1. NMR Relaxation Mo Mo B1 off… (or off-resonance) T1 & T2 relaxation B1
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-ne)t displacement from equilibrium value ne at time t
(n-ne)0 at time zero
Relaxation can be characterized by a time T
relaxation 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) z z z Mo Mo
B1 off…
(or off-resonance)
T1 & T2
relaxation x x x B1
Mxy w1 y y y w1

2. NMR Relaxation Mz = M0(1-exp(-t/T1))
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
Mz = M0(1-exp(-t/T1))

3. NMR Relaxation Spin-lattices or longitudinal relaxation
Relaxation process occurs along z-axis
Measure T1 using inversion recovery experiment

4. NMR Relaxation Spin-lattices or longitudinal relaxation
Collect a series of 1D NMR spectra by varying t
Measure T1 using inversion recovery experiment

5. NMR Relaxation Spin-lattices or longitudinal relaxation
Collect a series of 1D NMR spectra by varying t
Plot the peak intensities as a function of t fit to an exponential

6. NMR Relaxation 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

7. Field Intensity at any frequency
NMR Relaxation
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
Field Intensity at any frequency
tc represents the maximum frequency
10-11s = 1011 rad s-1 = MHz
All lower frequencies are observed

8. Spectral Density Function (J(w))
NMR 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
tc = 10-8 s-1
Spectral Density Function (J(w))
tc = 10-9 s-1
Increasing MW
tc = s-1
tc = s-1

9. NMR 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 width
T2 may be equal to T1, or differ by orders of magnitude
T2 can not be longer than T1
No energy change
T2 relaxation
(derived from Heisenberg uncertainty principal)

10. NMR Relaxation Mx = My = M0 exp(-t/T2)
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

11. NMR 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

12. NMR Relaxation Mx = My = M0 exp(-t/T2)
Spin-spin or Transverse relaxation
Collect a series of 1D NMR spectra by varying t
Plot the peak intensities as a function of t and fir to an exponential
Peaks need to be resolved to determine independent T2 values
Mx = My = M0 exp(-t/T2)
Biochemistry 1981, 20,

13. NMR Relaxation Mechanism for Spin-lattices and Spin-Spin relaxation
Relaxation is related to correlation time (tc)
Intensity of fluctuations in magnetic fields due to Brownian motion as a function of frequency
MW  radius  tc
where:
r = radius
k = Boltzman constant
h = viscosity coefficient
rotational correlation time (tc) is the time it takes a
molecule to rotate one radian (360o/2p).
the larger the molecule the slower it moves
T2 ≤ T1
small molecules (fast tc) T2 =T1
Large molecules (slow tc) T2 < T1

14. NMR Relaxation Mechanism for Spin-lattices and Spin-Spin relaxation
Illustration of the Relationship Between MW, tc and T2

15. Extreme narrowing limit:
NMR Relaxation
Mechanism for Spin-lattices and Spin-Spin relaxation
Relaxation is related to correlation time (tc)
intramolecular dipole-dipole relaxation rate of a nuclei being relaxed by n nuclei
Depends on nuclei type
Extreme narrowing limit:
Depends on distance
(bond length)

16. NMR Relaxation Mechanism for Spin-lattices and Spin-Spin relaxation
Relaxation is related magnetic field strength (w)
T1 minima and values increase with increasing field strength
T2 reduced at higher field strength for larger molecules leading to broadening

17. Extreme narrowing limit:
NMR 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 length
carbon: 13C > 13CH > 13CH2 > 13CH3
proton: dominated by relaxation with other protons in molecule
Same general trends as intramolecular relaxation
Extreme narrowing limit:

18. NMR Relaxation Typical 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 T1 values
T1 longer for smaller molecules
Differences in T1 values related to local motion
Faster motion  longer T1
Solvent can affect T1 values
Solvent Effects:
CH3OH
CD3OD

19. NMR Relaxation Chemical Shift Anisotropy Relaxation Remember:
Magnetic shielding (s) depends on orientation of molecule relative to Bo
magnitude of s varies with orientation Bo
Solid NMR Spectra
Orientation effect described by the screening tensor:
s11, s22, s33
If axially symmetric:
s11 = s22 = s||
s33 = s┴

20. NMR Relaxation 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 time
depends strongly on chemical shift ranges
results in line-broadening
increase in sensitivity and resolution at higher field strengths may be overwhelmed by CSA affects

21. NMR 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
Nature Structural Biology  5, (1998)

22. NMR 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)
Increasing Magnetic Field

23. NMR Relaxation 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 distributed
Different charge distribution  electric field gradient varies randomly with Brownian motion relaxation mechanism

24. NMR Relaxation Quadrupolar Relaxation Electric Field Gradient (EFG)
tensor quantity
can be reduced to diagonal values Vxx,Vyy,Vzz
Vxx + Vyy + Vzz = 0
asymmetry factor (h):
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

25. NMR Relaxation Quadrupolar Relaxation
Factors affecting quadrupolar relaxation
Depends strongly on nuclear properties
quadrupole moment (Q) and spin number (I)
Depends strongly on molecular properties
correlation time (tc)
increasing temperature increases tc and increases relaxation time and reduces resonance linewidth
shape (Vzz, h)
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

26. Very short T1 – average value Long T1 normal splitting
NMR 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
Very short T1 – average value
Increasing T1
Increasing T1
Long T1 normal splitting

27. NMR 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

28. NMR Relaxation Quadrupolar Relaxation
If the system is axially symmetric, h =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 f= o – “magic angle”
nuclei at center of a reqular tetrahedron, octahedron or cube have near-zero EFG
long relaxation time is source of structural information

29. Click on image to start dynamics simulation
NMR 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 Motions
Click on image to start dynamics simulation

30. NMR Dynamics and Exchange
Multiple Signals for Slow Exchange Between Conformational States
Two or more chemical shifts associated with a single atom/nucleus
Populations ~ relative stability
Rex < w (A) - w (B)
Exchange Rate
(NMR time-scale)
Factors Affecting Exchange:
Addition of a ligand
Temperature
Solvent

31. NMR Dynamics and Exchange
Different
environments
OH exchanges between different molecules and environments. Observed chemical shifts and line-shapes results from the average of the different environments.
Intermediate exchange:
Broad peaks
Slow exchange:
CH2-OH coupling is observed
Fast exchange:
Addition of acid
CH2-OH coupling is absent

32. NMR Dynamics and Exchange
Effects of Exchange Rates on NMR data
k = p Dno2 /2(W1/2)e – (W1/2)o)
k = p Dno / 21/2
k = p (Dno2 -  Dne2)1/2/21/2
k = p ((W1/2)e-(W1/2)o)
k – exchange rate
W1/2 – peak-width at half-height
– peak frequency
e – with exchange
o – no exchange
Dno

33. Increasing Exchange Rate
NMR Dynamics and Exchange
Equal Population of Exchange Sites
40 Hz
No exchange:
slow
k = 0.1 s-1
k = 5 s-1
k = 10 s-1
With exchange:
k = 20 s-1
k = 40 s-1
Increasing Exchange Rate
coalescence
k = 88.8 s-1
k = 200 s-1
k = 400 s-1
k = 800 s-1
fast
k = 10,000 s-1

34. NMR Dynamics and Exchange
Example of NMR Measurement of Chemical Exchange
Two different cyclopentadienyl groups in [Ti(h1-C5H5)2(h5-C5H5)2]
Exchange rate changes as a function of temperature
But, chemical shifts also change as a function of temperature

35. C3 & C4 separation smaller than C6 & C2
NMR Dynamics and Exchange
Example of NMR Measurement of Chemical Exchange
Multiple resonances may be affected by exchange
Rotation about N-C bond
different coalescence rates because of different na-nb
C3 & C4 separation smaller than C6 & C2

36. NMR Dynamics and Exchange
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 (d) Coupling Const. (J) T2 relaxation
Slow k << dA- dB k << JA- JB k << 1/ T2,A- 1/ T2,B
Intermediate k = dA - dB k = JA- JB k = 1/ T2,A- 1/ T2,B
Fast k >> dA - dB k >> JA- JB k >> 1/ T2,A- 1/ T2,B
Range (Sec-1) 0 – – 2 1 1
Slow exchange at -60o

37. Fast exchange, average of three slow exchange peaks
NMR Dynamics and Exchange
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.
dobs = f1d1 + f2d2
f1 +f2 =1
where:
f1, f2 – mole fraction of each species
d1,d2 – chemical shift of each species
Fast exchange, average of three slow exchange peaks
d ≈ 1.86 ppm ≈ 0.25 x 2.00 ppm
x 1.95 ppm
+ 0.5 x 1.75 ppm

38. Increasing Exchange Rate
NMR Dynamics and Exchange
Unequal Population of Exchange Sites
differential broadening below coalescence
lower populated peak broadens more
40 Hz 3
slow
k = 0.1 s-1 1
k = 5 s-1
Exchange rate depends on population (p):
k = 10 s-1
k = 20 s-1
k = 40 s-1
Increasing Exchange Rate
Above coalescence:
coalescence
k = 88.8 s-1
k = 200 s-1
k = 400 s-1
k = 800 s-1
fast
k = 10,000 s-1
Weighted average

39. NMR Dynamics and Exchange
Example of NMR Measurement of Chemical Exchange
Unequal populated exchange sites
exchange between axial and equatorial position
exchange rate can be measured easily up to -44oC. Can easily measure na-ne and peak ratios
again, different broadening is related to chemical shift differences between axial and equatorial positions
difficult to determine accurate na-ne
difficult to determine accurate k

40. NMR Dynamics and Exchange
Use 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)
decoupler or saturation pulse is set far from any peaks (reference spectrum)
subtract 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)
exchange from A to B A B
T1 relaxation
Exchange rate

41. NMR Dynamics and Exchange
Use of magnetization transfer to study exchange
at equilibrium (t=∞)
kB can be measured from MZB(∞), MZB(0) and T1B
assumes T1A = T1B
if T1A ≠ T1B,difficult to measure T1A and T1B partial average or

42. NMR Dynamics and Exchange
Use of magnetization transfer to study exchange
p – t – p/2 pulse sequence
exchange takes place during (t)
Saturate peak 2, exchange to peak 3
Saturate peak 1, exchange to peak 4

43. NMR Dynamics and Exchange
Use of magnetization transfer to study exchange
p – t – p/2 pulse sequence
exchange takes place during (t)
Selective 180o pulse
Saturation transferred during t
Fit peak intensities to determine average T1 and k (k=15.7 s-1 & T1 = s)

44. Different nuclei and magnetic field strengths
NMR Dynamics and Exchange
Activation Energies from NMR data
rate constant is related to exchange rate (k=1/tex)
Different nuclei and magnetic field strengths
Measure rate constants at different temperatures
Calculating DH and DS 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 DH and DS 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

45. NMR Dynamics and Exchange
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 peaks
usually opposite sign x l
Exchange peaks