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Some problems still need clarification. I will update them once we ask professor Cory. |
Dynamics
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{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 {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
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- ω
...
- is
...
- not
...
- a
...
- function
...
- of
...
- r
...
- (There
...
- are
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- no
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- interesting
...
- dynamics)
...
- Stationary
...
- -
...
- when
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- ω
...
- is
...
- not
...
- a
...
- function
...
- of
...
- time
...
- (the
...
- system
...
- can
...
- be
...
- refocused
...
- by
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- a
...
- π
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- pulse
...
- for
...
- any
...
- time)
...
- Incoherent
...
- -
...
- stationary
...
- and
...
- not
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- coherent,
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- explicitly
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- ω
...
- is
...
- a
...
- function
...
- of
...
- r
...
- (interesting
...
- question
...
- is
...
- the
...
- distribution
...
- of
...
- ω(r)
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- Decoherent
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- -
...
- when
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- ω
...
- is
...
- a
...
- function
...
- of
...
- time
...
- and
...
- r,
...
- and
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- the
...
- t
...
- dependence
...
- is
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- stochastic/Marchovian
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- (interesting
...
- dynamics:
...
- distribution
...
- of
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- ω(r),
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- spectral
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- density
...
- of
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- ω(r)
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- Periodic
...
- -
...
- ω
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- is
...
- a
...
- simple
...
- function
...
- of
...
- time
...
- (interesting
...
- dynamics:
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- distribution
...
- of
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- ω(r)
...
- at
...
- the
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- characteristic
...
- frequency)
Periodic
Frequency that an arbitrary location will see
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\\ h3. Periodic !p1.jpg! !p2.jpg! Frequency that an arbitrary location will see \\ {latex} $\omega(t) = \gamma r \frac{\partial B_{z}}{\partial x} cos(\omega _{s} t + \phi)$ { |
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} \\ {latex} $exp(i\int^{t}_{0}\omega(t')dt'=exp(i[\gamma \frac{\partial B_{z}/\partial x}{\omega_{s}}r sin(\omega_{s}t+\phi])$ { |
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} \\ {latex} $exp^{iRsin\alpha}=\sum J_{k}(R)e^{ik\alpha}$ {latex} |
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
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- dispersive
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- line-shape
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- (Show
...
- the
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- response
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- in
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- cylindrical
...
- coordinate)
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- Normal
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- shim:
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- x,y
...
- (first
...
- order
...
- spherical
...
- harmonic).
...
- If
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- there
...
- are
...
- terms
...
- x^2-y^2,
...
- xy,
...
- then
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- the
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- sideband
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- will
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- show
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- up
...
- at
...
- twice
...
- Ω
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- Calculate
...
- the
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- FID
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- and
...
- the
...
- spectrum
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- for
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- rotary
...
- vs
...
- non-rotary,
...
- then
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- plot
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- 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)
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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} |
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$H_{cs}=-\omega _{0}\sigma I_{z}${latex}
!p4.jpg!
PAS |
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
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?? !p6.jpg! {latex}$\sigma _{z} \sigma _{z}'${latex} |
=
...
secular
...
part
...
of
...
the
...
chemical
...
shift,
...
lead
...
to
...
small
...
rotation
...
in
...
x-y direction
Problem 2
- Show that chemical shift tensor
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direction !p5.jpg! *Problem 2* - Show that {latex} $\sigma = \sigma_{iso} + (\frac{\sigma}{2})(3 cos^{2}\theta -1)- \frac{\delta^{eta}}{4}sin^{2}\theta(e^{i2\phi}+e^{-i2\phi})$ { |
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} \\ {latex} $\sigma_{iso}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$ { |
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} \\ {latex} $\delta=\frac{2}{3}\sigma_{zz}-\frac{1}{3}(\sigma_{xx}+\sigma_{yy})$ {latex} \\ {latex} |
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$\eta=3(\sigma_{yy}-\sigma_{xx})/2(\sigma_{zz}-\sigma_{xx}-\sigma_{yy})$
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- Show that under random rapid motion spins
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{latex} \\ - Under random rapid motion spins {latex}$< \sigma > = \sigma _{iso}${latex} - When η = 0 \-> < |
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),
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- 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
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$e^{i\omega_{A}t_{1}}e^{i\omega_{A}t_{2}} , e^{i\omega_{A}t_{1}}e^{i\omega_{B}t_{2}} , e^{i\omega_{B}t_{1}}e^{i\omega_{A}t_{2}} , e^{i\omega_{B}t_{1}}e^{i\omega_{B}t_{2}}$
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then do phase cycle and collect data set
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$cos(\omega_{A/D}T_{1})e^{i\omega_{A/D}t_{2}} , sin(\omega_{A/D}T_{1})e^{i\omega_{A/D}t_{2}}$
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Then we get pure absorptive line-shape