116 IE Nuclear Physics

5.7 Fermi's Theory of Allowed B-decay

Enrico Ferni (1934)developed a theory for allowed B decay, based on Pauli's neutrino hypotb.
Feni used the result of Dirac's time-dependent perturbation theory according to which the ro
transition from an initial state ito a final statefis given by (Golden Rule 2)'
2n
W =
H pE) ..5.7-4
where H, is the matrix element of the perturbing interaction Hcausing the transition and is given k.
H, =|v,#y,dt . 5.12)
p(E) is the density of the final states. (See Quantum Mechanics $ 12.6, Vol. I.)
Equation (5.7-1)can be applied to the case of emission of electromagnetic radiation from an
atomic (or nuclear) system. An excited atom in an initial state makes a radiative transition to the fnol
state, which consists of the atom in the ground state plus the emitted photon.
According to Fermi, the B-emitting nucleus makes a transition such that the final state consisk
of the product nucleus plus the emitted electron and the neutrino. The initial state is of course the
parent nucleus. The transition can be written as
X(4, Z) -’ Y(4, Z± 1) +etv ..5.7-3)
where e is either the electron (ß) or the positron (B*) while v is either an antineutrino or a neutrino.
Since the emission of a particle is equivalent to the absorption of an antiparticle, we can
represent the B-transformation process by the following equation in place of Eq. (5.7.3):
X(4,Z) +v’ Y(A, Z+ 1)+e ..5.74)
As we have seen in § 5.6, during ß transformation, a neutron in the nucleus is transformed
into a proton with the creation of an electron-anti neutrino (V) pair, while in B* transformation a
proton in the nucleus is transformed into a neutron with the creation of a positron-neutrino pair.
Using Eq. (5.7-4) we thus see that the initial state wave function y, in Eq. (5.7-2) should consit
of the initial state nuclear wave function. u, and the leptonic wave function ¢, while the final state
wave function y, should consist of the final state nuclear wave function u, and the leptonic wave
function g
In the case of radiative transition, the perturbation causing the transition is the electromagnetie
field whose quantum manifestation is the photon. Its form is known. However in the case ot
B-emission, the nature of the perturbation is not known. It may be called the electron-neutrinofili.
Its quantum manifestation is the pair of these two particles.
The electron neutrino field interacts weakly with the nucleus, undergoing B-transformaton
and is known as the weak-interaction field. This follows from the fact the mean life tÉmes of ß-deca)
(minimum 10s) are long compared to the characteristic nuclear time, which is the time tor
nucleon to travel across the nucleus (10s).
B-decay theory is beset with another difficulty. The exact forms of the nuclear wave-functio
are not known.
Fermi chose the simplest form of the electron-neutrino field H which was relativisticay
invariant. The, relativistic form of the theory must be chosen, because the electron and the neutr
both move at relativistic velocities. The matrix element can be written as
-.(57.5)
H, -Pr,U;o,u,)dr
Here Fr) is a quantity which depends on the wave-functions o, and Í, of the emitted electron
neutrino (emitted field) at the position of the nth nucleon undergoing B-transformation. It sto
1 See § 12.6 in Vol. I
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117
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II118
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andthg .5.7-13, ..(5.7-12)
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-.(3.7-20) ".5.7-19) it
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as
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the in the second
of
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per probability
of (5.7-1)
the Eq. from get (5.7-20)
we and(5.7-14) Eqs. using bySo
119 Activity Beta Particles
and Beta