Formulas in the Fermi Theory of Beta-Decay, I

Abstract

28511-spectra are calculated by assuming the Fermi theory. All the effects due to the finiteness of the nuclear size are taken into account in an analytical form, and the screening effect by atomic electrons are included as additional factors though their values are not given explicitly. In ~ 2 the electron wave functions are calculated and in e 3 they are applied to get the correction factors for 11-spectra. The mixed type interaction consisting of scalar, tensor and pseudoscalar is especially examined. in detail. The final formulas are similar to those for light nuclei of Konopinski and Uhlenbeck. There is a difference, however, which lies in the fact that several "effective nuclear radii " are used in our formulas instead of the usual nuclear radius. Special attention is paid to the errors in our formulas, which seem to be inevitable from the practical view-point, and also to the insensitivity of the 11-spectra to the nuclear charge distribution. § I. IntroductionThe tJ-decay interaction first proposed by Fermill is a sum of products of the fourspinor field operators for the proton, neutron, electron and neutrino.The theory with.the above mentioned characteristics was extended afterwards to include all the possibilities. within the restriction of invariance under the proper and improper Lorentz transformations. This theory, which has been called "the Fermi theory of j3-decay ", is not onlysimple but also very powerful. There seemed to be no evidence of contradiction in it,. until recently the surprising experiment by Wu et ai.~l has revealed that the parity is. not conserved in the f9-decay. Although the original Fermi theory had to be abandoned in this respect, it is easy to extend it to explain the parity violation, still keeping the form of a sum of products of the four spinor field operators. Therefore, we still call this generalized theory the Fermi theory. Besides the Fermi theory a number of varieties.were proposed on the theory of tJ-decay but they either failed to explain experiments orwere too peculiar to be accepted widely. The Fermi theory is the ruling theory at present,. although we cannot neglect other theories which may be successful too. * On leave of absence from University of Tokyo. at Uniwersytet Warszawski Biblioteka Uniwersytecka on March 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from at Uniwersytet Warszawski Biblioteka Uniwersytecka on March 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from 2A. Unnormalized electron waye function inside the nucleus Eqs. (1) have suitable forms for obtainning the inner wave functions. The unnormalized (regular) solution of eqs. ( 1) is obtained by putting d1 = 0, d2 = 1 (these values are chosen only for convenience) for tc < b, and d 1 = 1, d 2 = 0 for tc > 0. The nuclear radius p is 0.01 to 0.02 in our unit for most nuclei, though the definition is somewhat ambiguous. The electron energy W is usually less than 10. V(r) is of the order of aZjp, and aZ is less than unity. If we regard W and aZ as of the order of unity, and r as of the order...

Formulas in the Fermi Theory of Beta Decay. II

“…(13) and (14) of Ref. 10) for arbitrary spherically symmetric charge distribution, and the large differences between the wave functions for the uniform and surface charge distributions and those for the point nuclear charge have been shown graphically in Figs. 3a-6b.…”

“…(5) of Ref. 10). If we use (3) and (5), the asymptotic forms of (1) at infinity are written as rf/C(r) "'"'-,,/WW isin(pr+(It), rglC(r)"'" J W;l cos (pr+C/C) .…”

“…Tl-, correction factor for the first forbidden transition is similarly expressed as: (10) (M, M) also gives the expressions for triple cascade transition.…”

Gross Theory of First Forbidden β-Decay

“…Several authors have investigated this effect. 35 >' 40 ), 4 ),e), 9 ) According to the theory of Matumoto and Yamada,35 ) this effect considerably reduces the radial dependence of the dipole operator as far as matrix elements involving M 0 and N 0 are concerned. This reduction will make the effective /1-decay operator different from the dipole one, and then the norm of the strength function will become somewhat smaller.…”

“…In order to take consideration of the finite-nuclear-size effect, we adopt the method of effective nuclear radius 35 ) in a simplified way. As far as the gross theory is concerned, the details are not necessary, so that we put 11 (r) =12 (r) = 1, F1 (p) /Fo (p) = 1 and neglect the screening effect in Eq.…”