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Problem 0 - Imaging on Cylinder

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

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). Show that 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 - Chemical Shift

• Show that chemical shift tensor

$\sigma = \sigma_{iso} + (\frac{\sigma}{2})(3 cos^{2}\theta -1)- \frac{\delta^{eta}}{4}sin^{2}\theta(e^{i2\phi}+e^{-i2\phi})$

$\sigma_{iso}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$

$\delta=\frac{2}{3}\sigma_{zz}-\frac{1}{3}(\sigma_{xx}+\sigma_{yy})$

$\eta=3(\sigma_{yy}-\sigma_{xx})/2(\sigma_{zz}-\sigma_{xx}-\sigma_{yy})$

• Show that under random rapid motion spins

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

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 - Decoherence

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

Problem 4 - Carl-Purcell Sequence

• Look at diffusive attenuation of water rotating in magnetic field gradient. Show that the faster you rotate it, the effective T2 is approaching T2.

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]

Problem 6 - Slow Exchange

• Show that by collect this terms in slow exchange

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

then do phase cycle and collect data set

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

Then we get pure absorptive line-shape