h2. Dynamics

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

h3. Periodic

!p1.jpg!

!p2.jpg!

Frequency that an arbitrary location will see
\\
{latex}
$\omega(t) = \gamma r \frac{\partial B_{z}}{\partial x} cos(\omega _{s} t + \phi)$
{latex}
\\
{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}
\\
{latex}
$exp^{iRsin\alpha}=\sum J_{k}(R)e^{ik\alpha}$
{latex}
for one location in the sample
\\

*Problem 1*

!p3.jpg!
- Show that for average over φ, we get absorptive line-shape, and for isochromat, φ in general has dispersive line-shape. Show the response in cylindrical coordinate ??
- Normal shim (x,y). If terms x^2-y^2, xy, then the sideband show up at twice Ω
- Calculate the FID and the spectrum, then plot them on top of each other

h3. Nuclear Spin

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

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

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

<MATRIX>
{latex}$\sigma _{z} \sigma _{z}${latex}
= secular part of the chemical shift, lead to small rotation in x-y direction

!p5.jpg!

!p6.jpg!

*Problem 2*
- Show that

{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}
\\
- Under random rapid motion spins
{latex}$< \sigma > = \sigma _{iso}${latex}

- When η = 0 \-> < 3cos(θ)^2 \-1 > = 0, average over sphere

- η = 0 ; calculate the line-shape for static powder, η ≠ 0 ; reduce to a summation over η

- Find σ(θ,φ), powder distribution of the sample

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

<EQUATIONS>

!p8.jpg!

!p9.jpg!

*Problem 3*
- Find the correlation time

h3. Carl-Purcell Sequence

!p10.jpg!

<EQ>

*Problem 4*
- Look at diffusive attenuation of water rotate in magnetic field gradient. (The faster you rotate it, you get closer to having the real T2 effect

- Is the underly process is truly stochastic

h3. Chemical Exchange

let <EQ>

!p11.jpg!

*Problem 5*
- Show plot

h3. Slow Exchange

!p12.jpg!

*Problem 6*

<EQ>
- Show that by collect this terms

<EQ>

then do phase cycle and collect data set

<EQ>

Then we get pure absorptive line-shape