nmr relaxation

NMR RelaxationNMR Relaxation After an RF pulse system needs to relax back to equilibrium condition

Related to molecular dynamics of system may take seconds to minutes to fully recovery usually occurs exponentially:

– (n-n(n-nee))tt displacement from equilibrium value n displacement from equilibrium value nee at time t at time t– (n-n(n-nee))00 at time zero at time zero

Relaxation can be characterized by a time T– relaxation rate (R): 1/Trelaxation rate (R): 1/T

No spontaneous reemission of photons to relax down to ground state probability too low cube of the frequency

Two types of NMR relaxation processes spin-lattice or longitudinal relaxation (T1) spin-spin or transverse relaxation (T2)

B1 off…

(or off-resonance)

Mo

z

x

B1

z

x

Mxy

y y1

1

Mo

y

z

xT1 & T2

relaxation

)/exp()()( 0 Ttnnnn ete

Spin-lattices or longitudinal relaxation Relaxation process occurs along z-axis transfer of energy to the lattice or solvent material coupling of nuclei magnetic field with magnetic fields created by the ensemble of

vibrational and rotational motion of the lattice or solvent. results in a minimal temperature increase in sample Relaxation time (T1) exponential decay

NMR RelaxationNMR Relaxation

Mz = M0(1-exp(-t/T1))

Spin-lattices or longitudinal relaxation Relaxation process occurs along z-axis

Measure T1 using inversion recovery experiment


NMR RelaxationNMR Relaxation Spin-lattices or longitudinal relaxation

Collect a series of 1D NMR spectra by varying Measure T1 using inversion recovery experiment

NMR RelaxationNMR Relaxation Spin-lattices or longitudinal relaxation

Collect a series of 1D NMR spectra by varying Plot the peak intensities as a function of fit to an exponential

Mechanism for Spin-lattices or longitudinal relaxation• Dipolar coupling between nuclei and solvent (T1)

interaction between nuclear magnetic dipoles depends on correlation time

– oscillating magnetic field due to Brownian motion– depends on orientation of the whole molecule

in solution, rapid motion averages the dipolar interaction –Brownian motion in crystals, positions are fixed for single molecule, but vary between molecules leading to range of frequencies and broad lines.

Tumbling of Molecule Creates local Oscillating Magnetic field


22241

2)(

c

c

vvK

c represents the maximum frequency– 10-11s = 1011 rad s-1 = 15920 MHz

All lower frequencies are observed

Field Intensity at any frequency

Mechanism for Spin-lattices or longitudinal relaxation• Solvent creates an ensemble of fluctuating magnetic fields

causes random precession of nuclei dephasing of spins possibility of energy transfer matching frequency



Mechanism for Spin-lattices or longitudinal relaxation• Intensity of fluctuations in magnetic fields due to Brownian motion as a function of frequency

causes random precession of nuclei dephasing of spins possibility of energy transfer matching frequency

Spectral Density Function (J())

c = 10-11 s-1

c = 10-10 s-1

c = 10-9 s-1

c = 10-8 s-1

Incr

easi

ng M

W

T2 relaxation


Spin-lattices or longitudinal relaxation Relaxation process in the x,y plane Related to peak line-width

– Inhomogeneity of magnet also contributes to peak widthInhomogeneity of magnet also contributes to peak width T2 may be equal to T1, or differ by orders of magnitude

– TT22 can not be longer than T can not be longer than T11

No energy change

(derived from Heisenberg uncertainty principal)

Spin-spin or Transverse relaxation exchange of energy between excited nucleus and low energy state nucleus randomization of spins or magnetic moment in x,y-plane related to NMR peak line-width relaxation time (T2)

Mx = My = M0 exp(-t/T2)

Please Note: Line shape is also affected by the magnetic fields homogeneity


Spin-spin or Transverse relaxation While peak width is related to T2, not an accurate way to measure T2

Use the Carr-Purcell-Meiboom-Gill (CPMG) experiment to measure “spin-echo”– Refocuses spin diffusions due to magnetic field inhomogeneiety


Biochemistry 1981, 20, 3756-3764

Mx = My = M0 exp(-t/T2)

NMR RelaxationNMR Relaxation Spin-spin or Transverse relaxation

Collect a series of 1D NMR spectra by varying Plot the peak intensities as a function of and fir to an exponential Peaks need to be resolved to determine independent T2 values

kT

rc 3

4 3

where: r = radiusk = Boltzman constant

= viscosity coefficient rotational correlation time (c) is the time it takes a molecule to rotate one radian (360o/2).

the larger the molecule the slower it movesthe larger the molecule the slower it moves T2 ≤ T1

small molecules (fast tsmall molecules (fast tcc) T) T22 =T =T11

Large molecules (slow tLarge molecules (slow tcc) T) T22 < T < T11

NMR RelaxationNMR RelaxationMechanism for Spin-lattices and Spin-Spin relaxation

• Relaxation is related to correlation time (c) Intensity of fluctuations in magnetic fields due to Brownian motion as a function of frequency MW radius c


Mechanism for Spin-lattices and Spin-Spin relaxation• Illustration of the Relationship Between MW, c and T2

NMR RelaxationNMR RelaxationMechanism for Spin-lattices and Spin-Spin relaxation

• Relaxation is related to correlation time (c)• intramolecular dipole-dipole relaxation rate of a nuclei being relaxed by n nuclei

Depends on distance(bond length)

n

jii ijcDDDD

DDDD raRR

TT )(16

421

21

110

11

n

jii ijc

c

c

ccDD

DD raR

T )(162222

42

2

1

41

2

53

1

n

jii ijc

c

c

cDD

DD raR

T )(162222

41

1

1

41

4

12

1

Depends on nuclei type

Extreme narrowing limit:

2/

,320/31

222

constantsPlanck'

sradinfrequencyNMRw

vacuumaoftypermeabilia oo


Mechanism for Spin-lattices and Spin-Spin relaxation• Relaxation is related magnetic field strength ()

T1 minima and values increase with increasing field strength

T2 reduced at higher field strength for larger molecules leading to broadening

n

jii ijcs

ccSI

DDDD

DD r

aRR

T )(1622

2212

2

1

1

64

32

1

n

jii ijcSI

c

cI

c

cSI

cSIDD

DD r

aR

T )(16222222

441

1

1

)(1

12

1

6

)(1

2

3

1


Mechanism for Spin-lattices and Spin-Spin relaxation• Different relaxation times (pathways) for different nuclei interactions

1H-1H ≠ 1H-13C ≠ 13C-13C– relaxation rates depend on the number of attached nuclei and bond lengthrelaxation rates depend on the number of attached nuclei and bond length– carbon: carbon: 1313C > C > 1313CH > CH > 1313CHCH22 > > 1313CHCH33

– proton: dominated by relaxation with other protons in moleculeproton: dominated by relaxation with other protons in molecule Same general trends as intramolecular relaxation

n

jii ijcSIDDDD

DDDD r

aRR

TT )(16

2221

21

1

3

2011 Extreme narrowing limit:

NMR RelaxationNMR RelaxationTypical Spin-lattices Relaxation Times

• T2 ≤ T1

• Examples of 13C T1 values number of attached protons greatly affects T1 value

– Non-proton bearing carbons have very long TNon-proton bearing carbons have very long T11 values values T1 longer for smaller molecules Differences in T1 values related to local motion

– Faster motion Faster motion longer T longer T11

Solvent can affect T1 values

CH3OH CD3OD

Solvent Effects:


Chemical Shift Anisotropy Relaxation• Remember:

Magnetic shielding () depends on orientation of molecule relative to Bo

magnitude of magnitude of varies with orientation varies with orientation

BBoo

Orientation effect described by the screening tensor:

11, 22, 33

If axially symmetric:

11 = 22 = ||

33 = ┴

Solid NMR SpectraSolid NMR Spectra

Chemical Shift Anisotropy (CSA) Relaxation• Effective Fluctuation in Magnetic field strength at the nucleus

Causes relaxation not very efficient in extreme narrowing region:

– depends strongly on field strength and correlation timedepends strongly on field strength and correlation time– depends strongly on chemical shift rangesdepends strongly on chemical shift ranges– results in line-broadeningresults in line-broadening– increase in sensitivity and resolution at higher field strengths may be increase in sensitivity and resolution at higher field strengths may be overwhelmed by CSA affectsoverwhelmed by CSA affects

15

21 ||22

1

coI

CSA

B

T


Nature Structural Biology 5, 517 - 522 (1998)


Chemical Shift Anisotropy (CSA) Relaxation• Line-shape increases as CSA increases with magnetic field strength

Two peaks in nitrogen doublet experience different CSA contributions

Can improve line shape if only select this peak


Chemical Shift Anisotropy (CSA) Relaxation• Line-shape increases as CSA increases with magnetic field strength

Peaks originating from 195Pt-1H2

coupling are broadened at higher field due to CSA (shortening of T1(Pt)

Increasin

g Magn

etic Field

Quadrupolar Relaxation• Quadrupole nuclei (I > ½)• Introduces a second and very efficient relaxation mechanism

a factor of 108 as efficient of dipole-dipole relaxation Distribution of charge is non-spherical ellipsoidal

– for I = ½, charge is spherically distributedfor I = ½, charge is spherically distributed

Different charge distribution electric field gradient varies randomly with Brownian motion relaxation mechanism


NMR RelaxationNMR RelaxationQuadrupolar Relaxation• Electric Field Gradient (EFG)• tensor quantity

can be reduced to diagonal values Vxx,Vyy,Vzz

Vxx + Vyy + Vzz = 0

asymmetry factor ():

Vxx,Vyy,Vzz are calculated from the sum of contributions from all charges qi at a

distance ri

Quadrupole relaxation times (T1Q,T2Q), where Q is quadrupole moment

zz

xxyy

V

VV

iiixx i

rxrqV )3( 225

czzQQQQ

Vh

Qe

II

IRR

TT

222

2

2

2121 3

11

)12(

)32(

10

311

NMR RelaxationNMR RelaxationQuadrupolar Relaxation

• Factors affecting quadrupolar relaxation

•Depends strongly on nuclear properties quadrupole moment (Q) and spin number (I)

• Depends strongly on molecular properties correlation time (c)

– increasing temperature increases increasing temperature increases cc and increases relaxation time and reduces and increases relaxation time and reduces

resonance linewidthresonance linewidth shape (Vzz, )

• Depends primarily on electric field gradient (EFG)

can vary from zero to very large numbers charge close to nucleus have predominating effect (distance dependence) movement of molecules in liquid reduces distance effect to zero solids with fixed distances have contributions from distant charges

czzQQQQ

Vh

Qe

II

IRR

TT

222

2

2

2121 3

11

)12(

)32(

10

311


Dipole nuclei (I=1/2) coupled to quadrupole nuclei (I>1/2) • Quadrupole relaxation significantly broadens nuclei

obscures spin-splitting pattern If quadrupole relaxation is slow, broadening is diminished and spin-splitting pattern is observed

Incr

easi

ng T

1 Increasing T1

Long T1 normal splitting

Very short T1 – average value


Dipole nuclei (I=1/2) coupled to quadrupole nuclei (I>1/2) • Quadrupole relaxation significantly broadens nuclei through scaler coupling

Lowering temperature can sharpen peaks broaden by quadrupole relaxation– lower temperature increase tc shorten T1Q

s

SCsc

TSSJTT 1

22

12

)1(3

41

2

11

NMR RelaxationNMR RelaxationQuadrupolar Relaxation

• If the system is axially symmetric, =0 and Vxx = Vyy

• Only need to determine Vzz equal distribution of three charges around the z-axis at a distance r from N

Vzz =0 if =54.7356o – “magic angle” nuclei at center of a reqular tetrahedron, octahedron or cube have near-zero EFG long relaxation time is source of structural information

3

2

5

222 )1cos3(3)cos3(3

r

q

r

rrqVzz

NMR Dynamics and ExchangeNMR Dynamics and Exchange

Despite the Typical Graphical Display of Molecular Structures, Molecules are Highly Flexible and Undergo Multiple Modes Of Motion Over a Range of Time-Frames

DSMM - Database of Simulated Molecular Motionshttp://projects.villa-bosch.de/dbase/dsmm/ Click on image to start dynamics simulationClick on image to start dynamics simulation

Populations ~ relative stability

Rex < (A) - (B)

Exchange Rate(NMR time-scale)

Multiple Signals for Slow Exchange Between Conformational States• Two or more chemical shifts associated with a single atom/nucleus

Factors Affecting Exchange: Addition of a ligand Temperature Solvent


NMR Dynamics and ExchangeNMR Dynamics and ExchangeEtOH + EtOH* EtOH + EtOH*H+

Slow exchange: CH2-OH coupling is observed

Fast exchange: Addition of acidCH2-OH coupling is absent

Intermediate exchange: Broad peaks

OH exchanges between different molecules and environments. Observed chemical shifts and line-shapes results from the average of the different environments.

Different environments

Effects of Exchange Rates on NMR data

k = ((W1/2)e-(W1/2)o)

k = (o2 - e

2)1/2/21/2

k = o / 21/2

k = o2 /2(W1/2)e – (W1/2)o)

k – exchange rateW1/2 – peak-width at half-height – peak frequency

e – with exchangeo – no exchange


o

coalescence


k = 0.1 s-1

k = 5 s-1

k = 200 s-1

k = 88.8 s-1

k = 40 s-1

k = 20 s-1

k = 10 s-1

k = 400 s-1

k = 800 s-1

k = 10,000 s-1

40 Hz

Incr

easi

ng E

xcha

nge

Rat

e

slow

fast

22/1

1

TW

No exchange:

With exchange:

exTW

11

22/1

ex

k1

Equal Population of Exchange Sites


Example of NMR Measurement of Chemical Exchange Two different cyclopentadienyl groups in [Ti(1-C5H5)2(5-C5H5)2] Exchange rate changes as a function of temperature

But, chemical shifts also change as a function of temperatureBut, chemical shifts also change as a function of temperature


Example of NMR Measurement of Chemical Exchange Multiple resonances may be affected by exchange

Rotation about N-C bondRotation about N-C bond different coalescence rates because of different different coalescence rates because of different aa--bb

})(){(2

)(

2/12/1

2

oex

BA

WWk

C3 & C4 separation smaller than C6 & C2

• Exchanges Rates and NMR Time Scale NMR time scale refers to the chemical shift time scale

– remember – frequency units are in Hz (sec-1) time scale– exchange rate (k)– differences in chemical shifts between species in exchange indicate

the exchange rate.

Time Scale Chem. Shift ( Coupling Const. (J) T2 relaxationSlow k << A- B k << JA- JB k << 1/ T2,A- 1/ T2,B

Intermediate k = A - B k = JA- JB k = 1/ T2,A- 1/ T2,B

Fast k >> A - B k >> JA- JB k >> 1/ T2,A- 1/ T2,B

Range (Sec-1) 0 – 1000 0 –12 1 - 20


1 1

2

Fe(CO)4

C C C

H2C

H2C CH2

CH2C C C

H2C

H2C CH2

CH2

Fe(CO)4

Slow exchange at -60o

• Exchange Rates and NMR Time Scale NMR time scale refers to the chemical shift time scale

– For systems in fast exchange, the observed chemical shift is the average of the individual species chemical shifts.

obs = f11 + f22

f1 +f2 =1where:

f1, f2 – mole fraction of each species1,2 – chemical shift of each species


Fast exchange, average of three slow exchange peaks

≈ 1.86 ppm ≈ 0.25 x 2.00 ppm + 0.25 x 1.95 ppm

+ 0.5 x 1.75 ppm

NMR Dynamics and ExchangeNMR Dynamics and ExchangeUnequal Population of Exchange Sites

differential broadening below coalescence- lower populated peak broadens more

k = 0.1 s-1

k = 5 s-1

k = 200 s-1

k = 88.8 s-1

k = 40 s-1

k = 20 s-1

k = 10 s-1

k = 400 s-1

k = 800 s-1

k = 10,000 s-1

Incr

easi

ng E

xcha

nge

Rat

e

slow

fast

40 Hz

31

A

BBA p

pkk

Exchange rate depends on population (p):

coalescenceAbove coalescence:

BA

BBAAavg pp

pp

Weighted average

NMR Dynamics and NMR Dynamics and ExchangeExchangeExample of NMR Measurement of Chemical

Exchange Unequal populated exchange sites

exchange between axial and equatorial exchange between axial and equatorial positionposition

exchange rate can be measured easily up exchange rate can be measured easily up

to -44to -44ooC. Can easily measure C. Can easily measure aa--e e and peak and peak

ratiosratios again, different broadening is related to again, different broadening is related to chemical shift differences between axial chemical shift differences between axial and equatorial positionsand equatorial positions

- difficult to determine accurate difficult to determine accurate aa--ee

- difficult to determine accurate difficult to determine accurate kk

NMR Dynamics and NMR Dynamics and ExchangeExchangeUse of magnetization transfer to study exchange

Lineshape analysis is related to the rate of leaving each site no information on the destination

problem for multisite exchange Saturation Transfer Difference (STD) Experiment

Collect two spectra:- one peak is saturated (decoupler pulse)one peak is saturated (decoupler pulse)- decoupler or saturation pulse is set far from any peaks (reference spectrum)decoupler or saturation pulse is set far from any peaks (reference spectrum)- subtract two spectrasubtract two spectra

If nuclei are exchanging during the saturation pulse, additional NMR peaks will exhibit a decrease in intensity due to the saturation pulse.

A B

Mz(0) Mz(0)

decouple site A

MzA = 0 Mz(0)

A Bexchange from A to B

)()()0()(

1

tMkT

tMM

dt

tdM BZBB

Bz

Bz

Bz

Exchange rateT1 relaxation


at equilibrium (t=∞)

kB can be measured from MZB(∞), MZ

B(0) and T1B

- assumes T1A = T1

B

- if T1A ≠ T1

B,difficult to measure T1A and T1

B partial average

)()()0(

01

BZBB

Bz

Bz Mk

T

MM

or

)(

)()0(1

1

BZ

Bz

Bz

BB M

MM

Tk

exchangebyunaffectedsignalBMMthenT

kif

decreasetosignalBcausestransferizationmagnetMthenT

kif

BZ

BZBB

BZBB

)0(~)(1

0~)(1

1

1


– – /2 pulse sequence- exchange takes place during ()

Saturate peak 2, exchange to peak 3

Saturate peak 1, exchange to peak 4


– – /2 pulse sequence- exchange takes place during ()

Selective 180o pulseSaturation transferred during

Fit peak intensities to determine average T1 and k (k=15.7 s-1 & T1 = 0.835 s)

Calculating H and S may not be reliable:

- temperature dependent chemical shifts- mis-estimates of line-widths in absence of exchange- poor temperature calibration- signal broadened by unresolved coupling

To obtain reliable H and S values:- obtain data over a wide range of temperature where coalescence points can be monitored- measure at different spectrometer frequencies- use different nuclei with different chemical shifts- use line-shape analysis software- use magnetization transfer

NMR Dynamics and NMR Dynamics and ExchangeExchangeActivation Energies from NMR data

rate constant is related to exchange rate (k=1/ex)

T

kRTG ln759.23‡

Measure rate constants at different temperatures

‡‡‡ STHG

Different nuclei and Different nuclei and magnetic field strengthsmagnetic field strengths

NMR Dynamics and NMR Dynamics and ExchangeExchange

x

Two-Dimensional Exchange Experiments Uses the NOESY pulse sequence (EXSY)

- uses a short mixing time (~ 0.05s)- exchange of magnetization occurs during mixing time- NOEs will also be present

• need to distinguish between NOE and exchange need to distinguish between NOE and exchange peakspeaks• usually opposite signusually opposite sign

Exchange peaksExchange peaks

nmr relaxation

Documents

longitudinal relaxationcollect

tt1 spinlattices

solvent t1 interaction

t1no energy

function of frequency

nmr relaxationafter

d nmr spectra

nuclear magnetic dipoles