Precessional frequency

The equation describes the manner in which the nuclear magnetization, M, at position r and time t processes about the magnetic flux density, B, in which it is found. The constant y is the magnetogyric ratio of the nuclides under study. The precessional frequency, co, is given by the Lannor equation. 

By examining the expression for Q ( equation (B1.16.4)). it should now be clear that the nuclear spin state influences the difference in precessional frequencies and, ultimately, the likelihood of intersystem crossing, tlnough the hyperfme tenn. It is this influence of nuclear spin states on electronic intersystem crossing which will eventually lead to non-equilibrium distributions of nuclear spin states, i.e. spin polarization, in the products of radical reactions, as we shall see below. 

In this simple case, there are just two nuclear spin states, a and (3. Equation (1.16.5) shows the calculation of the difference in electron precessional frequencies, Q, for nuclear spin states a (equation (B 1.16.5a)) and (3 (equation (B1.16.5h)). 

This velocity is called the Larmor precessional frequency. 

Now suppose that an additional small magnetic field is applied perpendicular to Ho in the plane formed by pi and Hq, call this field (see Fig. 4-4B). Field Hi will act upon pi to increase the angle 6. If field Hy is caused to rotate around Ho at the Larmor precessional frequency of Wq, the torque produced will steadily act to change the angle 6. On the other hand, if the frequency of rotation of Hi is not the same as the precessional frequency, the torque will vary depending upon the relative phases of the two motions, and no sustained effect will be produced. 

V (in Hz) by w = l-nv, comparison of Eqs. (4-43) and (4-44) shows that the Larmor precessional frequency is identical to the transition frequency that was calculated earlier. 

In the earlier treatment we reached the conclusion that resonance absorption occurs at the Larmor precessional frequency, a conclusion implying that the absorption line has infinitesimal width. Actually NMR absorption bands have finite widths for several reasons, one of which is spin-lattice relaxation. According to the Heisenberg uncertainty principle, which can be stated... 

In the presence of a field H, rotating at the precessional frequency the nuclear system can absorb energy, following which nuclear relaxation occurs. Thus, the equation of motion must include both the precessional and the relaxation contributions ... 

In Eq. (4-62) Wq is the Larmor precessional frequency, and Tc is the correlation time, a measure of the rate of molecular motion. The reciprocal of the correlation time is a frequency, and 1/Tc may receive additive contributions from several sources, in particular I/t, where t, is the rotational correlation time, t, is, approximately, the time taken for the molecule to rotate through one radian. Only a rigid molecule is characterized by a single correlation time, and the value of Tc for different atoms or groups in a complex molecule may provide interesting chemical information. 

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. 

After the 90° pulse is applied, all the magnetization vectors for the different types of protons in a molecule will initially come to lie together along the y -axis. But during the subsequent time interval, the vectors will separate and move away from the y -axis according to their respective precessional frequencies. This movement now appears much slower than that apparent in the laboratory frame since only the difference between the... 

Why in a decaying signal (FID) does the amplitude decay asymptotically toward zero while the precessional frequency remains unchanged ... 

This is because the precessional frequency of a nucleus depends on the magnetic field, and as the magnetic field varies, so will the precessional frequency. 

Measurements of the precessional frequencies in high field as a function of crystal orientation allows one to extract the hyperfine parameters and... 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

Fig. 11. (top) The field dependence of the Mu and Mu precessional frequencies in diamond on a magnetic field applied along the (110) direction, (middle and bottom) The Mu amplitudes as a function of field measured at T = 454 K and 494 K respectively. The resonant maximum at B establishes the signs of the Mu hyperfine parameters relative to Mil. (B in this figure is equivalent to H in the rest of this chapter.) From Odermatt et al. (1988). 

D. The Precessional Frequency The spinning frequency of the nucleus does not change at all, whereas the speed of precession does. Therefore, v °= Bo, i.e., the precessional frequency is directly proportional to the strength of the external field Bo. 

Table 23.1 Precessional Frequencies as a Function of Increasing Field Strength...

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