Subatomic Physics

336 The Weak Interaction
total transition rate wβα and the mean life τ can be obtained from Eq. (11.7) by
integration over the momentum:
w = 1
τ =
1
2π3c3�7 |〈N ′ eν|Hw|N〉| 2
� pmax
dpeF (∓,Z,Ee)p
0
2 e(Emax − Ee) 2 . (11.8)
For very large energies, where Emax ≈ cpmax, andforsmallZ, whereF ≈ 1, the
integral becomes
� pmax
0
dpep 2 e(Emax − Ee) 2 � 1
30c 3 E5 max. (11.9)
While this relation is useful for estimates, accurate values of the integral are
needed for a meaningful treatment of the data. Fortunately the integral has been
tabulated (3) as
� pmax
dpeF (∓,Z,Ee)p
0
2 e(Emax − Ee) 2 = m 5 ec 7 f(Emax). (11.10)
The factor m 5 e c7 has been inserted in order to make f dimensionless. With
Eqs. (11.10) and (11.8), the matrix element becomes
|〈N ′ eν|Hw|N〉| 2 = 2π3
fτ
�7 m5 . (11.11)
ec4 If τ is measured and f computed (3) then the square of the matrix element can be
obtained from Eq. (11.11). It is customary to use ft 1/2 and not fτ in tabulations.
ft 1/2 is called the comparative half-life. The name stems from the fact that all
beta-decaying states would have the same value of ft 1/2 if all matrix elements were
equal. Nature provides an enormous range of values of ft 1/2, fromabout10 3 to
10 23 sec. If such a variation were caused by the fact that the weak interaction, Hw,
were not universal but would change from decay to decay, an understanding of the
weak processes would be hopeless. It is assumed that Hw is the same for all decays
and that the nuclear wave functions that enter the calculation of 〈N ′ eν|Hw|N〉 are
responsible for the variations. The most fundamental decays have the “best” wave
functions and give rise to the largest matrix elements. A few cases are listed in
Table 11.1.
With ft 1/2 =(ln2)fτ (Eq. (5.34)) and with the numerical values of the constants,
Eq. (11.11) becomes
|〈N ′ eν|Hw|N〉| 2 = 43 × 10−6 MeV 2 fm 6 sec
. (11.12)
ft1/2(in sec)
Now consider the decay of the neutron. With the value of ft 1/2 given in Table 11.1,
the magnitude of the matrix element of Hw becomes
|〈pe¯ν|Hw|n〉| ≈ 2 × 10 −4 MeV fm 3 . (11.13)