Environmental Engineering Reference
In-Depth Information
lift the degeneracy of the modes at A on the Brillouin-zone boundary
of Fig 1.4. When q is parallel to the c -axis, a direct calculation of the
spin-wave energies (Jensen et al. 1975) shows that the two-ion terms in
H JJ lead to the following modifications of the earlier results (5.2.38) and
(5.3.22):
(i) The two-ion anisotropy may contribute to the parameters
A q ( T )
B q ( T ) at zero wave-vector.
(ii) B q ( T ) becomes dependent on q to leading order in 1 /J .
(iii) The q -dependent parts of A q ( T )
±
B q ( T ) may change when
the direction of magnetization is changed.
There are no direct ways of separating the single- and two-ion con-
tributions to the energy gap at zero wave-vector. However, a strong
q -dependence of B q ( T ) is only possible if the two-ion Hamiltonian is
anisotropic. One way to determine B q ( T ) is to utilize the dependence of
the neutron cross-section on this parameter, given by eqn (5.2.41). This
method requires accurate intensity measurements and is not straightfor-
ward. The other possibility is to measure the field dependence of the
spin-wave energies since, from (5.2.38) or (5.3.22),
±
α q ( T ) ≡ ∂E q ( T ) /∂ ( B H ) 2 A q ( T ) ,
(5 . 5 . 17)
when the field is parallel to the magnetization. This relation is only true
to first order in 1 /J , and corrections have to be made for the influence
of any field-dependent changes of the correlation functions σ and η ± .
Both A q ( T )and B q ( T ) may be determined from the energies and initial
slopes, since
2
± 2
4 E q ( T )] 2 .
[ α q ( T )
A q ( T )
±
B q ( T )
α q ( T )
(5 . 5 . 18)
This method was used by Jensen et al. (1975) for a comprehensive study
ot the two-ion anisotropy in Tb. The values of A q ( T )and B q ( T ), de-
duced from eqn (5.5.18), were parametrized in various ways, and clearly
the best least-squares fit was obtained with expressions of the form
( A q + B q )
( A 0 + B 0 )=
I
( q )+
K
( q )
−C
( q )cos6 φ
(5 . 5 . 19)
( A q
B q )
( A 0
B 0 )=
I
( q )
−K
( q )
−D
( q )cos6 φ,
where A 0 ±
B 0 were taken from the simultaneous measurements of
the magnetic anisotropy at q = 0 , discussed in the previous section.
The low-temperature isotropic coupling
I
( q ),whichintheabsenceof
anisotropy would just be J [
J
( 0 )
−J
( q )], and the φ -independent two-
ion anisotropy
K
( q )areshowninFig.5.10. The φ -dependent axial
Search WWH ::




Custom Search