110 INSTRUMENTAL METHODS
than 2.5 Å ), medium (atoms 2.5 – 3.5 Å apart), and weak (atoms 3.5 – 5 Å apart)
NOEs. Methods developed by the W ü thrich group use NOEs to determine
the amino acid sequence in any small protein; that is, nearly all resonances can
be assigned to a specifi c aa residue. (See Section 3.4.10 .)
3.4.7 Obtaining the NMR Spectrum,
Three parameters affect an NMR spectrum: the chemical shift, coupling, and
nuclear relaxation. These must be accounted for when obtaining the NMR
spectrum from the spectrometer ’ s output. Obtaining the NMR spectral plot
from the output (the free induction decay, FID) of a modern NMR spectrom-
eter involves the analysis of the mathematical relationship between the time
(t ) and frequency ( ω ) domains, known as the Fourier relationship:
Fftitdt( )ωω=−( ) exp( )
−∞∞
∫ (3.32)which is also written as
Ffttitdt( )ωωω=−( )[cos( ) sin( )]
−∞∞
∫ (3.33)The Fourier transform (FT) relates the function of time to one of frequency —
that is, the time and frequency domains. The output of the NMR spectrometer
is a sinusoidal wave that decays with time, varies as a function of time, and is
therefore in the time domain. Its initial intensity is proportional to Mz and
therefore to the number of nuclei giving the signal. Its frequency is a measure
of the chemical shift, and its rate of decay is related to T 2. Fourier transforma-
tion of the FID gives a function whose intensity varies as a function of fre-
quency and is therefore in the frequency domain.
When one applies the perturbing fi eld B 1 , the nuclei in the sample precess
as discussed above. The point at which maximum current has been induced
duringB 1 ’ s application is known as a 90 ° pulse. Short B 1 pulses — less than
100 μ s — ensure that all nuclei in a sample, whatever their chemical shift, are
swung aroundB 1 by an appropriate angle. Long B 1 pulses choose nuclei of a
particular chemical shift to precess aroundB 1 without affecting other nuclei in
the sample. The ideal Fourier transform experiment, allowing spins to relax to
equilibrium before successive pulses are applied, is illustrated in Figure 3.14.
Assuming that the spectrometer is stable, a series of isolated 90 ° pulses will
each give an identical nuclear response, and these can be added together in
computer memory to yield a strong total response. Actually, most pulsing
sequences omit the waiting time and have pulse sequences in the 40 ° to 30 °
range, depending on the nuclear isotope observed, the chemical shift range,
the relaxation timeT 1 , and the computer memory size. Different nuclei in dif-
ferent parts of the molecule may have different relaxation times, so that pulse