Elements of Modern Physics
332
agreement with predictions of Eq. (9.38) but do lie between the two values.
This suggest the need for a more detailed analysis including a mixing of states,
e.g. the states may contain components in which the pairs of nucleons do not
pair off to give zero angular momentum states.
Quadrupole moments: The predictions of the shell model for electric
quadrupole moments are not in good agreement with the experimental values.
If the quadrupole moment of an odd Z, odd A nucleus is due to the last proton,
it should be approximately of the order
Q ≈ R
2
(9.39)
where R is the radius of the nucleus. While this is the case for small nuclei,
some of the nuclei with large A, have Q as large as 10R
2
. Similar, large quadrupole
moments are observed for even Z, odd A nuclei as well. Many of these effects
are due to collective motions in nuclei, which are considered in the collective
model.
The shell model can be generalized by taking the average potential to be an
asymmetric harmonic oscillator potential. For example, in the Nilsson model
the force constant in the z-direction is taken to be different from those in the
x-and y-directions. This model retains the rotational symmetry in the z-direction
while being able to describe the observed large quadrupole moments of nuclei.
Collective Model
For nuclei with a closed shell plus one or a few nucleons, the elementary shell
model is quite successful in describing the nuclear properties. However, when
there are several nucleons outside the closed shell, the nucleus is significantly
deformed. The motion of the deformed nucleus gives rise to collective rotational
and vibrational levels of the nucleus.
In the deformed nucleus which is assumed to be ellipsoidal in shape, the
rotation can be of two types:
(i) it may be irrotational as in the case of tidal waves with no part of the
nucleus actually going around the nucleus,
(ii) the whole nucleus may rotate as a rigid body. Both these motions may
contribute to the rotational motion of a nucleus.
In even Z, even A nuclei, the angular momenta of the nucleons pair off to a
zero value, so that the total angular momentum is also the angular momentum
due to collective rotation. Accordingly, the rotational energy levels are given
by
E I =
2
( 1)
2
+
I I
I
(9.40)
where I is the moment of inertia and I is the total angular momentum quantum
number. However, since the remaining wave function (other than the rotational
332
agreement with predictions of Eq. (9.38) but do lie between the two values.
This suggest the need for a more detailed analysis including a mixing of states,
e.g. the states may contain components in which the pairs of nucleons do not
pair off to give zero angular momentum states.
Quadrupole moments: The predictions of the shell model for electric
quadrupole moments are not in good agreement with the experimental values.
If the quadrupole moment of an odd Z, odd A nucleus is due to the last proton,
it should be approximately of the order
Q ≈ R
2
(9.39)
where R is the radius of the nucleus. While this is the case for small nuclei,
some of the nuclei with large A, have Q as large as 10R
2
. Similar, large quadrupole
moments are observed for even Z, odd A nuclei as well. Many of these effects
are due to collective motions in nuclei, which are considered in the collective
model.
The shell model can be generalized by taking the average potential to be an
asymmetric harmonic oscillator potential. For example, in the Nilsson model
the force constant in the z-direction is taken to be different from those in the
x-and y-directions. This model retains the rotational symmetry in the z-direction
while being able to describe the observed large quadrupole moments of nuclei.
Collective Model
For nuclei with a closed shell plus one or a few nucleons, the elementary shell
model is quite successful in describing the nuclear properties. However, when
there are several nucleons outside the closed shell, the nucleus is significantly
deformed. The motion of the deformed nucleus gives rise to collective rotational
and vibrational levels of the nucleus.
In the deformed nucleus which is assumed to be ellipsoidal in shape, the
rotation can be of two types:
(i) it may be irrotational as in the case of tidal waves with no part of the
nucleus actually going around the nucleus,
(ii) the whole nucleus may rotate as a rigid body. Both these motions may
contribute to the rotational motion of a nucleus.
In even Z, even A nuclei, the angular momenta of the nucleons pair off to a
zero value, so that the total angular momentum is also the angular momentum
due to collective rotation. Accordingly, the rotational energy levels are given
by
E I =
2
( 1)
2
+
I I
I
(9.40)
where I is the moment of inertia and I is the total angular momentum quantum
number. However, since the remaining wave function (other than the rotational