Beta decay can involves three processes, in all the three processes atomic number of nucleus becomes one unecay dit greater or smaller but the mass number remains the same.

Problems in explaining beta decay

(1) non conservation of energy

(2) non conservation of angular momentum

To explain this Pauli suggested that a second particle is emitted with beta particles simultaneously but the sum of kinetic energies of two particles must be always equal to the energy difference between the parent and daughter nuclei. Thus the principle of conservation of energy is not voilated even the beta particle do not carry same energy.

The maximum beta particle energy is equal to the energy difference between the parent and daughter nuclei.

**Beta decay Theory (Fermi's Theory):-**

Electron, positron and ν does not as such exists inside the nucleus but are
formed at the time of the decay. Fermi assumed that β-decay results from some
form of nutrino, electrons and nucleus. This type of interaction is known as
weak interaction. Constant required to express its strength is
g=10^{-47}erg cm^{3} . It is because of the weakness of this
interaction that the β-decay does not take place instantly in the case when it
is energitically possible.

This theory must include some relationship between the particles of initial
and final nuclei. The relationship is expressed by means of matrix element. It
involves the wave function of initial and final nuclear states and hence their
spin, parities and the arrangement of the nucleons. When two states are very
different from each other |M_{if}|^{2} becomes smaller. The
total available β-decay energy E_{0} , which is the energy difference
between initial and final states, can be divided between electrons and ν in
large number of ways which affects the shape of beta ray spectrum.

The number of ways of distributing total available energy between electron
and ν per unit total energy E_{0} is dN/dE_{0}.

Now , number of ways in which electron may be given volume V and having
momentum between p_{e} and p_{e}+dp_{e} is given by

${\mathrm{dn}}_{e}=4\pi {p}_{e}^{2}V\frac{d{p}_{e}}{{h}^{3}}$

Similarly , number of ways in which neutrino can be arranged between volume
V having momentum between p_{ν} and p_{ν}+dp_{ν}
is

${\mathrm{dn}}_{\nu}=4\pi {p}_{\nu}^{2}V\frac{d{p}_{\nu}}{{h}^{3}}$

Number of ways in which β-decay can lead to an electron having momentum
between p_{e} and p_{e}+dp_{e} and ν having momentum
between p_{ν} and p_{ν}+dpν is

dN=dn_{e}.dn_{ν}

$\mathrm{dN}=\frac{16{\pi}^{2}{V}^{2}}{{h}^{6}}{p}_{e}^{2}{p}_{\nu}^{2}d{p}_{e}d{p}_{\upsilon}$

Relativistic momentum of particle of rest mass m is given by

$p=\frac{{\left[E\right(E+2m{c}^{2}]}^{1/2}}{c}$

since mass of neutrino is almost zero

p=$\frac{E}{c}$

Therefore momentum of neutrino is

${p}_{\nu}=\frac{{E}_{\nu}}{c}=\frac{{E}_{0}-{E}_{e}}{c}$

and hence,

${\mathrm{dp}}_{\nu}=\frac{d{E}_{0}}{c}$ for given E

This shows that ,$\frac{\mathrm{dN}}{d{E}_{0}}=\frac{16{\pi}^{2}{V}^{2}}{{h}^{6}{c}^{3}}{({E}_{0}-{E}_{e})}^{2}{p}_{e}^{2}d{p}_{e}$

We shall now also consider the role of coulomb's barrier in letting out
electron and positron against beta decay. The coulomb barrier aids the escape
of positron but hinders the escape of electrons. The effect of Coulomb barrier
depends on the atomic number Z and the energy of electron or positron. The
fermi factor represented by F(Z,E_{e}) is a complex function.

Now Fermi's theory finally gives the probability of decay with the emission
of an electron having a given momentum p_{e} by the expression which
involves the nucleon-beta-neutrino force constant g , the matrix element
|M_{if}|^{2} and the function of Fermi factor.

## No comments:

## Post a Comment