NMR Relaxation Contents CONTENTS ................................................................................................................................................. 1 RELAXATION .............................................................................................................................................. 1 LONGITUDINAL RELAXATION................................................................................................................................ 1 DECOHERENCE AND TRANSVERSE RELAXATION ........................................................................................................ 1 BLOCH EQUATIONS ........................................................................................................................................... 2 RELAXATION DUE TO VIBRATIONS AND ROTATIONS. ................................................................................................. 3 RELAXATION DUE TO PARAMAGNETIC IMPURITIES OR SUPERPARAMAGNETIC PARTICLES. ................................................. 4 FIELD INHOMOGENEITY ...................................................................................................................................... 4

Relaxation Spin interactions induce a coupling of spins between themselves and with the external world. The relaxation is the way taken by the spin system to go back to its equilibrium after a perturbation. We can separate two kind of relaxation. The longitudinal relaxation which characterize the evolution of the magnetization in the direction of the applied magnetic field and the transverse relaxation which characterize the evolution of the components of the magnetization perpendicular to the magnetic field. The main difference between these two relaxations is that the longitudinal relaxation corresponds to an exchange of energy with the external world so it is a real relaxation in terms of thermal exchanges. The evolution of the transverse components does not change the macroscopic energy of the spin system and characterize the internal evolution of the system to reach its equilibrium.

Longitudinal relaxation Longitudinal relaxation time is called T1. It implies energy exchange to another reservoir which can be the lattice, the vibration-rotation spectrum of the molecules or the paramagnetic impurities system. One main characteristic is the small coupling which induces T1 of the order of 1s. In liquids and in vivo, coupling with motions is the main source of relaxation.

Decoherence and transverse relaxation The decrease of the transverse components of the magnetization is different because it does not imply exchange of energy. In fact, we can distinguish two main effects: an effect of relaxation which is often related to the longitudinal relaxation and an effect of decoherence. If the different spins of the system are experiencing slightly different fields, there is a very little difference in their precession speed but it is enough to create a decoherence between spins and a decrease of the transverse magnetization. If this effect is created by the field homogeneity, one call T2* the apparent relaxation

time but it is not a relaxation process and there is usually a not exponential decrease but in time just of fourier transform of the field homogeneity distribution. This decoherence can have also internal sources like magnetic contents, shape of the system, partly solid systems. In solids, we forget the notion of transverse relaxation time which is meaningless and we speak about line widths and line shapes and the absorption signal which is far more relevant. The decrease in time of the transverse magnetization is then just the Fourier transform of the absorption line.

Bloch equations Felix Bloch has introduced phenomenological equations derived from Larmor equation to describe the motion of the magnetization ೣ ೣ ‫ ۓ‬ௗ௧ ൌ ߛൣ‫ܯ‬௬ ሺ‫ݐ‬ሻ‫ܤ‬௭ ሺ‫ݐ‬ሻ െ ‫ܯ‬௭ ሺ‫ݐ‬ሻ‫ܤ‬௬ ሺ‫ݐ‬ሻ൧ െ ்మ ۖௗெ ሺ௧ሻ ெ ሺ௧ሻ ೤ ൌ ߛሾ‫ܯ‬௭ ሺ‫ݐ‬ሻ‫ܤ‬௫ ሺ‫ݐ‬ሻ െ ‫ܯ‬௫ ሺ‫ݐ‬ሻ‫ܤ‬௭ ሺ‫ݐ‬ሻሿ െ ೤்
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Figure 6: 3D view of the evolution of magnetization and time evolution of their intensity.

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Relaxation due to vibrations and rotations. Bloembergen-Purcell-Pound Pound theory (BPP theory) gives the relaxation constant of a pure substance taking into account the effect of tumbling motion of molecules on the local magnetic field disturbance1. The relevant parameter is the correlation time of the molecular tumbling motion, τc. The initial model gives :

Where K is varying as

and

Taking for example the H2O molecules in liquid phase, and neglecting the 17O effect, the value of K is 1.02×1010 s-2 . We can then estimate using τc = 5×10-12 s: (dimensionless) We obtain independent of the frequency because τc is very small. More sophisticated theories take into account the nature of motion or the distribution of correlation times2. In vivo, a lot of motion involve large molecules and hence are slower correlation times comparable mparable to the inverse of the larmor frequency. The measurement of relaxation times as function of field is called relaxometry and is largely developed by using field cycling methods.

1

Bloembergen, E.M. Purcell, R.V. Pound "Relaxation Effects in Nuclear Magnetic Resonance Absorption" Physical Review (1948) v73. 7:679-746 746 2 Kubo et Tomita, J. Phys. Soc.. Jap. 9, (1954) p888, Davidson D.W., Cole R.H., J.Chem. Phys. 19, (1951) p 1484.

Figure 7: NMR profiles of 5% vol/vol calf cortical a-crystallin at 5 (A), 25 (0), and 35°C (-).3

Relaxation due to paramagnetic impurities or superparamagnetic particles. Paramagnetic impurities have a moment typically 100 times larger than the nuclear moment. Hence, they are creating a very strong local fluctuating field. The fluctuations are just thermal fluctuations as they are isolated and the corresponding correlation time is rather short As they are coupled to the nuclei by dipolar interaction, they can induce spin flips of nuclei at each fluctuation. This mechanism is very efficient. For example, 1% of Cu paramagnetic ions in water reduce the relaxation time from 4s to 100ms. Super-paramagnetic particles like ferrites or Gd based nanoparticles are very efficient to speed up the relaxation because they have a much bigger moment than an isolated paramagnetic ion and they still fluctuate rather fast.

Field inhomogeneity The field homogeneity should be related to the experiment targeted. We have seen that the extent of proton signal is about 20ppm and the separation of multiplets is much smaller than 0.1ppm. This implies that if one wants to do spectroscopy, the field homogeneity should be of this order. In high resolution magnets, shimming up to the fourth order are applied to achieve resolution of 10-8 to 10-9.

3

INTERMOLECULAR PROTEIN INTERACTIONS IN SOLUTIONS OF CALF LENS ALPHA-CRYSTALLIN - RESULTS FROM 1/T1 NUCLEAR MAGNETIC-RELAXATION DISPERSION PROFILES Author(s): KOENIG, SH; BROWN, RD; SPILLER, M, et al. Source: BIOPHYSICAL JOURNAL Volume: 61 Issue: 3 Pages: 776-785 Published: MAR 1992

In MRI, the problem is different if we exclude the MRSI approach, we do not need the high resolution necessary for spectroscopy. In practice, it is useless to have a field homogeneity far better than the smallest line width of the tissues, of the order of 2-4 Hz. For low field MRI, this is reasonably easy to achieve. If the field homogeneity is a limitation, spin echo can be used as we’ll see later.