Chemistry Reference
In-Depth Information
4f
10
ten 4f
states filled. It loses the two 6s and one 4f electrons to
form the Dy
3
+
ion. Show that the basic level of Dy
3
+
is
6
H
15
/
2
and
verify that the experimentally observed value for
p
of 10.6 is close to
the theoretically expected value.
(Note
L
(
)
=
3 for an f shell, which can hold 14 electrons. The notation
n
H
m
indicates
n
1, where
S
is total spin angular momen-
tum; the total orbital angular momentum is given by the capital
letter, with
S
,
P
,
D
,
F
,
G
,
H
,
=
2
S
+
...
≡
...
0, 1, 2, 3, 4, 5
; and the total angular
momentum
J
=
m
.)
6.6 We stated in Section 6.9 that it is impossible for a paramagnet to float
stably in amagnetic field. Anecessary condition for stability at point
P
is that the force
F
is always directed back towards
P
, so that
A
F
·
d
A
<
0, where
A
is a surface surrounding
P
. Hence
∇·
F
<
0.
2
B
2
a Show from eqs (6.45) and (6.46) that
∇·
F
<
0 requires
χ
∇
<
0.
b Show, using Maxwell's steady-state equations,
∇·
B
=
0 and
2
B
x
2
B
y
2
B
z
∇×
=
∇
=∇
=∇
=
B
0, that
0 in a steady magnetic
field.
2
B
2
2
2
2
c Hence show that
∇
=
2
[|∇
B
x
|
+|∇
B
y
|
+|∇
B
z
|
]≥
0, so that
a paramagnet will never float stably in a magnetic field.
6.7 By symmetry, the magnetic field points along the axis at the centre of
a circular solenoid,
B
k
, where
k
is the unit vector along the
z
-direction. Use Maxwell's equations and Taylor's theorem to show
that
B
=
B
(
0, 0,
z
)
(
x
,0,
z
)
is given at small
x
by
1
2
xB
(
1
)
(
1
4
x
2
B
(
2
)
(
(
)
=−
)
+
(
(
)
−
))
B
x
,0,
z
0, 0,
z
i
B
0, 0,
z
0, 0,
z
k
where
B
(
n
)
=
∂
n
B
z
n
.
(
0, 0,
z
)/∂
6.8 The sufficient condition for stability of a diamagnet at a point where
F
2
E
x
2
=
∂
/∂
>
0 along the axis of a solenoid is that
0 (horizontal
2
E
z
2
stability) and
0 (vertical stability). Themagnetic field along
the axis of a circular current loop varies with distance along the axis
as
B
∂
/∂
>
2
, where
a
is the radius of the loop.
Show that stable equilibrium can be achieved if
B
0
is such that the
gravita
ti
onal andmagnetic forces are balanced in the region bounded
by
a
2
3
/
(
)
=
/
[
+
(
/
)
]
0, 0,
z
B
0
1
z
a
√
7
1
/
2
a
.
/
<
z
<(
2
/
5
)
