In a geiger - marsden experiment. Find the distance of closest approach to the nucleus of a 7.7 me v `alpha`- particle before it comes momentarily to rest and reverses its direction. (z for gold nucleus = 79) .

Explanation

The key idea here is that throughout the scattering process, the total mechanical energy of the system consisting of an `alpha`-particle and a gold nucleus is conserved. The system's initial mechanical energy is `E_(i)` before the particle and nucleus interact, and it is equal to its mechanical energy `E_(f)` when the `alpha`-particle momentarily stops. The initial energy `E_(t)` is just the kinetic energy K of the incoming `alpha`-particle. The final_ energy `E_(f)` is just the electric potential energy U of the system. The potential energy U can be calculated from Equation.
`F=(1)/(4pi epsi_(0))((2e)(Ze))/(r^(2))`
Let d be the centre-to-centre distance between the `alpha`-particle and the gold nucleus when the `alpha`-particle is at its stopping point. Then we can write the conservation of energy
`E_(i)=E_(t)` as
`K=(1)/(4pi epsi_(0))((2e)(Ze))/(d)=(2Ze^(2))/(4pi epsi_(0)d)`
Thus the distance of closest approach d is given by
`d=(2Ze^(2))/(4pi epsi_(0)K)`
The maximum kinetic energy found in `alpha`-particles of natural origin is 7.7 MeV or `1.2xx10^(-2)J`. Since `1//4 pi epsi_(0)=9.0xx10^(9)N m^(2)//C^(2).` Therefore with `e=1.6xx10^(-19) C,` we have,
`d=((2)(9.0xx10^(9) Nm^(2)//C^(2))(1.6xx10^(-19)C)^(2)Z)/(1.2xx10^(-12)J)`
`=3.84xx10^(-16) Zm` The atomic number of foil material gold is Z = 79, so that
d (Au) `=3.0xx10^(-14)m=30 fm. ("1 fm (i.e. fermi) "=10^(-15)m)`
The radius of gold nucleus is, therefore, less than `3.0 xx 10^(-14) m`. This is not in very good agreement with the observed result as the actual radius of gold nucleus is 6 fm The cause of discrepancy is that the distance of closest approach is considerably larger than the sum of the radii of the gold nucleus and the `alpha`-particle. Thus, the `alpha` particle reverses its motion without ever actually touching the gold nucleus.