Dynamics

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

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)

Periodic

Frequency that an arbitrary location will see

$\omega(t) = \gamma r \frac{\partial B_{z}}{\partial x} cos(\omega _{s} t + \phi)$

$exp(i\int^{t}_{0}\omega(t')dt'=exp(i[\gamma \frac{\partial B_{z}/\partial x}{\omega_{s}}r sin(\omega_{s}t+\phi])$

$exp^{iRsin\alpha}=\sum J_{k}(R)e^{ik\alpha}$

for one location in the sample

Static Spectrum

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

Nuclear Spin

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

$H_{z}=\omega _{0}I_{z}$

$H_{cs}=-\omega _{0}\sigma I_{z}$

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

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

$\sigma _{z} \sigma _{z}'$

= secular part of the chemical shift, lead to small rotation in x-y direction

Problem 2

• 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 ?)

Decoherence

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

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

Problem 3

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

Carl-Purcell Sequence

Problem 4

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

Chemical Exchange

let

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]

Slow Exchange

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

Problem 6

• 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