Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• If we neglect the neutrino’s mass, the total kinetic energy, K, released in the β– decay is given by (17.22). In particular, the criterion that β− decay is possible is just that K must be positive, or

 matom(Z, N) ≥ matom(Z + 1, N − 1) (for β− decay)

In other words, one has only to compare the atomic masses involved to see if β− decay is possible. (a) Find an expression for the total kinetic energy released in the β+ decay

 (Z, N) → (Z − 1, N + 1) + e+ + v

and show that the condition for this process to be possible is

(b) Do the same for the electron capture

assuming that the original two particles are at rest, and show that the condition for electron capture is the same as (17.84) except that the term 2me is missing. (c) The mass of 231U is 231.03626 u, while that of 231Pa is 231.03588 u. Show that 231U can decay by electron capture but not by β+ emission.