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and external stresses by
N
∂
∂
γ
1
=
c
γ
γ
1
−
1
H
γ
O
2
−
B
γ
2
(
t
11
−
t
22
)=0
,
with
t
=(
V/N
)
T
,where
T
is the usual stress-tensor. Introducing the
equilibrium condition into the Hamiltonian, we get
B
γ
2
O
2
(
J
i
)
γ
1
+
O
−
2
(
J
i
)
γ
2
+
γ
,
H
γ
(sta) =
−
H
(7
.
3
.
2
a
)
2
i
where
γ
=
N
2
2
t
12
γ
2
.
c
γ
(
γ
1
+
γ
2
)
H
−
(
t
11
−
t
22
)
γ
1
−
(7
.
3
.
2
b
)
The thermal averages have to be calculated self-consistently, which im-
plies that the static magnetoelastic Hamiltonian, (7.3.2), must itself
be included in the total magnetic MF Hamiltonian, which determines
the thermal averages such as
O
2
in the equilibrium equation. The
magnetoelastic coupling changes the magnetic-excitation energies if the
crystal is strained, because the extra crystal-field term in (7
.
3
.
2
a
), in-
troduced by
H
γ
(sta), directly modifies
χ
o
(
ω
). In the (
J
=1)-model
corresponding to Pr,
O
±
2
(
J
i
) couples the two doublet states, and thus
the degeneracy of this level is lifted in proportion to the
γ
-strains.
Having included the contributions of
H
γ
(sta) to the single-ion sus-
ceptibility, we continue by discussing the influence of the coupling be-
tween the magnetic excitations and the phonons, as determined by the
dynamic part of the magnetoelastic Hamiltonian
H
γ
(dyn), given by eqn
(5.4.6) with
B
γ
4
= 0. As an example, we consider the coupling to the
transverse phonons propagating in the
a
-orthe
b
-direction, with the
polarization vector in the basal-plane, which is derived from
B
γ
2
i
O
−
2
2
i
O
−
2
2
∆
H
γ
(dyn) =
−
(
J
i
)
−
B
γ
2
i
=
−
N
νµ
a
νµ
(
i
)
i
,
(7
.
3
.
3)
νµ
where
i
is a shorthand notation for
γ
2
(
i
)
−
γ
2
,and
N
νµ
is the matrix
element of the Stevens operator between
<ν
µ>
, cf. eqns (3.5.11-
13). This Hamiltonian introduces an additional term on the l.h.s. of the
equation of motion (3.5.15) for the Green function
|
and
|
a
νµ
(
i
);
a
rs
(
i
)
:
B
γ
2
ξ
N
µξ
a
νξ
(
i
)
N
ξν
a
ξµ
(
i
)
i
;
a
rs
(
i
)
−
i
;
a
rs
(
i
)
B
γ
2
(
n
ν
−
n
µ
)
N
µν
,
(7
.
3
.
4)
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