Environmental Engineering Reference
In-Depth Information
zero for the
cc
-component, which implies that the induced quadrupolar-
interaction is a factor of about seven larger than the intrinsic value of
the electric-quadrupole hyperfine-interaction for the ion (
<
0
|H
Q
|
0
>
=
(5
/
7)
P
(
I
ξ
+
I
η
), with
P
=
0
.
128 mK, using the notation of Bleaney
(1972)). In any case, the quadrupole contribution to (7
.
3
.
25
b
)only
makes a 1.5% correction at the transition temperature
T
N
≈
−
50 mK in
Pr. The induced quadrupole interaction, due to the highly anisotropic
fluctuations of the electronic moments, may be important in nuclear-
magnetic-resonance (NMR) experiments. The most important effect
in NMR is, however, the strong enhancement of the Zeeman splitting
between the nuclear levels by the hyperfine coupling.
Introducing
H
I
(MF) =
−
g
N
µ
N
H
eff
·
I
in (7
.
3
.
25
a
), we find an enhancement
I
H
eff
I
|
/H
||
−
(
gµ
B
/g
N
µ
N
)
Aχ
zz
(
0
,
0)
|
,
(7
.
3
.
26)
1
which, for the hexagonal ions in Pr, gives a factor of about 40 in the low-
temperature limit, when the field is applied in the basal-plane, but unity
if
H
is along the
c
-axis. In addition to the hyperfine interactions consid-
ered above, the nuclear spins may also interact directly with the conduc-
tion electrons, leading to an extra
Knight shift
and Korringa broadening
of the NMR-levels. The most important NMR-linewidth effect is, how-
ever, due to the fluctuations of the localized electronic moment. If
J
=1,
corresponding to Pr, these fluctuations lead to a Lorentzian broadening,
so that
χ
ξξ
(0)
χ
ξξ
(0)
i
Γ
N
/
(
hω
+
i
Γ
N
)
,with
→
= 10(
n
0
n
1
/n
01
)
M
ξ
Im
K
(
ω
=∆
/h
)
,
Γ
N
to first order in 1
/Z
. In the case of Pr, this gives Γ
N
exp(
−
β
∆)
×
1
.
0
meV (Jensen
et al.
1987).
The magnetization and the neutron-scattering cross-section are de-
termined in the RPA by the usual susceptibility expression (7.1.2), with
χ
o
(
ω
) now given by (7.3.24), provided that we neglect the contribu-
tions of the small nuclear moments. This means that, even though the
electronic system has a singlet ground-state, the hyperfine interaction
induces an elastic contribution, and assuming the electronic system to
be undercritical, so that
R
(0)
<
1 in (7.1.6), we obtain in the low tem-
perature limit, where
k
B
T
∆,
∆
2
1+
A
2
χ
J
(0)
χ
I
(0)
E
q
−
E
q
)
A
2
χ
J
(0)
χ
I
(0)
χ
J
(0)
,
χ
ξξ
(
q
,
0) =
(7
.
3
.
27)
(∆
2
−
where
χ
J
(0) = 2
M
ξ
/
∆, and
E
q
is given by (7
.
1
.
4
b
), with
n
01
=1. If
we introduce the nuclear spin susceptibility, neglecting the quadrupo-
lar contribution, into this expression, it predicts a second-order phase
Search WWH ::
Custom Search