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NMR 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 e ) t displacement from equilibrium value n e at time t (n-n e ) 0 at time zero - PowerPoint PPT Presentation
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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
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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR 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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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:
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
Nature Structural Biology 5, 517 - 522 (1998)
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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 Relaxation
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
NMR RelaxationNMR Relaxation
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
NMR RelaxationNMR Relaxation
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 Exchange
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
NMR Dynamics and ExchangeNMR Dynamics and Exchange
o
coalescence
NMR Dynamics and ExchangeNMR Dynamics and Exchange
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
NMR Dynamics and ExchangeNMR Dynamics and Exchange
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
NMR Dynamics and ExchangeNMR Dynamics and Exchange
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
NMR Dynamics and ExchangeNMR Dynamics and Exchange
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
NMR Dynamics and ExchangeNMR Dynamics and Exchange
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
NMR Dynamics and NMR Dynamics and ExchangeExchangeUse of magnetization transfer to study exchange
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
NMR Dynamics and NMR Dynamics and ExchangeExchangeUse of magnetization transfer to study exchange
– – /2 pulse sequence- exchange takes place during ()
Saturate peak 2, exchange to peak 3
Saturate peak 1, exchange to peak 4
NMR Dynamics and NMR Dynamics and ExchangeExchangeUse of magnetization transfer to study exchange
– – /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
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