CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 347

temperatures but, in the high-temperature limit, is the dominant

term. Becker et al . (1977) have deduced the influence of the electron-
hole-pair scattering on the crystal-field excitations, with an accuracy
which corresponds to the results obtained here to first order in 1 /Z ,using
an operator-projection technique. They performed their calculations for
an arbitrary value of J , but without including the intrinsic damping
effects which, as pointed out above, may be more important, except in
the high-temperature limit.
The effects of the sf- exchange Hamiltonian on the effective mass and
the heat capacity of the conduction electrons in a crystal-field system
may be derived in an equivalent way to that used for the spin-wave
system. The mass-enhancement, m ∗ /m =1+ λ CF , is deduced to be
given by (White and Fulde 1981; Fulde and Jensen 1983):

ζ ( ω )

2 k F

dq d Ω q
4 π

2

1
2 k F

λ CF = N ( ε F ) q | j ( q ) | χ αα ( q ,ω → 0)

0

α

N
q

( ε F )
α

1 ζ ( q )

= χ αα ( q ,ω → 0) ,

2 π N

(7 . 3 . 21 a )
and is a generalization of eqn (5.7.50), valid in the paramagnetic phase.
The term χ αα ( q ,ω 0) is the zero-frequency susceptibility, omitting

possible elastic contributions, assuming the broadening effects to be
small. At non-zero temperatures, it is found that excitations with ener-
gies small compared to k B T do not contribute to the mass-enhancement,
and therefore, even in the low-temperature limit considered here, the
purely elastic terms in χ αα ( q ,ω ) do not influence the effective mass.
This is also one of the arguments which justifies the neglect to leading
order of the effect on m ∗ of the longitudinal fluctuations in a ferro-
magnet, which appear in χ zz ( q ,ω ). In contrast, the elastic part of the
susceptibility should be included in eqn (5.7.57), when the magnetic ef-
fects on the resistivity are derived in the general case, as in Section 5.7.
In systems like Pr, with long-range interactions, the dispersive effects
due to the q -dependence of χ ( q ,ω ) are essentially averaged out, when
summed over q . In this case, we may, to a good approximation, re-
place χ ( q ,ω )insumsover q by its MF value χ o ( ω ). The correction to
the MF value of the low-temperature heat capacity in Pr, for example,
is minute (Jensen 1982b). In the eqns (7.3.18-20) above, this means
that, to a good approximation,

→

N q ζ ( q )= ζ 0

1

ζ ( ω ) even at low

temperatures, and that the mass-enhancement parameter is

( ε F )

ζ 0

χ αα ( ω

λ CF → 0) . (7 . 3 . 21 b )

2 π N

α