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424 12MAGNETIC RESONANCE
0.000 0.001 0.002 0.003 0.004 0.005
−1.0
−0.5
0.0
0.5
1.0
t/ s
S(
t)
/S
(0
)
Figure 12.9
0.000 0.001 0.002 0.003 0.004 0.005
−1.0
−0.5
0.0
0.5
1.0
t/ s
S(
t)
/S
(0
)
Figure 12.10
where ω = 2πν is the angular frequency, ω0 is the resonance frequency,
S0 is a constant, and T2 is the transverse relaxation time. Inspection of the
function shows that it must be a maximum at ω = ω0 as this minimises
the denominator. At this point the function has the value IL(ω0) = S0T2.
�e peak reaches its half-height of S0T2/2 when
S0T2/2 =
S0T2
1 + T22 (ω1/2 − ω0)2
hence (ω1/2 − ω0)2 = 1/T22
It follows that ω1/2 = ω0 ± 1/T2. Converting from angular frequency
using ω = 2πν gives ν1/2 = ν0 ± 1/2πT2. Hence the width at half-height
is 2 × (1/2πT2) = 1/πT2 in Hz, or 2/T2 when expressed as an angular
frequency.
(b) �e NMR lineshape can also be approximated using a Gaussian function
of the form
IG(ω) = S0T2e−T
2
2 (ω−ω0)2