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  ne )t  (n  ne )0 exp( t / T )
–
–
(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
Mo
B1
w1
z
x
B1 off…
x
(or off-resonance)
y
z
y
Mxy w
1
Mo
T 1 & T2
relaxation
y
x
NMR Relaxation

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))
NMR Relaxation

Spin-lattices or longitudinal relaxation

Relaxation process occurs along z-axis

Measure T1 using inversion recovery experiment
NMR Relaxation

Spin-lattices or longitudinal relaxation

Collect a series of 1D NMR spectra by varying t

Measure T1 using inversion recovery experiment
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
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
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

2t c
K (v ) 
1  4 2v 2t c2
Field Intensity at any frequency
tc represents the maximum frequency
– 10-11s = 1011 rad s-1 = 15920 MHz
All lower frequencies are observed
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
Increasing MW
tc = 10-8 s-1
tc =
Spectral Density Function (J(w))
10-9 s-1
tc = 10-10 s-1
tc = 10-11 s-1
NMR Relaxation

Spin-lattices or longitudinal relaxation

Relaxation process in the x,y plane

Related to peak line-width
–

T2 may be equal to T1, or differ by orders of magnitude
–

Inhomogeneity of magnet also contributes to peak width
T2 can not be longer than T1
No energy change
T2 relaxation
(derived from Heisenberg uncertainty principal)
NMR Relaxation

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 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 Relaxation

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, 3756-3764
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

4r 3
tc 
3kT
where:
r = radius
k = Boltzman constant
 = viscosity coefficient

rotational correlation time (tc) is the time it takes a
molecule to rotate one radian (360o/2).


the larger the molecule the slower it moves
T2 ≤ T1
small molecules (fast tc) T2 =T1
Large molecules (slow t c) T2 < T1

NMR Relaxation
Mechanism for Spin-lattices and Spin-Spin relaxation
• Illustration of the Relationship Between MW, tc and T2
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
1
T1 DD
1
T2 DD
 R1 DD
 R2 DD

 n 1
tc
2t c

 a  3t c 

2 2
2 2  
6
5

w
t
1

4
w
t
r
c
c  i 1( i  j ) ij

1
Extreme narrowing limit:
T1 DD
4

a  3o2 2 / 320 2 , o  permeability of a vacuum
w  NMR frequency in rad s
  Planck' s constant / 2
1
 tc
 n 1
4t c

 2a 

2 2
2 2  
6
1

w
t
1

4
w
t
r
c
c  i 1( i  j ) ij

4
1
T2 DD
 R1 DD  R2 DD
n
1
 10a t c  6
i 1( i  j ) rij
Depends on distance
(bond length)
4
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
NMR Relaxation
Mechanism for Spin-lattices and Spin-Spin relaxation
• Different relaxation times (pathways) for different nuclei interactions
1
13C ≠ 13C-13C
1
1
 H- H ≠ H-
– 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
1
T1 DD
 R1 DD
 n 1
2t c
6t c
12t c
a 4 4 

  I S 


2 2
2 2
2 2  
6
3
 1  (w I  w S ) t c 1  w I t c 1  (w I  w S ) t c  i 1( i  j ) rij
1
T2 DD
 R2 DD
1
Extreme narrowing limit:
T1 DD

6t c  n 1
R1 DD a 2 2 


  I  S  4t c 
2 2  
6
2
3
1

w
t
r
i

1
(
i

j
)
s c 
ij

1
T2 DD
 R1 DD  R2 DD
n
20a 2 2
1

 I  St c  6
3
i 1( i  j ) rij
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
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┴
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:

1
T1CSA



2 I2 Bo2 s ||  s  t c
15
– 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
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, 517 - 522 (1998)
NMR Relaxation
Chemical Shift Anisotropy (CSA) Relaxation
• Line-shape increases as CSA increases with magnetic field strength
Increasing Magnetic Field
Peaks originating from 195Pt-1H2
coupling are broadened at higher
field due to CSA (shortening of
T1(Pt)
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

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 ():



V yy  V xx
Vzz
Vxx,Vyy,Vzz are calculated from the sum of contributions from all charges qi at a
distance ri
V xx   qi r 5 ( 3 x i2  ri2 )
i

Quadrupole relaxation times (T1Q,T2Q), where Q is quadrupole moment
1
1
3 2 ( 2 I  3)  e 2Q  2 
1 2



 R1Q  R2Q 
V
1

 t c

zz
2


T1Q T2Q
10 I ( 2 I  1)  h  
3 
NMR Relaxation
Quadrupolar Relaxation
• Factors affecting quadrupolar relaxation
1
1
3 2 ( 2 I  3)  e 2Q  2 
1 2



 R1Q  R2Q 
Vzz  1   t c
2


T1Q T2Q
10 I ( 2 I  1)  h  
3 
•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, )
• 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

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

Increasing T1
Increasing T1
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 through scaler coupling

Lowering temperature can sharpen peaks broaden by quadrupole relaxation
– lower temperature  increase tc  shorten T1Q
1
1 1
4

  2 J 2 S ( S  1)T1s
T2 sc 2 T1 SC 3
NMR Relaxation
Quadrupolar 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
3q( 3r 2 cos 2   r 2 ) 3q( 3 cos 2   1)
Vzz 

5
r
r3
Vzz =0 if f=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

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
http://projects.villa-bosch.de/dbase/dsmm/
Click on image to start dynamics simulation
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
NMR Dynamics and Exchange
EtOH + EtOH*
H+
EtOH + EtOH*
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
Different
environments
NMR Dynamics and Exchange
Effects of Exchange Rates on NMR data
k =  Dno2 /2(W1/2)e – (W1/2)o)
k =  Dno / 21/2
k =  (Dno2 - Dne2)1/2/21/2
k =  ((W1/2)e-(W1/2)o)
k – exchange rate
W1/2 – peak-width at half-height
n – peak frequency
e – with exchange
o – no exchange
Dno
NMR Dynamics and Exchange
Equal Population of Exchange Sites
No exchange:
With exchange:
W1/ 2 
1
1

T2 tex
k
1
t ex
slow
k = 0.1 s-1
k = 5 s-1
Increasing Exchange Rate
W1/ 2
1

T2
k = 10 s-1
k = 20 s-1
k = 40 s-1
coalescence
k = 88.8 s-1
k = 200 s-1
k = 400 s-1
k = 800 s-1
fast
k = 10,000 s-1
40 Hz
NMR 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 temperature
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

k
 ( A   B ) 2
2{(W1/ 2 ) ex  (W1/ 2 ) o }
C3 & C4 separation smaller than C6 & C2
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
Slow
Intermediate
Fast
Chem. Shift (d
k << dA- dB
k = dA - dB
k >> dA - dB
Range (Sec-1)
0 – 1000
Coupling Const. (J)
k << JA- JB
k = JA- JB
k >> JA- JB
T2 relaxation
k << 1/ T2,A- 1/ T2,B
k = 1/ T2,A- 1/ T2,B
k >> 1/ T2,A- 1/ T2,B
0 –12
1 - 20
2
H2C
C
C
H2C
C
CH2
CH2
H2C
C
H2C
C
C
CH2
CH2
Fe(CO)4
Fe(CO)4
Slow exchange at -60o
1 1
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
+ 0.25 x 1.95 ppm
+ 0.5 x 1.75 ppm
NMR Dynamics and Exchange
Unequal Population of Exchange Sites

differential broadening below coalescence
- lower populated peak broadens more
slow
k A  kB
pB
pA
Above coalescence:
n avg 
n A p A n B pB
p A  pB
1
3
k = 5 s-1
Increasing Exchange Rate
Exchange rate depends
on population (p):
k = 0.1 s-1
40 Hz
k = 10 s-1
k = 20 s-1
k = 40 s-1
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
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

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
Mz(0)
B
Mz(0)
decouple site A
MzA = 0
Mz(0)
exchange from A to B
A
B
T1 relaxation
dM zB (t ) M zB (0)  M zB (t )

 k B M ZB (t )
B
dt
T1
Exchange rate
NMR Dynamics and Exchange
Use of magnetization transfer to study exchange
 at equilibrium (t=∞)
M zB (0)  M zB ()
B
0

k
M
( )
B
Z
B
T1
or
1 M zB (0)  M zB ()
kB  B
T1
M ZB ()
if k B  1 B then M ZB () ~ 0 magnet ization transfer causes B signal to decrease
T1
if k B  1 B then M ZB () ~ M ZB (0) B signal unaffected by exchange
T1

kB can be measured from MZB(∞), MZB(0) and T1B
- assumes T1A = T1B
- if T1A ≠ T1B,difficult to measure T1A and T1B partial average
NMR Dynamics and Exchange
Use of magnetization transfer to study exchange

 – t – /2 pulse sequence
- exchange takes place during (t)
Saturate peak 2,
exchange to peak 3
Saturate peak 1,
exchange to peak 4
NMR Dynamics and Exchange
Use of magnetization transfer to study exchange

 – t – /2 pulse sequence
- exchange takes place during (t)
Selective 180o pulse
Fit peak intensities to
determine average T1 and k
(k=15.7 s-1 & T1 = 0.835 s)
Saturation transferred
during t
NMR Dynamics and Exchange
Activation Energies from NMR data

rate constant is related to exchange rate (k=1/tex)

 k 
DG ‡  RT 23.759  ln  
 T 

Different nuclei and
magnetic field strengths
Measure rate constants at
different temperatures

Calculating DH and DS may not be reliable:
DG ‡  DH ‡  TDS ‡
- 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
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