核壳层模型

考虑一个三维谐振子。这给出在第一、二层的核壳层（“l”是角量子数，对于质子或中子能量的影响力是主量子数的一半）

我们可以想像自己建立一个原子核，并开始对它加入质子和中子。质子和中子必会从能量较低的能级开始填入，因此，头两个质子先填入第0能级，下6个质子则填第一能级，依此类推。

在三维谐振子中，每个能级可填入的核子数量如为，由于自旋的简并性是加倍

因此

对所有的整数k，给出了以下的幻数2,8,20,40,70,112...

前六个能级是

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.

level 0 (n=0): 2 states (j = ¹⁄₂). 奇数对

level 1 (n=1): 2 states (j = ¹⁄₂) + 4 states (j = ³⁄₂) = 6. 偶数对

level 2 (n=2): 2 states (j = ¹⁄₂) + 4 states (j = ³⁄₂) + 6 states (j = ⁵⁄₂) = 12. 奇数对

level 3 (n=3): 2 states (j = ¹⁄₂) + 4 states (j = ³⁄₂) + 6 states (j = ⁵⁄₂) + 8 states (j = ⁷⁄₂) = 20. 偶数对

level 4 (n=4): 2 states (j = ¹⁄₂) + 4 states (j = ³⁄₂) + 6 states (j = ⁵⁄₂) + 8 states (j = ⁷⁄₂) + 10 states (j = ⁹⁄₂) = 30. 奇数对

level 5 (n=5): 2 states (j = ¹⁄₂) + 4 states (j = ³⁄₂) + 6 states (j = ⁵⁄₂) + 8 states (j = ⁷⁄₂) + 10 states (j = ⁹⁄₂) + 12 states (j = ¹¹⁄₂) = 42. 偶数对