Question

A neutron moving with speed a makes a heat on collision with a hydrogen atom in ground state kept at rest .Find the minimum kinetic energy of the neutron for which inelastic (completely or partially) collision may take plane at the mass of neutron = mass of hydrogen `= 1.67 xx 10^(-27) kg`

Answer 1

The collision will be inelastic if a part of kinetic energy is used to excite the atom i.e. `K_(i)ne K_(f)`
By momentum conservation
`mv=mv_(1)+mv_(2)`
`v=v_(1)+v_(2)`
`(1)/(2)mv^(2)=(1)/(2)mv_(1)^(2)+(1)/(2)mv_(2)^(2)+DeltaE`
`v^(2)=v_(1)^(2)+v_(2)^(2)+(2DeltaE)/(m)`
`(v_(1)+v_(2))^(2)=v_(1)^(2)+v_(2)^(2)+(2DeltaE)/(m)`
`2v_(1) v_(2)=(2DeltaE)/(m) implies v_(1) v_(2)=(DeltaE)/(m)`
`(v_(1)-v_(2))^(2)=(v_(1)+v_(2))^(2)-4v_(1)v_(2)=v^(2)-(4DeltaE)/(m)`
As `(v_(1)-v_(2))` must be real
`(v_(1)-v_(2))^(2) ge 0`
`v^(2)-4 (DeltaE)/(m) ge0`
`(1)/(2) mv^(2) ge 2DeltaE`
Minimum K.E. to excite electron
`DeltaE=13.6((1)/(1^(2))-(1)/(2^(2)))=10.2 eV`
`((1)/(2) mv^(2))_(min) ge 2xx10.2`
`((1)/(2) mv^(2))_(min) ge 20.4 eV`
Minimum K.E. of neutron =20.4 eV