The Nuclear Shell Model

What is a shell model?

A shell model is one in which the system is thought to consist of individual particles moving in bound orbitals in response to the remainder of the system.  Each orbital has a well designated energy, angular momentum, and parity associated with it.  In an atom, the electrons are bound to the highly charged nucleus, which contains most of the mass of the atom, by the electric force.  Since most of the mass is contained in the nucleus, the electrons can move in orbits relatively free of any obstacles and hence would suffer very few collisions in their eternal orbiting about the nucleus.  In such a model, treated quantum-mechanically, no two electrons, two protons, or two neutrons can occupy the same quantum state, ie have identical sets of quantum numbers.  This principle attributed to Pauli results in a finite number of such particles occupying a given energy level, and thus, leads to the concept of closed (or filled) levels (or shells).  When a shell is filled, any additional particles of that type must be put in a different level (shell).

Conceptual problem with a nuclear shell model

For nuclei we not a saturation of nuclear forces resulting in an approximate constant binding energy for each constituent nucleon, independent of the details of nuclear structure.  This is attributed to the fact that the size of the nucleus is basically proportional to the number of nucleons and hence the nucleus seems to be a rather compact object with nucleons basically touching each other.  Thus, one might question the validity of treating a nucleon as if it were moving in an orbital through this dense nuclear medium, keeping itself in a state of constant energy implying that collisions with the other nucleons were very infrequent. The Pauli Exclusion Principle perhaps provides a little philosophical support for proceeding with development of such a model.  Because the different nucleons must enter different energy and angular momentum states this perhaps forces the orbits to be enough different that all the nucleons can move in a harmonious fashion, suffering collisions frequently, but still remaining in discrete orbitals enough of the time to make the model have some validity.

Experimental evidence to support the shell model approach

Several kinds of experimental data support the concept that nuclei demonstrate a periodic structure, just like the elements, indicating that one major energy shell with particular energies, angular momenta, and parities can be filled, thus forcing an abrupt change in observed quantities.  The data indicate that closed shells occur at neutron or proton numbers of 2,8,20,28,50,82, and 126 (called Magic Numbers).  The types of data showing discontinuities at these magic numbers indicating a shell closure are discussed in Krane and include some of the following:
  1. Proton and Neutron Separation Energies
  2. Nuclear Spin and Parity Systematics
  3. Nuclear Binding Energy (increased stability of closed shell nuclei)
  4. Small variations of nuclear radii from the R = roA1/3 law.
  5. Alpha Decay Energies and Half-Life Systematics
  6. Neutron Absorption Cross Sections
  7. Magnetic Dipole and Electric Quadrupole Moments
The accumulation of data suggesting shell structure encourage us to pursue such a model, even in the face of the conceptual problem with the probable frequent collisions of nucleons in the nucleus.

The Nuclear Shell Model Potential

The nuclear shell model potential should represent the average interaction of all the other nucleons exerted on any one of the nucleons in the nucleus.  One can make quantum-mechanical calculations using square-well potentials and spherical harmonic oscillator potentials, with parameters adjusted to try to reproduce the binding energy of the nucleon and the proper nuclear radius.  Since the potential represents the average interaction with the other nucleons, and since the nucleon-nucleon interaction is short range, we might do better if we would use a potential that mimics the mass distribution of the nucleus.  Hence, modern shell model calculations frequently use the Woods-Saxon form of the shell model potential. 

This potential is relatively flat inside the nucleus and falls gradually to zero at the nuclear surface.  For a nucleus with 100 nucleons it has the following general shape:

Neither the infinite square well potential, the spherical harmonic oscillator potential, nor the Woods-Saxon produces energy eigenvalues with closed levels (shells) for all the magic numbers.  In particular the levels predicted by the model using the Woods-Saxon treatment is shown on the left side of Fig. 5.6 on p. 123 of Krane.  The lowest level is the 1s state which has only 1 ml substate (ml = 0) and two ms values to give a total degeneracy of 2.  Thus, two protons or two neutrons can fit into this lowest level.  A third neutron will have to go into the second level which is somewhat higher in energy, thus agreeing with the experimental fact that there is a major shell filling at 2.  The second level is the 1p state which has 3 possible ml values, and hence a total degeneracy of 6, ie 6 protons and 6 neutrons can occupy the 1p level.  When it is filled there will be a total of 8 protons and 8 neutrons in the nucleus.  Placement of another nucleon would require that it be placed in the next highest level, or the 1d level.  Thus, the model predicts a magic number at 8 in agreement with observation.  The 1d and 2s levels are predicted to be near each other, being almost degenerate.  These two levels can accomodate 12 neutrons and 12 protons, which bring us to the next major shell closure at 20.  Above Z or N = 20, the prediction of closed shells by this model breaks down; there are no level fillings at 28, 50, 82, or 126.

Spin Orbit Forces to the Rescue

In 1949 Maria Goeppert Mayer and independently, Haxel, Suess, and Jensen had a brilliant idea to split some of the energy levels by applying a small term to the potential which depends on the relative orientation of the spin of the nucleon with regard to the direction of the orbital angular momentum, ie a s.l or spin-orbit term. The total potential is then V(r) = VWS(r) + VSO(r)(l.s).
For a given l there are two values of j, ie j = l + 1/2 and j = l - 1/2, and (l.s) is positive for the first case and negative for the second case.  Thus, the energy level is lowered for the nucleons in l + 1/2 states and raised for the l - 1/2 states.  The splitting of a level is proportional to 2l+1 and is, thus, larger for the larger l states.  On the right side of Fig. 5.6 on page 123 the resulting level diagram is shown; now that ml and ms are not good quantum numbers, one must count mj values to get the degeneracy of each level.  It is observed that the other major shells at 28, 50, 82, and 126 are not reproduced and the theory predicts the next magic number will be 184.  The inclusion of the spin-orbit force was effective in reproducing all the major closed shells (magic numbers).

Side Proof of Dependence of Level Splitting on l

Extreme Single Particle Shell Model

In the extreme single particle model, one assumes that the ground state of an odd-A nucleus is determined by the quantum numbers of the odd nucleon after the A-1 other nucleons have coupled to spin 0.  Thus, 40Ca with 20 protons and 20 neutrons would have all its nucleons paired and the Ip should be 0+.  Indeed, the table in the back of Krane reports the experimental Ip is 0+.  By looking at the level predictions in Fig. 5.6 one would expect the next neutron to go into the f7/2 level and hence, she would expect the Ip of 41Ca to be 7/2-.  Likewise, since the last level filling before N = 20 was the d3/2 level, we would expect the ground state of 39Ca to have an  of 3/2+.  A glimpse in the  appendix shows that these expectations are satisfied by experimental values.  Another point of interest is the presence of two excited states in 41Ca at an energy of about 2 MeV which have  Ip of  3/2+ and 3/2-.  These first of these states is produced by raising one neutron from the d3/2 state to form a pair in the f7/2 level but leaving an unpaired neutron in the d3/2 state.  The second is produced by raising the odd neutron from the f7/2 level  to the p3/2 level.  These data indicate that the energy gap between the d3/2 state and the f7/2 level is about 2 MeV, and the gap between the f7/2 and p3/2 level is similar.(To be precise we would need to consider the difference in the pairing energy for two neutrons in the f and d states.)  Such observations seem to justify the shell model approach for nuclei near the closed shells.

Magnetic Moments in the Extreme Single Particle Model

The magnetic moment of a nucleon in an orbit has orbital and intrinsic parts.  The total moment is  predicted to be different for j = l + 1/2 and   j = l - 1/2 levels because of the difference in coupling of the orbital and spin parts.  The single-particle predictions are shown as solid lines in Fig. 5.9, page 127 and are called Schmidt limits.  It is noted that most of the measured magnetic moments are smaller than the Schmidt limits.  Recalling that the intrinsic component of the magnetic moment depends on the charge and current distributions within the nucleon.  We conclude that when other nucleons are present, the charge distribution of the interacting nucleons is affected, thus changing the intrinsic moment when the nucleon is bound to other nucleons to form a nucleus.  If one reduces the gs for bound nucleons to be 0.6 of the gs for free nucleons, then the Schmidt predictions are the dashed lines in Fig. 5.9, which do indeed kind of trace the experimental data. This result suggests that the polarizing effects is such as to make gs(bound) = 0.6 gs(free).

Electric Quadrupole Moments in the Single Particle Model

Electric Quadrupole Moments are not predicted very well by the spherical shell model.  For nuclei near closed shells, the calculations tend to give the correct sign, but the magnitude is usually missed by factors of about 2 or 3.  Away from closed shells where several valence nucleons may occupy a particular orbital, the quadrupole moments become very large demonstrating large collective effects in nuclear motion.

What happens as you move away from a closed shell?

Although the extreme single particle model works well near the closed shell, we quickly encounter problems as we move to several nucleons away from the major closed shell.  The assumption that all but the one odd particle pair to 0 angular momentum is too simple when we consider possible excited states.  For example, a good demonstration of this complication is shown in Fig. 5.12 on page 132.  41Ca and 41Sc each has one nucleon beyond a double closed shell nucleus 40Ca.  The ground states are 7/2- indicating that the 1 valence nucleon is in the f7/2 orbital, which agrees with the model predictions.  Note that both have excited p3/2 and d3/2 levels around 2 MeV, as you might expect if the odd f7/2 nucleon were moved to the adjacent shell model orbitals.  So the single particle model looks pretty good.

However, notice what happens when two additional neutrons are added to make 43Ca and 43Sc.  The ground states are still 7/2- as we would expect, but the excited states are completely changed.  So, the result is that we must expand the model to consider all the nucleons beyond the major closed shell as valence nucleons and consider all the different ways these valence nucleons can couple to make states.  Such shell model calculations become quite formidable and require super computers for numerical predictions of shell model properties.  For heavy nuclei, the shells contain too many possible orbitals to make detailed calculations, even with the most powerful computers.  Currently, nuclear theorists are devising many clever techniques to overcome these computational complications, including two in our own department, Drs. David Dean and Yang Sun.