Beta Decay: Energy Spectrum, Decay Constant, and Parity Conservation

Introduction
The concept of beta decay unites kinds of nuclear transformations: electron (β ̅) decay,
positron (β⁺) decay and electron capture. About 900 beta-radioactive isotopes are known.
Only about 20 of them are natural, the rest were obtained by artificial methods. The
overwhelming majority of these isotopes are subject to β ̅- decay. One electron is emitted in
each event of β ̅- decay. Double beta decay, with two electrons(positrons) emitted in each
event, is theoretically possible, but has not been observed experimentally as yet (Detlafand I︠*A+vorskiı̆- , 1980).
Energy Spectrum of Beta-DecayI︠
n beta decay the energy spectrum of the emitted electrons or positrons is continuous,
extending from E=0 to E=E0, where the quantity E0 is called the end-point energy of the beta
spectrum.
The average energy of the electrons emitted by heavy nuclei is Ē ≈ E0/3; for natural β ̅-
radioactive elements Ē =0.25 to 0.45 MeV. For light nuclei the energy spectrum of the
electrons (positrons) is more symmetrical: Ē ≈0.5E 0. The half-lives for beta decays are within
a wide time range: from 2.5 X 10-2 second to 4 X 1012 years, which is incommensurably
longer than the characteristic nuclear time ( ? 10-21 to 10-22 second). This indicates that beta
decay is due to weak interaction. Beta decay is commonly accompanied by the emission of
gamma rays which have a discrete energy spectrum (Detlafand I︠ * A+vorskiı̆ - , 1980).
(Figure 1: Energy specturum of Beta decay electron)
An electron antineutrino is emitted together with the electron in beta decay, and an electron
neutrino together with positron. The interaction between the electron neutrino (antineutrino)
and the nuclei is negligibly small in comparison with the interaction between the nucleons in
the nucleus (nuclear interaction). I︠ n beta decay the electron (positron) and electron
antineutrino (neutrino) have spins equal in magnitude and opposite in direction. Hence the

change in the spin of the nucleus equals zero. The continuous spectrum of beta decay is due
to the different distributions of energy between the electron (positron) and the electron
antineutrino (neutrino), the total energy of the two particles being equal to E0 (Detlafand I︠*A+vorskiı̆- , 1980).
According to the current concept, the electron (positron) and the electron antineutrino
(neutrino) do not exist in atomic nuclei but are formed at the instant of emission from the
nucleus as a result of weak interaction between the nucleons in the nucleus. Since new
particles are produced in the beta decay, the methods of non-relativistic quantum mechanics
are inapplicable to this process and the problem is dealt with the methods of quantum field
theory.I︠
n the theory of beta decay, the production of an electron and an electron neutrino (positron
and electron neutrino) is treated as the result of the interaction between a nucleon of the
nucleus and the electron (positron) and neutrino fields. Besides the production of e ̅ and υ̃ E e (or
e and υ̃⁺ e) particles, 0n1 is changed to 1p1 (or, on the contrary, 1p1 0n1). The intensity of this
interaction is characterised by the weak interaction constant g (coupling constant of the
nucleon and electron-positron fields); g≈1.4 X 10-49 erg-cm3. The probability of beta decay is
characterised by the nuclear matrix element of the transition |Hik| containing: a wave function
of the nucleon in the initial state i; wave functions of the nucleon, electron (positron) and
electron neutrino (anti-neutrino) in the final state k; interaction energy corresponding to the
transition i →k; and, finally, a quantity determining the density of the number of final states
of the system. The selection rules for beta decay establish a considerably higher probability
of allowed transitions and a low probability of so-called forbidden beta transitions (DetlafandI︠*A+vorskiı̆- , 1980).
Of essential significance in the study of beta decay is the analysis of the energy spectrum
N(E) , where N is the number of emitted electrons (or positrons). According to their N(E)
distribution curves, beta spectra are divided into allowed (Fermi-Spectra) and forbidden
spectra. Forbidden spectra, in turn are distinguished by their degree of forbiddenness. For
allowed beta spectra, assuming that mass of the neutrino equals zero,
N (E)dE ≅ F ( Z , E ) pE ( E0−E )
2
dE
where p and E = momentum and energy of the electrons in units of mec and mec2
me= rest mass of the electron
E0= end-point, or maximum, energy of the electrons (or positrons) of the beta spectrum.
The function F(Z,E) takes account of the influence of the nuclear field on the shape of the
N(E) curve. For forbidden beta spectra, N(E) contains a factor which depends on E0, E and
the degree of forbiddenness.
To decide whether a given beta spectrum belongs to the Fermi or forbidden kind, a Fermi-
Curie plot is made. Thus

K ( E )= [ N exp ( E )
pEF ( Z , E ) ]2
,
Where Nexp(E) is the observed curve of the beta spectrum. For the Fermi beta spectrum, K(E)
is a straight line which intersects the axis of abscissas at E=E0. Deviation of K(E) from a
straight line indicates that the given beta spectrum is of the forbidden kind.
Decay Constant
The decay constant λ for beta decay is
λ = C ∫
0
E
N ( E ) dE = CF (Z,E0)
λ=C∫
0
E
N ( E ) dE=CF (Z , E ₀)
The factor C is determined in the theory of beta decay as
C=
( g2 m ₑ5 c4
2 π h7 )| Hik|
2
Where g=weak interaction constant (coupling constant)
|Hik|=nuclear matrix element
Since λ=ln 2/T1/2, where T1/2 is the half-life, then
F ( Z , E0 ) T 1
2
=T 1
2 red=
2 π 3 ℏ7
g2 m ₑ5 c4 ( ln 2 )
Hik θ 2
The product F(Z,E0)T1/2=T0.5red is called the reduced half-life.I︠
t depends only on the character of the interaction between the nucleons of the nucleus and
the electron-neutrino field. Reduced half-life values, obtained from experimental data, enable
|Hik| to be determined.
According to their T0.5 red values, the types of beta decay can be classified as follows:
(a) Those with log T0.5 red ≈3.5 are superallowed transitions; they include: beta decay of
the neutron, 1H3, and transitions between mirror nuclei [(N-Z=1 for the initial nucleus
and N-Z=-1 for the final one), as well as transitions for which N-Z=±1 for one isobar
and N-Z=±2 for the other (where N is the number of neutrons in the nucleus)]; for
superallowed transitions, the | Hik | values are near to the maximum values;
(b) Those with logT0.5red≈5 are normally allowed transitions, and