Dynamics
$s(t)=e^{-t/T_{2}}\int P(r)e^{-i\int^
_
\omega(r,t')dt'}dr$
ω(r,t') = resonant frequency
P(r) = probability distribution
- Coherent - when ω is not a function of r (There are no interesting dynamics)
- Stationary - when ω is not a function of time (the system can be refocused by a π pulse for any time)
- Incoherent - stationary and not coherent, explicitly ω is a function of r (interesting question is the distribution of ω(r)
- Decoherent - when ω is a function of time and r, and the t dependence is stochastic/Marchovian (interesting dynamics: distribution of ω(r), spectral density of ω(r)
- Periodic - ω is a simple function of time (interesting dynamics: distribution of ω(r) at the characteristic frequency)
Periodic
Frequency that an arbitrary location will see
$\omega(t) = \gamma r \frac{\partial B_{z}}
cos(\omega _
t + \phi)$
$exp(i\int^
_
\omega(t')dt'=exp(i[\gamma \frac{\partial B_
/\partial x}{\omega_{s}}r sin(\omega_
t+\phi])$
$exp^
=\sum J_
(R)e^
$
for one location in the sample
Problem 1
- Show that for average over φ, we get absorptive line-shape, and for isochromat, φ in general has dispersive line-shape. Show the response in cylindrical coordinate ??
- Normal shim (x,y). If terms x^2-y^2, xy, then the sideband show up at twice Ω
- Calculate the FID and the spectrum, then plot them on top of each other
Nuclear Spin
- Zeeman interaction
- Chemical shift : ppm variation due to chemistry -> transform as a tensor (orientation of the molecule matter)
$H_
=\omega _
I_
$
,
$H_
=-\omega _
\sigma I_
$
PAS (Principle axis system) = coordinate system that leave the molecule in diagonal ??
$\sigma _
\sigma _
$
= secular part of the chemical shift, lead to small rotation in x-y direction
Problem 2
- Show that
$\sigma = \sigma_
+ (\frac
)(3 cos^
\theta 1) \frac{\delta^{eta}}
sin^
\theta(e^
+e^{-i2\phi})$
$\sigma_
=(\sigma_
+\sigma_
+\sigma_
)/3$
$\delta=\frac
\sigma_
-\frac
(\sigma_
+\sigma_
)$
$\eta=3(\sigma_
-\sigma_
)/2(\sigma_
-\sigma_
-\sigma_
)$
- Under random rapid motion spins
Unknown macro: {latex}
$< \sigma > = \sigma _
Unknown macro: {iso}$
- When η = 0 -> < 3cos(θ)^2 -1 > = 0, average over sphere
- η = 0 ; calculate the line-shape for static powder, η ≠ 0 ; reduce to a summation over η
- Find σ(θ,φ), powder distribution of the sample
Decoherence
Bloc = field that a test spin would see (every spin averagely see the same distribution of B)
average vector still pointing along y => |Bloc> of time or ensemble = 0
Problem 3
- Find the correlation time
Carl-Purcell Sequence
Problem 4
- Look at diffusive attenuation of water rotate in magnetic field gradient. (The faster you rotate it, you get closer to having the real T2 effect
- Is the underly process is truly stochastic
Chemical Exchange
let
Problem 5
- Show plot
Slow Exchange
Problem 6
- Show that by collect this terms
then do phase cycle and collect data set
Then we get pure absorptive line-shape