SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 209

The renormalization of the anisotropy parameters appearing in the
spin-wave energies, in the second order of 1 /J , is expected to be some-
what more important in the conical phase of Er than in Tb. In Er,
the moments are not along a symmetry direction (they make an angle
of about 28 ◦

with the c -axis) and the second-order modifications due

H in (5.2.12) might be expected to be important. The 1 /J -results
do not allow a precise estimate of the second-order contributions, but
by introducing two scaling parameters, one multiplying the exchange
terms by σ , and the other scaling the constant crystal-field contribution
in the 1 /J -expression for the spin-wave energies in the cone phase, it
is possible (Jensen 1976c) to give an accurate account of the excitation
energies derived by diagonalizing the MF Hamiltonian exactly, the rela-
tive differences being only of the order 10 − 2 . The two scaling parameters
are found to have the expected magnitudes, although σ turns out to be
slightly smaller (

to

0 . 94 in the model considered) than the relative mag-
netization predicted by the MF Hamiltonian ( σ MF

0 . 98). An analysis
of the MF Hamiltonian shows that the excitations can be described in
terms of an elliptical precession of the single moments, as expected, but
surprisingly the ellipsoid lies in a plane with its normal making an angle
(

33 ◦ )withthe c -axis which differs from the equilibrium cone-angle

28 ◦ ), so the polarization of the spin waves is not purely transverse.
In terms of the 1 /J -expansion, this modification of the excited states
can only be produced by

(

H has sig-
nificant effects in Er, since it explains the difference between σ and σ MF ,
as σ becomes equal to σ MF if the angle appearing in the renormalized
spin-wave energies is considered to be that defining the excited states,
i.e. 33 ◦ , rather than the equilibrium value.
We may conclude that the 1 /J -expansion is a valid procedure for
describing the low-temperature magnetic properties of the heavy rare
earth metals. This is an important conclusion for several reasons. To
first order in 1 /J , the theory is simple and transparent. It is therefore
feasible to include various kinds of complication in the model calcula-
tions and to isolate their consequences. This simplicity is retained in
the second order of 1 /J ,aslongas

H . This observation indicates that

H can be neglected, in which case
the first-order parameters are just renormalized. Accurate calculations
of the amount of renormalization of the different terms may be quite
involved, but because of the long range of the two-ion interactions in
the rare earth metals, the MF values of m o and b o utilized above nor-
mally provide good estimates. The spin-wave theory in the harmonic
approximation, to first order in 1 /J , has been employed quite exten-
sively in the literature, both for analysing experimental results and in
various theoretical developments. It is therefore fortunate that these
analyses are not invalidated, but only modified, or renormalized, by the