Spin Manipulation and Dipolar Relaxation · 1 Leibniz-Institut für Molekulare Pharmakologie NMR Spectroscopy and Imaging Department of Physics, FU Berlin Lecture 7: Spin Manipulation

Leibniz-Institut für Molekulare Pharmakologie1

NMR Spectroscopy and Imaging

Department of Physics, FU Berlin

Lecture 7:

Spin Manipulation and Dipolar Relaxation


Lecture 7

1. Hard Pulses

2. Selective Pulses

3. Dipolar Relaxation

4. Signal Enhancement: NOE


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


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


and similar for the other two cases

7.1 Hard Pulses


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


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


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


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


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


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)


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


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


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)


define a pulse shapewith truncation at the edgesdefined by


Gauss pulses

example: Gauss inversion pulse

increasing truncation yields decreased inversion bandwidth


define a pulse shape with truncation at the edges for 90° / 180°


Hermite pulses

example: Hermite inversion pulse

profiles are better, but higherpeak amplitudes are required


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


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


transition probabilities


these can be written as follows


transition probabilities


all transition probabilities fall of rapidly with spin distance r


important contributions

why is the contribution from 20

important?

local field is modulated with 20 when spin I2 rotates with 0


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




now combine rotation and precession

this yields component at 0 with correct sign




what happens to population of, e.g., state ?

it has three positive and three negative contributions, weighted by the populations


Solomon equations


summarizing all four populations in a vector, we get

and summarize the transition probabilities in the following matrix


Solomon equations

note: the negative contributions are on the diagonal, the positive ones coming from the other states are off-diagonal elements


an alternative basis to describe the state of the system is the Zeeman order vector

hence we need to transform the differential equation


Solomon equations

the vector transition is done by


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


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)


now we can express the formerly used population numbers with these operators


Solomon equations – changing the basis

this is why the transformation matrix reads

(factor ½ due to definition of density matrix entries in Levittnotation)


so we also need to transform the matrix using

to be used for


Solomon equations

in thermal equilibrium we have

where is just the tiny polarization due to the Boltzmann factor (see lecture 8)


we therefore rewrite the differential equation as

with approximating the matrix by omitting terms containing the small Boltzmann factor


Solomon equations

so we obtain for the center two elements of the Zeeman order vector the so-called Solomon equations


what do the entries describe?

the first rate is called the leakage rate constant or auto relaxation rate constant


Solomon equations

example for two protons, r = 0.2 nm, B0 = 11.47 T


what do the entries describe?

the second rate is called the cross relaxation rate constant


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


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


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


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


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


the transverse relaxation rate can be determined in a similar way by looking at the transverse components

the result is


T2 relaxation

T1 = T2 at short correlation times


often also plotted on logarithmic scale as relaxation dispersion

fast motion: ²tc² « 1

also called extreme narrowing regime both rates are identical


regimes of motion and B0

contrary to T1, T2 has almost no dependence on B0


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


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



look at old formula with I1 (now I) and I2 (now S)


this can also be written as

and yields

or using the gyromagnetic ratios as measures for the thermal equilibrium



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


hence, J has to be evaluated at four different positions

example: 1H and 13C at 500 MHz spectrometer


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


remember definition of the reduced density function J()


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:


Lecture 7 Summary

1. Hard Pulses

2. Selective Pulses


Lecture 7 Summary

3. Dipolar Relaxation

4. Signal Enhancement: NOE


NMR Spectroscopy and Imaging

Department of Physics, FU Berlin

Lecture 8:

(Hyper-)Polarization, SEOP


Lecture 8

1. Thermal Polarization

2. Brute Force Approach

3. Optical Pumping

4. Polarization Transfer in SEOP

5. SEOP Applications


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?


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


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


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


Eth kBT 10-21 J at room temperaturewhereas E 10-26 -10-25 J


intrioducing the Boltzmann factor,the two exponentials read

high temperature approximation


since the Boltzmann factor is « 1, this can be expanded into a power series

the denominator then reads

and the two populations


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



conventional solution to the problem



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



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)


Curie‘s law

the partition function is given by


the expectation value becomes

and the macroscopic magnetization


Curie‘s law

an alternative parameter that is often used is the polarization P


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


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



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




stray field problems already at 1.5T:




stray field problems already at 1.5T:




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



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


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


this is an even more extreme case:

9T whole body MRI system carries ca. 151 MJ of field energy

safety issues – 9.4 T



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


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


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


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


rubidium properties

Daniel A. Steck, “Rubidium 85 D Line Data”, http://steck.us/alkalidata

8.3 Optical Pumping


rubidium properties


8.3 Optical Pumping


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


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


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


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


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


Rb D1 transition


magnetic sublevels:even smaller corrections (another factor 4000)

field for quantization axis:ca. 20G

8.3 Optical Pumping


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


polarization optics

8.3 Optical Pumping


oven box

gas out gas in

collimatorfor optical

spectrometer

hot sidecold side

Helmholtz coils, ca. 20 G

8.3 Optical Pumping


oven box

8.3 Optical Pumping


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


impact of increasing xenon concentration


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)


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



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


void space imaging

Mugler et al., J. Magn. Reson. Imag. 37: 313 (2013)

129Xe


increasing anatomical resolution + functional information


void space imaging

129Xe

3He-enabled lung metastases detection

Branca et al.; Proc. Natl. Acad. Sci. USA 107: 3693-3697 (2010)


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


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)



Lecture 8 Summary

1. Thermal Polarization

2. Brute Force Approach

3. Optical Pumping


Lecture 8 Summary

4. Polarization Transfer in SEOP

5. SEOP Applications

Documents

Spin Manipulation and Dipolar Relaxation · 1 Leibniz-Institut für Molekulare Pharmakologie NMR Spectroscopy and Imaging Department of Physics, FU Berlin Lecture 7: Spin Manipulation