HALF-LIFE OF H³ AND THE NEUTRINO MASS

In 1940 O’Neal and Goldhaber¹ determined that the half-life of H³ is \(31 \pm 8\) years. As follows from more recent work²˒³, this value is considerably overestimated. Goldblatt, Robinson, and Spence investigated the change with time in the ionization-chamber current from a mixture of ordinary hydrogen and H³ (tritium).

Measurements were made every 18 days. The dependence of the logarithm of the voltage drop on time is represented by a straight line, from whose slope the half-life was determined. However, the authors were unable to take into account the errors resulting from the absorption of tritium in the chamber and from the slight increase in pressure arising when tritium molecules are converted into He³ atoms. The half-life reported by the authors is \(10.7 \pm 2.0\) years.

Novick studied, on two samples, the formation of He³ from tritium. Isotopic analysis of the samples was carried out twice. One of the methods consisted in converting a known volume of hydrogen into water. The water was weighed, and in this way the ratio of tritium to hydrogen was determined, with atomic weights 3 and 1 respectively assigned to them. The other method consisted in comparing the intensities of the \(T_\alpha\) and \(H_\alpha\) lines of the radiation spectrum. The results of the analyses agreed to within 5% and corresponded to data obtained by measuring the ionization-chamber current of the \(\beta\)-activity of each sample. The ratio \(T_2/(H_2+T_2)\) was 0.71 for one sample and 0.74 for the other. After a certain time had elapsed, He³ was separated from the hydrogen. For complete purification, the helium was passed through copper oxide at a temperature of \(400^\circ\) C and through a trap with activated charcoal at the temperature of liquid nitrogen. The helium obtained was examined spectroscopically; there were no traces of He⁴ in the emission spectrum. In the first sample, after 51 days, the ratio of tritium to He³ was 0.00815, and in the second sample, after 197 days, it was 0.03025. The half-life for the first sample

HALF-LIFE OF H$^3$ AND THE NEUTRINO MASS

was 11.85 years, and for the second 12.35. Thus the authors obtained for the half-life of tritium \(12.1 \pm 0.5\) years.

The maximum energy of the electrons in the \(\beta\)-decay of tritium is unusually small and is equal to \(11 \pm 2\) KeV. Konopinski\(^5\) used this circumstance to determine the mass of the neutrino. As is known\(^6\), the product \(|M|^2 f(E_0)T\) is constant for all allowed \(\beta\)-transitions. Here \(|M|^2\) is the so-called nuclear matrix element, \(T\) is the half-life in seconds, and

\[ f(E_0)=\int_0^{E_0} dE\,(1+E)(2E+E^2)^{1/2}(E_0+\mu-E)(E_0-E)^{1/2}(2\mu+E_0-E)^{1/2}. \]

where \(\mu\) is the ratio of the rest mass of the neutrino to the electron mass. \(E_0\) is expressed in units of \(mc^2\), and in the present case \(E_0=0.0215\).

For comparison the author uses He\(^6\), regarding it as a specimen of allowed \(\beta\)-transitions. The half-life of He\(^6\) is 0.8 sec., and the maximum energy \(E_0=7.25\).

Because of the large energy, the role of \(\mu\) in the estimate \(f(\mathrm{He}^6)=1200\) is negligibly small. For \(|M|^2\) the value 6 is adopted\(^6\). Thus for (He\(^6\)) one obtains \(|M|^2 fT=5760\) sec. Similar values are obtained for all allowed \(\beta\)-transitions. The expected value of \(|M|^2\) for tritium is 3.

If one assumes that the neutrino has no mass, then

\[ f(\mathrm{H}^3)\simeq 0.216\,E_0^{7/2}=3.2\cdot 10^{-7}. \]

It follows that \(|M|^2 fT=350\) sec.

This value is too small. Under this assumption, for the half-life of tritium one obtains a time of the order of \(\sim 200\) years.

Let us suppose that the rest mass of the neutrino is \(\mu m\) and that \(|M|^2 fT\) for H\(^3\) is also of the order of \(\simeq 5700\) sec.

If

\[ E_0<\mu<1,\quad \text{then}\quad f\simeq 5\left(\frac{\pi}{4}\right)\mu^{3/2}E_0^2\left(1+\frac{5E_0}{8\mu}+\cdots\right). \]

Konopinski carries out calculations for two cases, assuming that the half-life is \(\simeq 20\) and 30 years, and correspondingly obtains \(\mu \simeq 1/30\) and \(1/45\). Taking the new data for the half-life, we find that \(\mu \sim 1/15\).

It is necessary to note that the rest mass of the neutrino has been determined from very simple formulas. If one does not take into account the factor 2 arising from the change in the magnitude of \(|M|^2\), then the only theoretical assumption was the statistical formula for \(f\), expressing the law of distribution of energy between the electron and the neutrino.

G. M. Budyanskii

CITED LITERATURE

1. O’Neal a. Goldhaber, Phys. Rev. 58, 574 (1940).
2. Goldblatt, Robinson a. Spence, Phys. Rev. 72, 973 (1947).
3. Novick, Phys. Rev. 72, 972 (1947).
4. Watts a. Williams, Phys. Rev. 70, 640 (1946).
5. Konopinski, Phys. Rev. 72, 518 (1947).
6. Konopinski, Rev. Mod. Phys. 15, 209 (1943).