Chapter 5 – Detection and Imaging Tools that Use Nonoptical Waves  187

a, to the external B-​field), indicating a total of 2² of 4 possible spin combinations (p–​p, p–​a,

a–​p, a–​a) with a proton of an ¹H atom bound to C1. Low and high energy states are p–​p and

a–​a, respectively, but p–​a and a–​p are energetically identical; therefore, the single chemical

shift peak for the C1 protons is split into a triplet with the central peak amplitude higher by a

factor of 2 compared to the smaller peaks due to the summed states of p–​a and a–​p together,

so the amplitude ratio is 1:2:1. Similarly, the C1 atom is covalently bound to three ¹H atoms,

which results in 2³ or eight possible spin combinations with one of the protons of the ¹H atom

bound to C2, which can be grouped into four energetically identical combinations as follows

(from low to high energy states):

{

,

p

p

p} , {p

p

a, p

a

p, a

p

p} ,{a

a

p, a

p

a, p

a

a}

a

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

1

2

3

a

a

−

{

}4

Thus, the single chemical shift peak for the C2 protons is split into a quartet of relative amp­

litude 1:3:3:1. The ratio of amplitudes in general is the (n +​ 1)th level of Pascal’s triangle (so a

quintet multiplicity would have relative amplitudes of 1:4:6:4:1). Note also that J-​coupling can

also be detected through H-​bonds, indicating some covalent character of hydrogen bonding

at least. Observant readers might note from Figure 5.4a that there appears to be only a single

peak corresponding to the ¹H atom attached to the O atom of the –​OH group, whereas from

the simple logic earlier, one might expect a triplet O is covalently bonded to C2. However, the

effects of J-​coupling in this instance are largely lost and are similar for all ¹H atoms in general,

which are bound to heteroatoms (specifically, –​OH and –​NH groups) due to a rapid chem­

ical transfer of a proton H^+​, allowing it to exchange with another proton from OH or NH in

aqueous solution, which can occur even with tiny traces of water in a sample. This exchange

process results in line broadening of all peaks in the hypothetical triplet, resulting in the

appearance of a single broad peak.

5.4.4  NUCLEAR RELAXATION

The transition from a high to low magnetic nuclear spin energy state in general is not a radia­

tive process since the probability of spontaneous photon reemission varies as ν³, which is

insignificant at radio frequencies. The two major processes, which affect the lifetime of an

excited state, are spin–​lattice relaxation and spin–​spin relaxation, which are important in

practical NMR spectroscopy since the absence of relaxation mechanisms would imply rapid

saturation to high energy states and thus a small equivalent resonance absorption signal in a

given sample.

Spin–​spin relaxation (also known as “transverse relaxation”) involves coupling between

nearby magnetic atomic nuclei, which have the same resonance frequency but which differ

in their magnetic quantum numbers. It is also known as the “nuclear Overhauser effect” and,

unlike J-​coupling, is a free-​space interaction not mediated through chemical bonds. A transi­

tion can occur in which the two magnetic quantum numbers are exchanged. There is there­

fore no change in the occupancy of energy states; however, this does decrease the “on” time

probability of the excited state since an exchange in magnetic quantum number is equiva­

lent to a transient misalignment of the magnetic dipole with the external B-​field, which also

broadens absorption peaks. Transient misalignment can also be caused by inhomogeneity in

the B-​field. The mean relaxation time associated with this process is denoted as T2. Solids can

have T2 of a few milliseconds, while liquids more typically tens to hundreds of milliseconds.

Spin–​lattice relaxation (also known as “longitudinal relaxation”) is due to a coupling

between a spinning magnetic atomic nucleus and its surrounding lattice, for example, to

collisions between mobile sample molecules and the solvent. This results in energy loss from

the magnetic spin state and a consequent rise in temperature of the lattice. The mean relax­

ation time taken to return from an excited state back to the thermal equilibrium state is

denoted T1. In general, T1 > T2, such that in normal nonviscous solvents at room temperature

T1 ranges from ~0.1 to 20 s.