Question

Obtain the maximum kinetic energy of β-particles, and the radiationfrequencies of γ decays in the decay scheme shown in Fig. 13.6. Youare given thatm(198Au) = 197.968233 um(198Hg) =197.966760 u

Answer 1

Given: The mass of A 198 u is 197.968233 u and the mass of H 198 g is 197⋅966760 u.

The diagram for the decay is given as,

From the above diagram, the energy corresponding to γ 1 -decay is,

E 1 =1.088−0 =1.088 MeV =1.088×1.6× 10 −13  J

The frequency corresponds to γ 1 -decay is given as,

E 1 =h ν 1

Where, the frequency of radiation is ν 1 and Planck’s constant is h.

By substituting the given values in the above equation, we get

1.088×1.6× 10 −13 =6.626× 10 −34 × ν 1 ν 1 =2.627× 10 20  Hz

The energy corresponding to γ 2 -decay is,

E 2 =0.412−0 =0.412 MeV =0.412×1.6× 10 −13  J

The frequency corresponds to γ 2 -decay is given as,

E 2 =h ν 2

By substituting the given values in the above equation, we get

0.412×1.6× 10 −13 =6.626× 10 −34 × ν 2 ν 2 =9.949× 10 19  Hz

The energy corresponding to γ 2 -decay is,

E 3 =1.088−0.412 =0.676 MeV =0.676×1.6× 10 −13  J

The frequency corresponds to γ 3 -decay is given as,

E 3 =h ν 3

By substituting the given values in the above equation, we get

0.676×1.6× 10 −13 =6.626× 10 −34 × ν 3 ν 3 =1.632× 10 20  Hz

Thus, the frequencies corresponding to γ-decays are 2.627× 10 20  Hz, 9.949× 10 19  Hz and 1.632× 10 20  Hz respectively.

The energy of the highest level is given as,

E=[ m( A 78 198 u )−m( H 80 190 g ) ] c 2

By substituting the given values in the above equation, we get

E=[ 197.968233−197.966760 ] c 2 =0.001473 u× c 2 =0.001473×931.5 =1.3720995 MeV

The maximum kinetic energy of the β 1 particle is given as,

E β1 =1.3720995−1.088 =0.2840995 MeV

The maximum kinetic energy of the β 2 particle is given as,

E β2 =1.3720995−0.412 =0.9600995 MeV

Thus, the kinetic energies of β particles are 0.2840995 MeV and 0.9600995 MeV.