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For the b+ (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 is emitted). e + ""_(Z)^(A)X to ""_(Z - 1)^(A)Y + v Show that if beta+ emission is energetically allowed, electron capture is necessarily allowed but not vice–versa. |
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Answer» Solution :CONSIDER the competing processes : `""(Z)^(A)RARR""(Z-1)^(A)Y+e^(+)+v_(e)+Q_(1)"(poitron emission)"` `e^(-)+""(Z)^(A)Xrarr""(Z-1)^(A)Y+v_(e)+Q_(2)"(electron capture)"` `Q_(1)=[m_(N)(""(Z)^(A)X)-m(N)(""(Z-1)^(A)Y)-m(e)]c^(2)` `=[m(""(Z)^(A)X)-Zm(e)-m(""(Z-1)^(A)Y)-(Z-1)m(e)-m_(e)]c^(2)` `[m(""(Z)^(A)X)-m(""(Z-1)^(A)Y)-2m_(e)]c^(2)` `Q_(2)=[m_(N)(""(Z)^(A)X)+m(e)-m_(N)(""(Z-1)^(A)Y)]c^(2)=[m(""(Z)^(A)X)-m(""(Z-1)^(A)Y)]^(2)` This means `Q(1) GT 0` implies `Q_(2) GT0` but `Q_(2) gt0` does not necessarily mean `Q_(1)gt0`. Hence the result. |
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