see Class Notes Lecture 5 for now
Problem 0 - Imaging on Cylinder
Flow is along y-axis
What are the acquired image and the velocity, position, diffusion signatures?
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
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 ?)
Problem 3
- What is the contribution of the chemical shift anisotropy to T2?
Problem 4
- Look at diffusive attenuation of water rotating in magnetic field gradient. (The faster you rotate it, the effective T2 is approaching T2)
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]
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