Environmental Engineering Reference
In-Depth Information
7.3.3 Coupling to the nuclear spins
The hyperfine coupling to the nuclear spins normally has a negligible
influence on the properties of the electronic magnetic moments. How-
ever, in the special case of a crystal-field system with a singlet ground-
state, where the two-ion coupling is smaller than the threshold value for
magnetic ordering, this minute coupling may become of decisive impor-
tance. Under these circumstances, the hyperfine interaction may induce
a cooperative ordering of the combined system of the electronic and
nuclear magnetic moments at very low temperatures. The Hamiltonian
describing the hyperfine interaction in a rare earth ion has been compre-
hensively discussed by Bleaney (1972) and McCausland and Mackenzie
(1979), and the leading-order term is
H
hf
=
A
I
·
J
,
(7
.
3
.
22)
where
I
is the nuclear spin. For the isotope of Pr with mass number 141,
which has a natural abundance of 100%,
I
=5
/
2and
A
=52
.
5mK= 4
.
5
µ
eV. This coupling modifies the MF susceptibility
χ
o
(
ω
) of the single
ion, and since
A
is small, we may derive this modification by second-
order perturbation theory. In order to simplify the calculations, we as-
sume that the MF ground-state of the electronic system is a singlet, and
that
k
B
T
is much smaller than the energy of the lowest excited
J
-state,
so that any occupation of the higher-lying
J
-states can be neglected.
Considering first a singlet-singlet system, with a splitting between the
two states
1
>
is
non-zero, and denoting the combined electronic and nuclear states by
|
|
0
>
and
|
1
>
of ∆
|
A
|
,whereonly
M
z
=
<
0
|
J
z
|
0
,m
I
>
and
|
1
,m
I
>
,where
I
z
|
p, m
I
>
=
m
I
|
p, m
I
>
, we find that the
H
hf
only non-zero matrix elements of
are
<
0
,m
I
|H
hf
|
1
,m
I
>
=
<
1
,m
I
|H
hf
|
0
,m
I
>
=
m
I
M
z
A,
yielding the following modifications of the state vectors:
0
,m
I
>
=
|
|
0
,m
I
>
−
(
m
I
M
z
A/
∆)
|
1
,m
I
>
1
,m
I
>
=
|
|
1
,m
I
>
+(
m
I
M
z
A/
∆)
|
0
,m
I
>,
to leading order. If we neglect the shifts in energy of the different levels,
due to the hyperfine coupling, and the change of the inelastic matrix
element,
<
0
,m
I
|
1
,m
I
>
=
M
z
{
2(
m
I
M
z
A/
∆)
2
J
z
|
1
−
}
M
z
,
the susceptibility is only modified by the non-zero matrix-element,
<
0
,m
I
|
0
,m
I
>
=
2
m
I
M
z
A/
∆
,
J
z
|
−
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