变形的谐振子近似模型

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）

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 ( n + 1 ) ( n + 2 ) 2 {\displaystyle {(n+1)(n+2) \over 2}} ，由于自旋的简并性是加倍 ( n + 1 ) ( n + 2 ) {\displaystyle (n+1)(n+2)}

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.

自旋核轨域

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

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

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

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

level 4 ( n =4): 2 states ( j = ⁄ 2 ) + 4 states ( j = ⁄ 2 ) + 6 states ( j = ⁄ 2 ) + 8 states ( j = ⁄ 2 ) + 10 states ( j = ⁄ 2 ) = 30. 奇数对

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

核轨域

核壳层与预测的幻数

核轨域能级交错的情形

参考文献

Talmi, Igal; de-Shalit, A.Nuclear Shell Theory. Academic Press, (reprinted by Dover Publications). 1963. ISBN 0-486-43933-X.

Talmi, Igal.Simple Models of Complex Nuclei: The Shell Model and the Interacting Boson Model. Harwood Academic Publishers. 1993. ISBN 3-7186-0551-1.

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