Subatomic Physics

11.2. Beta Decay Lifetimes 335
The spectrum will be distorted: There will be more positrons of high energy and
more electrons of low energy than predicted by Eq. (11.5). The Coulomb correction
introduces an additional factor in Eq. (11.5), and for a decay N → N ′ eν it becomes
dwβα =
1
2π 3 c 3 � 7 |〈N ′ eν|Hw|N〉| 2 F (∓,Z,Ee)p 2 e(Emax − Ee) 2 dpe. (11.7)
Figure 11.2: Kurie plot for the neutron decay. [From J.
M. Robson, Phys. Rev. 83, 349 (1951).] Here N(p) corresponds
to dw/dpe in Eq. (11.6).
F (∓,Z,Ee) is called the
Fermi function; (3) the sign indicates
whether it applies to
electrons or positrons. The
Fermi function also corrects
Kurie plots for the Coulomb
distortion, and the momentum
dependence of the matrix
element can be tested in many
decays. It turns out that the
matrix element is essentially
momentum-independent in all
cases of interest, for decay energies
up to a few MeV. The
shape of the electron spectrum
in beta decay is dominated
by phase–space considerations
and not by properties
of the matrix element.
However, the high-energy end of the beta spectrum can provide information
about the mass of the neutrino. In deriving Eq. (11.7) we have assumed a massless
electron neutrino. If the mass is not zero, the Kurie plot will deviate from a straight
line at the upper limit; the deviation will be most pronounced for decays with small
maximum energy, for instance 3 H → e − ¯νe 3 He. Searches with this nucleus indicate
that the electron neutrino rest energy is smaller than approximately 3 eV. (4)
11.2 Beta Decay Lifetimes
Information about the magnitude of the matrix element can be obtained from the
lifetimes of beta emitters. If the matrix element is momentum-independent, the
3 H. Behrens and J. Jänecke, Numerical Tables for Beta Decay and Electron Capture, Landolt-
Börnstein, New Series, Vol. I/4, Springer, Berlin, 1969. H. Behrens and W. Bühring, Electron
Radial Wave Functions and Nuclear Beta-Decay, Clarendon Press, Oxford, 1982.
4 J. Bonn et al., Nucl. Phys. Proc. Suppl. 110, 395 (2002); see also http://www-ik.fzk.de/ katrin/index.html
for an experiment under preparation.