184 5.4 NMR and Other Radio Frequency and Microwave Resonance Spectroscopies
Typical values of γ are equivalent to ~107 T−1 s−1 but are often quoted as these values divided
by 2π and are given for a few atomic nuclei in Table 5.2. The bulk magnetization M of a
sample is the sum of all the atomic nuclear magnetic moments, which average out to zero in
the absence of an external magnetic field.
However, in the presence of an external magnetic field, there is a nonzero net magnet
ization, and each atomic nuclear magnetic state will also have a different energy E due to
the coupling interaction between the B-field and the magnetic moment (also known as the
Zeeman interaction), which is given by the dot product of the external magnetic field B with
the atomic nucleus magnetic moment:
(5.17)
E
B
B
B hm
m
z
z
z
= −⋅
= −
= −
µ
µ
γ
π
2
Therefore, the presence of an external magnetic field splits the energy into (2I + 1) discrete
energy levels (Zeeman levels), a process known as Zeeman splitting, with the lower energy
levels resulting from the alignment of atomic nuclear magnetic moment with the external
B-field and higher energies with alignment against the B-field. The transition energy between
each level is given by
(5.18)
∆E
B h
z
= −γ
π
2
If a photon of electromagnetic energy hv matches ΔE, it can be absorbed to excite a nuclear
magnetic energy level transition from a lower to a higher state; similarly, a higher energy state
can drop to a lower level with consequent photon emission, with quantum selection rules
permitting Δm = ±1, which indicates 2I possible reversible transitions. An absorbed photon
of frequency ν can thus result in a resonance between the different spin energy states. This
resonance frequency is also known as the Larmor frequency and is identical to the classically
calculated frequency of precession of an atomic nucleus magnetic moment around the axis
of the external B-field vector.
The value of ν depends on γ and on B (in most research laboratories, B is in the range of
~1–24 T, ~106 times the strength of Earth’s magnetic field), but is typically ~108 Hz, and it is
common to compare the resonance frequencies of different atomic nuclei under standard ref
erence conditions in relation to a B-field, which would generate a resonance frequency of 400
MHz for 1H (B ~ 9.4 T), some examples of which are shown in Table 5.1. For magnetic atomic
nuclei, these are radio frequencies. For example, the resonance frequency of 13C is very close
to that of a common FM transmission frequency of ~94 MHz for New York Public Radio.
A typical value of ΔE, for example, for 1H in a “400 MHz NMR machine” (i.e., B ~ 9.4 T) is ~3
× 10−25 J. Experiments in such machines are often performed at ~4 K, and so kBT/ΔE ~ 180,
hence, still a significant proportion of occupied lower energy states at thermal equilibrium.
The occupational probability pm of the mth state (see Worked Case Example 5.2) is given
by the normalized Boltzmann probability of
(5.19)
p
E
k T
E
k T
B hm
k T
m
m
B
all m
m
g
z
B
all
=
−
−
=
∑
exp
/
exp
/
exp
/
γ
π
2
m
z
B
B hm
k T
∑
exp
/
γ
π
2
The relative occupancy N of the different energy levels can be predicted from the Boltzmann
distribution:
(5.20)
N
N
E
k T
Bh
k T
m
m I
B
B
=
= +
=
−
=
−
1
1
2
exp
exp
∆
γ
π