100 INSTRUMENTAL METHODS
Substitution and rearrangement lead to equation 3.16 :
νγ
π=
2
B 0 (3.16)
Values of the magnetogyric ratio, γ , are such that frequencies lie within the
radio range (frequencies, ν , = < 1000 MHz; see Figure 3.2 ).
The low - energy, low - frequency range for NMR transitions corresponds to
a small change in energy, ΔE. This has implications for the population of
excited states, the Boltzmann distribution. For a spin - 1/2 nuclei with ΔE = μ
B 0 / I and I = 1/2, equation 3.17 applies. Since N+ ≈ N − , one can write equation
3.18.
N
NE
kT kT+
−=−exp( )=−exp( )Δ 2 μB 0
(3.17)N
NkT+
−=−^1 ( )2 μB 0
(3.18)For the proton ( I = 1/2, ΔE = 2 μ B 0 ) in a magnetic fi eld of 9.39 T (tesla) —
resonance frequency of 400 MHz — and at a temperature of 300 K, the quantity
2 μB 0 / kT has a value of about 6 × 10 − 5. In words the excess population in the
lower energy state is extremely small, approximately one nucleus in 300,000.
As will be seen in the following, the NMR experiment monitors the relaxation
of molecules from an excited state to a ground state. This is in contrast to the
EPR technique and many other instrumental techniques that detectabsorp-
tion of energy in a species moving from a ground state to an excited state.
If one defi nes the length of the vector as in Figure 3.9B as [ I ( I + 1)] 1/2 , the
angleθ is given by equation 3.19 :
cos
[( )]/θ=
+m
II 1 12(3.19)
It is possible to translate these, in three dimensions, as the motion of a magnet
of moment μ in an applied magnetic fi eld. The magnet axis becomes inclined
to the fi eld, precesses around it, and is defi ned as the Lamour precession. The
half - apex angle of the cone equals θ and the angular velocity is γB 0 , so that
one fi nds equation 3.16 now rewritten as equation 3.20 :
νγ
π=( )
2
B 0 (3.20)
Equation 3.20 defi nes the nuclear resonant frequency. The precession cone of
excess low - energy nuclei is shown in Figure 3.10 , with the magnetic axis indi-
cated as thez axis leading to a net magnetization Mz along the z axis. The x -
andy - axis components average to zero, that is, Mx = My = 0.