Subatomic Physics

232 Angular Momentum and Isospin
8.7 Isospin in Nuclei
A nucleus with A nucleons, Z protons, and N neutrons, has a total charge Ze.
The total charge can be written (12) as a sum over all A nucleons with the help of
Eq. (8.14):
Ze =
A�
i=1
qi = e(I3 + 1
2A), (8.31)
where the third component of the total isospin is obtained by summing over all
nucleons,
A�
I3 = I3,i. (8.32)
i=1
The isospin � I behaves algebraically like the ordinary spin J, and the total isospin
of the nucleus A is the sum over the isospins from all nucleons:
�I =
A�
�Ii. (8.33)
i=1
Do these equations mean something? All states of a given nuclide are characterized
by the same values of A and Z. What are the values of I and I3? According to
Eq. (8.31), all states of a nuclide have the same value of I3, namely
I3 = Z − 1 1
2A = 2 (Z − N). (8.34)
The assignment of the total isospin quantum number I is not so simple. There are
A isospin vectors with I = 1
2 , and, since they add vectorially, they can add up to
many different values of I. The maximum value of I is 1
2A, and it occurs if the
contributions from all nucleons are parallel. The minimum value is |I3|, because a
vector cannot be smaller than one of its components. I therefore satisfies
1
1
2 |Z − N| ≤I ≤ 2 A. (8.35)
Can a value of I be assigned to a given nuclear level, and can it be determined
experimentally? To answer these questions, we return to a world where all but the
hadronic interactions are switched off, and we consider a nucleus formed from A
nucleons. I is a good quantum number in a purely hadronic world, and each state
of the nucleus can be characterized by a value of I. Equation (8.35) shows that I is
integer if A is even and half-integer if A is odd. The state is (2I +1)-fold degenerate.
12E. P. Wigner, Phys. Rev. 51, 106, 947 (1937); Proc. Robert A. Welch Confer. Chem. Res.
1, 67 (1958).