Use our NMR service to measure NMR relaxation times.

NMR relaxation is the processes by which an excited magnetic state returns to its equilibrium distribution. NMR relaxation measurement can be used for spectral assignment and the study of quadrupolar and paramagnetic interactions, and exchange dynamics.

# What you should already know before continuing to read this:

Before you continue reading about NMR relaxation, you should have some knowledge of:

To understand the section on exchange processes, you should have some knowledge of:

If you want to read about these subjects first, please go to the links above.

Use our NMR service to measure NMR relaxation times.

# Types of magnetic relaxation

Any excited magnetic moment (generally excited by a radio-frequency pulse) relaxes
back to equilibrium on the *z*-axis. There are two mechanisms
involved: spin-lattice and spin-spin. Spin-spin
relaxation is also referred to as transverse
relaxation or *T*_{2} and describes
the decay of the excited magnetization perpendicular to the applied
magnetic field (fig.1). The observed spectral line-width is related
to the spin-spin relaxation but is also affected by magnetic
inhomogeneity. This combination of relaxation and inhomogeneity is
referred to as the dephasing time or *T*_{2}*. Spin-lattice relaxation
is also referred to as longitudinal relaxation or *T*_{1}
and describes the return to equilibrium in the direction of the
magnetic field. The spin-lattice relaxation (*T*_{1}) can be measured from the
buildup of magnetization along the static applied magnetic field
(conventionally the *z*-axis, fig. 1). Alternatively, the
spin-lattice relaxation can be measured by the decay of a signal
under spin-lock conditions that generate a rotating magnetic field
near the resonant frequency perpendicular to the static magnetic
field (conventionally the *x*-*y*-plane, fig. 1), in which
case it becomes spin-lattice relaxation in the
rotating frame or *T*_{1ρ}.

In anisotropic systems, the spin-lattice relaxation also becomes
anisotropic. The isotropic component is then termed the Zeeman
relaxation whose time constant is *T*_{1,2Z}.
There is also a dipolar relaxation, *T*_{1,2D},
and, for quadrupolar nuclei, a quadrupolar relaxation *T*_{1,2Q}.
As this website is mainly concerned with isotropic solution NMR, the
anisotropy of *T*_{1} and *T*_{2} will not
be discussed further.

Fig. 1. Relaxation of magnetization vectors after a 90° pulse. *T*_{1} relaxation along the
*z*-axis in green, *T*_{2}
relaxation in the *x*-*y*-plane in red and *T*_{1ρ} with spin-lock in
the *x*-*y*-plane in purple.

# Ranges of relaxation times

Most relaxation times observed in routine NMR are between 0.1 and 10 seconds. Longer relaxation times, tens or hundreds of seconds, are observed in the absence of oxygen for deuterated solvents, quaternary carbon signals, medium-mass spin ½-nuclei and in the gas phase. Shorter relaxation times, milli- or microseconds, are observed when there is medium-to-fast chemical exchange, heavy spin-½ nuclei, paramagnetism and for quadrupolar nuclei.

# Analysis and presentation

At the very least, relaxation measurements may be carried out by
comparing two spectra acquired using different evolution times (*τ*)
or, in the case of *T*_{1}, a single
spectrum that finds a null point. It is more accurate to measure
many (typically 32) spectra with different evolution times (fig. 2)
and fit the intensities to the relevant mathematical function (fig.
3). These functions are generally non-linear exponentials but can be
linearized by using a logarithm function on the intensity. However,
a linear fit in this manner is less accurate than the direct
non-linear fit because too much weight is attributed to the noisier
points of low intensity (fig. 4). These fitting methods only address
one point at a time. It is much more efficient to process the whole
spectrum at once and represent it as a contour plot of relaxation
time against chemical shift. This is performed by carrying out a
non-linear fit on all the columns in the spectrum in an operation
called an inverse of the Laplace transform. The relaxation time is
presented here as increasing from top to bottom (fig. 5) as is
customary for DOSY spectra even
though this is the opposite sense to what is used for most other
applications.

Fig. 2. Stack plot representation of a relaxation measurement: *T*_{2}
measurement of the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C

Fig. 3. A non-linear fit to a relaxation decay gives the most
accurate values for relaxation times: Exponential *T*_{2}
decay curves for the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C

Fig. 4. A linear fit to a linearized function is a simple method of
analysis although it is less accurate than the non-linear method:
Linear fits to the logarithms of exponential *T*_{2}
decay curves for the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C

Fig. 5. Inversion of the Laplace transform gives a plot of
relaxation time against chemical shift: Contour plot for the *T*_{2}
spectrum of the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C

#
Longitudinal or spin-lattice relaxation, *T*_{1}

Longitudinal or spin-lattice relaxation (*T*_{1}) is
the mechanism by which an excited magnetization vector returns to
equilibrium along the axis of the static applied magnetic field
(conventionally shown along the *z*-axis, fig. 6).

Fig. 6. Relaxation *via* *T*_{1} along the
magnetic-field (*z*) axis

The inversion recovery (*T*_{1}) pulse sequence (fig. 7) inverts the
magnetization on the *z*-axis so that the second pulse yields a
signal of intensity directly proportional to the relaxing
magnetization along that axis: negative for short evolution times
and positive for long evolution times (fig. 8). The intensity of the
signal is described by the equation: . The
intensity is zero at *T*_{1}ln(2) (0.693*T*_{1})
so the value of *T*_{1} can be measured by running
single experiments, changing the value of *τ* until a null
intensity is found. If the intensity is positive then *τ* needs
to be reduced and if it is negative then *τ* needs to be
increased. The value of *T*_{1} is 1.443*τ*_{null}.
In the example below (fig. 9), *τ*_{null}'s are 0.40
and 1.01 s so *T*_{1}'s are 0.58 and 1.45 s.

Fig. 7. Inversion recovery pulse sequence for measuring *T*_{1}

Fig. 8. Inversion recovery stack plot for the *T*_{1}
spectrum of the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C. The peak on the left relaxes
slower than the peak on the right.

Fig. 9. Non-linear fit showing the null points for the *T*_{1}
spectrum of the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene
under vacuum at 500 MHz at 25°C.

The fitting function can be made by plotting the log of the
intensity after a long evolution time minus the intensity against
time. However, a linear fit in this manner is less accurate than the
direct non-linear fit because too much weight is attributed to the
noisier points of low intensity (fig. 10). The slopes (1.92 and 0.73
s^{-1}) are the reciprocals of the relaxation times (0.52
and 1.37 s, respectively).

Fig. 10. Linear fit for the *T*_{1} spectrum of the ^{t}butyl
protons of 12,14-di^{t}butylbenzo[g]chrysene under
vacuum at 500 MHz at 25°C.

An inversion of the Laplace transform processed the whole spectrum
to give a result (*T*_{1}'s of 0.59 and 1.47 s) in the
form of a contour plot (fig. 11) that is easier to read and
visualize than the stack plot (fig. 8) and is unaffected by possible
bias in the choice of the position in the spectrum for processing.

Fig. 11. Contour fit of the *T*_{1} spectrum of the ^{t}butyl
protons of 12,14-di^{t}butylbenzo[g]chrysene under
vacuum at 500 MHz at 25°C.

For extremely long *T*_{1}
relaxation times such as in the gas phase it may not be possible to
prevent diffusion and convection into and out of the coil area. In
such cases, the sample should be removed from the magnet for at
least five *T*_{1} which may be several hours. The
sample can then be reinserted and a regular 1D acquisition with a
very small pulse angle applied every minute or so for a period of a
few *T*_{1}'s. The build-up of intensity follows the
equation where *τ* is the time since
insertion. In the case of ^{3}He
gas at a pressure of two atmospheres, the *T*_{1}
appears to be about 1 s when measured by inversion recovery due to
diffusion out of the receiver coil area while the true value is a
little more than 1000 s (fig. 12).

Fig. 12. Signal buildup of ^{3}He gas (2 atm) as a function
of time after insertion into the magnet on a 400 MHz instrument
(304 MHz ^{3}He) at 25°C. The *T*_{1} is just
over 1000 s.

The long relaxation delay required by the inversion recovery
experiment is a problem when sensitivity is low and when the
relaxation time is long or when there is moderate convection as in
the example of ethylbenzene. In
such cases, the DESPOT method (driven equilibrium single pulse
observation of *T*_{1} also known as the variable
nutation angle method) can be used. The intensity is plotted as a
relation of pulse angle (fig. 13).

Fig. 13. DESPOT curve for the methyl protons of ethylbenzene (0.1%)
in CDCl_{3} at 400 MHz at 25°C

Two or more regular 1D experiments are carried out with different
pulse angles. There is no need for a long relaxation delay but the
experiment must start with dummy scans that continue for at least 5*T*_{1}.
The pulse angle must be calibrated accurately taking into account
the rise time of the transmitter, typically 0.1 μs (larger in old
spectrometers). A plot (fig. 14) of *I* cosec(*φ*) against
*I* cot(*φ*) gives a straight line of slope where *φ* is the pulse
angle and *T _{r}* is the time between acquisitions. In
the case of the methyl of ethylbenzene,
the

*T*

_{1}is 17.4 s.

Fig. 14. Linear DESPOT plot for the methyl protons of ethylbenzene
(0.1%) in CDCl_{3} at 400 MHz

If only two angles are used then the relaxation time can be
expressed analytically as . If the two angles, *φ*_{1}
and *φ*_{2}, are 45° and 90° respectively then this
simplifies to .

In the case of the methyl protons
of ethylbenzene, *I*_{1}
= 3.4469, *I*_{2} = 3.1118 this gives at *T*_{1}
of 15.0 s which is reasonably close for a two point experiment and
certainly good enough if the purpose is to select a reasonable
relaxation delay in another experiment.

#
Transverse or spin-spin relaxation, *T*_{2}

Transverse or spin-spin relaxation (*T*_{2}) is the
mechanism by which the excited magnetization vector (conventionally
shown in the *x*-*y*-plane) decays (fig. 15).

Fig. 15. *T*_{2} relaxation mechanism

The magnitude of the magnetic moment in the *x*-*y* plane
decays according to . The CPMG *T*_{2}
experiment (fig. 16) yields signals of intensity where *τ* is the total
evolution time (2Δ). This pulse
sequence is best for singlets but is very sensitive to instrumental
phase and environmental instability for long relaxation times.

Fig. 16. Spin-echo pulse sequence for measuring *T*_{2}

The spin-echo sequence does not refocus spin-spin coupling.
Therefore, multiplets require a multipulse sequence called
CPMG-PROJECT (Carr-Purcell-Meiboom-Gill with Periodic Refocusing of
*J* Evolution by Coherence Transfer, fig. 17) to suppress the
coupling artifacts, thereby allowing an accurate measurement of *T*_{2}
to be made. The mixing time, *τ*, is 2*n*Δ. The value of Δ
in the pulse sequence should be much shorter than the reciprocal
coupling constant, , but long enough that the sample
should not heat up significantly while keeping the pulse time to
less than 1% of the duty-cycle (proportion of the mixing time that
the pulse is on) in order to minimize spectral instability and the
contribution from *T*_{1ρ}, and to prevent
overheating. A value for Δ of 5 ms is usually appropriate for ^{1}H spectra with
homonuclear coupling. The experiment is repeated many times with
different values of *τ* and the resulting intensities are used
to find the value of *T*_{2}. Like the basic *T*_{2}
spin-echo pulse sequence this one is still very sensitive to
environmental changes during acquisition, especially for long
relaxation times.

Fig. 17. CPMG-PROJECT pulse sequence for measuring *T*_{2}

Fig. 2 above shows a stack plot of the measurement of the *T*_{2}
for the ^{t}butyl protons
of 12,14-di^{t}butylbenzo[g]chrysene.
The peak on the left decays more slowly and therefore has a longer
*T*_{2} than the right-hand peak. A plot of peak height
against mixing time (*τ*) gives exponential decays (fig. 3
above) with rate constants of 1.68 and 0.68 s^{-1} that
correspond to *T*_{2} values of 0.60 and 1.47 s. The
curves can be linearized by using the logarithm of the intensity
(fig. 4 above) yielding linear fits that correspond to *T*_{2}
values of 0.63 and 1.47 s. The discrepancy in the value for ^{t}butyl14
is due to the attribution of too much weight to noise-affected
points of low intensity in the linearized fit. An inversion of the
Laplace transform for the mixing time gives a contour plot with *T*_{2}
values of 0.59 and 1.47 s (fig. 5 above).

#
Spin-lattice relaxation in the rotating frame, *T*_{1ρ}

Spin-lattice relaxation in the rotating frame (*T*_{1ρ})
is the mechanism by which the excited magnetization vector
(conventionally shown in the *x*-*y*-plane) decays (fig.
18) while under the influence of spin-lock radio-frequency
irradiation that is effectively a weak magnetic field in the *x*-*y*-plane
that rotates at a similar frequency to the magnetization vector. The
result is much the same as one would get by measuring the *T*_{1} at very low magnetic fields
with a proton resonance of tens of kHz.

Fig. 18. *T*_{1ρ} relaxation mechanism

*T*_{1ρ} is measured using a spin-lock pulse sequence (fig. 19) to yield a
signal intensity of where *τ*
is the spin-lock pulse length. The experiment is repeated many times
with different values of *τ* and the resulting intensities used
to find the value of *T*_{1ρ}. If the sample is
very concentrated then the relaxation time will appear shorter than
it really is due to saturation. In such a case, off-tune the probe,
recalibrate the pulse widths and repeat the experiment.

Fig. 19. Pulse sequence for measuring *T*_{1ρ}

The *T*_{1ρ} experiment is generally noisier
than the *T*_{2} experiment because
it causes more sample heating, especially for large values of *T*_{1ρ}.

Fig. 20 shows a stack plot of the measurement of the *T*_{1ρ}
for the ^{t}butyl protons
of 12,14-di^{t}butylbenzo[g]chrysene
. The peak on the left decays more slowly and therefore has a longer
*T*_{1ρ} than the right-hand peak. A plot of
peak height against mixing time (*τ*) gives exponential decays
(fig. 21) with rate constants of 1.57 and 0.65 s^{-1} that
correspond to *T*_{1ρ} values of 0.64 and 1.55
s. The curves can be linearized by using the logarithm of the
intensity (fig. 22) yielding linear fits that correspond to same *T*_{1ρ}
values of 0.64 and 1.55 s. An inversion of the Laplace transform for
the mixing time gives a contour plot with *T*_{2}
values of 0.63 and 1.54 s (fig. 23).

Fig. 20. *T*_{1ρ} stack plot of the ^{t}butyl
protons of 12,14-di^{t}butylbenzo[g]chrysene

Fig. 21. Exponential *T*_{1ρ} decay curves for
the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene

Fig. 22. Linear fits to the logarithms of exponential *T*_{1ρ}
decay curves for the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene

Fig. 23. Contour plot for the *T*_{1ρ} spectrum
of the ^{t}butyl protons of 12,14-di^{t}butylbenzo[g]chrysene

#
Dephasing time, *T*_{2}*

The dephasing time, *T*_{2}*, also known as the
effective transverse relaxation time, is a combination of transverse
relaxation and magnetic field inhomogeneity. In a perfectly
homogeneous magnetic field *T*_{2}* = *T*_{2}
but *T*_{2}* is shorter when the field is
inhomogeneous, *i. e.*, when the shimming is not perfect. The
dephasing time is the reciprocal of *π* times the line-width at
half height (*w*_{½}): . For a well shimmed and
stabilized spectrometer with a homogeneous sample in a good quality
NMR tube the contribution to the linewidth of inhomogeneity and
field instability is about 0.1 Hz. For example, the line-width of
the signal of TMS in fig. 24 is 0.157 Hz, giving a *T*_{2}*
of 2.02 s. The *T*_{2} of TMS is 13.1 s and contributes
0.024 Hz to the line-width, the remaining 0.133 Hz being from
inhomogeneity and field instability. (The satellite signals in the
spectrum are real signals arising from heteronuclear coupling.)

Fig. 24. Measurement of the *T*_{2}* of TMS in CDCl_{3}

# Improving experimental accuracy

To maximize accuracy and reliability of relaxation measurements, one must be aware of the effects of noise, saturation, thermal convection and, in gasses, diffusion.

Noise should be kept to a minimum. For insensitive samples, more
than one scan per row may be needed. Care must be taken to ensure
that the experimental conditions (temperature, *etc.*) are
stable and that the pulses have been calibrated carefully. The *τ*
values should be chosen to cover the range zero to twice the
expected relaxation time if the relaxation times are similar. If the
relaxation times are not similar then the *τ* values should not
be distributed evenly but at least four should be concentrated
within the minimum relaxation time. For *T*_{1}
measured by inversion recovery, one measurement should be carried
out with a *τ* of five to seven times the maximum *T*_{1}
should be made and a few measurements between that value and twice *T*_{1} may improve accuracy by
confirming that the decay curve is truly monoexponential.

If the sample is very concentrated then the relaxation time will appear shorter than it really is due to saturation. In such a case, off-tune the probe, recalibrate the pulse-widths and repeat the experiment.

If the relaxation time is very long (greater that about 5 s in a
well stabilized NMR probe at room temperature) and/or the sample has
low viscosity and/or the experiment is being carried out at elevated
temperatures, then convection is likely to affect the results. The
effect of convection is evident as a biexponential decay. In the
example below, the fitted curve (fig. 25) does not match the last
three points well, suggesting such a problem. A double exponential
decay fits better (fig. 26) indicating a *T*_{1} closer
to 12 s. It is often possible to measure relaxation times of
hundreds of seconds in spinning liquid samples.

Fig. 25. Inversion recovery curve with a monoexponential fit. The
imperfection in the fit is due to convection. The spectrum is of
the methyl protons of ethylbenzene (0.1%) in CDCl_{3} at
400 MHz at 25°C.

Fig. 26. Biexponential fit to the inversion recovery curve fits
much better than the monoexponential fit. The spectrum is of the
methyl protons of ethylbenzene (0.1%) in CDCl_{3} at 400
MHz at 25°C.

Convection effects can be reduced by spinning the sample and using a narrower tube. However, spinning introduces noise that reduces the accuracy of the measurement. So, for short relaxation times, it is better not to spin the sample. For gaseous samples with very long relaxation times, diffusion is also a problem so the measurement should be made in a restricted chamber using the buildup method.

Experimental conditions such as temperature and magnetic field
affect the relaxation time. The relaxation time is also magnetic
field dependent (fig. 27) so the relaxation time measured on a 300
MHz NMR instrument will be different than that measured on a 600 MHz
instrument. In addition, *T*_{1ρ}
is dependent on the spin-lock intensity.

Fig. 27. Effect of magnetic field on the ^{1}H *T*_{1}
relaxation of water in CDCl_{3}.

# Differences within multiplets

The relaxation of different peaks in a multiplet may be different
due to cross-correlation (as shown for the H1 AA'BB' multiplet
of diphenylanthracene, fig. 28).
For greatest accuracy all the peaks in a multiplet should be measured
separately. The above applies to all
relaxation measurements: *T*_{1}, *T*_{2}, *T*_{1ρ}
and *T*_{2}*.

Fig. 28. Contour plot for the H1 AA'BB' multiplet of diphenylanthracene in
THF-*d*_{8} at 400 MHz at 25°C showing *T*_{1} values
ranging from 3.7 to 4.85 s

# Applications of relaxation measurements:

# Application of relaxation times to the study molecular motion

The relaxation times for a spin ½ nucleus are approximately related
to the correlation time (*τ _{c}*, fig. 29) which is the
time that a molecule takes to tumble by one radian (57°). The
three relaxation times (

*T*

_{1},

*T*

_{2}and

*T*

_{1ρ}) are similar for small diamagnetic molecules in regular (not especially viscous) solvents at room temperature (fig. 30 and left side of fig. 29). Since fig. 29 only approximates the true relaxation time,

*T*

_{2}and

*T*

_{1ρ}are sometimes larger than

*T*

_{1}(fig. 30) but never exceed twice its value under normal conditions. For large entities (

*e. g.*, protiens or microemulsion droplets) or low temperatures

*T*

_{2}and

*T*

_{1ρ}are shorter than

*T*

_{1}(fig. 31 and middle of fig. 29). It is usually impractical for

*T*

_{1ρ}to become much larger than

*T*

_{2}(right side of fig. 29) because the motion whould have to be very slow such as in cryogenic solids or the spin-lock would need to be very strong.

Fig. 29. Approximate dependence of relaxation time on correlation
time. This diagram is drawn for water (H_{2}^{16}O)
at 500 MHz with a 26 kHz spin-lock and is within an order of
magnitude of being correct for most samples.

Fig. 30. Relaxation times for a small molecule
(2,4-dichloroquinazoline) showing similar values for *T*_{1}
(red), *T*_{2} (blue) and *T*_{1ρ}
(green)

Fig. 31. Relaxation times for large entities (surfactants and oil
in the nanodroplets of an O/W microemulsion) showing *T*_{1}
(red) much longer than *T*_{2} (blue) and *T*_{1ρ}
(green)

The relaxation times are approximately related to the correlation
time by the following equations according to
Bloembergen-Purcell-Pound (BPP) theory where *ω*_{0} is
the rotational frequency of the signal (2*πf*),*µ*_{0}
is the magnetic permeability of free space (4*π*×10^{−7}
H m^{−1}), *ħ* is the reduced Planck constant
1.054571726×10^{−34} J s, *γ* is the gyromagnetic ratio
of the nucleus (for ^{1}H it is 2.67513×10^{8} rad s^{−1}
T^{−1}) and *r* is the distance between magnetically
active spin-½ nuclei.

Consider the example of 2,4-dichloroquinazoline (fig. 32). The
distance between H5 and H6 is 2.36×10^{−10} m. The value of
*K* is therefore 1.0×10^{−9} s^{−2}. The value
of *ω*_{0}^{2} is 3.14×10^{9} rad s^{−1}.
*T*_{1} was measured to be 3.8 s
(fig. 30). This corresponds to a correlation time of 59 ps (5.9×10^{−11}
s). H6 and H7 interact with two protons each so their *T*_{1}'s
would be expected to be half of this (1.9 s) and were in reasonable
agreement (1.8 and 2.1 s, respectively). H8 has a *T*_{1}
of 1.9 s instead of the expected 3.8 s due to its proximity to the
quadrupolar ^{14}N nucleus. The shorter *T*_{1}
that indicates proximity to the nitrogen can be used to confirm the
assignment. BPP theory predicts that *T*_{2}
should be the same as *T*_{1} for
these protons but this is not the case due to spin diffusion. The *T*_{2} values all fall between 1.8 and
2.2 s.

Fig. 32. Molecular structure of 2,4-dichloroquinazoline

Consider the example of the microemulsion (fig. 31) where the
signals shown arise mainly from the aliphatic chains of Tween80. The
signal at 2.84 ppm appears to correspond to a CH signal because of
its slower relaxation while the others arise from CH_{2} signals.
The internuclear distance for CH_{2} is 1.75×10^{−10}
m. The value of *K* is therefore 6.2×10^{−9} s^{−2}.
The average *T*_{1} of the CH_{2}
signals is 0.62 s and the average *T*_{2}
is much shorter at 0.12 s. Clearly this is to the right of the
minimum *T*_{1} in fig. 32. The
correlation time that matches this result is 700 ps (7×10^{−10}
s), an order of magnitude slower than the previous example.

# Effect of steric hindrance on relaxation

Relaxation is faster when there is steric hindrance due to the
proximity of other magnetically active nuclei. In the case of
12,14-di^{t}butylbenzo[g]chrysene (fig. 33), the
relaxation times *T*_{1} (fig. 11), *T*_{2} (fig. 5) and *T*_{1ρ}
(fig. 23) are shorter by a factor of 2½ for ^{t}butyl14
(0.92 ppm) than for ^{t}butyl12 (1.40 ppm). There is
strong steric hindrance on ^{t}butyl14 which puts its
protons close to H1 and H2. Given that the correlation time is
similar for the two groups, this means that the internuclear
distance between methyl protons and H1 and/or H2 is a factor of 1.16
(^{6}√2.5) closer (1.5×10^{−10} m) than between the
methyl protons themselves (1.75×10^{−10} m). This obvious
reduction in relaxation time can be used to assign the NMR signals
in sterically hindered portions of the molecule.

Fig. 33. Molecular structure of 12,14-di^{t}butylbenzo[g]chrysene

# Effect of dissolved air and other paramagnetic species

The relaxation time can be used to detect complexation with paramagnetic species. This is so sensitive that the effect of dissolved oxygen from the air that speeds up relaxation even in the normal concentrations found in solution. This need not be important if the relaxation measurement is a means to another end, such as spectroscopic assignment. However, when the main purpose of the measurement is to determine the relaxation properties of a compound or to determine the correlation time, the sample must usually be degassed with an inert gas or sealed under vacuum.

The paramagnetic effect of oxygen is linearly related to the
pressure: .
The value of *k* is typically 0.06 s/atm of air but is higher
when there is stronger than usual binding between the molecule and
oxygen. This is the case for *β*-cyclodextrin but for
calix[4]arene value of *k* is around 0.13 for the non-polar
part of the molecule (fig. 34), indicating weak binding to molecular
oxygen.

Fig. 34. Difference between the effect of oxygen on relaxation
times of *β*-cyclodextrin and calix[4]arene. The larger
difference for calix[4]arene indicates weak binding to oxygen.

# Relaxation weighting for assignment

Although the relaxation time is a property of the nucleus that
reflects its properties and environment, it is mostly used to decide
on the length of the relaxation delay between acquisitions. For
quantitative work this needs to be at least five times *T*_{1}
to achieve a 1% accuracy although seven times *T*_{1}
is recommended to ensure accuracy. For maximum (non-quantitative)
sensitivity within a limited time for a regular
1D experiment, the repetition rate should be *T*_{1}
with a pulse angle of 68°. The mixing time in NOESY and EXSY experiments is
dependent on *T*_{1} while the mixing
time in ROESY and TOCSY is dependent on *T*_{1ρ}. In principle, the
preferred acquisition time is dependent on *T*_{2}*
although in practice this is determined by observing the fid
directly.

Relaxation times can be used for assignment. In general, relaxation
is faster when signals are strongly coupled or close to quadrupolar
nuclei. For example, quaternary carbons (not attached to protons)
relax slower than carbons attached to protons. This can be easily
observed in the ^{13}C
NMR because the slower relaxing nuclei give sharper signals as
shown in fig. 35 below.

Fig. 35. Part of the ^{13}C NMR spectrum of 12,14-di^{t}butylbenzo[g]chrysene
showing the difference between proton attached and non-proton
attached carbons

Under usual ^{13}C NMR
acquisition (not overlong relaxation delay) and processing (exponential apodization with 1 Hz
line broadening) conditions as shown in the spectrum below (fig. 36), the
carbons attached to protons appear higher than those not attached to
protons due to the difference in relaxation times in addition to NOE factors. This is the
spectroscopic equivalent to the *T*_{1} weighting
technique used in magnetic resonance imaging (MRI).

Fig. 36. Aromatic Part of the ^{13}C NMR spectrum of
12,14-di^{t}butylbenzo[g]chrysene

# Exchange processes

Dynamic exchange processes
reduce the relaxation time, be it chemical or conformational
exchange. This is especially significant for medium-fast exchange on a
timescale of milliseconds to seconds. The *T*_{2}
relaxation time is related to and can be used to determine the
exchange rate and consequently the thermodynamic free energy of activation
(Δ*G*^{‡}).
Using measurements at several temperatures, the activation enthalpy and entropy can also be
determined. However, it is very time consuming to measure *T*_{2}, so the value of *T*_{2}* is usually measured from
the line-widths and used instead. For more details see the page on dynamic NMR.