Duringnuclear magnetic resonance observations,spin–lattice relaxation is the mechanism by which the longitudinal component of the totalnuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium state tothermodynamic equilibrium with its surroundings (the "lattice"). It is characterized by thespin–lattice relaxation time, a time constant known asT₁.

There is a different parameter,T₂, thespin–spin relaxation time, which concerns the exponential relaxation of the transverse component of the nuclear magnetization vector (perpendicular to the external magnetic field). Measuring the variation ofT₁ andT₂ in different materials is the basis for somemagnetic resonance imaging techniques.^[1]

T₁ characterizes the rate at which the longitudinalM_z component of the magnetization vector recovers exponentially towards its thermodynamic equilibrium, according to equation$${\displaystyle M_z(t)=M_{z,\mathrm{e}\mathrm{q}}-\left[M_{z,\mathrm{e}\mathrm{q}}-M_z(0)\right]e^{-t/T_1}}$$ Or, for the specific case that${\displaystyle M_z(0)=-M_{z,\mathrm{e}\mathrm{q}}}$$${\displaystyle M_z(t)=M_{z,\mathrm{e}\mathrm{q}}\left(1-2e^{-t/T_1}\right)}$$

It is thus the time it takes for the longitudinal magnetization to recover approximately 63% [1-(1/e)] of its initial value after being flipped into the magnetic transverse plane by a 90° radiofrequency pulse.

Nuclei are contained within a molecular structure, and are in constant vibrational and rotational motion, creating a complex magnetic field. The magnetic field caused by thermal motion of nuclei within the lattice is called the lattice field. The lattice field of a nucleus in a lower energy state can interact with nuclei in a higher energy state, causing the energy of the higher energy state to distribute itself between the two nuclei. Therefore, the energy gained by nuclei from the RF pulse is dissipated as increased vibration and rotation within the lattice, which can slightly increase the temperature of the sample. The namespin–lattice relaxation refers to the process in which the spins give the energy they obtained from the RF pulse back to the surrounding lattice, thereby restoring their equilibrium state. The same process occurs after the spin energy has been altered by a change of the surrounding static magnetic field (e.g. pre-polarization by or insertion into high magnetic field) or if the nonequilibrium state has been achieved by other means (e.g.,hyperpolarization by optical pumping).^{[citation needed]}

The relaxation time,T₁ (the average lifetime of nuclei in the higher energy state) is dependent on thegyromagnetic ratio of the nucleus and the mobility of the lattice. As mobility increases, the vibrational and rotational frequencies increase, making it more likely for a component of the lattice field to be able tostimulate the transition from high to low energy states. However, at extremely high mobilities, the probability decreases as the vibrational and rotational frequencies no longer correspond to the energy gap between states.

Different tissues have differentT₁ values. For example, fluids have longT₁s (1500-2000 ms), and water-based tissues are in the 400-1200 ms range, while fat based tissues are in the shorter 100-150 ms range. The presence of strongly magnetic ions or particles (e.g.,ferromagnetic orparamagnetic) also strongly alterT₁ values and are widely used asMRI contrast agents.

Magnetic resonance imaging uses the resonance of the protons to generate images. Protons are excited by a radio frequency pulse at an appropriate frequency (Larmor frequency) and then the excess energy is released in the form of a minuscule amount of heat to the surroundings as the spins return to their thermal equilibrium. The magnetization of the proton ensemble goes back to its equilibrium value with an exponential curve characterized by a time constantT₁ (seeRelaxation (NMR)).^{[citation needed]}

T₁ weighted images can be obtained by setting shortrepetition time (TR) such as < 750 ms andecho time (TE) such as < 40 ms in conventionalspin echo sequences, while in Gradient Echo Sequences they can be obtained by using flip angles of larger than 50^o while setting TE values to less than 15 ms.

T₁ is significantly different betweengrey matter andwhite matter and is used when undertaking brain scans. A strongT₁ contrast is present between fluid and more solid anatomical structures, makingT₁ contrast suitable for morphological assessment of the normal or pathological anatomy, e.g., for musculoskeletal applications.

Spin–lattice relaxation in the rotating frame is the mechanism by whichM_xy, the transverse component of the magnetization vector, exponentially decays towards its equilibrium value of zero, under the influence of aradio frequency (RF) field innuclear magnetic resonance (NMR) andmagnetic resonance imaging (MRI). It is characterized by the spin–lattice relaxationtime constant in the rotating frame,T_1ρ. It is named in contrast toT₁, thespin-lattice relaxation time.^[2]

T_1ρ MRI is an alternative to conventionalT₁ andT₂ MRI by its use of a long-duration, low-powerradio frequency referred to as spin-lock (SL) pulse applied to the magnetization in the transverse plane. The magnetization is effectively spin-locked around an effectiveB₁ field created by the vector sum of the appliedB₁ and any off-resonant component. The spin-locked magnetization will relax with a time constantT_1ρ, which is the time it takes for the magnetic resonance signal to reach 37% (1/e) of its initial value,${\displaystyle M_{xy}(0)}$. Hence the relation:${\displaystyle M_{xy}(t_{\mathrm{S}\mathrm{L}})=M_{xy}(0)e^{-t_{\mathrm{S}\mathrm{L}}/T_{1\rho }}}$, wheret_SL is the duration of the RF field.

T_1ρ can be quantified (relaxometry) bycurve fitting the signal expression above as a function of the duration of the spin-lock pulse while the amplitude of spin-lock pulse (γB₁~0.1-few kHz) is fixed. QuantitativeT_1ρ MRI relaxation maps reflect the biochemical composition of tissues.^[3]

T_1ρ MRI has been used to image tissues such as cartilage,^[4][5] intervertebral discs,^[6] brain,^[7][8] and heart,^[9] as well as certain types of cancers.^[10][11]

• Relaxation (NMR)
• Spin–spin relaxation time
• Ernst angle

• McRobbie D., et al.MRI, From picture to proton. 2003
• Hashemi Ray, et al.MRI, The Basics 2ED. 2004

• Magnetic resonance imaging
• Nuclear magnetic resonance

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