Thermal equilibrium magnetization

B1.15.2.2 THERMAL EQUILIBRIUM, MAGNETIC RELAXATION AND LORENTZIAN UNESHAPE... 

So far the magnetization has been treated for a single NMR frequency Hq without consideration of the chemical shift and a space-dependent spin density. The amplitude Afo of the thermal equilibrium magnetization is the sum over all magnetization components which differ in space and frequency. 

We now consider an example to illustrate the various methods for evaluating discussed above. The figure at left shows the usual decay curve for protons in (CHg NB Hg at room temperature and 30.6 MHz. The straight line is a best fit by eye and leads to a T of 1.4 seconds. Eight more data points were taken at different delays past the last point shown to try to determine the thermal equilibrium magnetization Mq. Since the points plotted are 1-M(t)/MQ, Mq should have been determined as accurately as M(t) so we expect the uncertaintly in to be the major uncertainty in Tj. 

Therefore, the effect resulting of the application of a static magnetic field is the appearance of a nuclear magnetization parallel to that field. The thermal equilibrium magnetization for an ensemble of nuclei with 7 = 1/2, is given by [4,7] ... 

It is helpful to define the ratio A = which is a dimensionless parameter that measures the deviation of populations from a uniform value due the application of the static magnetic field. It is this parameter (typically around 10 ) that determines the magnitude of the thermal equilibrium magnetization characteristic of nuclear paramagnetism. 

In the absence of an external magnetic field, the 21 + 1 energy states of a nucleus are of identical energy (they are said to be degenerate) and, therefore, are equally populated at thermal equilibrium in any assemblage of such nuclei. In the presence of an applied steady field Ho, these 21 + states will assume different energy... 

Figure 4-6 illustrates the relaxational eontribution to the motion. Figure 4-6A shows moment vectors for a spin system in the absenee of the rf field (Hi = 0) the magnetization eomponents are = Mq, = 0, My = 0, beeause in the xy plane the magnetization eomponents caneel. In the presenee of the rf field at the resonanee frequency the spin system absorbs energy, increasing the angle between Ho and M and perturbing the thermal equilibrium so that and My components are induced and M < Mo (Fig. 4-6B). With the passage of time (comparable to the relaxation times Tj and Tj), relaxation back to the equilibrium configuration takes place, so M. increases toward Mo, whereas nd My decrease toward zero as a consequence of the gradual loss of coherence of the moment vectors.

When nuclei with spin are placed in a magnetic field, they distribute themselves between two Zeeman energy states. At thermal equilibrium the number (N) of nuclei in the upper (a) and lower (j8) states are related by the Boltzmann equation (1) where AE=E — Ep is the energy difference between the states. In a magnetic field (Hq), E = yhHo and... 

Here, is the magnetization of spin i at thermal equilibrium, p,j is the direct, dipole-dipole relaxation between spins i and j, a-y is the crossrelaxation between spins i and j, and pf is the direct relaxation of spin i due to other relaxation mechanisms, including intermolecular dipolar interactions and paramagnetic relaxation by dissolved oxygen. Under experimental conditions so chosen that dipolar interactions constitute the dominant relaxation-mechanism, and intermolecular interactions have been minimized by sufficient dilution and degassing of the sample, the quantity pf in Eq. 3b becomes much smaller than the direct, intramolecular, dipolar interactions, that is. 

Figure 1.13 (a) Bulk magnetization vector, M°, at thermal equilibrium, (b) Magnetization vector after the application of a radiofrequency pulse. 

---