Basic Concepts
T₁ - Relaxation time

Spin-Lattice relaxation - T₁
Ernst angle
Relaxation and molecular motion
Relaxation mechanisms: 1- Dipole-Dipole interaction "through space" | 2-Electric Quadrupolar Relaxation | 3- Paramagnetic Relaxation | 4- Scalar Relaxation | 5- Chemical Shift Anisotropy Relaxation | 6- Spin Rotation

Spin-lattice Relaxation time T₁ (longitudinal)

The relaxation time T₁ represents the "lifetime" of the first order rate process that returns the magnetization
to the Boltzman equilibrium along the +Z axis.
T₁ relaxation time can be measured by various techniques describe in the table below.

Name	Pulse Sequence	signal evolution vs T₁
Inversion Recovery (IRFT)	D1-180-tau-90-Acq {D1+Acq>5*T₁}	M(tau)/M0= 1-2*exp(-tau/T₁)
Progressive Saturation (PSFT)	(preceded by dummy pulses) - D1-90-Acq {tau=D1+Acq}	M(tau)/M0= 1-exp(-tau/T₁)
Saturating Comb (Mainly useful in solid) require: T₂*<<T₁	{n90-t}-tau-90-Acq t: pulse spacing during Comb. :T₂< t <T₁ tau: delay for magnetization recovery	M(tau)/M0= 1-exp(-tau/T₁)

The inversion recovery technique is presented in more details in the following animation..
After a delay of 1*T₁, 63% of the magnetization is recovered along the +Z axis.
To recover 99% of the magnetization a delay of 5*T₁ need to be used.

The magnitude of the relaxation time depends highly on the type of nuclei (nuclei with spin 1/2 and low magnetogyric ratio have usually long relaxation time whereas nuclei with spin>1/2 have very short relaxation time) and on other factors
like the physical state (solid or liquid state), on the viscosity of the solution, the temperature ... etc.
in other words the relaxation time depends on the motion of the molecule.

The longitudinal relaxation process (T₁) governs the time interval between 2 transients.

If the interval between 2 transients is shorter that 5*T₁, the accuracy of the integration might be questionable.
If the interval between 2 pulses is shorter that 5*T₁, as for obtaining routine NMR spectra, the pulse width must be adjusted to accommodate the different length of the relaxation process to obtain the best sensitivity for the NMR experiment.
The length of the pulse that provide the best sensitivity for a given relaxation time is called the "Ernst angle".
For example: if 1 sec. acquisition time is used, you can find below the best pulse angle to use for different relaxation time.

Relaxation time (sec)	Ernst Angle (with 1 sec repetition time)
100 (very slow T1)	8 degree
10	25 degree
4	33 degree
2	53 degree
1	68 degree
0.4	86 degree
0.1 (rapid T1)	90 degree

Relaxation and Molecular Motion

The relaxation process is induced by field fluctuation due to molecular motion. (The local field experienced by a molecule changes when the molecule reorients)

A few definitions:
The correlation time -tc (Tau-c): represents the time it takes for a molecule to reorient by 1 degree ("tumbling time").
The spectral density - J(w): describes the ranges of frequency motion that are present. Not all molecules tumbles at a unique rate: molecules tumbles, collide, change direction... at a range of rates up to the maximum rate of (1/tc). The concentration (or intensity) of fields at a given frequency of motion (w) is known as the spectral density J(w).

There are several relaxation mechanism:

Interaction	Range of interaction (Hz)	relevant parameters
1- Dipolar coupling	10⁴ - 10⁵	- abundance of magnetically active nuclei - size of the magnetogyric ratio
2- Quadrupolar coupling	10⁶ - 10⁹	- size of quadrupolar coupling constant - electric field gradient at the nucleus
3- Paramagnetic	10⁷ -10⁸	concentration of paramagnetic impurities
4- Scalar coupling	10 - 10³	size of the scalar coupling constants
5-Chemical Shift Anisotropy (CSA)	10 - 10⁴	- size of the chemical shift anisotropy - symmetry at the nuclear site
6- Spin rotation

All of them (except scalar mechanism) involves the magnetogyric ratio of the nucleus. The first 3 mechanisms are much stronger and efficient than the other 3 mechanisms.

There are different approach to distinguish the various relaxation mechanisms:

by the strength of the interaction
by the use of isotopic substitution to identify the dipolar mechanism
by the field dependence:
CSA is proportional the B0 (applied field).
Quadrupole interaction is inversely proportional the B0 (applied field)
by their temperature dependence

you will find below a very brief description of those mechanisms. In general terms, the relaxation rate R₁ (1/T₁) depends on the strength of the interaction and on a correlation function.

1- Dipole-Dipole interaction "through space"

This relaxation mechanism is particularly important for molecules containing protons (high natural abundance nuclei equipped with a large magnetogyric ratio).

This interaction depends on the strength of the dipolar coupling (depends on gamma), on the orientation/distance between the interacting nuclei and on the motion.

The distance dependence is very large as can be seen in Carbon-13 (protonated carbons relax more rapidly than quaternary)
The dependence on gamma is also very large. Proton relaxation is dominated by dipolar relaxation. X-H (hetero nuclei directly substituted by proton) is also dominated usually by dipolar relaxation due to the short distance and due to the fact that proton has a strong gamma-ratio. If proton is replaced by deuterium, in the X-D bond, the X-nuclei relax much slower than the corresponding X-H due to the lack of dipole-dipole relaxation. (gamma-H is 6.5 time larger than gamma-D).

2- Electric Quadrupolar Relaxation

If a nuclei has a spin>1/2, it is characterized by a non-spherical distribution of electrical charges and possesses an electric magnetic moment. The quadrupole coupling constant is in the MHz range (very efficient). As this relaxation process is very large, it dominates over the other mechanisms.

This relaxation depends on:

"eQ" - the quadrupole Moment of the nucleus
e.g. Deuterium: eQ=.003 and 55Mn has eQ=0.55
"eq" - the Electric Field Gradient (EFG).
The Quadrupole coupling vanish,in a symmetrical environment.
e.g. for symmetrical [NH4}+ : eq * eQ = 0 and therefore has very long T₁ =50 sec.
whereas CH3CN : eq * eQ about 4 MHz and T₁=22 msec.

3- Paramagnetic Relaxation

The molecular motion modulates the electric field from unpaired electron spin.

There is dipole relaxation by the electron magnetic moment.
There is also a transfer of unpaired electron density to the relaxing nucleus.

4- Scalar Relaxation (due to coupling with fast relaxing quadrupolar nuclei)

The effect of scalar coupling relaxation on T₁ is significant only if the two interacting nuclei have very close frequency. This condition occur very rarely!

It occur for example for Carbon-13 (75.56 MHz with B1=7.06 T) and Br-79 (75.29 MHz with B1=7.06 T) which are very close in frequency.

Scalar relaxation is more important for the T₂ relaxation as with this mechanism the quadrupolar nuclei can broaden lines significantly on nuclei that are coupled to it.

5- Chemical Shift Anisotropy Relaxation

The magnetic field sense by the nucleus depends on the chemical shift tensor in the molecule.

The chemical shift is in fact dependent by the orientation of the molecule in the magnetic field. This effect, called the chemical shift anisotropy (CSA), is very well known in solid state NMR as it is responsible (in part) for the very wide line width observed on a static sample.

In solution, CSA is averaged out by molecular tumbling and a sharp isotropic shift is observed; but the modulation of the shielding can provide a relaxation mechanism in absence of other mechanism. This mechanism is field dependent.
CSA is an important relaxation mechanism for nuclei with large chemical shift scale as for example on Phosphorus-31 and on Cadmium-113.

6- Spin Rotation

Intramolecular dynamic process (like the rotation of methyl group) can also contribute to longitudinal relaxation.

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Basic Concepts T1 - Relaxation time

Spin-lattice Relaxation time T1 (longitudinal)