Angular Momentum of Nucleus | Nuclear and Particle Physics for GATE - GATE Physics PDF Download

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1(d). Angular Momentum of Nucleus 
The hydrogen nucleus
1
1
H consists of a single proton and its total angular momentum is 
given by ?
2
3
? S . A nucleon in a more complex nucleus may have orbital angular 
momentum due to motion inside the nucleus as well as spin angular momentum. The total 
angular momentum of such a nucleus is the vector sum of the spin and orbital angular 
momenta of its nucleons, as in the analogous case of the electrons of an atom.  
When a nucleus whose magnetic moment has z component 
z
? is in a constant magnetic 
field B , the magnetic potential energy of the nucleus is  
Magnetic energy            B U
z m
? ? ? 
The energy is negative when 
z
? is in the same direction as B and positive when 
z
? is 
opposite to B . In a magnetic field, each angular momentum state of the nucleus is 
therefore split into components, just as in the Zeeman Effect in atomic electron states. 
Figure below shows the splitting when the angular momentum of the nucleus is due to the 
spin of a single proton.  
  
 
 
 
 
 
 
 
 
down Spin
up Spin
2
1
? ?
s
m
2
1
? ?
s
m
0
E
B E
pz
? ?
0
B E
pz
? ?
0
B E
pz
? 2 ? ?
0 ? B 0 ? B
Figure: The energy levels of a proton in a magnetic field are split into spin-up 
and spin-down sublevels.   
Page 2


 
   
 
  
 
    
 
 
1(d). Angular Momentum of Nucleus 
The hydrogen nucleus
1
1
H consists of a single proton and its total angular momentum is 
given by ?
2
3
? S . A nucleon in a more complex nucleus may have orbital angular 
momentum due to motion inside the nucleus as well as spin angular momentum. The total 
angular momentum of such a nucleus is the vector sum of the spin and orbital angular 
momenta of its nucleons, as in the analogous case of the electrons of an atom.  
When a nucleus whose magnetic moment has z component 
z
? is in a constant magnetic 
field B , the magnetic potential energy of the nucleus is  
Magnetic energy            B U
z m
? ? ? 
The energy is negative when 
z
? is in the same direction as B and positive when 
z
? is 
opposite to B . In a magnetic field, each angular momentum state of the nucleus is 
therefore split into components, just as in the Zeeman Effect in atomic electron states. 
Figure below shows the splitting when the angular momentum of the nucleus is due to the 
spin of a single proton.  
  
 
 
 
 
 
 
 
 
down Spin
up Spin
2
1
? ?
s
m
2
1
? ?
s
m
0
E
B E
pz
? ?
0
B E
pz
? ?
0
B E
pz
? 2 ? ?
0 ? B 0 ? B
Figure: The energy levels of a proton in a magnetic field are split into spin-up 
and spin-down sublevels.   
 
   
 
  
 
    
 
 
The energy difference between the sublevels is  
                                                           B E
pz
? 2 ? ? 
A photon with this energy will be emitted when a proton in the upper state flips its spin to 
fall to the lower state. A proton in the lower state can be raised to upper one by absorbing 
a photon of this energy. The photon frequency 
L
? that corresponds to E ? is  
Larmor frequency for photons              
h
B
h
E
pz
L
?
?
2
?
?
? 
This is equal to the frequency with which a magnetic dipole precesses around a magnetic 
field.