NMR

NMR: see magnetic resonance.

Selective absorption of very high-frequency radio waves by certain atomic nuclei subjected to a strong stationary magnetic field. Nuclei that have at least one unpaired proton or neutron act like tiny magnets. When a strong magnetic field acts on such nuclei, it sets them into precession. When the natural frequency of the precessing nuclear magnets corresponds to the frequency of a weak external radio wave striking the material, energy is absorbed by the nuclei at a frequency called the resonant frequency. NMR is used to study the molecular structure of various solids and liquids. Magnetic resonance imaging, or MRI, is a version of NMR used in medicine to view soft tissues of the human body in a hazard-free, noninvasive way.

Learn more about nuclear magnetic resonance (NMR) with a free trial on Britannica.com.

In nuclear magnetic resonance (NMR) spectroscopy and magnetic resonance imaging (MRI) the term relaxation describes several processes by which nuclear magnetization prepared in a non-equilibrium state return to the equilibrium distribution. In other words, relaxation describes how fast spins "forget" the direction in which they are oriented. The rates of this spin relaxation can be measured in both spectroscopy and imaging applications.

T₁ and T₂

Different physical processes are responsible for the relaxation of the components of the nuclear spin magnetization vector M parallel and perpendicular to the external magnetic field, B₀ (which is conventionally oriented along the z axis). These two principal relaxation processes are termed T₁ and T₂ relaxation respectively.

T₁

The longitudinal (or spin-lattice) relaxation time T₁ is the decay constant for the recovery of the z component of the nuclear spin magnetization, M_z, towards its thermal equilibrium value, $M\_\{z,mathrm\{eq\}\}$. In general,

$M\_z(t)\; =\; M\_\{z,mathrm\{eq\}\}\; -\; [M\_\{z,mathrm\{eq\}\}\; -\; M\_z(0)]e^\{-t/T\_1\}$

In specific cases:

• If M has been tilted into the xy plane, then $M\_z(0)=0$ and the recovery is simply

$M\_z(t)\; =\; M\_\{z,mathrm\{eq\}\}left(1\; -\; e^\{-t/T\_1\}\; right)$

i.e. the magnetisation recovers to 63% of its equilibrium value after one time constant T₁.

• In the inversion recovery experiment, commonly used to measure T₁ values, the initial magnetisation is inverted, $M\_z(0)=-M\_\{z,mathrm\{eq\}\}$, and so the recovery follows

$M\_z(t)\; =\; M\_\{z,mathrm\{eq\}\}left(1\; -\; 2e^\{-t/T\_1\}\; right)$

T₁ relaxation involves redistributing the populations of the nuclear spin states in order to reach the thermal equilibrium distribution. By definition this is not energy conserving. Moreover, spontaneous emission is negligibly slow at NMR frequencies. Hence truly isolated nuclear spins would show negligible rates of T₁ relaxation. However, a variety of relaxation mechanisms allow nuclear spins to exchange energy with their surroundings, the lattice, allowing the spin populations to equilibrate. The fact that T₁ relaxation involves an interaction with the surroundings is the origin of the alternative description, spin-lattice relaxation.

Note that the rates of T₁ relaxation are generally strongly dependent on the NMR frequency and so may vary considerably with magnetic field strength, B.

T₂

The transverse (or spin-spin) relaxation time T₂ is the decay constant for the component of M perpendicular to B₀, designated M_xy, M_T, or $M\_\{perp\}$. For instance, initial xy magnetisation at time zero will decay to zero (i.e. equilibrium) as follows:

$M\_\{xy\}(t)\; =\; M\_\{xy\}(0)\; e^\{-t/T\_2\}\; ,$

i.e. the transverse magnetization vector drops to 37% of its original magnitude after one time constant T₂.

T₂ relaxation is a complex phenomenon, but at its most fundamental level, it corresponds to a decoherence of the transverse nuclear spin magnetization. Random fluctuations of the local magnetic field lead to random variations in the instantaneous NMR precession frequency of different spins. As a result, the initial phase coherence of the nuclear spins is lost, until eventually the phases are disordered and there is no net xy magnetization. Because T₂ relaxation involves only the phases of other nuclear spins it is often called "spin-spin" relaxation.

T₂ values are generally much less dependent on field strength, B, than T₁ values.

T₂* and magnetic field inhomogeneity

In an idealized system, all nuclei in a given chemical environment in a magnetic field spin with the same frequency. However, in real systems, there are minor differences in chemical environment which can lead to a distribution of resonance frequencies around the ideal. Over time, this distribution can lead to a dispersion of the tight distribution of magnetic spin vectors, and loss of signal (Free Induction Decay). In fact, for most magnetic resonance experiments, this "relaxation" dominates. This results in intra-voxel dephasing.

However, decoherence because of magnetic field inhomogeneity is not a true "relaxation" process; it is not random, but dependent on the location of the molecule in the magnet. For molecules that aren't moving, the deviation from ideal relaxation is consistent over time, and the signal can be recovered by performing a spin echo experiment.

The corresponding transverse relaxation time constant is thus T₂^*, which is usually much smaller than T₂. The relation between them is:

$frac\{1\}\{T\_2^*\}=frac\{1\}\{T\_2\}+frac\{1\}\{T\_\{inhom\}\}\; =\; frac\{1\}\{T\_2\}+gamma\; Delta\; B\_0$

where γ represents gyromagnetic ratio, and ΔB₀ the difference in strength of the locally varying field.

Unlike T₂, T₂* is influenced by magnetic field gradient irregularities. The T₂* relaxation time is always shorter than the T₂ relaxation time and is typically milliseconds for water samples in imaging magnets.

The reason that T₁ is slower than T₂

The following always holds true: T₁ > T₂ > T₂*.

If T₂ were to be slower than T₁, then the magnetizations perpendicular to the initial direction would have not dephased by the time the sample had returned to equilibrium. This is physically impossible, as once the sample has returned to equilibrium, there is no magnetization perpendicular to the original direction. Hence, T₁ must be greater than or equal to T₂.

Bloch equations

Are used to calculate the nuclear magnetization M = (M_x, M_y, M_z) as a function of time when relaxation times T₁ and T₂ are present. Bloch equations are phenomenological equations that were introduced by Felix Bloch in 1946.

$frac\; \{partial\; M\_x(t)\}\; \{partial\; t\}\; =\; gamma\; (bold\; \{M\}\; (t)\; times\; bold\; \{B\}\; (t)\; )\; \_x\; -\; frac\; \{M\_x(t)\}\; \{T\_2\}$

$frac\; \{partial\; M\_y(t)\}\; \{partial\; t\}\; =\; gamma\; (bold\; \{M\}\; (t)\; times\; bold\; \{B\}\; (t)\; )\; \_y\; -\; frac\; \{M\_y(t)\}\; \{T\_2\}$

$frac\; \{partial\; M\_z(t)\}\; \{partial\; t\}\; =\; gamma\; (bold\; \{M\}\; (t)\; times\; bold\; \{B\}\; (t)\; )\; \_z\; -\; frac\; \{M\_z(t)\; -\; M\_0\}\; \{T\_1\}$

Where γ is the gyromagnetic ratio and B(t) = (B_x(t), B_y(t), B₀ + B_z(t)) is the magnetic flux density experienced by the nuclei. The z component of the magnetic flux density B is typically composed of two terms: one, B₀, is constant in time, the other one, B_z(t), is time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal. M(t) × B(t) is the cross product of these two vectors.

The equation listed above in the section on T₁ and T₂ relaxation can be derived from Bloch equations.

Common relaxation time constants in human tissues

Following is a table of the approximate values of the two relaxation time constants for nonpathological human tissues, just for simple reference.

At a main field of 1.5 T

Tissue Type	Approximate T1 value in ms	Approximate T2 value in ms
Adipose tissues	240-250	60-80
Whole blood (deoxygenated)	1350	50
Whole blood (oxygenated)	1350	200
Cerebrospinal fluid (similar to pure water)	2200-2400	500-1400
Gray matter of cerebrum	920	100
White matter of cerebrum	780	90
Liver	490	40
Kidneys	650	60-75
Muscles	860-900	50

Following is a table of the approximate values of the two relaxation time constants for chemicals that commonly show up in human brain magnetic resonance spectroscopy (MRS) studies, physiologically or pathologically.

At a main field of 1.5 T

Signals of Chemical Groups	Relative resonance frequency	Approximate T1 value (ms)	Approximate T2 value (ms)
Creatine (Cr) and Phosphocreatine (PCr)	3.0 ppm	gray matter: 1150-1340, white matter: 1050-1360	gray matter: 198-207, white matter: 194-218
N-Acetyl group (NA), mainly from N-Acetylaspartate (NAA)	2.0 ppm	gray matter: 1170-1370, white matter: 1220-1410	gray matter: 388-426, white matter: 436-519
—CH3 group of Lactate	1.33 ppm (doublet: 1.27 & 1.39 ppm)	(To be listed)	1040

Microscopic mechanism

In 1948, Nicolaas Bloembergen, Edward Mills Purcell, and R.V. Pound proposed the so-called Bloembergen-Purcell-Pound theory (BPP theory) to explain the relaxation constant of a pure substance in correspondence with its state, taking into account the effect of tumbling motion of molecules on the local magnetic field disturbance . The theory was in good agreement with the experiments for pure substance, but not for complicated environment such as human body.

From this theory, one can get T₁、T₂:

$frac\{1\}\{T\_1\}=K[frac\{tau\_c\}\{1+omega\_0^2tau\_c^2\}+frac\{4tau\_c\}\{1+4omega\_0^2tau\_c^2\}]$

$frac\{1\}\{T\_2\}=frac\{K\}\{2\}[3tau\_c+frac\{5tau\_c\}\{1+omega\_0^2tau\_c^2\}+frac\{2tau\_c\}\{1+4omega\_0^2tau\_c^2\}]$,

where $omega\_0$ is the Larmor frequency in correspondence with the strength of the main magnetic field $B\_0$. $tau\_c$ is the correlation time of the molecular tumbling motion. $K=frac\{3mu^2\}\{160pi^2\}frac\{hbar^2gamma^4\}\{r^6\}$ is a constant with μ being the magnetic dipole moment of the spin-1/2 nuclei, $hbar=frac\{h\}\{2pi\}$ the reduced Planck constant, γ the gyromagnetic ratio of such species of nuclei, and r the distance between the two nuclei carrying magnetic dipole moment.

Taking for example the H₂O molecules in liquid phase without the contamination of oxygen 17, the value of K is 1.02×10¹⁰ s^-2 and the correlation time $tau\_c$ is on the order of picoseconds = $10^\{-12\}$ s, while hydrogen nuclei ¹H (protons) at 1.5 teslas carry an Larmor frequency of approximately 64 MHz. We can then estimate using τ_c = 5×10^-12 s:

$omega\_0tau\_c\; =\; 3.2times\; 10^\{-5\}$(dimensionless)

$T\_1=(1.02times\; 10^\{10\}[frac\{\; 5times\; 10^\{-12\}\; \}\{1\; +\; (3.2times\; 10^\{-5\}\; )^2\}\; +\; frac\{\; 4cdot\; 5times\; 10^\{-12\}\; \}\{1\; +\; 4cdot\; (3.2times\; 10^\{-5\}\; )^2\}])^\{-1\}$= 3.92 s

$T\_2=(frac\{1.02times\; 10^\{10\}\}\{2\}[3cdot\; 5times\; 10^\{-12\}\; +\; frac\{5cdot\; 5times\; 10^\{-12\}\; \}\{1\; +\; (3.2times\; 10^\{-5\}\; )^2\}\; +\; frac\{\; 2cdot\; 5times\; 10^\{-12\}\; \}\{1\; +\; 4cdot\; (3.2times\; 10^\{-5\}\; )^2\}])^\{-1\}$= 3.92 s,

which is close to the experimental value, 3.6 s. Meanwhile, we can see that at this extreme case, T₁ equals T₂.

References

See also

• basics of NMR
• MRI - Magnetic resonance imaging Animation made by bigs.eu; contents are: spin, spin modification, induction, relaxation and precession, spin echo sequence, gradient echo sequence, inversion recovery sequence.
• Relaxation in high-resolution NMR spectroscopy
• Field-cycling NMR relaxometry
• Relaxometry