The Chemistry Maths Book, Second Edition

440 Chapter 15Orthogonal expansions. Fourier analysis

of the results of diffraction experiments; the observed diffraction pattern is the

representation of the structure of the system in momentum space and the Fourier

transformation gives the structure in ordinary (coordinate) space. Of even greater

importance to the chemist is the time to frequency transformation because it

forms the basis for Fourier transform magnetic resonance spectroscopy. In such an

experiment, the molecules of a sample (or other species with excitable degrees of

freedom) are excited by a short pulse of radiation, and the system is observed as it

relaxes to its thermodynamically stable state.

Relaxation is essentially a first-order kinetic process and in the simplest case, when

the molecules can undergo transitions at only one frequency,ω

0

say, the intensity of

the output signal shows exponential decay as a function of time:

I(t) 1 = 1 e

−t 2 T

(15.84)

where Tis called the relaxation time. This is an example of case (c) in Figure 15.14,

with xas time tand yas frequency ω. The Fourier transform of the decay curve is

therefore a Lorentzian curve (see Example 15.10)

(15.85)

The width of this ‘spectral line’ is inversely proportional to the relaxation time T.

Its position (at ω

0

) is determined in practice by the fine structure of the output, as

illustrated in Example 15.11.

EXAMPLE 15.11Find the real part of the Fourier transform of the exponentially

damped harmonic wave,

f(x) 1 = 1 e

−ax

1 cos 1 bx forx 1 > 1 0anda 1 > 10

By equation (15.82),

or, because cos 1 bx 1 = 1 (e

ibx

1 + 1 e

−ibx

) 22 ,

=

+−

−−













1

22

11

π a ib iy a ib iy

gy e e

aibiyx aibiyx

()

()()

=+

−+− −−−











1

22

0

π

Z

∞





dx

=

−−

1

2

0

π

Z

∞

ebxedx

ax ixy

cos

gy fxe dx

ixy

()= ()

−

+

−

1

2 π

Z

∞

FItedt

T

it

()ω ()

ω

ω

==













−

Re

1

2

1

2 1

0

22

ππ

Z

∞

....

.....

...

....

.....

....

...

.....

....

.....

....

...

.....

....

.....

....

...

.....

....

.....

...

....

.....

....

.....

....

.....

....

...

....

.....

....

.....

....

.....

....

...

......

....

.....

.......

.....

....

.....

....

.....

....

.....

....

.....

....

.....

....

.....

....

...

......

...

....

.....

....

.....

...

.....

....

.....

....

......

....

............

......

.....

....

.....

....

.....

....

.....

....

.....

....

.....

....

.....

....

...

.....

......

...

.....

....

......

.......

.....

....

.....

....

.....

....

.....

....

.....

...

.....

....

.....

...............

......

.....

....

.....

....

.......

.....

....

.......

............

.......

....

......

.....

......

.....

......

........

.............

........

.......

.....

.......

.....

.......

.........

............

......

.......

..............

.......

..............

.........

..........

.............................

............

.............

......................

Figure 15.15