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Some problems still need clarification. I will update them once we ask professor Cory. |
Dynamics
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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
...
- no
...
- 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
...
- 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
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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
...
- (interesting
...
- dynamics:
...
- distribution
...
- of
...
- ω(r),
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- spectral
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- density
...
- of
...
- ω(r)
...
- Periodic
...
- -
...
- ω
...
- is
...
- a
...
- simple
...
- function
...
- of
...
- time
...
- (interesting
...
- dynamics:
...
- distribution
...
- of
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- ω(r)
...
- at
...
- the
...
- characteristic
...
- frequency)
Periodic
Frequency that an arbitrary location will see
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Periodic <PIC> 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}\\ {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}\\ \\ {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
...
- 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
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- them
...
- on
...
- top
...
- of
...
- each
...
- other
...
Nuclear
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Spin
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- Zeeman
...
- interaction
...
- Chemical
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- shift
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- :
...
- ppm
...
- variation
...
- due
...
- to
...
- chemistry
...
- ->
...
- transform
...
- as
...
- a
...
- tensor
...
- (orientation
...
- of
...
- the
...
- molecule
...
- matter)
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}$H_{z}=\omega _{0}I_{z}$ |
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{latex} , {latex}$H_{cs}=-\omega _{0}\sigma I_{z}${latex} <PIC> 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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?? <MATRIX> {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
<PIC>
h4. 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} \\ {latex} $\sigma_{iso}=(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})/3$ { |
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latex} \\ {latex} $\delta=\frac{2}{3}\sigma_{zz}-\frac{1}{3}(\sigma_{xx}+\sigma_{yy})$ { |
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} \\ {latex} $\eta=3(\sigma_{yy}-\sigma_{xx})/2(\sigma_{zz}-\sigma_{xx}-\sigma_{yy})$ {latex} \\ - Under random rapid motion spins {latex} |
- Show that under random rapid motion spins
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$< \sigma > = \sigma _{iso}${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),
...
- 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