The book in my series of trying to make use of the books on my shelf for November is Elementary Theory of Nuclear Shell Structure by Mayer and Jensen. It was published in 1955 and so is much closer to the start of the discipline of nuclear physics than it is to today. In fact, if you believe Blatt and Weisskopf's textbook, they say:
Well, in any case, it's an old book and now partly of historical interest, but what a history! Maria Goeppert-Mayer was the second woman to win a Nobel prize in physics, and it was for work that is described in this textbook. She won the prize along with her coauthor Hans Jensen jointly for their work on the nuclear shell model, and with Eugene Wigner for his related work on subatomic physics.
When reading through the book, I decided to concentrate on the bits most to do with figuring out the nuclear shell structure, which is by now an old tale and a staple of most, if not all, introductory books on nuclear structure. Indeed, in a recent post I came across a moaning Denys Wilkinson complaining of the endless rehashing of the slight differences in single particle orderings coming from different assumptions about the nuclear potential. With apologies to Denys (or at least his memory, since he died in 2016), I have ended up looking at the same material, and present some of it here.
These days, the standard single particle potential at the level of an analytic form that can be used to describe the mean (average) field in which nucleons move is the Woods-Saxon potential. Woods and Saxon published their potential in 1954, just before Mayer and Jensen published their book, and it was too recent an innovation to make it in.
Mayer and Jensen started their discussion of potentials with the finite "square well" - a spherically-symmetric region of constant binding potential –V0 in a region with r<R, with a potential 0 outside. They didn't even bother proposing the infinite square well potential as is so often done now, except as an approximate way of finding solutions for the finite square well potential for bound states. They go on to do the same with the simple harmonic oscillator as an approximation to a truncated oscillator which they give as their kind of optimum potential:
V(r) = -V0(1-(r/R)2), r<R
= 0, r>R
Since this truncated oscillator is not much used these days, Woods-Saxon being superior, I thought I'd work out the energy levels, so I wrote a short Python code to calcuate the energy levels with this truncated potential. There are a couple of parameters to play with: V0 and R, and with a little fiddling around with those, I produced the following plot:
You can see the shape of the oscillator potential which goes flat for R=7.0 fm and greater. The choice of R and V0 was made to ensure enough levels woudl be bound in the potential to see what was going on. The result shows a close approimation to the infinite oscillator: equally-spaced levels with degeneracy (same energy) for states of the same parity (here s,p,d,f,g,h ... label the orbital angular momentum of particles in those levels, starting with s=0, p=1 etc. The parity is the odd/evenness of those numbers). You can see the degeneracy being lifted for the highest energy states, closest to the energy where the infinite oscillator is truncated.
Because the states in this truncated oscillator so closely resemble the infinite oscillator states, the potential is not responsible for the "intermediate form" plotted in Fig IV.1 of the book, which is probably a literal interpolation between the infinite oscillator and infinite square well result. Back in the 50's it wasn't at all easy to do what I have done with my short Python code, and produce results for an arbitrary potential expressed numerically.
For fun, we can see what happens to the structure if we play with the shape of the potential. Let's also flatten the potential in a simple way in the interior: if we flatten the potential for r<3 to be the r=3 value in the above plot we get the following
Now the degeneracy is lifted within the shells, in the order expected - the larger the angular momentum value, the deeper the level, as they are kept away from the new flat bit of the nuclear potential by the centrifugal barrier.
If you want to play with the code, I am happy to stick it up somewhere online. Let me know!
update: I did stick it online: 
