Physical Chemistry Third Edition

24.5 Fourier Transform NMR Spectroscopy 1033

7.0 7.0

d

c

e

5.0

5.0

(H)

ppm

(H), ppm

2.0

2.0

Ha 3 Ca

Cb(O)

Cd

Cc

Hc

He Hd

Figure 24.15 The Two-Dimensional COSY Spectrum of Methyl Vinyl Ketone in Deuterated Benzene.From T. C. Farrar,Introduction
to Pulse NMR Spectroscopy, Farragut Press, Chicago, 1989, p. 181.

b.Construct an accurate graph of the Fourier transform

from part a for several different values ofaandb. Tell

in words how the spin–spin relaxation time affects the

width of the spectral line.

24.38There is a useful theorem for the Fourier transform of a
product of two functions, called theconvolution theorem
or theFaltung theorem(Faltung is German for “folding”).
The convolution of two functionsf(x) andg(x) is defined
as the integral

1

√

2 π

∫∞

−∞

f(y)g(x−y)dy

This integral is a function ofx, and its Fourier transform

is equal toF(k)G(k) whereF(k) is the Fourier transform

off(x) andG(k) is the Fourier transform ofg(x).^9 Use

this theorem to find the inverse Fourier transform of

I(t)e−at

2

sin (bt). Verify that your answer is the same

as that of Problem 24.36.

24.39The Fourier transform ofeiatis proportional to the Dirac

delta function. Argue that the integral

∫∞

−∞

eiateiωtdt

diverges (becomes infinite) ifaωand vanishes

ifaω.

Summary of the Chapter

Magnetic resonance spectroscopy is absorption spectroscopy in which the sample sub-

stance is placed in a magnetic field, causing spin states that are degenerate in the

(^9) Philip M. Morse and Herman Feshbach,Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 1953, p. 464ff.