CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 331

ries, we confine ourselves to the (1 /Z ) 2 -corrections which can be de-
termined within this approximation. This provides a better estimate
of the effects due to the single-site fluctuations, but neglects the pos-
sible q -dependence of the self-energy. The correct (1 /Z ) 2 -terms in the
effective-medium theory are obtained by introducing

(6) in the third
term of the single-site series in eqn (7.2.5). This calculation has been
carried out by Jensen et al. (1987) for the ( J = 1)-singlet-doublet case,
and the most important effect of the second-order terms is to replace
the MF population-factors in (7 . 2 . 7 b ) by approximately the actual pop-
ulation of the excitonic states. Furthermore, Γ q ( ω ) becomes non-zero
outside the excitation band, and it stays non-zero (although small) in
the T = 0 limit.
The ( J = 1)-case has been analysed by Yang and Wang (1975), to
first order in 1 /Z , and Bak (1975) independently derived the linewidth
and applied the result to Pr. Psaltakis and Cottam (1982) have consid-
ered the ( J = 1)-model in the ordered phase, in the presence of uniaxial
anisotropy, where the 'kinematic' effects cannot be neglected. In the
paramagnetic singlet-doublet XY -model, the (1 /Z )-results are close to
those derived above for the Ising model. If the xx -and yy -couplings are
assumed to be equal, it is found, to a good approximation, that n 0 + n 1
in eqn (7 . 2 . 7 b ) is replaced by n 0 +2 n 1 = 1, and that the frequency sum
in this equation is multiplied by a factor 3 / 2. If

S

J zz ( q ) is non-zero,
it gives rise to additional contributions to the average q -independent
self-energy. Furthermore, it also leads to a q -dependent contribution,
even in the first order of 1 /Z . This occurs because the odd-rank cu-
mulants (corresponding to half-integral p in (7.2.4)) involving all three
components may be non-zero. The lowest-rank odd cumulant which is
non-zero is 0 . Although this formally leads to

a(1 /Z )-contribution to the q -dependent part of Σ( q ,ω ), which is not
immediately compatible with the effective-medium results above, this
should be a minor term in systems with long-range interactions and, if
∆ is positive, its importance is much reduced at low temperatures under
all circumstances.
The results of calculations of the lifetimes of the long-wavelength
magnetic optical-modes in Pr, based on eqn (7.2.7), are compared with
the experimental results of Houmann et al . (1979) in Fig. 7.4. This
theory predicts very nearly the same temperature dependence of the en-
ergies as does the self-consistent RPA; the excitation depicted in Fig.
7.4 is the uppermost mode in Fig. 7.3. The theory to first order in
1 /Z accounts very well for the temperature dependence of the energies,
lifetimes, and intensities of these excitations, without adjustable param-
eters. The low temperature results are similar to those of Bak (1975),
but the experiments at the highest temperatures in Fig. 7.4 are more

T τ J ix ( τ 1 ) J iy ( τ 2 ) J iz ( τ 3 )