{info:title=Useful Information}
See more information about the problems in [Class Notes Lecture 5]
{info}

h4. Problem 0 - Imaging on Cylinder

Flow is along y-axis

!p01.jpg!

!p02.jpg!

What are the acquired image and the velocity, position, diffusion signatures?

\\

h4. Problem 1 - Periodic

- 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

\\

h4. Problem 2 - Chemical Shift

- Show that chemical shift tensor

\\
{latex}
$\sigma = \sigma_{iso} + (\frac{\sigma}{2})(3 cos^{2}\theta -1)- \frac{\delta^{eta}}{4}sin^{2}\theta(e^{i2\phi}+e^{-i2\phi})$
{latex}
\\
{latex}
$\sigma_{iso}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$
{latex}
\\
{latex}
$\delta=\frac{2}{3}\sigma_{zz}-\frac{1}{3}(\sigma_{xx}+\sigma_{yy})$
{latex}
\\
{latex}
$\eta=3(\sigma_{yy}-\sigma_{xx})/2(\sigma_{zz}-\sigma_{xx}-\sigma_{yy})$
{latex}
\\
- Show that under random rapid motion spins

\\
{latex}$< \sigma > = \sigma _{iso}${latex}
\\

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 ?)

\\

h4. Problem 3 - Decoherence

- What is the contribution of the chemical shift anisotropy to T2?

\\

h4. Problem 4 - Carl-Purcell Sequence

- Look at diffusive attenuation of water rotating in magnetic field gradient. (The faster you rotate it, the effective T2 is approaching T2)

\\

h4. Problem 5 - Chemical Exchange

- Show the plot of the chemical exchange (when τ\|ΔωA-ΔωB\| approaching 1, the 2 peaks merge at the center) \[Hint: check out appendix F\]

\\

h4. Problem 6 - Slow Exchange

- Show that by collect this terms in slow exchange

\\
{latex}
$e^{i\omega_{A}t_{1}}e^{i\omega_{A}t_{2}} , e^{i\omega_{A}t_{1}}e^{i\omega_{B}t_{2}} , e^{i\omega_{B}t_{1}}e^{i\omega_{A}t_{2}} , e^{i\omega_{B}t_{1}}e^{i\omega_{B}t_{2}}$
{latex}
\\

then do phase cycle and collect data set

\\
{latex}
$cos(\omega_{A/D}T_{1})e^{i\omega_{A/D}t_{2}} , sin(\omega_{A/D}T_{1})e^{i\omega_{A/D}t_{2}}$
{latex}
\\

Then we get pure absorptive line-shape