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12 Magnetic resonance
12A General principles
Answers to discussion questions
D12A.2 �e energy levels associated with the interaction between magnetic nuclei and
an applied magnetic �eld scale directly with the �eld, with the constant of
proportion depending on the identity of the nucleus.�e interaction between
an electron and an applied magnetic �eld behaves in the same way, but the
interaction is much greater, by a factor of the order of 1000.
Solutions to exercises
E12A.1(b) �e nuclear g-factor gI is given by [12A.4c–489], gI = γNħ/µN, where µN is the
nuclear magneton (5.051 × 10−27 J T−1) and γN is the nuclear magnetogyric
ratio, the value of which depends on the identity of the nucleus.�e units of ħ
are J s and gI is a dimensionless number, so the nuclear magnetogyric ratio γN
has units (J T−1)/(J s) = T−1 s−1.
In SI units, 1 T = 1 kg s−2 A−1 hence γN has units (kg s−2 A−1)−1 × (s−1) =
Askg−1 .
E12A.2(b) �e magnitude of the angular momentum is given by [I(I + 1)]1/2ħ where
I is the nuclear spin quantum number. For a 14N nucleus, I = 1, hence the
magnitude of the angular momentum is [1(1 + 1)]1/2ħ =
√
2ħ .
�e component of the angular momentum along the z-axis ismIħ wheremI =
I, I−1, ...,−I. For a 14Nnucleus, the components along the z-axis are 0,±ħ and
the angle between angular momentum vector and the z-axis takes the values
θ = 0,± cos−1 ( ±ħ√
2ħ
) = 0,±0.7854 rad (or 0○ ,±45.00○)
E12A.3(b) �e NMR frequency is equal to the Larmor precession frequency, νL, which
is given by [12A.7–489], νL = γNB0/2π, where B0 is the magnitude of the
magnetic �eld and γN is the nuclear magnetogyric ratio. Use Table 12A.2 on
page 289 in the Resource section for the value of γN. Hence,
νL =
γNB0
2π
= (25.177 × 107 T−1 s−1) × (17.1 T)
2π
= 6.85 × 108 Hz = 685 MHz