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)

<PIC>

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
- 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