Rotating vectors

At equilibrium, the transverse magnetization equals zero. A net magnetization vector rotated off the -axis creates transverse magnetization. 

The second equation merely shows that the radius vector rotates with a constant angular velocity. As to (6-14) it is integrated by the standard procedure which gives ... 

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

In pulsed NMR, the magnetic field is turned on for the time necessary to rotate the magnetization vector into a plane called the 90° rotation or 90° pulse. The field is turned off and the magnetization vector rotates at a nuclear precession frequency relative to the coil. This induces an NMR signal that decays with time as the system returns to equilibrium. This signal is called the free induction decay (FID). 

The radiofrequency, or frequency with which the spin vector rotates around the central axis Oz that induces a change in states, is called the Larmor precession frequency. The Irish physicist Larmor, whose work preceded NMR, has shown independently that to, the angular rotation frequency around a central axis Oz, has a value of ... 

The interaction of a nucleus with the oscillating magnetic field B0, created by the electromagnetic wave, can be understood if it is assumed that it results from the composition of two half vectors, rotating in opposite directions in the xOy plane with identical angular velocities (Fig. 9.6). The vector rotating in the same direction as the precession is the only one that can interact with the nucleus. 

Circular dichroism arises from the same optically active transitions responsible for the Cotton effects observed in ORD curves, but unlike ORD it is an absorption, not a dispersion, phenomenon. Hence, the CD effect is restricted to the region of the transition and can be interpreted more straightforwardly. Both ORD and CD can best be understood if one imagines the incident plane-polarized beam resolved into two in-phase circularly polarized beams whose vectors rotate in opposite directions. A difference in index of refraction between the left and right circularly polarized beams results in rotation of the transmitted plane polarized beam while differential absorption of the two circularly polarized beams results in depolarization of the transmitted beam, so that an incident plane-polarized beam whose frequency is within that of an optically active absorption band becomes both rotated and elliptically polarized upon passage through the sample. This depolarization effect is CD, and the measured parameter is (et — er), the difference in extinction coefficient between the left and right circularly polarized beams. The data is usually recorded as the specific ellipticity, defined as ... 

In a general hydrodynamic system, the vorticity w is perpendicular to the velocity field v, creating a so-called Magnus pressure force. This force is directed along the axis of a right-hand screw as it would advance if the velocity vector rotated around the axis toward the vorticity vector. The conditions surrounding a wing that produce aerodynamic lift describe this effect precisely (see Fig. 2). 

Turning now to the case of polarization/rotation modulation, or continuous rotation of iT i 2rl2 corresponding to a continuous rotation of Cj through 20, there is a rotation of the resultant through 0. This correspondence is a consequence of the A 1A = 7 relation, namely, that if the unitary transformation of A or A 1 is applied separately the identity matrix will not be obtained. However, if the unitary transformation is applied twice, then the identity matrix is obtained and from this follows the remarkable properties of spinors that corresponding to two unitary transformations of, for example, 27i, namely, 471, one null vector rotation of 271 is obtained. This is a bisphere correspondence and is shown in Fig 3b. This figure also represents the case of polarization/rotation modulation—as opposed to static polarization/rotation. 

FIGURE 6.2 Trajectories of the electric vector when the electromagnetic wave travels along the r-axis. Projections of the trajectories on the xy plane are a line (a) corresponding to oscillations in a plane and a circle (b) when the vector rotates around the z-axis. 

There is a simple and well-known model for the description of DNMR phenomenon, provided no scalar coupling is involved.11 In Section 3.2.1, the macroscopic magnetisation vector of the vector model gives the detected signal (the FID). This vector rotates in the xy plane perpendicular to the direction of the external magnetic field during the detection its frequency is determined by the shielding effect of the chemical environment of the nuclei. 

If, after the material has been magnetically saturated to the value Bs, the field is reduced to zero, the magnetization vectors rotate out of line with the field towards the nearest preferred direction which is determined in part by magnetocrystalline anisotropy. The magnetization is thus prevented from complete relaxation to the virgin curve and hence, for zero field, there is a remanent induction Bx. In order to reduce the induction to zero a reverse field II,. has to be applied. The coercive field or coercivity II,. depends in part on crystalline anisotropy, as might be expected. 

---