68 ISOTOPIC TRACERS AND NUCLEAR RADIATIONS [Chap. 3 



spend more time within and near the nucleus than do the L and M electrons 

 and, hence, have a higher probability for capture when the process is ener- 

 getically possible. K capture leads to a nucleus one charge unit less and to an 

 atomic weight that is smaller than the initial atom by only the mass equiva- 

 lence of the increase in binding energy. 



The energy of the absorbed K electron has a definite value which is given 

 by its rest energy minus its atomic binding energy, i.e., E = (1 — a 2 Z 2 )^ 

 in units of m c 2 . Thus the neutrino must be ejected with a kinetic energy 

 just equal to the sum of the total electron energy, and the nuclear transition 

 energy and is therefore monoenergetic. Its value is given by 



E n = Ez — -E(z-i) + E e — 1 = E + E e 



where E = transition energy in units of m c 2 



Ez = energy equivalent of exact atomic weight of initial nucleus in 

 units of m c 2 

 -E(z-i) = energy equivalent of exact atomic weight of final nucleus in 



units of m c 2 

 Although K capture frequently competes with positron emission in the 

 same transition, it is in some instances the only process energetically possible. 

 Thus, only K capture is possible if the transition involves less energy than 

 that equivalent to the rest mass of an electron (0.5 mev), i.e., when 



1 — E c < E z — E(z-i) < 2 (units of m c 2 ) 



However, both positron emission and K capture are possible when E > 1. 



The theoretical treatment and calculation of mean life and the transition 

 probability is essentially the same as for beta decay. The probability of a 

 transition depends principally, as in beta decay, on the accompanying change 

 in total angular momentum AJ of the nucleus, i.e., A/ = 0, allowed; ±1, 

 first forbidden; ±2, second forbidden. The mean life is given by [15,19] 



I- j! 



T 27T 3 



' \Q\ 2 h 



where Q = matrix element (~ 1 for light elements; see Beta Decay, Sec. 3.11) 



g = constant 

 For allowed transitions [15], A/ = 0, the function fk is approximately 



f k « 2x(aZ) 3 (£ + l) 2 



The probabilities of the first forbidden to the allowed transition will have 

 the ratio (aZ/2) 2 , and successive higher orders of forbiddenness will have 

 probability ratios of (E /R) 2 , where a is the fine structure constant, K37> and 

 R is the nuclear radius. 





