Nuclear matrix element

If the nuclear matrix element does not depend on the electron kinetic energy, as we have assumed so far, then a plot of the reduced spectral intensity, the left-hand side, versus the electron kinetic energy will be a straight line that intercepts the abscissa at the Q value. Such a graph is called a Kurie plot, and an example is shown in Figure 8.3. This procedure applies to allowed transitions (see below). There are correction terms that need to be taken into account for forbidden transitions. 

The differential form of the spectrum can be integrated over all electron momenta to obtain the total decay constant. The expression, for a constant nuclear matrix element, to be integrated is... 

The decay constant is now reduced to an expression with the nuclear- matrix element, M(= Mjf ), and the strength parameter, g, written ... 

The left-hand side of this equation is called the comparative half-life, or ft value because this value can be readily measured in experiments and should only depend on the nuclear matrix element and the (3-decay strength constant. Recall that (3 decay half-lives span many orders of magnitude so the ft values will span a similarly large range. It is therefore convenient to use the common logarithm of the ft value (with t]/2 in seconds) to characterize observed (3 decays. 

The quantities A and B are usually called the constants of the magnetic dipole and electric quadrupole interactions. The nuclear matrix elements in (22.27) for k = 1 and 2 are proportional to the magnetic dipole p and electric quadrupole Q nuclear momenta, respectively ... 

Hamilton operator or Hamilton matrix (general, electronic, nuclear) Matrix element of a Hamilton operator between Slater determinants Exchange type matrix elements in semi-empirical theory x,y = s,p,d) Summation indices for occupied MOs ... 

Hif = VMg/y/VjWuVrg edr = [Pg.230]

It is obvious that more sophisticated relativistic many-body methods should be used for correct treating the NEET effect. Really, the nuclear wave functions have the many-body character (usually, the nuclear matrix elements are parameterized according to the empirical data). The correct treating of the electron subsystem processes requires an account of the relativistic, exchange-correlation, and nuclear effects. Really, the nuclear excitation occurs by electron transition from the M shell to the K shell. So, there is the electron-hole interaction, and it is of a great importance a correct account for the many-body correlation effects, including the intershell correlations, the post-act interaction of removing electron and hole. 

The nuclear matrix elements which enter into first-forbidden beta decay in the impulse approximation in normal order are formed from the following two classes of operators,... 

Note that we have introduced nucleon field operators N instead of quark operators to express simply the nucleon contribution to The first term in (10.3.12) is negligible since the nuclear matrix element will be proportional to the nuclear spin and therefore small, whereas the second term, which non-relativistically is dominated by the 70 component, gives rise to a coherent nuclear contribution, with matrix element proportional to... 

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