Gyromagnetic equation

A. Ferromagnetic Materials The Purpose of this Review The Gyromagnetic Equation... 

We may summarize the contents of this chapter in more detail as follows. In Section I we demonstrate how the explicit form of Gilbert s equation describing Neel relaxation may be written down from the gyromagnetic equation and how, in the limit of low damping, this becomes the Landau-Lifshitz equation. Next the application of this equation to ferrofluid relaxation is discussed together with the analogy to dielectric relaxation. 

In the case of proton-proton interactions, both nuclei S and /will have the same gyromagnetic ratios, and an implication of the Equation (4) then is that there is an upper limit of 50% on the nOe obtainable, whatever the distance between nuclei S and I. This means that the observation of an nOe between two nuclei does not necessarily mean they are spatially close to one another, and nOe results must therefore be interpreted with caution. Similarly, as will be seen later, the absence of nOe between two nuclei does not necessarily mean they are far apart. In the case of heteronuclear nOe, since the gyromagnetic ratio of proton (yv) is four times the gyromagnetic ratio of carbon y,), js/ jt can be four times greater than that obtainable in homonuclear nOe. 

This equation is called the Curie law and relates the equilibrium magnetization M0 to the strength of the magnetic field B0. The constants have the following meaning I is the nuclear spin quantum number (see below), y is the gyromagnetic ratio specific for a given isotope, h is Planck s constant, kB is Boltzmann s constant, N is the number of nuclei and T is the temperature. 

Here, S0 is the signal when G = 0, D is the self-diffusion coefficient, yG is the gyromagnetic ratio and am are roots of the Bessel function equation amaf3 2(ama) — (l/2)J3/2(a ma) = 0. If the system is polydisperse, the signal decay is due to contributions from droplets of different sizes. Then, the signal attenuation is given by the volume average over all sizes as... 

The energy difference between these levels is very small, which means that the population difference is also small. The NMR signal arises from this population difference and hence the signal is also very small. There are several factors which influence the population difference and these include the nature of the nucleus (its gyromagnetic ratio ) and the strength of the magnetic field that they are placed in. The equation that relates these factors (and the only one in this book) is shown here ... 

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Since the microwave frequency and gyromagnetic ratio are constant, changes in the applied stress will result in a shift in the resonant magnetic field. Indeed, the shift equation is... 

These equations can be also expressed as functions of the strength H of the main magnetic field by using a relation of = yH, were y is the nuclear gyromagnetic ratio. Such expressions may be more useful for the usual broad-line NMR spectrometry in which the main field is slowly swept under a constant rotating subfield. 

In this equation, AEe is the approximated electronic excitation energy yc and yx are the gyromagnetic ratios of the coupled nuclei, 13C and X pp is the Bohr magneton, h the Planck constant. 

If the primary isotopic effect is neglected, very accurate values may be obtained for the gyromagnetic constant ratio y("7Sn)/y(119Sn) [equation (10)1 from the ratio of the tin resonance frequencies, vTMs(U7Sn)/vTMs(U9Sn)- The ratios of resonance frequencies measured for pairs of tin isotopes with different accuracies by various authors are compared in Table XIV. 

The Overhauser effect has been widely employed as an NMR analysis method in many disciplines ranging from medical to chemical sciences, and broadly refers to the motion-mediated transfer of spin polarization from a species with a higher gyromagnetic ratio (y) to one with a lower gyromagnetic ratio. Because molecular motion is critical for efficient transfer, the Overhauser effect is most commonly observed in liquid samples. The Overhauser effect can be divided into two categories the nuclear Overhauser effect (NOE), where both species are nuclear spins, and Overhauser DNP, where the higher y spin is an unpaired electron. As Overhauser DNP is the focus of this review, some of the terminology and equations are specific to the Overhauser DNP effect. 

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