Beta decays take place when the ratio of protons and neutrons is not optimal. Beta decays tend to allow the nucleus to approach the optimal proton/neutron ratio. When there are too many neutrons related to the protons, negative beta decay occurs; when there are too many protons related to the neutrons, positive beta decay takes place. As a result of beta decays, the mass number of the atoms remains the same, but the atomic number changes: the atomic number increases in the negative beta decay and decreases in the positive beta decay, respectively. Besides the beta particle, another particle is also emitted: antineutrino in the nega­tive beta decay and neutrino in the positive beta decay.

Negative beta decay: | + 1M 1 в-+ V (4.99)

Positive beta decay: A-M! A — 1M 1 в+ 1 v (4.100)

In Eqs. (4.99) and (4.100), в — are в+ are the negative and positive beta particles, i. e., electrons and positrons. It is important to note that the term “beta particles” means only electrons (positive or negative) emitted from nuclei. Electrons emitted from the extranuclear shell are called “electrons” and designed by e2.

Similar to alpha decay, the emitted energy of beta decays can be calculated from the rest masses of the parent and daughter nuclide plus the emitted particles:

Negative beta decay: E = (£m — A + 1M) 931 MeV (4.101)

Positive beta decay: E = (AM — A __ 1M — 2me) 931 MeV (4.102)

The rest mass of the neutrino can be ignored because its rest mass is about 10,000 times lower (150 eV at most) than the rest mass of the electron (0.51 MeV). As seen in Eqs. (4.101) and (4.102), besides the differences between the rest masses of the parent and daughter nuclides, there are differences between the rest masses of two electrons since the increase of the atomic number in the negative beta decay requires the uptake of another electron, while the decrease of the atomic number in the positive beta decay causes the emission of another electron. This means that positive beta decay can take place only if the rest mass of the parent nuclide is at least two electron masses (1.02 MeV) heavier than the rest mass of the daughter nuclide.

Since the radioactive decay always releases energy (in the exothermic process), it takes place only if the rest mass of the parent nuclide is greater than the rest mass of the daughter nuclide + the emitted particle(s). (As mentioned previously, the rest mass of the neutrino can be ignored.) For negative beta decay, this can be

expressed as:

Am — Zme > A + jM — (Z 1 1)me 1 me (4.103)

Similarly, for positive beta decay:

AM — Zme > A — 1M — (Z — 1)me 1 me (4.104)

The solution of Eqs. (4.103) and (4.104) is:

AM > A і M (4.105)

Am > A — 1M 1 2me (4.106)

As seen in Eqs. (4.105) and (4.106), the differences in the rest masses give dis­crete values for the emitted energy. The spectrum of the beta radiation, however, is continuous (Figure 4.10), and the calculated energy is equal to the maximum energy. (The electrons with discrete energy are emitted from the electron shells.)

The continuous beta spectra can be interpreted by the two emitted particles, the beta particle and the neutrino. The energy of beta decay is divided into two parts: both beta particles and neutrinos have some energy. The emission of two particles explains the changes of the spin of the nucleus as a result of the decay: the spin of the nucleus changes by 1, the spin of both beta particle and neutrino is 1/2 (see Table 2.3).

Figure 4.10 General shape of beta spectra: the number of beta particles with a given energy (N(E)) versus beta energy (E).

The elementary process of the beta decay can be described as follows:

Negative beta decay:

n! p+ + (Г+ V (4.107)

Positive beta decay:

p1! n 1 (3+ + v (4.108)

It is important to note that the processes in Eqs. (4.107) and (4.108) do not mean the free nucleons, but bound in the nucleus. Since the rest mass of the neutron is larger than the rest mass of the proton, the difference of masses in the process of Eq. (4.107) produces energy. The negative beta decay is obviously exothermic. In positive beta decay, however, a proton is transformed to a neutron. This requires energy because of the differences between the rest masses (1.3 MeV; see Table 2.1), which is provided by the decrease of the mass of the nucleus. In addition, the emittion of the positron requires more 0.51 MeV energy, which is also to be provided by the decrease of the mass of the nucleus. The sum of the two energies is 1.8 MeV.

The neutrino emitted in the beta decays cannot be detected directly because it is neutral and its rest mass is very small. However, because of the conservation of lin­ear momentum at beta decay, the momentum vectors (i. e., the pathways of the par­ticles) of the daughter nuclide and the beta particle should be at an angle of 180°. However, as photographed in a cloud chamber in the beta decay of 6He by Csikai and Szalay in 1957 (Figure 4.11), another particle (neutrino) has to be released during the decay as well.

The antineutrino can be detected using the following reaction: p 1V! n 1 в1 (4.109)

Since the cross section of the reaction (4.109) is very low (as discussed in Chapter 6), the high flux of antineutrinos is required similar to those present in nuclear reactors. When an aqueous solution of CdCl2 is placed into a nuclear

reactor, antineutrinos react with the protons of water in the reaction (4.109). The two products, namely, the positive beta particle and the neutron, can be detected simultaneously in the following way. The positive beta particles and electrons are annihilated, and as a result, photons of 0.51 MeV are emitted (see Section 5.3.3). The neutrons are thermalized in a few microseconds and initiate the nuclear reac­tion 113Cd(n, Y)114Cd. The gamma photons emitted in this nuclear reaction of 113Cd follow the emission of the photons with 0.51 MeV after a few microseconds. The two photons can be detected by coincidence measurements.

In beta decays, the nuclei usually emit one beta particle. However, two beta par­ticles are emitted in a single process in some cases. This process is called double beta decay. Theoretically, two types of double beta decays can exist: in the first, two beta particles and two neutrinos are emitted [(3(3(vv)], in the other, only two beta particles (no neutrinos) are formed [(3(3(0v)]. In the first case, the two neutrinos annihilate each other; and in the second, the emitted neutrino is absorbed by another one.

Decay products of the double beta decay [(3(3(vv)] (by extraction of crypton and xenon from very old selenium and tellurium minerals) in geological samples were detected in 1950. Under laboratory conditions, double beta decay was observed in 1986 when the double beta decay of 82Se was measured:

82Se!82Kr 1 2e2+ 2v (4.110)

In the laboratory experiments, 1.1 X 1020 years was obtained for the half-life of the double beta decay of 82Se. This value is similar to the results obtained in geochemical measurements.

More than 60 naturally occurring isotopes are capable of undergoing double beta decay. Only 10 of them were observed to decay via the two-neutrino mode: 48Ca, 76Ge, 82Se, 96Zr, 100Mo, 116Cd, 128Te, 130Te, 150Nd, and 238U.

The neutrinoless double beta decay [(3(3(0v)] has not been demonstrated beyond any doubt.