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426 12MAGNETIC RESONANCE
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
x
I(
x)
/S
0T
2
IL
IG
Figure 12.12
�erefore the Fourier transform of a cosω1 t e−t/τ + b cosω2 t e−t/τ will consist
of two Lorentzians at ω = ±ω1, with height scaled by a, and two Lorentzians at
ω = ±ω2, with height scaled by b; all the peaks will have width 2/τ.
P12C.8 �e di�erential equationwhich describes the time-dependence of the z-magnetization
is given in Problem P12C.7
dMz(t)
dt
= −Mz(t) −M0
T1
Separate the di�erential equation and integrate between t = 0 and t = τ, and
the corresponding limitsMz(0) = −M0 andMz(τ).
∫
Mz(τ)
−M0
1
Mz(t) −M0
dMz(t) = −∫
τ
0
1
T1
dt
∣ln (Mz(t) −M0)∣Mz(τ)
−M0 = ∣− t
T1
∣
τ
0
ln(Mz(τ) −M0
−2M0
) = − τ
T1
hence Mz(τ) = M0(1 − 2e−τ/T1)
�e time τnull at whichMz(τnull) = 0 is found by solving
M0(1 − 2e−τnull/T1) = 0 hence τnull = T1 ln 2
With the given data, T1 ln 2 = 0.50 s, hence T1 = (0.50 s)/ ln (2) = 0.72 s .
P12C.10 Figure 12.13 shows the motion of magnetization vectors from typical ‘fast’ and
‘slow’ spin packets (relative to the rotating frame frequency) during the spin
echo sequence. �e 90○ pulse rotates the magnetization onto −y, and then
during the delay τ the vectors from the spin packets fan out. �e 180○ pulse