Leibniz-Institut für Molekulare Pharmakologie1
NMR Spectroscopy and Imaging
Department of Physics, FU Berlin
Lecture 7:
Spin Manipulation and Dipolar Relaxation
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Lecture 7
1. Hard Pulses
2. Selective Pulses
3. Dipolar Relaxation
4. Signal Enhancement: NOE
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remember: M precesses about B1 only; simple relation for flip angle holds for negligible off-resonance effects |B1| «
7.1 Hard Pulses
effective flip angles
otherwise we have to consider the effective field, which is tilted by angle out of the transverse plane towards the z’ axis
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the flip angle is then defined as
7.1 Hard Pulses
describing rotations
what is the exact description for the precession about the (tilted) effective field?
A-D: increasing off-resonant effectsA: pure –y magnetizationB-D: increasing x component
rotation can be described in terms of matrices, e.g. by angle about x axis
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and similar for the other two cases
7.1 Hard Pulses
describing rotations
now for a tilted axis, characterized by
(assume = 0 for now)in essence, the off-resonant effect is equivalent to first rotating B1 about y towards z and then rotating M around B1
this can be replaced by rotating M by – about y’, doing the rotation about x’ and then about y’ again
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extreme example:instead of rotating /4 about y, the axis is tilted back onto z (i.e. axis tilted by +/2 about x)
7.1 Hard Pulses
describing rotations
this is equivalent to the following 3 rotations, with the middle step representing the ‘old’ rotation axis y
this idea allows to decompose any arbitrary rotation into a combination of the three first mentioned matrices
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now consider rotation about x for duration T
this generates mainly y-magnetization, but also – depending on the off-resonance – unwanted x-magnetization
7.1 Hard Pulses
unwanted magnetizations
example:T set to achieve = 90° on resonant
Q: how to keep this effect small?
use short but strong (B1!) pulses
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knowing the pulse shape B1(t) in time domain can give the excitation profile B1() in frequency domain after FT (within the so-called small flip-angle approximation)
assume pulse about x we get y magnetization and x contributions
depending on the offset
the FT of a box function yields a sinc function (sinc(x) = (sin x)/x)
7.1 Hard Pulses
pulse profiles and the FT: predicting transverse magnetization
problem: NMR flip angle response is nonlinear but FT is a linear operation
linear response only for small flip angles
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consequence:wrong linear phase response and too narrow excitation profile around ‘on-resonance’ condition
rotation matrices: blackFT: grey
7.1 Hard Pulses
pulse profiles and the FT
profiles for small pulse angles are relatively uniform
large flip angles are only achieved in small bandwidths
this offset-dependence eventually necessitates 1st order phase correction (e.g. in 31P data or 129Xewith large chemical shift)
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7.1 Hard Pulses
phase corrections after hard pulses
1st order phase correction (e.g. in 31P data or 129Xewith large chemical shift)
Q: why is this especially problematic for the X-nuclei 13C and 129Xe?
they have large chemical shifts (!) but low which makes even strong pulses relative weak
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avoid deviation from ideal behavior by using strong pulses B1 >
an ideal 90°x pulse followed by delay t would give x-component for magnetization with offset of
this yields
7.1 Hard Pulses
approximating finite pulses
we can describe the real, finite pulse of duration T by an ideal infinite short pulse followed by a corresponding evolution delay t
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define a pulse shapewith truncation after 3, 5, ... n lobes
7.2 Selective Pulses
sinc pulses
note: • wiggles in x and z-component have different amplitude• the inversion pulse generates a considerable amount of transverse
magnetization
representation of finite shaped pulse more complicated then decomposition for hard pulse into ideal pulse + free precession
detailed analysis can be done with solving Bloch equations (not done here)
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define a pulse shapewith truncation at the edgesdefined by
7.2 Selective Pulses
Gauss pulses
example: Gauss inversion pulse
increasing truncation yields decreased inversion bandwidth
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define a pulse shape with truncation at the edges for 90° / 180°
7.2 Selective Pulses
Hermite pulses
example: Hermite inversion pulse
profiles are better, but higherpeak amplitudes are required
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components beyond the fluctuating local field
secular part of dipolar Hamiltonian:averages to zero and hence does not influence the shape of the spectrum
but non-secular part:has measurable impact for relaxation of ensembles of spin-1/2 nuclei
effect is also linked to molecular dynamics
7.3 Dipolar Relaxation
relaxation of interacting spin systems
tumbling by rotational correlation time, i.e. roughly the time to rotate by 1 rad
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four level energy system of weakly coupled homonuclear AX system
eight single-quantum transition probabilities
and two double quantum as well as two zero-quantum transition probabilities
7.3 Dipolar Relaxation
transition probabilities
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these can be written as follows
7.3 Dipolar Relaxation
transition probabilities
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all transition probabilities fall of rapidly with spin distance r
7.3 Dipolar Relaxation
important contributions
why is the contribution from 20
important?
local field is modulated with 20 when spin I2 rotates with 0
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precession of spin I1 at Larmor frequency causes rotation of local field for I2 to happen with –0
this is ineffective (as seen from decompostion for B1 of excitation pulse)
hence precession by itself does not induce relaxation, butmotion /rotation of molecule is also required
7.3 Dipolar Relaxation
important contributions
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now combine rotation and precession
this yields component at 0 with correct sign
7.3 Dipolar Relaxation
important contributions
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what happens to population of, e.g., state ?
it has three positive and three negative contributions, weighted by the populations
7.3 Dipolar Relaxation
Solomon equations
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summarizing all four populations in a vector, we get
and summarize the transition probabilities in the following matrix
7.3 Dipolar Relaxation
Solomon equations
note: the negative contributions are on the diagonal, the positive ones coming from the other states are off-diagonal elements
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an alternative basis to describe the state of the system is the Zeeman order vector
hence we need to transform the differential equation
7.3 Dipolar Relaxation
Solomon equations
the vector transition is done by
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so why is this Zeeman order vector useful?
instead of describing the system by 4 population numbers, we can also work with population differences
we know the total number is constantand has to be distributed ontothe four levels
7.3 Dipolar Relaxation
Solomon equations – changing the basis
first, we use the population differences that correspond to I1zmagnetization and the same for I2z
another combination is the difference between the magnetizations used in I1z (or I2z)
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now we can express the formerly used population numbers with these operators
7.3 Dipolar Relaxation
Solomon equations – changing the basis
this is why the transformation matrix reads
(factor ½ due to definition of density matrix entries in Levittnotation)
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so we also need to transform the matrix using
to be used for
7.3 Dipolar Relaxation
Solomon equations
in thermal equilibrium we have
where is just the tiny polarization due to the Boltzmann factor (see lecture 8)
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we therefore rewrite the differential equation as
with approximating the matrix by omitting terms containing the small Boltzmann factor
7.3 Dipolar Relaxation
Solomon equations
so we obtain for the center two elements of the Zeeman order vector the so-called Solomon equations
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what do the entries describe?
the first rate is called the leakage rate constant or auto relaxation rate constant
7.3 Dipolar Relaxation
Solomon equations
example for two protons, r = 0.2 nm, B0 = 11.47 T
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what do the entries describe?
the second rate is called the cross relaxation rate constant
7.3 Dipolar Relaxation
Solomon equations
example for two protons, r = 0.2 nm, B0 = 11.47 T
this function has a zero-crossing since it is the difference of two terms
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from the Solomon equations we know the equations of motion from the individual spins
this yields the change for the net longitudinal magnetization
problem with 1/4
7.3 Dipolar Relaxation
T1 relaxation
assuming no additional relaxation pathways, this yields(holds only for identical spins)
this can be used to predict the net magnetization at later time tb when it is known at t = ta
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this is just the exponential recovery as it was described in the phenemenological Bloch equations
and we see the relation between T1 and the spectral density functions
7.3 Dipolar Relaxation
T1 relaxation
note: this applies only to the sum of longitudinal magnetizations; individual components are more complicated
T1 dispersion shows minimum for certain correlation times
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the transverse relaxation rate can be determined in a similar way by looking at the transverse components
the result is
7.3 Dipolar Relaxation
T2 relaxation
T1 = T2 at short correlation times
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often also plotted on logarithmic scale as relaxation dispersion
fast motion: ²tc² « 1
also called extreme narrowing regime both rates are identical
7.3 Dipolar Relaxation
regimes of motion and B0
contrary to T1, T2 has almost no dependence on B0
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the connectivities in the dipolar coupled system can be used to increase signal intensity of one species through manipulation of the other
A) regular 31P in vivo spectrum
B) decoupled from protons; increased resolution
C) with NOE; increased signal intensity
useful in the context of poor NMR sensitivity – how does it work?
7.4 Signal Enhancement: NOE
Nuclear Overhauser effect
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consider a heteronuclear system I, Swith saturation of spin S which reaches steady state through cw irradiation
remember:since the spins are not identical any more, the derived relaxation of the total z component does not apply any more as before; relaxation will NOT be mono-exponential after perturbation
so we said S should be saturated,i.e. n1 = n2 and n3 = n4
7.4 Signal Enhancement: NOE
Nuclear Overhauser effect
look at old formula with I1 (now I) and I2 (now S)
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this can also be written as
and yields
or using the gyromagnetic ratios as measures for the thermal equilibrium
7.4 Signal Enhancement: NOE
Nuclear Overhauser effect
remember the transition probabilities
strictly speaking, these have to be evaluated not any more as before for identical spins but nor for I and S
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hence, J has to be evaluated at four different positions
example: 1H and 13C at 500 MHz spectrometer
7.4 Signal Enhancement: NOE
NOE for extreme narrowing
remember from lecture 3:the spectral density function becomes very broad for short correlation times
hence, the evaluation at four different frequencies yields (nearly) the same result
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remember definition of the reduced density function J()
7.4 Signal Enhancement: NOE
NOE for extreme narrowing
in the extreme narrowing limit ²tc² « 1, the reduced spectral density function J becomes just 2c
and the magnetization enhancement becomes
experiment:
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Lecture 7 Summary
1. Hard Pulses
2. Selective Pulses
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Lecture 7 Summary
3. Dipolar Relaxation
4. Signal Enhancement: NOE
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NMR Spectroscopy and Imaging
Department of Physics, FU Berlin
Lecture 8:
(Hyper-)Polarization, SEOP
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Lecture 8
1. Thermal Polarization
2. Brute Force Approach
3. Optical Pumping
4. Polarization Transfer in SEOP
5. SEOP Applications
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from lecture 3:random field fluctuations cause transverse component that acts similar to an RF pulse
why do these contributions do not cause the magnetic moments to orient completely randomly in the end?
T1 relaxation and thermal equilibrium
8.1 Thermal Polarization
assume initial preparation to be –M0
why do we not end up with M() = 0?
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the lowest energy state for the spins would be aligned with the z axis
however, the spins are in thermal contact with their environment, the “lattice”
function of the lattice
8.1 Thermal Polarization
rotating the bulk magnetization away from z increases energy of the spin system
magnetic moments rotating towards z release energy into the lattice
the (large!) lattice reservoir is at thermal equilibrium its lower energy levels are higher populated, i.e. it is more likely that it
absorbs energy from the added spin system than release into spin system
energy of interaction between spins and B0 is miniscule compared with the thermal energy of the lattice
hence, the asymmetry in probabilities causes a slight orientation of the magnetic moments towards the z-axis and not randomly
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consider a system with given eigenstates and energy levels
for the thermal equilibrium at temperature T we know that all coherences have vanishedand that the populations are given by the Boltzman distributions
the lower energy state is slightly higher populated
describing the thermal equilibrium
8.1 Thermal Polarization
Eth kBT 10-21 J at room temperaturewhereas E 10-26 -10-25 J
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intrioducing the Boltzmann factor,the two exponentials read
high temperature approximation
8.1 Thermal Polarization
since the Boltzmann factor is « 1, this can be expanded into a power series
the denominator then reads
and the two populations
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in matrix notation
the population difference is extremely small, only 1 in 105
multiplied with the magnetic moments of nuclei, this gives a small net magnetization
high temperature approximation
8.1 Thermal Polarization
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conventional solution to the problem
8.1 Thermal Polarization
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working at 300 K with small magnetic moments requires
large number of nuclei
conventional MRI with protons is a great tool
water proton concentration: ca. 110 M
aim: intense, specific signal with noor negligible background
MEca. 5 in a million
@ 1.5 T, room temperature
conventional solution to the problem
8.1 Thermal Polarization
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so what is the magnetization at other temperatures?
the two magnetic moments of a spin can have two energies
and the expectation value is(here, is the Boltzmann factor)
8.1 Thermal Polarization
Curie‘s law
the partition function is given by
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the expectation value becomes
and the macroscopic magnetization
8.1 Thermal Polarization
Curie‘s law
an alternative parameter that is often used is the polarization P
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conventional 1H detection
8.2 Brute Force Approach
104-fold improvement yieldsunique powerful combination:
high sensitivity of pre-polarized nuclei+
high specificity of NMR signal
manipualation of magnetic moments outside the scanner
pushes P = 10-5 10-1...1
EM
E M
intrinsic problem: ħB0 << kBT
potential of hyperpolarized nuclei
high-field approach: P 0.003% ca. 1 in 30 000 protons contributes
~10-fold improvement in P
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1E-3 0.01 0.1 1 10 1000.0
0.2
0.4
0.6
0.8
1.0
13C @ 21.2 T
1H @ 21.2 T
Pola
rizat
ion
Temperature [K]
e- @ 21.2 T
but even at low temperature …
cooling down the sample or the agent ...
… only electrons yieldgood polarization under moderate conditions
(P > 90 % @ 1K and 3 T)
DNP as one option
exterme thermal polarization
8.2 Brute Force Approach
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this approach has some side effects:• very high costs• the resonance frequency increases and the spectral density
function has to be evaluated at higher frequencies for the relaxation mechanisms
• T1 usually increases for high fields and can make signal accumulation extremely time-consuming
• also, unwanted susceptibility effects become more serious since they are directly related to the applied field
extreme high B-fields
8.2 Brute Force Approach
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8.2 Brute Force Approach
stray field problems already at 1.5T:
extreme high B-fields
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8.2 Brute Force Approach
stray field problems already at 1.5T:
extreme high B-fields
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8.2 Brute Force Approach
massive shielding neededexample: 8T/800 mm bore
active shielding too expensive
only passive shielding used
room made entirely of annealed low-carbon steel with joints internally welded
contains 200–500 tons of steel and reduces the extent of the 5-gauss contour by a factor two
extreme high B-fields
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from lecture 1:energy in a 21.1 T NMR spectrometer of 65 mm bore ~ 27 MJ
consider dimensions for whole body imaging and the volume the field has to penetrate:
7 T whole body MRI system: ca. 80 MJ (here: 7T 900 mm bore)
has to be dissipated safely in the event of a quench
safety issues – 7 T
8.2 Brute Force Approach
quench: current can drop from ~200 A 0within a few tens of seconds,
keep in mind: magnet inductance can be over 1000 H possibility of generating voltages in
excess of 10 kV harmful to personnel and magnet
windings
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this is an even more extreme case:
9T whole body MRI system carries ca. 151 MJ of field energy
safety issues – 9.4 T
8.2 Brute Force Approach
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why optical pumping?
aim: condensation into one spin statemost simple case: spin-1/2 system, 2 energy levels
problems:weak magnetic dipoles (even for ‘large’ magnetic moments of electrons)tiny thermal polarizationminor energy splitting of optical transitions in external fields
Nobel Prize in Physics 1966 to Alfred Kastler:"for the discovery and development of optical methods forstudying Hertzian resonances in atoms".
8.3 Optical Pumping
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why optical pumping?
award ceremony speech by Professor Ivar Waller:
The starting point of the work was research intoHertzian resonances. These are produced when atomsinteract with radio waves or microwaves, i. e. withelectromagnetic radiation having a frequency at least athousand times lower than visible light. Such waves aretherefore well suited to the study of fine details inspectra, which, though observable by opticalspectroscopy, could not be measured with satisfactoryprecision by this method....
Kastler was the first to propose a method ofinvestigating Hertzian resonances by optical methods,indicating the possibility of exciting selectively magneticsublevels from excited states by polarized light havingthe resonance frequency.
8.3 Optical Pumping
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basics of optical pumping
access spin sublevels indirectly by optical transition
manipulate light and apply selection rules for dipole transitions: +/- light to selectively drive one transition in an external field
one sublevel is continuously depletedwhile the other one becomesoverpopulated
transfer angular momentum from photonto electrons
8.3 Optical Pumping
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atomic system for efficient electron spin pumping
keep energy levels simple single electron system of alkali metals
achieve high vapor pressure at ‘reasonable’ temperatures heating up to ca. 200°C
optical transition in the rage of available laser wavelengths powerful (ca. 100W) diode lasers in the NIR range
our candidate: rubidium(method 1 to burn your lab)
8.3 Optical Pumping
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rubidium properties
Daniel A. Steck, “Rubidium 85 D Line Data”, http://steck.us/alkalidata
8.3 Optical Pumping
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rubidium properties
Daniel A. Steck, “Rubidium 85 D Line Data”, http://steck.us/alkalidata
8.3 Optical Pumping
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laser profile
vendor information:
line narrowing through volume Bragg grating, reduces line width to ca. 0.6 nm -> efficient pumping of D1 transition
8.3 Optical Pumping
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793.0 793.5 794.0 794.5 795.0 795.5 796.0 796.5 797.00.0
2.0x103
4.0x103
6.0x103
sign
al [a
.u.]
[nm]
laser profile
laser profile:2 Voigt lines of ca. 0.243 nm width
D1 line width: 0.038 nm/amagat-> use pressure broadening
vendor information:
line narrowing through volume Bragg grating, reduces line width to ca. 0.6 nm -> efficient pumping of D1 transition
8.3 Optical Pumping
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light absorption through rubidium vapor
793.0 793.5 794.0 794.5 795.0 795.5 796.0 796.5 797.0
1000
2000
3000
4000
5000
cold hot
inte
nsity
[a.u
.]
[nm]
symmetric absorption over ca. 0.53 nm withchiller: 17.0°Cdiode: 22.2°Ccurrent: 40 Apower: 130 W
793 794 795 796 797
0
20
40
60
80
abso
rptio
n [%
]
[nm]
8.3 Optical Pumping
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light absorption through rubidium vapor
better always know where the power goes!
(method 2 to burn your lab)
793.0 793.5 794.0 794.5 795.0 795.5 796.0 796.5 797.0
1000
2000
3000
4000
5000
cold hot
inte
nsity
[a.u
.]
[nm]
symmetric absorption over ca. 0.53 nm withchiller: 17.0°Cdiode: 22.2°Ccurrent: 40 Apower: 130 W
8.3 Optical Pumping
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Rb D1 transition
Daniel A. Steck, “Rubidium 85 D Line Data”,http://steck.us/alkalidata
resolved fine structure:D2 line to 52P3/2at ca. 780nm
unresolved HFS splitting:87Rb: I = 3/2-> F = I ± J = 5/2 ± 1/2for D1 states
energy corrections < 2 GHz (0.0005%)
F = 2 -> F = 3changes by -0.004 nm
F = 3 -> F = 2 changes by +0.003 nm
8.3 Optical Pumping
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Rb D1 transition
Daniel A. Steck, “Rubidium 85 D Line Data”, http://steck.us/alkalidata
magnetic sublevels:even smaller corrections (another factor 4000)
field for quantization axis:ca. 20G
8.3 Optical Pumping
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polarization optics
optical fiber
polarizing beamsplitter cube blocking
ordinary beam
/4 plateiriscollimator
Galileantelescope
mirror for ordinary beam
optical elements are temperature monitored
8.3 Optical Pumping
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polarization optics
8.3 Optical Pumping
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oven box
gas out gas in
collimatorfor optical
spectrometer
hot sidecold side
Helmholtz coils, ca. 20 G
8.3 Optical Pumping
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oven box
8.3 Optical Pumping
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gas phasetransition
bloodtransfer of photon polarization
SEOP: Spin Exchange Optical Pumpingphoton electron nucleus
16000-fold signal enhancementSchröder; Physica Medica,
in press (2011)
8.4 Polarization Transfer in SEOP
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impact of increasing xenon concentration
8.4 Polarization Transfer in SEOP
answer: well, it depends...
build-up of Xe polarization:
definition of ~ <PRb>SEXe density-normalized figure for SEOP efficiency
temperature conditions are crucial for optimized pumping
if T is included in optimization, more Xe reduces SEOP efficiency
Whiting et al. J. Magn. Reson. 208:298-304 (2010)
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late HH field ramp; what determines cell temp
different test run diode 0 40A
Xe gas mixture already causes heat production(Xe inflow causes loss in PRb)
even more serious Rb runaway when HH field is turned on late
remaining absorption drives temperature towards 190° for cell center
early HH turn on causes continuous increase towards equilibrium
Rb burn off happens twice (slightly higher T compared to 30A diode current)
0 5 10 15 20 25 30 35 40 450
50
100
150
200
cooling onHelmholtzfield on
Ar to Xe mix
cell heateron
Tem
pera
ture
[°C
]
time [min]
cell center exit window entrance window
laser on
8.4 Polarization Transfer in SEOP
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pure 3He/129Xe or functionalized 129Xe for• imaging of void space (lung imaging)• imaging different types of tissue (e.g., grey/white matter)• display distribution of functionalized
contrast agent
T1 has to outlast deliverydetection of further target molecule possible
concentrations: ca. 30 %vol. in lungs, mM in brain tissue
Bachert et al.; Magn. Reson. Med. 36, 192-196 (1996) Zhou et al.; NMR Biomed. 21: 217–225 (2008)
noble gas MRI
8.5 SEOP Applications
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void space imaging
Mugler et al., J. Magn. Reson. Imag. 37: 313 (2013)
129Xe
8.5 SEOP Applications
increasing anatomical resolution + functional information
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void space imaging
129Xe
3He-enabled lung metastases detection
Branca et al.; Proc. Natl. Acad. Sci. USA 107: 3693-3697 (2010)
8.5 SEOP Applications
high resolution anatomical illustration +
gas exchange information by probing the molecular
environment
Driehuys et al.; Proc. Natl. Acad. Sci. USA 103: 18278-18283 (2006)
129Xe Tissue barrier RBC
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temporary binding of 129Xe in functionalized molecular host for
solution state NMR
more in lecture 19
xenon biosensors
129Xe
Taratula et al.; Curr. Opin. Chem. Biol. 14: 97-104 (2010)
8.5 SEOP Applications
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Lecture 8 Summary
1. Thermal Polarization
2. Brute Force Approach
3. Optical Pumping
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Lecture 8 Summary
4. Polarization Transfer in SEOP
5. SEOP Applications