CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 342

and the results, corresponding to Pr in the limit T =0,aregivenby
Jensen (1976a).
In the present approximation, the sound velocities are not affected
by the interaction between the dipoles, in the paramagnetic phase at
zero magnetic field. However, in the vicinity of a second-order transi-
tion to a ferromagnetic phase, strong softening of the long-wavelength
phonons may be observed, depending on the symmetry properties, and
this behaviour cannot be explained within the RPA. We have seen that,
according to eqns (5.4.15) and (5.4.38), c 66 vanishes in the basal-plane
ferromagnet when a field equal to the critical field H c is applied along
the hard basal-plane direction. When T C is approached from below, H c
vanishes rapidly, resulting in a strong softening of c 66 even in zero field,
and it seems likely that similar behaviour should be observed when T C
is approached from above, considering that just above T C there will be
large domains of nearly constant magnetization, allowing an 'RPA' cou-
pling between the dipole moments and the sound waves similar to that
occurring in the ferromagnetic phase. Clear indications of this kind of
behaviour have been seen in for example Tb (Jensen 1971b), indicating
that the RPA is not even qualitatively trustworthy when the fluctuations
are a dominating feature of the system.

7.3.2 Conduction-electron interactions
The sf -exchange Hamiltonian (5.7.6) was derived without making any
special assumptions about the rare earth metal involved, and it there-
fore applies equally well to a metallic crystal-field system. For the
weakly-anisotropic ferromagnet considered in Section 5.7, this Hamil-
tonian leads to a Heisenberg two-ion coupling,

J

( q ,ω ), which to a
first approximation is instantaneous, and is thus effectively J ( q )=

(1 /N ) q J

J

( q , 0), as given by eqn (5.7.28). This remains
true in crystal-field systems, as may be demonstrated by expanding the
angular-momentum operators in (5.7.6) in terms of the standard-basis
operators, and then calculating the corresponding Green functions which
determine χ ( q ,ω ), utilizing an RPA decoupling of the coupled Green
functions.
In the ordered phase,

( q , 0) −

J zz ( q ,ω ) may actually differ from the two
other components of the exchange coupling, due to the polarization of
the conduction electrons. However, in the paramagnetic phase in zero
field, the coupling is isotropic, within the approximation made in Sec-
tion 5.7. This may be seen by analysing the full expression (5.7.27) for
J

( q ,ω ), or the simpler result (5.7.26), in which the susceptibility of the
conduction electrons becomes a scalar:

c . el . ( q ,ω )= 2

χ αβ χ + −

c . el . ( q ,ω ) δ αβ .

(7 . 3 . 13)