SEPTEMBER 15, 1935 PHYSICAL REVIEW VOLUM E 48 Double Beta-Disintegration M. GOFPPERT-MAvER, The Johns Hopkins Um'versity (Received May 20, 1935} From the Fermi theory of P-disintegration the probability of simultaneous emission of two The result is that this process occurs suffi- electrons (and two neutrinos) has been calculated. ciently rarely to allow a half-life of over 10" years for a nucleus, even if its isobar of atomic different by 2 were more stable by 20 times the electron mass. number only isobars of even difference in atomic number INTRODUCTION occur. N a table showing atomic nuclei the existing A metastable isobar can, however, change into - - it is observed that of isobars many groups a more stable one by simultaneous emission of occur, the term isobar referring to nuclei of the two electrons. It is generally that the assumed same atomic weight but different atomic number. of such a process frequency is very small. In It is unreasonable to assume that all isobars have of a disintegration this paper the propability one of them therefore exactly the same energy; of that kind has been calculated. will have the lowest energy, the others are un- The only method to attack processes involving stable. The question arises the unstable why is that of the emission of electrons from nuclei that is, why, io nuclei are in reality metastable, Fermi' which associates with the emission of an geologic time, they have not all been transformed electron that of a neutrino, a chargeless particle into the most stable isobar by consecutive p-dis- mass. Thereby it is possible to ex- of negligible integrations. to plain the continuous P-spectrum and yet The explanation has been given by Heisen- have the conserved in each individual energy lies in the fact that of berg' and the energies process adjusting the momentum of the by of fixed atomic nuclei weight, plotted against In the treatment of a neutrino. this theory atomic number, do not lie on one smooth curve, P-disintegration is very similar to that of the but, because of the peculiar stability of the emission of light by an excited. atom. cx-particle are distributed alternately on two emis- A disintegration with the simultaneous smooth curves, displaced by an approximately sion of two electrons and two neutrinos will then constant amount against each other (the mini- be in strong to the Raman effect, or, analogy mum of each curve is therefore at, roughly, the even more closely, to the simultaneous emission of two light quanta, ' and same atomic number). For even atomic weight can be calculated in the nuclei of even atomic number lie on the lower from the essentially the same manner, namely, curve, those odd atomic number on the with second-order terms in the perturbation theory. a one. One P-disintegration then brings higher oc- The process will appear as the simultaneous from a point nucleus on the lower curve into currence of two transitions, each of which does one of the upper curve, or vice versa. The nuclei the of conservation of energy not fulfill law on the curve are all of them unstable. upper separately. But it may happen that a nucleus on the lower is a calculation The following investigation of curve, in the neighborhood of the minimum, due to the inter- the second-order perturbation, even though it is not the most stable one, cannot by Fermi action potential introduced between emit a single p-particle, since the resuitant isobar, electrons neutrinos. neutrons, protons, and As whose energy lies on the upper curve, has higher far as the notation used is that of possible energy. This nucleus would then be metastable, Fermi. For a more detailed discussion and justi- since it cannot go over into a more stable one by the as- fication of this mathematical form and consecutive emission of electrons. This two be made to sumptions involved reference must is borne out by the fact that almost explanation Fermi's paper. 2 E. Fermi, Zeits'. f. Physik SS, 161 (1934}. ' M. Goeppert-Mayer, ' W. Heisenberg, Ann. d. Physik (V) 9, 2/3 (1931). Zeits, f. Physik 78, 156 (1932}. 512
DOU B LE 8 ETA — D I SI NTEG RATION 2. THE transition from neutron to proton MATHEMATICAL APPARATUS is necessarily accompanied by the emission of an electron and The nucleus not to contain is assumed any of II a neutrino and vice versa. A matrix element electrons and neutrinos but to be built up out of to the transition of a neutron corresponding with neutrons and protons only. Neutron and proton eigenfunction I„ to a proton with eigenfunction are regarded as not essentially different from one is different from 0 only if at the same time two v but to represent another, two different quantum numbers N„M, change from 0 to 1, and is then states of the heavy particle. The two kinds of given by particles outside of the nucleus, the elec- light according to the trons and neutrinos are treated nl ~ ~ 0 ~ ~ ~ 0 ~ ~ ~ IIm". 1s "lo' IM of superquantization. The stationary method states of the electrons are taken to be those of — l )Ny+ "+Pa — i+My+ "+M~ l, t, g V s~'Po' » nmy rr 0 — (2) ( energy II, in the Coulomb field of the positive nucleus, described four Dirac functions by with the abreviation P. = g, ', P, ', P, ', P, '). Since the neutrinos are not $41+82+p42+vl+p43 84 ys4 Qg (3)Q affected by the field of the nucleus their eigen- functions are represented by plane Dirac waves H. =g J's *u. dr. energies Z, . The p, =(p, l, rp, ', y, 3, y, 4) with The functions q are taken at the place of to for both P and Pauli principle is assumed hold electrons and neutrinos, so that the number N, of the nucleus, the assumption that being made do not vary electrons in a stationary state s and the number they considerably over its range. 3'. of neutrinos J'v *N„dr, taken o. can be For an "allowed" transition in a stationary state over the volume of the nucleus, 0 or 1 only. has the order of 1. The value state N of the total system is then magnitude of the proportionality A quantum by these numbers N„3f, and by the constant determined g can be roughly obtained by equating state n, with energy 8'„of the nucleus; calculated quantum and observed intensity of P-disinte- states gration. Fermi determines it to be n means in this case not only the quantum sense but includes in the ordinary the quantum g=4&10 "cm' ergs. states of all possible isobars. The energy of a state N is given by quantum 3. CALCULATIONS Z~ — — PN. H, + P M. E. + W. . ' The probability of simultaneous emission of two electrons and is obtained two neutrinos The interaction the heavy energy between par- from the second order terms of the perturbation ticles within the nucleus and the light particles In this approximation theory. the amplitude of in such a way that without is constructed the the transition probability is given by (E&yg — e(2~i/h) (L~'~ — NII K E~) t e(2~i/h) JlN) t 1 (6) E~ — E~ We want to consider of such a type transitions is a quantum state of the isobar nucleus with state, that in the beginning N, no electrons or atomic number larger by 1 than the original state, 3', the neutrinos are present; in end nucleus. The that P-dis- assumption ordinary the charge of the nucleus has been increased by integration is energetically impossible means that S'I, „— 8"/, — S' & — — 2, that is two neutrons have been transformed mc'; the first denomina- tor Z~ — into protons, and two electrons have been emitted E~ in therefore be (6) will always into states s and t, two neutrinos into states 0. positive. The process of double P-disintegration state E then W „ 7. . The intermediate must be and however, if is, energetically permissible = W — S' & — such that one neutron has changed into a proton, 2mc'. In this case the second have been emitted. one electron and one neutrino This means that the intermediate state k of the ~ A star denotes the conjugate complex of a quantity.
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