons that mediate the in- direct coupling, it seems important to study a role played by them. White and Flude [5] have pointed out that virtual excitations of the f-electrons from the singlet ground state contribute to the effective mass of conduction electrons. Recently we have derived [6] the effective Hamiltonian of the system by eliminating the f-electrons since they always remain in the singlet ground state at low temperature. On the basis of this work, we study in this paper the nuclear relaxation rate and the electrical resistivity near the nuclear ordering temperature Tc in the mean field (MF) approximation. Both of these quantities reflect the critical behaviour near
Tc
through the conduction electrons.We summarize briefly the result of [6]. After elimi- nating the f-electron freedom, the system is described by the following Hamiltonian,
grals. The van Vleck paramagnetism is represented by A, =
l(nl
J, lg)12/(En
-
E,) where J, isn
the angular momentum for the f-electrons and
U
= (J:/
2) A,. For other notations see [B]. The sys-Y
tem described by equation (1) is that in which nuclear spins are immersed in the conduction electron sea with repulsive interaction
U
which corresponds to the self- energy correction studied in [5]. Electronic properties of this system are described in terms of the dynamic susceptibility X e (q, w ),
which is given in RPA byx,
(q, W ) = xo (9, W )/
[ I - (PFUI N ) Xo (9, w )/
XPI
where X p is the Pauli susceptibility. (2) Further elimination of conduction electrons from equation (1) yields an indirect coupling between nu- clear spins,
where
JL
=.(Ac+
2 J d f ~ f h , ) ~ N - ~ XThe expressions for the enhancement factor 1
+
K and nuclear anisotropy energy are given in [6]. Nuclear spins order in the direction for which D, is maxi- mum with wave vector q for which Jy (q) the Fourier transform of J$ is maximum. It is not easy to calcu- late Jr (q) for real substances so that in this paper, we consider the ferromagnetic case without D y and measure temperature or energy in the unit of ~ B T , = J ( 0 ) I (I+
1)/
6 of the MF relation.Nuclear relaxation rate T;~ is calculated for T
>
T, where ck. is the annihilation operator for the con-by applying the formula by Iaggdt and Vuorio [71 to duction electron (k, a), and its Wannier represen-
equation (I) in the presence of weak field Hor tation is denoted as cj,. nj, = cJ,cja and sj =
(1/2) C;,U,,~C~,~ with Pauli matrices u. I,, de-
TI-'
= (2h2/3/
~ c N N ~ ) (Ac+
xaa 03 Y=x, Y
notes the f t h nuclear spin (7 = x, y, z ) , Af and w
dw
Ac are the hyperfine constants for f- and conduc- 4 sinh2 (Pfw/2) Im X N ~ Y ( Q W ) Im Xe ( Q W )
tion electrons, Jcf
/
( g ~-
1) the s-f exchange inte- (5)C8 - 2056 JOURNAL DE PHYSIQUE
where C N is the nuclear specific heat and X N is the dynamic susceptibility in the unit of ( g ~ p ~ ) 2 for the nuclear spin systems. We have put
p
= 1/
LBT. First we represent Im X N (qw) as a product of X N ( q ) and the Fourier transform of the time correlation function. The former is given by the MF theory as X N (q) =I ( I + 1 ) / 3 ( T - T , ) with T , = T c J ( q ) / J ( 0 ) . The latter is assumed merely to be
G
(w-
wo) withfiwo = ~ N P N ( 1
+
K ) HO representing Zeeman pre- cession without decay. This may be allowed if one compares it with the broad spectrum of Im X , (qw)in equation ( 5 ) . From equation ( 2 ) we approximate ImXe (qw) by ImXo (qw)
/
[ I - ( P U / N ) X o ( q )/
x p I 2 .After these are inserted into equation ( 5 ) and using the expression of xo (qw) for the free electron band, we ob- tain 1 / T 1 = ( 1
/
T I ) , ( 2 N k ~ r ( I+
1 )/
3CN) (2kFfiwO/
~ B T ) ~ X [I - ( P F U I N ) XO ( q ) / x P ] - ~ ( 6 ) with ( 1/
T i ) f = X ~ B T/
f i ( P F/
N)' (Ac+
Z J ~ ~ A ~ A , ) ~,
( 7 )where 7 = 2 h ~ h w o
/
E F and ( 1/
T I ) , is the relaxation rate for free nuclear spins obtained by Tsarevskii [8].The factor [ I - ( ~ F U
/
N )xo
( q )/
~ p ] - ~ in the inte-grand of equation ( 6 ) is an enhancement factor due to repulsion U studied by Moriya [9]. The characteristic of TT' near Tc is determined by C N and the q-averaged
X N ( 4 )
In order to see the behaviour we evaluate equa- tion ( 6 ) numerically for the model Jij = J
>
0 for nearest neighbour pairs and vanishing otherwise, as it is not easy to do for the RKKY interaction. First we average Tq or J ( q ) over the direction of q to obtainJ ( q ) = J ( 0 ) sin (qa)
/
qa with the lattice constant a.Then the integrand in equation ( 6 ) is calculated nu- merically. Together with C N obtained by solving the MF equation in the presence of Ho, we obtain the re- sult of TT', which is shown in figure 1 for the cases
~ F U / N = 0.5 with k F a = 1 and 5. Although it is clear from equation ( 6 ) that TT' increases linearly with T at high temperature, the result of figure 1
shows that it deviates from this law with decreasing
T and vanishes near Tc. This is due to the increase
g
m
e
+-
I TIT.
of C N . In fact the divergence of X N at Tc is weakened as log ( 1
-
Tc/
T+
77) with 77 (hwo/
E F ) ~ by the q- integration. This feature holds in the MF approxima- tion as far as J ( q ) decreases quadratically from q = 0.It seems interesting to proceed beyond the MF theory as the TT' near Tc strongly depends on the critical
behaviour of X N and C N .
Next we study the electrical resistivity R. It re-
sults from the s-f exchange interaction and the hyper- fine interaction besides the scattering by phonons. In the system such as PrNis both contributions from f- electrons and nuclear spins are completely separated in the temperature regions. At high T, R due to the
inelastic scattering between crystalline levels decreases gradually from the constant value. When T is low- ered (T 5 1 K) , only the singlet ground state is oc- cupied and the scattering by this channel is hard to occur. Then the scattering by nuclear spins turns dom- inant. Although its contribution t o R may be about
( A ~ A , ) ~ times
(N
as small as that due to the f-electrons as is inferred from equation ( I ) , this part of R is expected to show some sign characteristic tothe nuclear ordering. Correspondence of the nuclear spins to the localized electronic moments for the re- sistivity may allow us to deduce the answer for the present case. Following de Gennes and Friedel [ l o ] , we may expect a cusp-like peak arising from the short- ranged order effect near T,, which is manifested by the factor (114 k ~[ 2 k F ) dqq3 ~
/
( 1-
Tq/
T ) just likeJ!'
the integral in equatlon ( 6 ) . As to the repulsion U
in equation ( I ) , it is expected that its effect appears to increase R linearly with m*
/
m, the mass enhance ment.
I wish to acknowledge helpful discussions with A. Sakurai and M. Kubota.
[ l ] For a review see Andres, K. and Lounasmaa,
0. V., Progress in Low Temperature Physics Vol. VIII, Ed. D. F. Brewer (North Holland Pub. Co) 1982, Chapter 4. .
[2] Grover, B., Phys. Rev. 140 (1965) A1944. [3] Landesmann, A., J. Phys. Fmnce 32 (1971) 671. [4] Murao, T., J. Phys. Soc. Jpn 39 (1975) 50. [5] White, R. M. and Flude, P., Phys. Rev. Lett. 39
(1981) 1540.
[6] ~shii, H . , Jpn J. Appl. Phys. Suppl. 26-3 (1987) 429.
[7] Leggett, A. J . and Vuorio, M. J., J. Low Temp.
Phys. 3 (1970) 359.
[8] Tsarevskii, S. L., Fiz. Tverd. TeIa 12 (1970) 2047; translation Sou. Phvs. Solid State 12 (1970) 1625.
Fig. 1.
-
Nuclear relaxation rate T;' for p~ U/
N = 0.5and I = 512 with (1) kF a = 1 and (2) kF a = 5. For the [9] Moriya, T., J. Phys. Soc. Jpn 18 (1963) 516. sake of comparison (3) the asymptotic line at high T limit [lo] De Gennes, P. G. and Friedel, J., J. phys. ~ h e r n .
