Dynamics
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$s(t)=e^{-t/T_{2}}\int P(r)e^{-i\int^
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_
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\omega(r,t')dt'}dr$
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
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$\omega(t) = \gamma r \frac{\partial B_{z}}
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cos(\omega _
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t + \phi)$
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$exp(i\int^
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_
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\omega(t')dt'=exp(i[\gamma \frac{\partial B_
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/\partial x}{\omega_{s}}r sin(\omega_
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t+\phi])$
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$exp^
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=\sum J_
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(R)e^
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$
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