The Nucleus
323
may be ascribed to the two different charges. A model calculation with assumed
charge distribution then provides an estimation for the nuclear radius. All these
approaches are in essential agreement with Eq. (9.11) with r 0 ≈ 1.2 fm.
An important consequence of Eq. (9.11) is that the volume per nucleon is
the same for all nuclei:
V 1 =
1/3 3
0
4 (
) /
3
r A
A
π
=
3
0
4
3
r
π
(9.12)
Thus, the nuclear density is the same for all nuclei. The result is in agreement
with what might be expected from the strong, short-range forces in nuclei.
Furthermore, it implies that the nuclear forces are independent of the charge of
the nucleons. This is known as charge independence on nuclear forces.
Angular Momentum and Magnetic Moment
The total angular momentum of nuclei is made up of the spins and orbital angular
momenta of the constituent nucleons. Associated with the angular momentum
is a magnetic moment.
The angular momentum of the nuclei can be deduced from the hyperfine
interaction between the magnetic moments of the nuclei and of the electrons.
This interaction is of the form
H = A I.J
(9.13)
with I and J being the angular momenta of the nucleus and of the electrons
respectively. The atomic states are characterized by the total angular momentum
F = I + J
(9.14)
and the corresponding quantum number takes on the values
F = J + I, J + I – 1, ..., | J – I |
(9.15)
The shift in the energy of these states is given by
∆E =
2
1
[ (
1) ( 1)
(
1)]
2
A F F
I I
J J
+ −
+ −
+
(9.16)
which leads to a separation of
E F –E F – 1 = A
2
(I + J), A
2
(I + J – 1), ..., A
2
(| I – J | + 1)
(9.17)
between the successive states in Eq. (9.15). The analysis of the spectral lines
corresponding to these levels gives the value of I (and also of J).
The spin of the nuclei can also be determined from the spectra of
homonuclear molecules. It was observed in Chapter 5 that the transitions in
323
may be ascribed to the two different charges. A model calculation with assumed
charge distribution then provides an estimation for the nuclear radius. All these
approaches are in essential agreement with Eq. (9.11) with r 0 ≈ 1.2 fm.
An important consequence of Eq. (9.11) is that the volume per nucleon is
the same for all nuclei:
V 1 =
1/3 3
0
4 (
) /
3
r A
A
π
=
3
0
4
3
r
π
(9.12)
Thus, the nuclear density is the same for all nuclei. The result is in agreement
with what might be expected from the strong, short-range forces in nuclei.
Furthermore, it implies that the nuclear forces are independent of the charge of
the nucleons. This is known as charge independence on nuclear forces.
Angular Momentum and Magnetic Moment
The total angular momentum of nuclei is made up of the spins and orbital angular
momenta of the constituent nucleons. Associated with the angular momentum
is a magnetic moment.
The angular momentum of the nuclei can be deduced from the hyperfine
interaction between the magnetic moments of the nuclei and of the electrons.
This interaction is of the form
H = A I.J
(9.13)
with I and J being the angular momenta of the nucleus and of the electrons
respectively. The atomic states are characterized by the total angular momentum
F = I + J
(9.14)
and the corresponding quantum number takes on the values
F = J + I, J + I – 1, ..., | J – I |
(9.15)
The shift in the energy of these states is given by
∆E =
2
1
[ (
1) ( 1)
(
1)]
2
A F F
I I
J J
+ −
+ −
+
(9.16)
which leads to a separation of
E F –E F – 1 = A
2
(I + J), A
2
(I + J – 1), ..., A
2
(| I – J | + 1)
(9.17)
between the successive states in Eq. (9.15). The analysis of the spectral lines
corresponding to these levels gives the value of I (and also of J).
The spin of the nuclei can also be determined from the spectra of
homonuclear molecules. It was observed in Chapter 5 that the transitions in