W iso ¼ 8p=3 w 0
ð Þ
2 l ez l nz
where w 0 is the wave function evaluated on the nucleus.
l ez and l nz are the components of electronic and nuclear magnetic moments
along the H direction.
4.3.2 Hyperfine Interaction by Quantum Mechanical
Approach
By using quantum mechanical operators:
M ez ¼ ÀgbS z
l nz ¼ Àg n b n I z
H ¼ 8p=3gbg n b n w 0
ð Þ
2 S z I z ¼ hA 0 S z I z
where A 0 is the isotropic hyperfine coupling constant (hertz). In the case of the
hydrogen atom, the complete Hamiltonian operator is
H ¼ gb HS z þ hA 0 S z I z
By associating the eigenvalues of S z and of I z to the four spin and nuclear states
M S ¼ 1=2 a e M S ¼ À1=2 b e
M I ¼ 1=2 a N M I ¼ À1=2b N
four different states are generated a e a N
ð
Þ b e a N
ð
Þ a e b N
ð
Þ b e b N
ð
Þ
Operating by the H operator on each one of the states
W ae aN ¼ \a e a N jgbHS z þ hA 0 S z I z j e a N [ ¼ 1=2gbH þ 1=4 hA 0
W aebN ¼ 1=2gbH À 1=4hA 0
W bebN ¼ À1=2gbH þ 1=4hA 0
W beaN ¼ À1=2gbH À 1=4hA 0
The microwave-induced energy transition undergoes the following selection
rules
DM s Æ 1DM I ¼ 0
The energy transition is consequently
76
4 Magnetism