For the `beta^(+)` (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K–shell, is captured by the nucleus and a neutrino isemitted).
`e^(+)+""_(Z-1)^(A)Y+V`
Show that if `beta^(+)` emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

For one-electron species, the wave number of radiation emitted during the transition of electron from a higher energy state (n_(2)) to a lower energy state (n_(1)) is given by: bar v =(1)/(lamda)=R_(H) xx Z^() ((1)/(n_(1)^(2)) (1)/(n_(2)^(2))) where R_(H)=(2 pi m_(s) k^(2) c^(4))/(h^(3) c) is Rydberg constant for hydrogen atom. Now, considering nuclear motion, the accurate measurement would be obtained by replacing mass of electron (m_(e)) by the reduced mass (mu) in the above expression, defined as mu =(m_(n)xx m_(e))/(m_(n) +m_(e)) where m_(n) = mass of nucleus. For Lyman series, n_(t) =1 (fixed for all the lines) while n_(2) = 2, 3, 4 .... For Balmer series: n_(1) = 2 (fixed for all the lines) while n_(2) = 3,4,5 .... If proton in hydrogen nucleus is replaced by a positron having the same mass as that of an electron but same charge as that of proton, then considering the nuclear motion, the wavenumber of the lowest energy transition of He+ ion in Lyman series will be equal to

For one-electron species, the wave number of radiation emitted during the transition of electron from a higher energy state (n_(2)) to a lower energy state (n_(1)) is given by: bar v =(1)/(lamda)=R_(H) xx Z^() ((1)/(n_(1)^(2)) (1)/(n_(2)^(2))) where R_(H)=(2 pi m_(s) k^(2) c^(4))/(h^(3) c) is Rydberg constant for hydrogen atom. Now, considering nuclear motion, the accurate measurement would be obtained by replacing mass of electron (m_(e)) by the reduced mass (mu) in the above expression, defined as mu =(m_(n)xx m_(e))/(m_(n) +m_(e)) where m_(n) = mass of nucleus. For Lyman series, n_(t) =1 (fixed for all the lines) while n_(2) = 2, 3, 4 .... For Balmer series: n_(1) = 2 (fixed for all the lines) while n_(2) = 3,4,5 .... The ratio of the wave numbers for the highest energy transition of electron in Lyman and Balmer series of hydrogen atom is