190 5.4  NMR and Other Radio Frequency and Microwave Resonance Spectroscopies

decay (FID). The signal damping in this mode is exponential with a decay time referred to as

T2

* . This basic mode is the essence of all pulsed NMR methods, and in a biomedical setting,

single-​pulse methods such as this form the basis of a common pulsing sequence used in mag­

netic resonance imaging (see Chapter 7) called “gradient recalled echos.”

A pulse here is a superposition of oscillating radiofrequency waves (or spin packets) with

a broad range of frequencies, which are used to rotate the bulk magnetization of the sample,

which is set by the external B-​field vector. Pulses are described as having a specific phase in

terms of the angle of rotation of the bulk magnetization. So, for the simplest form of FID, a

90° pulse (referred to commonly as a “90” or “pi over two” pulse) is normally applied initially,

with decay then resulting in dephasing 90° back to realignment of the bulk magnetization

to its original orientation. This simple pulse sequence will normally be repeated to improve

the signal-​to-​noise ratio. For analysis, this time-​resolved repeating signal is usually Fourier

transformed, resulting in a signal amplitude S, which depends on the relaxation time T1 as

well as the time between pulse loops called the “repetition time” (TR):

(5.24)

S

k

T

T

=

−

−















^



^

ρ 1

1

exp

R

where k is a constant of proportionality with a density of spin nuclei in the sample given by ρ.

The spin-​echo (SE) pulse sequence (also known as Hahn echo) is another common mode.

Here, a sample is stimulated with two or more radio frequency pulses with subsequent detec­

tion of an echo resonance signal at some time after these initial pulses. Usually, this involves

an initial 90° pulse, a wait period known as the echo time TE, then a 180° refocusing pulse,

another wait period TE, then observation of the energy peak of the SE signal (Figure 5.5b)—​

the 180° pulse causes the magnetization to at least partially rephase, which results in the echo

signal. On top of this are T1 and T2 relaxation processes, so the Fourier transformed signal is

described by

(5.25)

S

k

T

T

T

T

=

−

−















^



^

−













ρ 1

1

2

exp

exp

R

E

The enormous advantage of SE pulsing is that normally inhomogeneous relaxation processes

will ultimately cause dephasing following repeating 90° pulsing, that is, different spin nuclei

from the same atom types will start to precess at noticeably different rates, whereas the 180°

pulse largely resets this dephasing back to zero, allowing more pulse loops before dephasing

effects dominate, resulting in large effective increases in the signal-​to-​noise ratio.

The inversion recovery sequence is similar to SE pulsing, but here a 180° radio frequency

pulse is initially applied. After a given time period known as the “inversion time” TI, during

which time the bulk magnetization undergoes spin–​lattice relaxation aligned 180° from the

original vector, a 90^o pulse is applied, which rotates the longitudinal magnetization into the

FIGURE 5.5  NMR spectroscopy pulse profiles. (a) Simple free induction decay. (b) Spin-​echo

pulse sequence (which is repeated multiple times on the same sample to allow averaging to

improve the signal-​to-​noise ratio).