{note:title=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.
{note}

h2. Dynamics

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{latex}
$s(t)=e^{-t/T_{2}}\int P(r)e^{-i\int^{t}_{0}\omega(r,t')dt'}dr$
{latex}
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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)

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h3. Periodic

!p1.jpg!

!p2.jpg!

Frequency that an arbitrary location will see
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{latex}
$\omega(t) = \gamma r \frac{\partial B_{z}}{\partial x} cos(\omega _{s} t + \phi)$
{latex}
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{latex}
$exp(i\int^{t}_{0}\omega(t')dt'=exp(i[\gamma \frac{\partial B_{z}/\partial x}{\omega_{s}}r sin(\omega_{s}t+\phi])$
{latex}
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{latex}
$exp^{iRsin\alpha}=\sum J_{k}(R)e^{ik\alpha}$
{latex}
for one location in the sample
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Static Spectrum

!p3.jpg!

*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

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h3. Nuclear Spin

- Zeeman interaction
- Chemical shift : ppm variation due to chemistry \-> transform as a tensor (orientation of the molecule matter)

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{latex}$H_{z}=\omega _{0}I_{z}${latex}
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{latex}$H_{cs}=-\omega _{0}\sigma I_{z}${latex}
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!p4.jpg!

PAS (Principle axis system) = coordinate system that leave the molecule in diagonal ??

!p5.jpg!

ω in transverse plane (slow) can be suppressed if rotation around z-axis is fast

!p6.jpg!\\
{latex}$\sigma _{z} \sigma _{z}'${latex}
= secular part of the chemical shift, lead to small rotation in x-y direction
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*Problem 2*
- Show that chemical shift tensor

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{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}
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{latex}
$\sigma_{iso}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$
{latex}
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{latex}
$\delta=\frac{2}{3}\sigma_{zz}-\frac{1}{3}(\sigma_{xx}+\sigma_{yy})$
{latex}
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{latex}
$\eta=3(\sigma_{yy}-\sigma_{xx})/2(\sigma_{zz}-\sigma_{xx}-\sigma_{yy})$
{latex}
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- Show that under random rapid motion spins

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

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h3. Decoherence

Bloc = field that a test spin would see (every spin averagely see the same distribution of B)

!p7.jpg!

average vector still pointing along y => \|Bloc> of time or ensemble = 0

!eq1.jpg!

!p8.jpg!

!p9.jpg!

*Problem 3*
- What is the contribution of the chemical shift anisotropy to T2?

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h3. Carl-Purcell Sequence

!p10.jpg!

!eq2.jpg!

*Problem 4*
- Look at diffusive attenuation of water rotating in magnetic field gradient. (The faster you rotate it, the effective T2 is approaching T2)

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h3. Chemical Exchange

let

!eq3.jpg!

!p11.jpg!

*Problem 5*
- Show the plot of the chemical exchange (when τ\|ΔωA-ΔωB\| approaching 1, the 2 peaks merge at the center) \[Hint: check out appendix F\]

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h3. Slow Exchange

!p12.jpg!

choose Δ ≥ τ exchange, Δ << T1, Δ > T2

*Problem 6*
- Show that by collect this terms in slow exchange

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{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}
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then do phase cycle and collect data set

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{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}
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Then we get pure absorptive line-shape