442 14 Fundamental Aspects of Nucleon Interactions
If the emitted electrons are not atomic electrons, the process must originate in the
nucleus itself and since the nuclei do not contain electrons, the process corresponds
to a variation of the nucleus. In 1933, Sargent analyzed the dependence of the decay
lifetime on the emitted electrons energy. He observed that, for energies E
max
m
e
c
2
, the lifetime is proportional to E
5
max
(Sect. 8.4.2).
To conserve energy and momentum in the ˇ decay, energy must be emitted in
the form of neutral radiation in addition to the electron and the nucleus Y .Thisis
the Pauli hypothesis for the neutrino, see Sect. 8.2. The ˇ decay is then a three-body
process:
ˇ
decay W
A
Z
X !
A
ZC1
Y C e
C
e
(14.35)
ˇ
C
decay W
A
Z
X !
A
Z1
Y C e
C
C
e
: (14.36)
A ˇ decay conserves the atomic number A constant, see Eqs. 14.35, 14.36.The
phenomenon can be explained in terms of the binding energy BE (14.15). All
systems tend to the value of minimum energy. The mass of a nucleus (14.5)is
the sum of the constituents masses minus the binding energy. If we consider the
Weizsacker mass formula as a function of Z, for each given value of A,thereisa
parabola (if A is odd) or two parabolas (if A is even) with the concave side opening
up, that is,
M.A;Z/ D Nm
n
C Zm
p
BE.Z; A/ D .b
0
C b
1
A ˙ a
4
=A
1=2
/ b
2
Z C b
3
Z
2
(14.37)
[b
0
; ::: ; b
3
are constants depending on a
0
; ::: ; a
3
of (14.5)]. The ˇ decay transforms
the nucleus .Z; A/ in a nucleus which differs by one unit of Z. The nucleus moves
to a location in the parabola closer to the minimum which corresponds to a more
stable position.
Nuclei with A odd. If A is odd, a
4
D 0 and the nuclei are located on a single mass
parabola, such as, in Fig. 14.11a which refers to A D 101. There is a stable nucleus,
101
44
Ru, with the highest binding energy. All isobars have smaller binding energies
and higher masses with respect to the stable nucleus. Decays occurring on the left
side of the minimum are ˇ
, while those on the right side are ˇ
C
.Theˇ
C
decay
is only possible within a nucleus because the rest mass of the neutron is larger than
that of the proton.
Nuclei with A even. The isobars with even mass number form two separate
parabolas: one for even-even nuclei, the other for odd-odd nuclei. They are separated
by twice the energy of the configuration term a
4
D 12:6 MeV. Sometimes, there is
more than one ˇ stable even-even nucleus. For example, in the case of A D 106
(as shown in Fig. 14.11b), there are
106
46
Pd and
106
48
Cd. The first is truly stable,
as it corresponds to the minimum of the parabola. The isotope Cd could instead
transform via double ˇ decay in
106
48
Cd !
106
46
Pd C 2e
C
C 2
e
.However,the
probability for this process is so small (second order in the weak interaction