Pulse and FT NMR
In practice bulk magnetization is observed
For isolated spins behavior is according to
classical Bloch equations
dM/dt = γ M X B – (Mz – M0) (1/T1) – Mx,y (1/T2)
Precession in a Static Field
Torque on system with angular momentum (L) causes
precession of the momentum vector.
dL/dt = T. Torque is given by M X B0 and M = γL.
Bloch equations give precession at ω = -γB0.
dM/dt = γ M(t) X B0
B0
L = M/γ
T
x
y
i
dM/dt = γ Mx
0
j
My
0
k
Mz
B0
dMx/dt = γMyB0
dMy/dt = -γMxB0
dMz/dt = 0
Solution: Mx(t) = M0xcos(-γB0t) - M0ysin(-γB0t)
= M0exp(-i γB0t) = M0exp(iω0t)
Spin precession is like precession of a gyroscope
The Rotating Frame Simplifies Analysis of RF
Pulses and Small Frequency Offsets
ω0 is very high (500 MHz x 2π) – differences due to
chemical shifts etc. are small (ppms x 500 Hz x 2π)
Therefore, use reference frame moving near ω0 .
Signal in lab frame
t
x’
x
y
y’
- ω0
ωobs = ω0 - ω’
Signal in rotating frame
t
RF Pulses are Required to Establish
Initial Transverse Magnetization
z
z
90° pulse at γB1t = π/2
M0
x
B1
y
x
M0
B1
y
Note: diagram is in the “rotating frame”
B1 = B01cos(B0γt) to keep pace with precessing magnetization
Maximum signal is obtained with a 90° pulse t90 = π/(2γB1)
Small Modifications in Precession Frequency (or
decay of magnetization) are Primary Observables
in High Resolution NMR
H = -γB0∑i (1-σi)IZi + ∑j>i 2π J Ii · Ij + ∑j>i 2π Ii · D · Ij
chemical shift scalar coupling dipolar coupling
1/T1,2 = ∑ij Ji (ωi) | Dij|2 ,
spin relaxation
Ji (ωi) = 2τc/(ωi2 τc2 + 1)
Measurement of Chemical Shift
υi (Hz) = γB0/2π (1-σi)
σi: shielding constant dependent on electronic structure, ~10-6.
Measurements are made relative to a reference peak: TMS.
Offsets given in terms of δ (parts per million, ppm, downfield).
δi = (σref - σi ) x 106
or
δi = (( υi - υref )/υref) x 106
Ranges: 1H, 2H, 10 ppm; 13C, 15N, 31P, 300 ppm; 19F, 1000
ppm
Pulse FT NMR
M0
M0
lab frame
y
rotating frame
y’
90° pulse
B1
x’
x
B1
B0
B0
M0
y’
precession
x’
x’
T2
Δν = (πT2)-1
FT
Ey
time
ν
Inversion-Recovery Measures T1s
Using a 180º Pulse
180 x
90 x
B1
τ
y
x B1
z
M
M
M
x B1
z
z
z
t
y x B1
M
y
x B1
y
FT and observe as a function of τ
M(τ) = M0(1-2exp(-τ/T1))