The 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. 

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. 

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 ... 

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. 

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. 

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

Finally, local field fluctuations along z for the first spin can also be caused by fluctuations of the xy component of the second spin. This originates a term of the type of 7(0) but offset by the precessional frequency of the second spin, (02 ... 

The precessional frequency (g>0) of these aligned nuclei is, in addition to the quantum mechanical limitations, also controlled by the magnetogyric ratio (y) of the nucleus being studied. The precessional frequency w0 is referred to as the Larmorfrequency. 

The fundamental principle governing the NMR technique centres on the induction of transitions between different nuclear Zeeman levels of a particular nucleus. To cause these transitions, a variable radiofrequency (RF), referred to as B, acts perpendicular to the applied magnetic field (B0), which is causing the nuclear alignments. When the frequency of the applied RF is identical to the precessional frequency (w0) of the nuclei being observed, a transition between nuclear spin states occurs. 

When an applied rf (vL) is equal to the precessional frequency of the equivalent protons (Larmor frequency in MHz), the state of nuclear magnetic resonance is attained, and the basic NMR relationship can be written ... 

Thus, as we expected from Chapter 1, for resonance to occur, the radiation frequency must exactly match the precessional frequency. 

Remember from Section 2.3 that to tip M off the z axis, so it has a component in the x,y plane, we need an irradiating magnetic field (B[) that oscillates at exactly the precessional frequency of the nuclei of interest and is oriented perpendicular to B0. How are we going to generate such a precessing magnetic field ... 

The precessional frequency (v) of an unpaired electron is directly proportional to the applied magnetic field strength (B) and can be expressed by the equation... 

Passage of a brief but powerful pulse of rf current through the transmitter coil (whose axis lies in the xy plane) gives rise to a secondary oscillating magnetic field (Bi) perpendicular to and precessing around B0. The frequency of B is set to match the precessional frequency of the specific isotope (e.g., I3C) to be observed. 

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