Interaction Hamiltonian nuclear

Chemical shift Hamiltonian Dipolar interaction Hamiltonian Nuclear-nuclear interaction Hamiltonian Quadrupolar interaction Hamiltonian... 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

The electron coupled interaction of nuclear magnetic moments with themselves and also with an external magnetic field is responsible for NMR spectroscopy. Since the focus of this study is calculation of NMR spectra within the non-relativistic framework, we will take a closer look at the Hamiltonian derived from equation (76) to describe NMR processes. In this regard, we retain all the terms, which depend on nuclear magnetic moments of nuclei in the molecule and the external magnetic field through its vector potential in addition to the usual non-relativistic Hamiltonian. The result is... 

When the electric field gradient at the nucleus exerted by the electrons is nonzero, the nuclear levels will be split. The eigenvalues of the quadrupolar interaction Hamiltonian are given by... 

The events related to the suspace PaP > would correspond to direct collisions between reactants that may either leave the system as it is or it may interconvert into products via the interaction hamiltonian W. The component Q LP > contains events involving states in the orthogonal complement that includes supermolecule states. To populate these states require physical excitation processes these states may have finite lifetimes. The Q-component is called the time-delaying component by Feshbach in the context of nuclear reactions [29]. The Q T > component is given by... 

In our subsequent development we shall take the origin of coordinates to be at the centre of mass of the two nuclei, although we could equally well have chosen the molecular centre of mass as origin. Setting aside the translational motion of the molecule, we use equation (2.28) to represent the kinetic energy of the electrons and nuclei. To this we add terms representing the potential energy, electron spin interactions, and nuclear spin interactions. We subdivide the total Hamiltonian Xx into electronic and nuclear Hamiltonians,... 

The Zeeman interaction and nuclear hyperfrne terms have to be added to this effective Hamiltonian. The Zeeman terms are, as in previous examples,... 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

We now turn to the last of the tree main magnetic interactions, the nuclear Zeeman term. The general approach (7) is to divide the spin Hamiltonian into two distinct parts ... 

In the case of dipole-dipole interactions between nuclear spins, the Hamiltonian can be separated into an uncorrelated product of a spin part, and the dipole-dipole interaction tensor. 

Although we are not primarily interested in peiramagnetic resonance here, it is worth pointing out that the nuclear hyperfine interaction may be treated in an exactly analogous manner. If the nuclear interaction Hamiltonian is... 

We have just examined the rotational-electronic interaction and the external field-electronic interaction with the resultant rotational Hamiltonian in Eq. (6), which is valid in the absence of nuclear-electronic interactions. We now add in the effects of the nuclear-electronic interactions that give rise to the spin-rotation interaction and nuclear magnetic shielding. 

If the carrier frequency of the RF pulses is close to one of the resonance transitions of the irradiated sample, the effect of the multi-pulse sequence leads to the establishment of a new time independent effective Hamiltonian Heff, now also containing the interaction of nuclear-spins with the RF field (that the effective Hamiltonian is time independent should be imderstood that it is independent of the number of the pulse interval but not independent of the time inside the definite pulse interval). A part of the dipole Hamiltonian Ha no longer commutes with the new effective Hamiltonian, which after the T2 time results in heat mixing of the quadrupole reservoir and the reservoir of the components of... 

Because the temperature of a realistic nuclear spin system is not at zero degrees absolute, the internal interaction Hamiltonians always contain two parts one is stationary or coherent, the other fluctuating or incoherent, or random. The former part usually determines the positions of the peaks in a spectrum, the latter part governs the dynamics of the system. However, they may become entangled under certain conditions. For spin-1/2 systems, r.f. interactions can be made larger than the internal interactions in most cases, thus the manipulation of the interactions with r.f. pulses is realizable. This is an advantage of NMR over many other spectroscopies. In fact, most experimental methods in NMR spectroscopy correspond to certain manipulations of the internal interactions. For quadrupolar spin systems, the internal... 

The predominant isotope of cesium is Cs which has a nuclear spin /] of 7/2 its quadrupole moment and g-factor will be denoted by Q and gi. The F nucleus has spin /2 of 1 /2 (and therefore no quadrupole moment) and a nuclear g-factor denoted gi The nuclear hyperfine Hamiltonian used by Enghsh and Zorn [51] was the sum of five terms representing the Cs quadrupole interaction, the Cs nuclear spin-rotation interaction, file nuclear spin-rotation interaction, the dipolar (tensorial) interaction between the Cs and F nuclear spins, and the scalar interaction between the two nuclear spins. Consistent with the conventions in use at the time, this Hamiltonian was written in the following form ... 

Xemr can calculate (ESR) EPR transitions using the first order simulation or the solution of fully numerical spin Hamiltonian. In the latter case the numerical transition moments can also be calculated. The first order simulation is restricted to 5 = Vi and to electron Zeeman and hyperfine interaction whereas the numerical method can handle electron and nuclear Zeeman, hyperfine interaction, electron-electron interaction, and nuclear quadrupole interaction. The latter method can simulate both (ESR) EPR and ENDOR spectra. In addition a simple 1st order ENDOR simulation is also possible, so that the parameters can be extracted from the ENDOR spectra with better accuracy. 

The terms 7 2, 3, H4 (crucial for the NMR experiment) correspond to the magnetic dipole-dipole interaction involving nuclear spins (the term H5 of the Breit Hamiltonian) the classical electronic spin - nuclear spin interaction (7 2) plus the corresponding Fermi contact term (Tia) and the classical interaction of the nuclear spin magnetic dipoles... 

The through space dipole-dipole nuclear magnetic moment interaction (the nuclear analog of the Hs term in the Breit Hamiltonian) 5Za[Pg.768]

In quadrupolar nuclei, the situation differs notably the quadrupolar interaction only affects spins with I>% and is created by electric field gradient resulting from the asymmetry of charge distribution around the nucleus of interest. The quadrupolar interaction is characterized by the nuclear quadrupolar coupling constant Cq (from 0 in symmetrical environments to tens or hundreds of MHz) and an asymmetry parameter T]q. NMR spectra are usually recorded when Cq Vl the Larmor frequency of the quadrupolar spin. In such a case, the NMR spectrum can easily be simulated First, the first-order quadrupolar Hamiltonian, which is the quadrupolar interaction Hamiltonian truncated by the Larmor frequency, has to be taken into account. The first-order quadrupolar interaction (or the zeroth-order term in perturbation theory) is an inhomogeneous interaction and is modulated by MAS and does not affect symmetrical transition —m +m. Therefore, in half-integer spins, the single-quantum central transition (CT, i.e., —1/2 +1/2) is not affected by the first-order quadrupolar inter-... 

EFG (a property of a sample) is called the qnadrupole interaction. The nuclear quadrupole coupling is expressed by the Hamiltonian... 

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