*3.2. Isospin-Forbidden Decays*

Observation of transitions violating isospin selection rules, pointed to in the Introduction, signifies that the states are not pure in isospin. To predict theoretically the magnitudes of isospin impurities based on a fully microscopic calculation represents quite a complicated task. This can be understood as follows. In the shell model discussed here, isospin impurities arise from mixing of states of the same spin and parity but different isospin, if charge-dependent forces are present. Let us consider the simplest case of just two eigenstates of a charge-independent Hamiltonian, <sup>|</sup>*Jπ*, *<sup>T</sup>* and <sup>|</sup>*Jπ*, *<sup>T</sup>* , of isospin *T* and *T* , respectively. Inclusion of a charge-dependent interaction will result in new eigenstates, being linear combinations of unperturbed states, as

$$\begin{array}{c} |a, f^{\pi}\rangle = \sqrt{1 - \mathbf{x}^{2}} |f^{\pi}, T\rangle + \mathbf{x} |f^{\pi}, T'\rangle \\ |b, f^{\pi}\rangle = \sqrt{1 - \mathbf{x}^{2}} |f^{\pi}, T'\rangle - \mathbf{x} |f^{\pi}, T\rangle \end{array}$$

The mixing amplitude, *x*, in the first order is given by the ratio of the isospin-mixing matrix element and the energy difference between the two states:

$$\propto \sim \langle J^{\pi}, T | V\_{\text{INC}} | J^{\pi}, T' \rangle / \Delta E \dots$$

It is known that it is difficult to predict theoretically the energy difference between states, especially for an odd–odd nucleus. Uncertainties of a few hundred keV may result in huge uncertainty on the mixing probability. However, we would like to require from theory to robustly predict the value of the mixing matrix element, *V*INC. In practice, there could be many-state mixing, and the theory should able to deal with such a problem.

Mixing matrix elements depend strongly on the structure of the states considered, and therefore require in each case a dedicated calculation. Systematic calculations of *V*INC, and distinction between its long-range (Coulomb) and short-range contributions, may bring interesting information, especially when compared to available experimental data (see Refs. [16,49] for an earlier study). From various specific calculations, it seems that typical values of *V*INC are between a few keV to a few tens of keV. Maximum values are 150 keV in *p*-shell nuclei [23,100], around 100 keV for *sd*-shell nuclei [16] and about 50 keV in the *p f* shell [101]. These estimations are in agreement with the largest observed values reported until now: −145(20) keV for 8Be in the *<sup>p</sup>* shell [100], 106(40) keV for 24Mg [102] in the *sd* shell and 40(23) keV for <sup>56</sup> Cu [103] in the *p f* shell. Although theoretical uncertainty on energy differences between admixed states hampers direct predictions of isospin impurities from theory, it often turns out that combining calculations with experimental data may be sufficient to constrain predictions, as illustrated in Section 3.2.3 below.
