2 e

and acquires kinetic energy

K = 2 e (20 kV) = 40 keV

. For Coulomb head-on closest approach with target recoil, the initially stationary beryllium nucleus is free to recoil. Approximating the nuclear masses by

4 u

and

9 u

, the initial relative-motion kinetic energy is

K_{*} = \frac{m _{Be}}{m _{α} + m _{Be}} K = \frac{9}{13} (40 keV) = 27.6923 keV .

(1)

At closest separation the relative radial velocity vanishes, although the centre of mass continues moving. Conservation of energy for the relative motion gives

\frac{( 2 e ) ( 4 e )}{4 π ε _{0} r _{m i n}} = K_{*}, r_{m i n} = \frac{8 e ^{2}}{4 π ε _{0} K _{*}} ≃ 4.16 \times 1 0^{- 13} m .

(2)

Using

e^{2} / (4 π ε_{0}) = 1.43996 MeV fm

gives

r_{m i n} ≃ 416 fm

. If the nucleus were externally held fixed, the relevant energy would be

K

, yielding

288 fm

instead. Initially stationary is not the same as constrained stationary; the given proton and neutron counts supply the recoil correction. Nuclear sizes and strong interactions are negligible at this separation.