Precessional motion

An angular momentum vector L, associated with a magnetic moment will precess in a magnetic field, as illustrated in Fig.2.3. Frequently we encounter this phenomenon in the description of atomic and molecular processes. To demonstrate the phenomenon, imagine magnetic poles q in analogy with the electrical case, for which we have the same mathematical description. For the mechanical moment M we then have 

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F tre 1.3 Precessional motion of an NMR active nucleus in magnetic field B. ... 

C. The Precessional Motion The proton appears to be behaving as spinning magnet and therefore, not only can it align itself with or oppose an external field, but also may move in a characteristic manner under the influence of the external magnet. 

To investigate multispin systems, the so-called electron spin transient nutation (ESTN) spectroscopy is recently elaborated. This is a version of pulsed ESR. Nutation is the precessional motion of spin. The method and its applications are detailed in the paper of Itoh et al. (1997). Chapters 1 and 8 describes that the determination of spin multiplicity becomes a very important problem in organic chemistry of ion-radicals. 

The precessional motion can be maintained by a suitable radio frequency field superimposed on the steady field. For example, in Fig. 9.38(b), when a steady field Hz is applied along the z axis and a radiofrequency field //,., is applied in the x-y plane and rotates in the same sense and at the same frequency as the precession, resonance occurs. Gyromagnetic resonance as outlined above is in principle the same as ferrimagnetic resonance referred to earlier (Section 9.3.1), except that in the former case the material is magnetically saturated by a strong applied field. In practice the steady field, which determines the Larmor frequency, is made up of the externally applied field, the demagnetizing field and the anisotropy field, and is termed the effective field He. Figure 9.39 shows the He values at which resonance occurs in some of the important communications and radar frequency bands. 

Fig. 9.38 Precessional motion of magnetization (a) Ms spiralling into line with H as the precessional energy is dissipated (b) precession maintained by an applied radio frequency field.

Derive an expression for the angular frequency of the precessional motion of an electron situated in a magnetic field. 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

This precessional motion causes the tip of the magnetic moment vectors (either up or down) to trace out a circular path, as shown in Figure 2.4. Note also that the precession fre-... 

Frequently in this book we will want to depict nuclear spin orientations like those shown in Figure 2.7. More often than not, we will focus our attention on the net nuclear magnetic moment (M) rather than on the individual nuclear spins. Because M will sometimes precess around B0 (i.e., the z axis), we need a more convenient way than the dashed ellipses used so far to depict M as it precesses and changes orientation. Henceforth we will use another convention to represent this precessional motion of M, the rotating frame of reference, which is designed to show the effects of B on M. 

Acremann Y, Back CH, Buess M, Portmaim O, Vaterlaus A, Pescia D, Melchior H (2000) Imaging precessional motion of the magnetization vector. Science 290 492-495 Aharoni A (1969) Effect of a magnetic field on the superparamagnetie relaxation time. Phys Rev 177 793-796... 

The precessional motion of the magnetic moment around Bq occurs with angular frequency wq, called the Larmorfrequency, whose units are radians per second (rad s ). As Bq increases, so does the angular frequency that is, coq cx Bq, as is demonstrated in Appendix 1. The constant of proportionality between o>o and Bq is the gyromagnetic ratio 7, so that wq = Bq. The natural precession frequency can be expressed as linear frequency in Planck s relationship AE = Hvq or angular frequency in Planck s relationship AE = h(x)Q (coq = 2 rrvo). In this way, the energy difference between the spin states is related to the Larmor frequency by the formula... 

More recently, Gilbert (1955) [6,7] proposed an equation describing the dynamic behavior of M which incorporated the collision damping incurred by the precessional motion, in an effective damping field term. He assumed that the damping field is... 

The last term in Gilbert s equation above is the aligning term and consequently the term of interest in relaxation of the magnetization with respect to a magnetic field. For low damping (a< l), we see that precessional motion is the dominant motion and that the relaxation time is approximately given by... 

We see that in both the limits a O and a->°o, the relaxation time approaches infinity and alignment does not take place. This is due in the former case to infinitely persisting undamped precessional motion and in the latter to the absence of all motion as a result of the high damping. The minimization of the Neel relaxation time was a goal of Kikuchi s 1956 paper [23] and clearly occurs at a = 1 where neither motion is dominant and... 

The arrows indicate the possible transitions, i.e. give the positions of the lines in the Zeeman effect. Here the selection rules for the magnetic quantum number m must be taken into account. These can be deduced from the correspondence principle in exactly the same way as at p. 110 (see also Appendix XXI, (p. 308)). As m denotes the precessional motion about the direction of the field, the transition Am = +1 corresponds to the classical vibrations at right angles to H ... 

Fig. 3.2 If the magnetization vector is tilted away from the z axis it executes a precessional motion in which the vector sweeps out a cone of constant angle to the magnetic field direction. The direction of precession shown is for a nucleus with a positive gyromagnetic ratio and hence a negative Larmor frequency.

In the laboratory frame, the Y component corresponds to a magnetization vector rotating in the XY plane. The magnetization vector rotates in the XY plane because the individual nuclear magnetization vectors are processing about Z (the principal field axis). Before the pulse, individual nuclei have random precessional motions and are not in phase. The pulse causes phase coherence to develop, so that all of the vectors process in phase (see Fig. 10.7). Because all of the individual vectors process about the Z axis, M, the resultant of all of these vectors, also rotates in the XY plane. 

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