JEE Physics/ Lakshmikanta Satapathy/ Beta decay of a nucleus and the subsequent emission of a Gamma ray photon with the related concepts of mass defect and excitation energy

1. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
79
β2
β1
Au198

Hg
1.088 MeV
0.412 MeV
12 0
80
198
-Decay of Nucleus

2. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
79
1β
β2
Au198

80 Hg198
1.088 MeV
0.412 MeV
12

Question :
Two gold nuclei undergo β-decay and
convert to mercury in ground state
through different excited states with
subsequent emission of -rays as per the
decay scheme shown in the figure. Find
(a) The maximum possible kinetic
energies of the β - particles
(b) the frequencies of the  - rays.
Given : The atomic masses are
198
79( ) 197.968233m Au u
198
80( ) 197.966760m Hg uand

3. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
Concepts
Eqn. for β – decay [electron emission] :  
  1
A A
Z ZX Y
Energy is released when ( Final Mass ) < ( Initial Mass )
Decrease in 1 amu of mass releases 931.5 MeV of energy
Decrease in mass = Initial mass – Final mass ( = ∆m )
 Energy released = ∆m  931.5 MeV
Planck’s constant 
  34
6.625 10 .h J s

4. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
 KE of β particle = Energy released – Excitation energy
Here ∆E is to be expressed in JOULE using
Part of the energy released is retained by the product nucleus as excitation
energy and the remaining energy is emitted in the form of a -ray.
A -ray photon is emitted when the product nucleus jumps down from excited
state to ground state.
 Frequency of  - ray is 

 
' .
changeinenergy E
f
Planck s Const h

  13
1 1.6 10MeV J

5. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
Initial mass = mass of the decaying nucleus
     198 198
79 80[ ( ) 79 ] [ ( ) 80 ]e e em m Au m m Hg m m
  197.968233 197.966760 0.001473u
 Energy released = 0.001473  931.5 = 1.372 MeV
 198 198
79 80( ) ( )m Au m Hg
For both nuclei , the decrease in mass is the same
Final mass = mass of the product nucleus + mass of -particle
 198
79( ) 79 em Au m
  198
80( ) 80 e em Hg m m

6. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
For the 1st nucleus energy released = 1.372 MeV


  
   

13
20
1 34
1.088 1.6 10
2.627 10
6.62
[
5 10
]
E
f Hz Ans
h
 KE of β1 = 1.372 – 1.088 = 0.284 MeV [Ans]
Its excitation energy = 1.088 MeV
When the nucleus jumps down to ground state ,
energy of the emitted -ray = the excitation energy
 Frequency of the emitted -ray is given by

7. Physics Helpline
L K Satapathy
Modern Physics - 1
-Decay of Nucleus
 KE of β2 = 1.372 ─ 0.412 = 0.960 MeV [Ans]


  
   

13
20
2 34
0.412 1.6 10
0.995 10
6.62
[
5 10
]
E
f Hz Ans
h
For the 2nd nucleus also, energy released = 1.372 MeV
Its excitation energy = 0.412 MeV
When the nucleus jumps down to ground state ,
energy of the emitted -ray = the excitation energy
 Frequency of the emitted -ray is given by

8. Physics Helpline
L K Satapathy
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