Deformed Harmonic Oscillator Approximated Model
Consider a three-dimensional harmonic oscillator. This would give, for example, in the first two levels ("l" is angular momentum)
| level n | l | ml | ms |
|---|---|---|---|
| 0 | 0 | 0 | 1⁄2 |
| −1⁄2 | |||
| 1 | 1 | 1 | 1⁄2 |
| −1⁄2 | |||
| 0 | 1⁄2 | ||
| −1⁄2 | |||
| −1 | 1⁄2 | ||
| −1⁄2 |
We can imagine ourselves building a nucleus by adding protons and neutrons. These will always fill the lowest available level. Thus the first two protons fill level zero, the next six protons fill level one, and so on. As with electrons in the periodic table, protons in the outermost shell will be relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore nuclei which have a full outer proton shell will have a higher binding energy than other nuclei with a similar total number of protons. All this is true for neutrons as well.
This means that the magic numbers are expected to be those in which all occupied shells are full. We see that for the first two numbers we get 2 (level 0 full) and 8 (levels 0 and 1 full), in accord with experiment. However the full set of magic numbers does not turn out correctly. These can be computed as follows:
- In a three-dimensional harmonic oscillator the total degeneracy at level n is . Due to the spin, the degeneracy is doubled and is .
- Thus the magic numbers would be
- for all integer k. This gives the following magic numbers: 2,8,20,40,70,112..., which agree with experiment only in the first three entries. Note that these numbers are twice the tetrahedral numbers (1,4,10,20,35,56...) from the Pascal Triangle.
In particular, the first six shells are:
- level 0: 2 states (l = 0) = 2.
- level 1: 6 states (l = 1) = 6.
- level 2: 2 states (l = 0) + 10 states (l = 2) = 12.
- level 3: 6 states (l = 1) + 14 states (l = 3) = 20.
- level 4: 2 states (l = 0) + 10 states (l = 2) + 18 states (l = 4) = 30.
- level 5: 6 states (l = 1) + 14 states (l = 3) + 22 states (l = 5) = 42.
where for every l there are 2l+1 different values of ml and 2 values of ms, giving a total of 4l+2 states for every specific level.
These numbers are twice the values of triangular numbers from the Pascal Triangle: 1,3,6,10,15,21....
Read more about this topic: Nuclear Shell Model
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