Nuclear Physics PHY303
3 Nuclear Models
There are two basic types of simple nuclear model
Collective body with no individual particle states.
An example is the Liquid Drop Model which is the basis of the semi-empirical mass formula.
Individual particle model with nucleons in discrete energy states
for example the Fermi Gas Model or the Shell Model.
3.1 Shell model - This model very much builds on the success of the
atomic shell model which explains the periodic properties of atoms in terms of
the filling of electron energy levels.
When the group of levels associated with a shell are all occupied we have particularly
stable (chemically inert) atoms - the rare gases.
In the nuclear case we will first summarise the evidence that there are particular
values of Z and N (so called magic numbers) which are significant with
regard to the structure of nuclei.
There are a large number of isotopes, isotones
at these particular values of Z,N.
This is also supported by the natural abundances of elements shown in the figure below.
The stable elements coming at the end of the principal radioactive series all
have a 'magic number' of neutrons or protons
thorium series - 82Pb208
uranium series - 82Pb206
actinium series - 82Pb207
neptunium series - 83Bi209.
As can be seen in the figure below (note the log scale) the neutron
cross-sections for different nuclei are about two orders of magnitude lower when
the neutron number N is a 'magic number'.
This indicates that these nuclei are much less likely to absorb an additional neutron.
Clear evidence of effects at particular values of N can also be seen
in a plot of the binding energy of the last neutron versus neutron number.
In this figure the measured binding energy is plotted relative to the value predicted
by the Semi-Empirical mass formula.
There are discontinuities of about 2 MeV at the shell closures.
Electric Quadrupole moments should be zero for closed shell nuclei since
they are spherically symmetric.
The next figure shows this to be so.
The figure is based on measurements for odd A nuclei.
The measured moments have been normalised with respect to the size and charge of
each nucleus and these so called reduced quadrupole moments are plotted against
the number of protons or neutrons - depending upon which is odd.
It is also clear from the figure that in some cases the moments are relatively large.
This points to some nuclei having shapes which are strongly non-spherical.
Nuclei with closed shells are relatively stable and hence it requires
considerable energy to excite them out of their ground state.
The figure below shows the ground and first excited states of the even A isotopes of lead.
For A = 208 both the number of protons (Z = 82) and the number of
neutrons (N = 126) are 'magic' which means this is a double closed shell
nucleus and it takes more than 2 MeV to raise it to its first excited state.
We have referred above to some of the significant or 'magic' numbers associated
with nuclear structure.
The complete list for Z or N is
2, 8, (14), 20, 28, 50, 82, 126
Any model of the structure of the nucleus has to provide a reasonable explanation
of these characteristic numbers.
We will examine how this is done in the Shell Model.
The Shell Model is based on the assumption that nucleons inside the nucleus are
in definite states of energy and angular momentum.
The notion of nucleons moving on orbits is somewhat at odds with a strongly
interacting many particle system.
It would be expected that the nucleons were continually scattering off each other,
gaining and losing energy and thus changing state.
However it should be realised that in an energy level model the nucleons will
fill the levels up to a particular energy and although there are levels vacant at
higher energy there are no lower levels into which a nucleon can be scattered.
Thus it can be said that the Pauli exclusion principle effectively suppresses
scattering and ensures reasonably well defined orbits.
This restriction does not apply to a higher energy nucleon coming from outside.
This can readily scatter and raise the energy of the constituents of the nucleus.
The Shell Model is also based on the single particle approximation in which the
effect of all other nucleons is smoothed out and represented by a potential
which is then used in the Schrödinger equation.
Neglecting the coulomb repulsion between protons for the moment, the potential
can be based on the nuclear matter distribution function
V(r) = - V(0)/(1 + e(r - b)/a)
with V(0) ~ 60 MeV, b ~ 1.25 x A1/3 fm, a ~
0.65 fm as obtained from scattering experiments.
For a central potential (ie spherical symmetry) the variables can be
separated and the wavefunction written as
.
The functions Ylm are the same spherical harmonics that
appear in the wavefunctions for the hydrogen atom, and the equation for the
radial function Rnl can be written as
where E is the sum of the kinetic and potential energies and
the term 
represents the centrifugal effect
that the angular momentum has.
It appears in the radial equation as an additional potential energy and is
illustrated as such below in the sketch of the energies and wave functions for a square well.
Note how this term tends to raise the energy levels and push the wavefunction away from the origin.
For zero angular momentum a simple approximation to the solutions for the finite square well
V(r) = - V0 for r < b
V(r) = 0 for r > b
is of the form
for r < b
for r > b
where k is approximately equal to n
/b
(exactly so for the infinite square well) and K is
(-2mE)1/2/
.
Then employing the usual technique of normalisation and application of the condition
of continuity to the wavefunction and its derivative at r = b we can find
the constants A, B and E.
As expected only certain discrete values of E are allowed.
The ordering of the levels is
| | 1s | 1p
| 1d | 2s | 1f
| 2p | 1g | 2d
| 3s |
|---|
2(2 +1)
| 2 | 6 | 10
| 2 | 14 | 6
| 18 | 10 | 2 |
| Total
| 2 | 8 | 18
| 20 | 34 | 40 | 58
| 68 | 70 |
As can be seen these totals are not in very good agreement with the 'magic' numbers
that we have listed above.
The use of other shapes for the potential does not improve matters, and this was
for a long time a puzzle in the development of the theory of the structure of nuclei.
The missing ingredient in the potential is spin-orbit coupling which
leads to a term of the form V(r).s which is
attractive for j = + 1/2
( and s parallel) and repulsive for
j = - 1/2 (
and s anti-parallel).
The proposal of this extra term in the nuclear potential was made in 1949 by
Maria Goeppert Mayer and Johannes Jensen.
The critical contribution that they made to the development of nuclear physics
was recognised with the award of the Nobel Prize in 1963 - together with Eugene Wigner.
The spin-orbit force is also required in order to explain polarization
effects in nucleon-nucleon scattering.
The effect on the potential shape is shown schematically below.
Note that by using the cosine rule and recalling that the
magnitude of the vector is
[( + 1)]1/2
etc it can be shown that:
s. = [j(j + 1) -
( + 1) -
s(s + 1)]/2
For j = + 1/2 ,
s. =()/2
and for j = - 1/2 ,
s. = - ( + 1)/2 .
Thus the splitting in the energy levels due to this difference in the potential
is proportional to (2 + 1).
The development of the understanding of the nuclear potential is illustrated
below in a diagram which traces the energy levels of a system through from the
three dimensional harmonic oscillator potential to an approximation to the nuclear
potential including the spin-orbit contribution.
The total numbers of protons or neutrons are given in brackets if all the energy
levels up to that point are filled and the values of
j = + s for each nuclear level are given to the right.
The number of protons or neutrons in each of these individual levels is given by (2j + 1).
3.2 Predictions of the Shell model
- Angular Momentum:
As with all single particle energy level
models, the Shell Model predicts that all even-even nuclei will have zero angular momentum.
An odd A nucleus will have the angular momentum of the odd nucleon.
For example -
| Nuclide | Z
| N | Shell Model
| Observed Ground State |
|---|
| 17O
| 8 | 9 | l = 2; j = 5/2
| I = 5/2; + parity |
| 17F
| 9 | 8 | l = 2; j = 5/2
| I = 5/2; + parity |
| 43Se
| 21 | 22 | l = 3; j = 7/2
| I = 7/2; - parity |
| 209Pb
| 82 | 127 | l = 4; j = 9/2
| I = 9/2; + parity |
| 209Bi
| 83 | 126 | l = 5; j = 9/2
| I = 9/2; - parity |
Excited states of such single particle nuclei follow the Shell Model up to about
2 MeV above the ground state but then excitations of the core complicate the
energy level picture.
The model makes no predictions for odd-odd nuclei.
- Magnetic Moments:
Odd A nuclei should have the moment of the odd nucleon.
µ/µN = (j - 1/2)gl +
gs if
= j - 1/2 for odd nucleon
µ/µN = {(j + 3/2)gl - gs}[j/(j
+ 1)] if
= j + 1/2 for odd nucleon
gl = 1 for the proton, 0 for the neutron:
gs = 2.79 for the proton, -1.91 for the neutron.
These are the so called Schmidt Limits for nuclear magnetic moments and most
actual values lie between them.
This can be explained in terms of a mixing of states such as that we have already
seen in the case of the deuteron.
- Electric Quadrupole Moments:
If this moment were just due to the odd proton
it should be given by Q ~ -R2(2j - 1)/(2(j + 1))
where j is the angular momentum quantum number of the odd particle.
For even-even nuclei it should be about zero and should change sign on going
through the shell closure.
Some examples are
| Nuclide | Z
| N | Character
| Qobs(QSM) | Ratio |
|---|
| 17O
| 8 | 9 | +1 neutron; j = 5/2
| -2.6(-0.1) | 20 |
| 39K
| 19 | 20
| - 1 proton; j = 3/2 | +5.5(+5.0)
| 1 |
| 175Lu
| 71 | 104
| mid shell; j = 7/2 | +560(-25)
| -20 |
| 209Bi
| 83 | 126
| +1 proton; j = 9/2 | -35(-30)
| 1 |
The Q are expressed in (fm)2 thus the numbers have to be
multiplied by the electronic charge to obtain the actual quadrupole moment.
There are clear deviations from the Shell Model predictions (QSM):
Odd neutron nuclei have about the same Q as odd proton ones.
Nuclei with atomic mass number in the ranges 150 to 190 and
greater than 200 have very large quadrupole moments.
This is an outstanding failure of the model.
3.3 Collective model - To explain the large quadrupole moments we
return to the picture of the nucleus as a collective body.
The basic idea of this Model is that interactions between the outer nucleons
and the closed shell core lead to permanent deformation.
This is expected to be a particularly strong effect midway between shell closures.
In the case of a permanent deformation the single particle states have to be
calculated in a non-spherical potential.
The spacing of the energy levels then depends upon the magnitude of the distortion.
This marriage of the single particle and the corporate models presents very difficult
theoretical problems so we will just concentrate on the qualitative features.
As noted above doubly closed shell nuclei are very stable with a first excited
state well removed from the ground state.
One nucleon more or one less than this very stable configuration will exhibit
single particle states.
Nuclei further away from the closed shells should be easily deformed leading to
excited states due to the vibrational motion of the core.
In the region of half filled shells the nuclei are permanently deformed and
consequently have large quadrupole moments and also rotational energy levels.
These points are illustrated in the table below which lists the type of energy
levels observed for a wide range of even-even nuclei.
The examples given cover those nuclei in the region just below 208Pb
(Z = 82, N = 126).
As additional evidence the approximate sizes of the quadrupole moments of the
odd A nuclei in this same region are also listed (in fm2).
| Nuclides
| 150Sm...152Gd
| 152Sm...190Os
| 200Pt...200Hg
| 206Pb | 207Pb
| 208Pb |
|---|
| | | |
| | | |
| Spectra
| vibrational | rotational
| vibrational | two particle
| one particle | "magic" |
| Q(odd A)
| 50 | 200-700-200 | 50
| - | - | - |
3.4 Rotational states of deformed nuclei - We will just consider
even-even nuclei and recall that the ground state is always 0+ in this case.
Defining the z direction as the symmetry axis of the deformed nucleus and
recalling that the angular momentum operator for the component along this axis is
equal to -i 
,
then there are no rotational states about this axis because
/
is zero.
For rotational angular momentum R, the energy is just
R2/(2
) where
is the effective moment of inertia.
The energies are given quantum mechanically by the Schrödinger equation of
the form
(R2/2)op
= E and since the operator
(R2)op 
(L2)op, the normal angular momentum operator, the
eigenvalues and eigenfunctions are given by
where J can normally be 0, 1, 2, ... but in this
case since the ground state is even parity, only even J values
are admissable (odd J spherical harmonics have odd or negative parity).
Thus the energy levels are as given below.
EJ = J(J +
1)2/(2)
with J = 0, 2, 4, ...
For example E2 =
62/(2)
and other energy levels can be expressed in terms of this as
EJ = J(J + 1)E2/6 (see table of states given below in MeV).
| Spin parity
| 164Er | 166Yb
| 170Hf | 172W
| J(J + 1)/6 |
|---|
| | | |
| | |
|---|
| 16+
| - | - | 3.15(31.5)
| - | 45.3 |
| 14+
| - | - | 2.56(25.6)
| 2.68(21.8) | 35.0 |
| 12+
| 2.08(22.8) | 2.17(21.3)
| 2.01(20.1) | 2.13(17.3) | 28.2 |
| 10+
| 1.52(16.5) | 1.60(15.8)
| 1.50(15.0) | 1.62(13.2)
| 18.3 |
| 8+
| 1.02(11.2) | 1.10(10.8)
| 1.04(10.4) | 1.15(9.3) | 12.0 |
| 6+
| 0.61(6.7) | 0.68(6.5)
| 0.64(6.4) | 0.73(5.9) | 7.0 |
| 4+
| 0.30(3.3) | 0.33(3.2)
| 0.32(3.2) | 0.38(3.1) | 3.3 |
| 2+
| 0.09(1.0) | 0.10(1.0)
| 0.10(1.0) | 0.12(1.0) | 1.0 |
| 0+
| 0 (0) | 0 (0) | 0 (0)
| 0 (0) | 0 |
The predicted values are given in the last column of the table, normalised to the
lowest interval 0+ 2+.
They should be compared to the entries in brackets.
The discrepencies at larger J values are due to the centrifugal stretching
of the nucleus and the consequent increase in the moment of inertia
. Since
EJ = J(J +
1)2/(2),
the higher J values have lower energies than expected.
The extent to which these rotational states are intermingled with those due to
other excitations is illustrated in the energy level diagram for
164Er below.
3.5 Vibrational states - Although we have a picture of the nucleus as
being like a drop of incompressible fluid it is still possible for the system
to perform shape oscillations without change of density.
These displacements from spherical symmetry take the form of surface standing waves
which are proportional to the Legendre polynomials 
with oscillating coefficients.
Examples of quadrupole, sextupole and octupole oscillations are sketched below.
The quadrupole is the lowest order vibration and has
= 2.
Thus there are five (2 + 1) possible values
for m (ie five degrees of freedom).
This is an eigenfunction of the total angular momentum and we say that this
state has a single phonon of angular momentum 2.
The second excited state has two phonons which can couple to form
0,2,4 units of angular momentum.
Starting with the normal even-even nuclei ground state (0+)
the excited states are (2+); (0+ 2+ 4+);
( 0+ 2+ 3+ 4+ 6+).
The energies are E = (N +
5/2)
where N is the number of phonons.
Note that the levels are equi-spaced just like the harmonic oscillator.
This simple picture is by no means exact since energy levels due to different
forms of excitation often overlap.
In general vibrational levels are identifiable only up to excitation energies of
1 to 2 MeV.
As an example the lowest energy levels for 64Zn and
122Te are sketched in the final figure below.
As expected the excitation energy from the ground state to the two-phonon
triplet (0+ 2+ 4+) is about
twice that to the one phonon singlet (2+).
The content and information presented here is for the academic session 2005-2006.
In the case of difficulties with this course contact
Dr. Chris Booth.
If you find this material at all helpful please let me know!
* * * * Example problems * * * *
© 1999 - FH Combley; 2005 - CN Booth