Nuclear Magnetic Resonance (NMR) - Relaxation Mechanism
Nuclear Magnetic Resonance (NMR) - Relaxation Mechanism
Let us consider a solid or liquid sample
in which there is at least one type of isotope which have odd number of proton
or neutron, or in other words the nuclei have a net spin magnetic moment. When
this sample is placed in a strong magnetic field (let’s consider the field is
in z direction of a right-hand co-ordinate system) the bulk magnetic moment stars
precessing around the field direction in a cone. This build up of a net
magnetization along the static field direction is referred to as spin-lattice
relaxation.
Now if a r.f. field is applied in
a direction perpendicular to the applied static magnetic field, i.e. in the x-y
plane then the magnetic moment is perturbed to rotate in the x-y plane. When
the r.f. field is turned of then the bulk magnetic moment of the sample relaxes
back to equilibrium state in z direction.
Basically, relaxation is a
mechanism in which nuclear spin come back to its equilibrium state when the
perturbation is withdrawn from the system. It is used to study the dynamic
properties of the materials.
Longitudinal or spin-lattice
relaxation
Longitudinal or spin-lattice relaxation (T1) 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. 1).
 |
| Fig. 1. Relaxation via T1 along the magnetic-field (z) axis |
In the absence of any external
magnetic field, the spin polarizations of the sample are uniformly distributed
along all possible direction making the net magnetic moment of the sample zero.
When a external magnetic field is applied then the
tiny magnetic moments starts spinning (Larmor precession) around the external
field without changing their position. Total magnetic moment of the sample
however does not change because of the isotropic distribution of the spin
polarization.
Beside this external applied field there are other internal
magnetic fields which arises due to the orbital motion of atomic electrons and
intrinsic magnetic moments of electron and nuclei present in the molecule. The
nuclear spins precesses about a net magnetic field which is sum of the static
external magnetic field and the internal fields. The tiny internal fields
fluctuate rapidly due to the thermal motion of the environment. As a result,
the total magnetic field also has a fluctuating magnetic field which causes the
gradual breakdown of the cone at which the nuclear spins are revolving around
the net magnetic field.
The net magnetic field which is the sum of the
external and internal fields has a slightly fluctuating magnitude, and also a
slightly fluctuating direction. Therefore, at any given time, the local
magnetic field experienced by each nuclear spin is slightly different than its
neighbor, both in magnitude and direction.
After sufficiently long period of time, the tiny fluctuations of the local magnetic field cause the isotropy of the nuclear spin polarization to be broken and, hence, a macroscopic nuclear magnetic moment to develop. Without the fluctuating molecular fields, nuclear magnetism would be unobservable. Magnetic energy of any dipole is lowest when the dipole is oriented along the field direction and this is the direction preferred by magnetic dipoles. The nuclear spins also have the tendency to orient along low magnetic energy as a result an eventually a net magnetic moment along external field direction is developed with time.
 |
| Fig. 2. Development of net magnetic moment along B field |
When an external field is suddenly turned on, the macroscopic nuclear magnetic moment starts building up along the field direction. Let the magnetic field is turned on at a certain time ton and the field is applied in z direction. At any later time, t ≥ ton, the value of longitudinal magnetization can be defined as,
 |
| Fig. 3.: Development of net magnetic moment along field direction after the field is turned on |
So, the building up of net magnetization is exponential in nature. The exponential time constant T1 in the equation is called spin-lattice relaxation time constant or the longitudinal relaxation time constant.
Relaxion means re-establishment of equilibrium after a perturbation is applied in the system. When there is no external field is applied to the system resides in a thermal equilibrium state with spin polarization distributed isotropically. The equilibrium is destroyed when an external field is applied and the system relaxes to a new equilibrium state with anisotropic distribution of spin polarization along applied field direction.
Now, if the magnetic field is again turned off at any time toff then the nuclear spins polarizations again start redistributing themselves to a isotropic state. In this case the net magnetization of the system is given by
Transverse or spin-spin relaxation
Transverse or spin-spin relaxation is the mechanism by which the excited magnetization vector which is developed by application of a r.f. pulse perpendicular to the direction of static filed (along the x-y-plane) decays.
 |
| Fig. 4. T2 relaxation mechanism |
The longitudinal nuclear spin magnetization is about four orders of magnitude less than the typical diamagnetism of the sample developed due to the motion of the electrons. Due to the presence of this diamagnetism, instead of measuring the nuclear spin magnetization along the field, the magnetization perpendicular to the field is measured. At first, the spin system is allowed to reach thermal equilibrium in the presence of strong static magnetic field (generally applied along z axis). In this condition there is no net magnetization along the perpendicular direction of the applied static field.
Now the polarization of every single spin is suddenly rotated by π/2 radians around the x-axis, by the application of r.f. pulse (oscillating magnetic field) of appropriate frequency and time duration. If a spin polarization is initially along the z-axis and is rotated by π/2 about the x-axis, then the result is a spin polarization along the negative y-axis i.e. along an axis perpendicular to the magnetic field. This net magnetization developed perpendicular to the magnetic field is called transverse magnetization.
After the applied r.f. pulse is turned off, each individual spin starts their precessional motion, so as the bulk magnetic moment of the sample. In this condition the bulk magnetic moment precesses in the x-y plane at a frequency equal to the Larmor frequency of the isotope.
The bulk magnetic moment of the sample along x and y axis at any later time t, after applying the r.f. pulse is given by the equations below –
My(t) = - M0 cos(ω0t) exp(- t/T2) ………… (3)
Mx(t) = M0 sin(ω0t) exp(- t/T2) .………..(4)
Here, ω0 is Larmor frequency
The transverse magnetization decays slowly because the microscopic magnetic fields fluctuate slightly both in direction and magnitude. As a result, the precessing tiny nuclear magnets gradually get out of phase with each other. This decay process is irreversible. Once the transverse magnetization is gone, it cannot be recovered. This type of process is called homogeneous decay. This time constant T2 called the transverse relaxation time constant or coherence dephasing time constant or coherence decay time constant or the spin–spin relaxation time constant. The magnitude of the magnetic moment in the x-y plane decays following the equation M=M0exp(-τ/T2).
1. Spin Dynamics by Malcolm H. Levitt
2.Principles of Nuclear Magnetic Resonance in One and Two Dimensions, by Richard R. Ernst, et al.
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