Be Careful
I didn't have enough time to type all the equations, so I just scanned them up for now. Still need to add more details to the problems to make them clear.
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
Static Spectrum
Problem 1
- Show that for average over φ, we get pure absorptive line-shape, and for a particular isochromat, average over φ in general has dispersive line-shape (Show the response in cylindrical coordinate)
- Normal shim: x,y (first order spherical harmonic). If there are terms x^2-y^2, xy, then the sideband will show up at twice Ω
- Calculate the FID and the spectrum for rotary vs non-rotary, 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 ??
ω in transverse plane (slow) can be suppressed if rotation around z-axis is fast
$\sigma _
\sigma _
'$
= secular part of the chemical shift, lead to small rotation in x-y direction
Problem 2
- Show that chemical shift tensor
$\sigma = \sigma_
+ (\frac
)(3 cos^
\theta 1) \frac{\delta^{eta}}
sin^
\theta(e^
+e^{-i2\phi})$
$\sigma_
=(\sigma_
+\sigma_
+\sigma_
)/3$
$\delta=\frac
\sigma_
-\frac
(\sigma_
+\sigma_
)$
$\eta=3(\sigma_
-\sigma_
)/2(\sigma_
-\sigma_
-\sigma_
)$
- Show that under random rapid motion spins
$< \sigma > = \sigma _
$
It average out any non-isometric parts, so we have a homogeneous sample. So the result does not depend on the orientation of the sample.
When η = 0 -> < 3cos(θ)^2 -1 > = 0, average over sphere
- η = 0 ; calculate the line-shape for static powder (constant orientation with magnetic field), η ≠ 0 ; reduce to a summation over η. [Hint: can be written in elliptical integral, check out appendix I ]
- Find σ(θ,φ), powder distribution of the sample (when spinning at the magic angle ?)
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
- What is the contribution of the chemical shift anisotropy to T2?
Carl-Purcell Sequence
Problem 4
- Look at diffusive attenuation of water rotating in magnetic field gradient. (The faster you rotate it, the effective T2 is approaching T2)
Chemical Exchange
let
Problem 5
- Show the plot of the chemical exchange (when τ|ΔωA-ΔωB| approaching 1, the 2 peaks merge at the center) [Hint: check out appendix F]
Slow Exchange
choose Δ ≥ τ exchange, Δ << T1, Δ > T2
Problem 6
- Show that by collect this terms in slow exchange
$e^{i\omega_
t_{1}}e^{i\omega_
t_{2}} , e^{i\omega_
t_{1}}e^{i\omega_
t_{2}} , e^{i\omega_
t_{1}}e^{i\omega_
t_{2}} , e^{i\omega_
t_{1}}e^{i\omega_
t_{2}}$
then do phase cycle and collect data set
$cos(\omega_
T_
)e^{i\omega_
t_{2}} , sin(\omega_
T_
)e^{i\omega_
t_{2}}$
Then we get pure absorptive line-shape