Relaxation Processes and Constraints on Distance Measurements

1
Relaxation Processes and Constraints on Distance
Measurements
Gareth R. Eaton, University of Denver
ACERT Workshop, August 7, 2004
Funding EB002807
2
Scope

• Why should you care about relaxation times?
• Solids and frozen solutions
• Spin echo decays
• Spin diffusion
• Spin-lattice relaxation
• Relaxation processes and mechanisms
• Results for metals and organic radicals
• Summary of contributions to T1 and Tm
• Summary of relaxation mechanisms
• Fluid solutions
• Relaxation mechanisms
• Results for nitroxyl radicals
• Results for triarylmethyl radicals
• Implications for distance measurements

3
Why understanding of relaxation is important for
distance measurements
T1

• Affects the temperature range in which the study
  can be performed.
• Limits pulse repetition rates
• Selects for sub-populations
• Identifies optimum frequency for experiment

T2

• Determines the time window during which any
  method based on observation of an echo or FID
  can be measured.

Understanding the mechanisms of relaxation guides
selection of paramagnetic centers to use.
4
What can you learn by measuring spin echo
dephasing in solids?

• Feasibility of doing pulse experiments that
  depend upon
• echo detection
• Local spin concentration
• Librational motion
• Proton spin concentration
• Methyl group types and concentrations
• Dynamic processes that average inequivalent
  nuclei
• Enhancement of echo dephasing by neighboring fast
  relaxing
• spin

5
A Spins and B Spins
The spins that are excited in an experiment are
called the A spins. All other spins are
designated as B spins. As a function of time A
spins become B spins due to spectral diffusion,
which decreases the number of A spins that can be
observed.
6
Analysis of Spin Echo Decays
1. Fit to a stretched exponential
Eq. (1)

• is the time between pulses
• Tm is the dephasing time constant
• x is a phenomenological parameter that depends
  
• on dephasing mechanism
• Values of x are between 0.5 and about 2.5

For nuclear spin diffusion, x gt 2 For a
process that averages inequivalent environments,
x varies from 2 to 0.5 to 1 as the rate of the
process increases. For instantaneous diffusion x
1 If dephasing is T1 driven, x 1
7
Analysis of Spin Echo Decays

• Fit to a model appropriate for a particular
• dephasing mechanism
• a. Nuclear spin diffusion
• b. Dynamic averaging of inequivalent nuclei
• c. Dipolar interaction with rapidly relaxing
  metal

8
Nitroxyl Radicals Tm
330 K
Values of Tm for tempone in 85 glycerol15
water obtained by (O) two-pulse spin echo data
(D) 2D- ESE contour plots The three lines show
the calculated values based on jump diffusion (
__ ), free diffusion ( _ _ ) and Brownian
diffusion (_ . _ . ).
In the intermediate tumbling regime Tm for a
nitroxyl radical is too short to measure by
current spin echo techniques.
G. L. Millhauser and J. H. Freed, J. Chem. Phys.
81, 37 (1984).
? Temperature
9
Nuclear Spin Diffusion and Instantaneous Diffusion
Protons in the solvent dominate dephasing
When protons are replaced by deuterons,
instantaneous diffusion makes a
larger contribution.
Instantaneous diffusion is decreased by making
the turning angle smaller.
Note the difference in x-axis scales for the 3
plots and the much longer Tm in deuterated
solvent.
S. S. Eaton and G. R. Eaton, Biol. Magn. Reson.
19, 29 (2000).
10
Nuclear Spin Diffusion
Frequently, an unpaired electron is dipolar
coupled to many surrounding nuclear spins. If
one of these nuclear spins flips, the dipolar
coupling to the unpaired electron is changed,
which changes the precession frequency for the
unpaired electron. Although the spin flip rate
for an individual nuclear spin is relatively
slow, the probability of some nuclear spin
flipping is large, because of the large number of
nuclear spins in typical biological and organic
materials, and water. The most common nuclear
spin flip process is a flip-flop, I1I2-. The
rate of this process increases proportional to
the nuclear spin concentration. Dipolar coupling
is proportional to g so proton spin diffusion is
a more effective dephasing process than is
deuteron spin diffusion.
A. Zecevic, G. R. Eaton, S. S. Eaton, and M.
Lindgren, Mol. Phys. 95, 1255 (1988).
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Effects of Methyl Groups
Echo decay curves are substantially different for
tempone in these 3 solvents with
approximately the same concentration of methyl
protons. Because the barrier to rotation is
lower, the aromatic methyl groups in xylene are
less effective in echo dephasing than the
methyls in n-propanol or 2,5-Me2-THF.
40 K
12
Nuclear Spin Diffusion
In the absence of methyl groups, 1/Tm increases
monotonically with proton concentration.
The effects of methyl groups on dephasing depend
upon the type of methyl group.
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Librations and Anisotropy in Tm
Vanadyl porphyrin in 91 tolueneTHF
The change in resonance field per degree of
rotational reorientation is largest at
intermediate orientations of the molecule.
1/Tm is largest at intermediate orientations and
smallest along the principal axes.
Field-swept echo detected spectrum at 50 K
First-derivative CW spectrum at 20 K
J.-L. Du, K. M. More, S. S. Eaton, G. R. Eaton,
Isr. J. Chem. 32, 351 (1992).
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Spin Lattice Relaxation

• As temperature increases, 1/T1 increases and
  eventually becomes
• the dominant contribution to 1/Tm. The
  contribution from nuclear spin diffusion is
  smaller in deuterated solvent so 1/T1 dominates
  echo dephasing at lower temperature in
• deuterated solvent than in natural abundance
  solvent.
• Saturation recovery
• ? Inversion recovery

Vanadyl ion in 11 waterglycerol
G. R. Eaton and S. S. Eaton, J. Magn. Reson. 136,
63 (1999).
15
Any process that takes spins off resonance can
contribute to a recovery curve. The key to
measuring T1e is to separate its contribution
from those of spectral diffusion processes. Most
EPR spectra are so broad that only a small
fraction of the spins are saturated or inverted
in a pulse experiment.
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Spin-Lattice Relaxation Processes
Direct Process - There is an exact match of the
spin transition energy with a phonon energy so
there can be direct transfer of energy from the
spin system to the lattice phonon bath. A phonon
mode involves collective motion of lattice atoms.
Orbach Process is a two-phonon process in which
the energy to be transferred to the lattice is
the difference between the energies absorbed and
emitted for a specific low-lying excited state.
Raman Process is a two-phonon process in which
the energy to be transferred to the lattice is
the difference between the energies absorbed and
emitted for a virtual excited state at any energy
less than the Debye temperature. Local
Vibrational Modes also can contribute to
relaxation. Thermally-activated process is
characterized by a correlation time and an
activation energy. Each process has a
characteristic temperature dependence.
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Temperature Dependence
1
T is temperature in Kelvin Adir is the
coefficient for the contribution from the direct
process ARam is the coefficient for the
contribution from the Raman process ?D is the
Debye temperature J8 is the transport integral,
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Temperature Dependence
Aloc is the coefficient for the contribution from
a local vibrational mode ?loc is the energy for
the local mode in units of Kelvin
Aorb is the coefficient for the contribution from
the Orbach process Dorb is the energy separation
between the ground state and the excited state
for the Orbach process Atherm is the
coefficient for the contribution from the
thermally-activated process
?c is the correlation time for the
thermally-activated process
Ea is the activation energy for the
thermally-activated process
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Raman Process
The dashed lines are the contributions from the
Raman process. An additional process contributes
at higher temperatures. The faster relaxation for
tempone is attributed to its greater spin-orbit
coupling. The faster relaxation in sucrose
octaacetate is attributed to a lower Debye
temperature.
20
Local modes vanadyl ion d1
Relaxation is faster for the more flexible aquo
vanadyl ion than for the more rigid vanadyl
porphyrin. The lines were calculated to fit the
data.
21
Frequency Dependence - Thermally activated process
T1e for tempol doped into 4-hydroxy-tetramethyl-pi
peridinol
The relaxation rate is frequency dependent. This
indicates a contribution from a
thermally-activated process, which may be
rotation of the nitroxyl ring methyl groups.
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Thermal Process low spin Fe(III), S1/2
The solid lines were calculated to fit the data.
23
Orbach Process high-spin Fe(III), S5/2
______   4D ? E? ______ 2D
? ______
The zero-field splitting of the high-spin iron
results in low-lying excited states at 2D and 6D.
The value of D can be determined by analyzing
the temperature dependence of T1e in the range
where the Orbach process dominates.
Hb-F, 2D 17 K Hb-aquo, 2D 30 K
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Cu2 - d9
T1 values between about 10 and 120 K for Cu(II)
complexes with a variety of geometries. The
dominant processes are the Raman process (at
lower temperatures) and a local mode (at higher
temperatures).
25
g-irradiated 2,4,6-tri(t-butyl)-phenol
linewidth 7 G
26
T1e for g-irradiated L-alanine
aH 25 G
The barrier to rotation of methyl group in
irradiated L-alanine is much higher than for
4-Me-phenol, so the impact of methyl rotation on
T1e occurs at higher temperature. The lines were
calculated to fit the data, using Eq. 1.
27
Summary
When measuring T1e it is crucial to check the
experimental data for effects of spectral
diffusion. The "right" technique to measure T1e
for a particular sample is strongly dependent on
the nature of competing spectral diffusion
processes. The temperature and microwave
frequency dependence of T1e provides insight into
the dominant relaxation process. Relaxation
processes shed light on electronic structure and
molecular motion.
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Summary of Temperature Dependence
Temperature dependence of relaxation rates
observed when Tm is dominated by A nuclear
spins in the surroundings B collapse of
couplings to methyl groups or rapidly relaxing
spin C averaging of g- or A-anisotropy D
T1 T1 is dominated by E direct process
and/or high spin concentration F, G Raman or
Orbach process H interaction with a more
rapidly- relaxing electron spin I local
mode, thermally-activated process, modulating
spin-orbit coupling or ZFS
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Contributions to T1

•   vibrations anisotropically modulate
  spin-orbit coupling, and mix spin and orbital
  angular momentum
•   hence, for axial symmetry, T1 is shortest in
  the xy plane, and longest along z
• vibrations of the molecule exchange energy
  with the lattice
•   collisions of the molecule with solvent
  (including in frozen solution) excite vibrations
  of the molecule
•   at ca. 100 K in a "frozen" solution a molecule
  undergoes low-amplitude molecular motions many
  times during T1
•   in a "rigid" lattice energy can be exchanged
  with the lattice via
• a single resonant phonon direct process
• two phonons Raman process
• excited state electronic energy level within the
  phonon bath Orbach process
• excited state vibrational energy level within the
  phonon bath Murphy process
• local vibrational mode
• special vibration/phonon interactions also are
  possible
• temperature dependence can be complicated, and
  includes linear (T1 ? T-1) to T1 ? T-9,
  but is commonly T1 ? T-n with n 2 to 3 from ca.
  10 K to the melting point of the solvent.
•   spin-spin interaction with a faster-relaxing
  electron spin

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Contributions to Tm

• T2 electron spin-spin relaxation
• Instantaneous diffusion. The second pulse flips a
  neighboring spin. The process
  has larger impact for
• higher spin concentrations,
• narrower EPR signals, and
• larger microwave B1
• Intramolecular dynamic processes
• small amplitude motion
• rotation of methyl groups to which the unpaired
  electron is coupled
• relaxation of spin-coupled systems
• Averaging of g and A anisotropy by molecular
  tumbling
• Nuclear spins in the surroundings
• T1 when it becomes short.

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Summary of Relaxation Mechanisms
The spin angular momentum of the A spins is mixed
with other angular momenta by spin-orbit coupling
or spin-rotation coupling. The time-dependence
of the mixing of angular momenta, occurs by
rotational motion motional modulation of
dipolar coupling mixing of excited states (e.g.
by conformational changes) thermal motions
modulating zero-field splitting This mixing of
angular momenta is thermally activated, and
provides a mechanism for coupling the spin
angular momenta to the thermal energy of the
lattice.
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Relaxation Mechanisms
33
Fluid Solution Relaxation Mechanisms - Spin
Rotation
This mechanism is important for nitroxyls and
semiquinones when the tumbling correlation time
is less than about 0.03 ns.
P. W. Atkins, in Electron Spin Relaxation in
Liquids, L. T. Muus and P. W. Atkins, Eds.,
Plenum, N. Y., 279-312 (1972).
J. S. Hwang, R. P. Mason, L.-P. Hwang, and J. H.
Freed, J. Phys. Chem. 79, 489-511 (1975).
The contribution from this mechanism to nitroxyl
T1e was calculated from experimental values of
tumbling correlation times and g values.
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Modulation of g and Hyperfine Anisotropy
? is the resonant frequency B is the external
magnetic field ?R is the molecular tumbling
correlation time ?B is the Bohr magneton I and
mI are the nuclear spin quantum numbers ?g gz
½ (gxgy) ?g ½ (gx - gy) ?A Az ½ (AxAy) ?A
½(Ax-Ay)
Schneider et al., Biol. Magn. Reson. 8, 1
(1989) Robinson et al., J. Phys. Chem. B. 103,
5881 (1999).
The contribution from this mechanism, was
calculated from experimentally-determined g
values and the nitroxyl nitrogen hyperfine
interaction.
35
Modulation of g and Hyperfine Anisotropy

• For the g and A values of nitroxyl radicals, the
  relative importance of modulation of A
  anisotropy, and g anisotropy at X-band is
• 1.0 0.02
• Impact of replacing 14N by 15N
• (7I(I1)-mI2)(?A)2 for 15N/14N 0.70
• Thus, when the difference in nuclear spin and
  magnetogyric ratios is taken into account, we
  predict that 1/T1e for 15N will be 70 of that
  for 14N.
• Replacement of 1H by 2H in the nitroxyl caused a
  decrease in 1/T1e. This contribution was treated
  empirically because we do not know the anisotropy
  of the couplings to individual protons.

36
Nitroxyl Radicals

• Relaxation times were measured at L-band, S-band,
  and X-band as a function of tumbling correlation
  time.
• The dominant contributions to T1 are modulation
  of nitrogen hyperfine anisotropy and a
  thermally-activated process, with a small
  contribution from spin rotation at the shortest
  tumbling correlation times.
• The relative importance of these contributions
  varies with frequency and tumbling correlation
  time.

() natural isotope abundance, (x) d17-tempol
(?) 15N-d17 tempol
R. Owenius, G. E. Terry, M. J. Williams, S. S.
Eaton, and G. R. Eaton, J. Phys. Chem. B 108,
9475-9481 (2004).
37
Conclusions concerning relaxation for nitroxyl
radicals

• At tumbling times longer than about 2 ns,
  thermally-activated process(es) dominates the
  relaxation.
• At tumbling correlation times between about 0.2
  and 2 ns, modulation of hyperfine anisotropy
  dominates the relaxation at X-band and S-band.
• The effects of thermally-activated processes are
  larger at S-band than at X-band because the
  rotation frequency at room temperature is closer
  to the Larmor frequency at S-band than at X-band.
• When the tumbling correlation rate is closer to
  the S-band frequency, modulation of hyperfine
  anisotropy has a greater effect at S-band than at
  X-band.

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Triarylmethyl Radicals
For these radicals g-values are close to 2, and
the hyperfine coupling is weak. As a result, spin
rotation and modulation of g and/or A anisotropy
are not expected to be very effective relaxation
mechanisms.
sym-trityl-CH3 R H, M Na sym-trityl-CD3 R
D, M K
Values of T1e in mixtures of waterglycerol
showed little dependence on viscosity or on
microwave frequency between 1.5 GHz and 9.5 GHz,
which indicates that spin rotation and modulation
of g and A anisotropy do not dominate the
relaxation, in this frequency range.
L. Yong et al., J. Magn. Reson. 152, 156 (2001).
39
Triarylmethyl Radicals
Values of T1e measured by saturation recovery or
inversion recovery were the same, within
experimental uncertainty, which indicated the
absence of spectral diffusion processes. The 11
waterglycerol glass melts at 200 K, but there is
no change in the slope of the plot of 1/T1e vs.
T, consistent with the room temperature
measurements that indicated no dependence on
tumbling correlation time.
The temperature dependence is consistent with a
Raman process at low temperature plus a local
vibrational mode at higher temperature. The
energy for the local mode is consistent with that
for the C-S stretch.
40
Trityls
OX63

• Trityl T1 values are more strongly dependent on
  solution viscosity at 250 MHz than at X-band
• At 250 MHz trityl tumbling correlation times are
  1/?
• Deuteration of solvent or trityl causes the
  relaxation rates to decrease.
• Modulation of electron-proton dipolar coupling
  dominates the relaxation at 250 MHz.

VHF
OX31
CH3
CD3
X-band
41
Modeling of Trityl Relaxation at 250 MHz
(?) waterglycerol (?) D2Oglycerol-d8
Relaxation rates were modeled with contributions
from a local mode, and modulation of intra- and
intermolecular electron-proton dipolar
interaction ( ___ ).
42
Conclusion

• For nitroxyls 50 K is a good tradeoff between
  long T1, Boltzmann population and repetition
  rate.
• Need to be below about 60 K to avoid shortening
  of Tm by rotation of methyl groups at rate
  comparable to inequivalences in proton hyperfine.
  
• Boltzmann population difference varies
  approximately proportional to T.
• Temperature dependence of T1 is at least T2.
• S/N is better the faster the rep rate.

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Further Information
Biological Magnetic Resonance , vol. 19, 2001
Distance Measurements by EPR Biological
Magnetic Resonance, vol. 23, 2004
Biomedical ESR - Part A Free Radicals, Metals,
Medicine, and Physiology Biological Magnetic
Resonance, vol. 24, 2004 Biomedical ESR -
Part B Methodology, Instrumentation and Dynamics