Hamiltonian dipolar interaction

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

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

The Hamiltonian term for the electron-electron dipolar interaction is ... 

When we include the Zeeman interaction term, gpBB-S, in the spin Hamiltonian a complication arises. We have been accustomed to evaluating the dot product by simply taking the direction of the magnetic field to define the z-axis (the axis of quantization). When we have a strong dipolar interaction, the... 

We now will show that spin-orbit coupling can give a spin Hamiltonian term identical to that we obtained from the electron dipolar interaction. Consider the... 

Aniosotropic hyperfine coupling results primarily from dipolar interactions between a magnetic nucleus and an unpaired electron in a p, d, or f orbital. Such interactions give rise to a Hamiltonian... 

However, in the presence of a strong quadrupolar interaction, the second-order cross-terms between the quadrupolar interaction and the CSA [27, 28] or the heteronuclear dipolar interaction [29, 30] also have to be taken into account. For instance, upon including the CSA in the expression of the time-dependent Hamiltonian in (9), the second-order term becomes... 

In an organic solid representative broadenings are 150 ppm for aromatic carbon chemical shift anisotropy and 25 kHz (full width at half-height) for a rather strong carbon-proton dipolar interaction. At a carbon Larmor frequency of 15 MHz, the shift anisotropy corresponds to 2.25 kHz. In high magnetic fields the forms of the respective Hamiltonians are... 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

The dipolar interaction Hamiltonian under the conditions of MAS NMR is given as... 

If i and j are different nuclei (for instance 13C and H) the hetero-nuclear dipolar interaction, which is often strong, can be removed by dipolar decoupling, which consists of irradiation nucleus j (say H) at its resonance frequency while observing nucleus i (say 13C). The time-averaged value of the Hamiltonian is then zero. 

Hamiltonian Eq. (1.24) is formally equivalent to that describing the dipolar interaction between two spins s and S2 whose sum is S. Actually, in organic radicals where spin-orbit interactions are negligibly small, the dipolar interaction between the two electron spins in an S = 1 system causes ZFS. 

Contact interactions also give rise to relaxation. The perturbing Hamiltonian for contact interaction, H (last term in Eq. (III.l)), is analogous to the first term of the perturbing Hamiltonian of the dipolar interaction (see Appendix V, Eq. (V.10)) except that the part containing the ladder operators is multiplied by 1 /2 instead of — 1/4. The transition probability wq (see Fig. 3.8) is provided by (see Appendix V)... 

It is largely accepted that the dominant mechanism of nuclear spin relaxation in condensed polymers is due to dipolar interactions between the spins. The truncated homonuclear dipolar Hamiltonian has the form [15] ... 

For abundant nuclei with spin V2, the spectrum is often dominated by heteronuclear or homonuclear dipolar interactions, i.e., the interactions between the magnetic moments of two neighbouring spins. In this case there is no isotropic contribution and q is zero, so that Equation 14.1 simplifies correspondingly. For a two-spin system one obtains a spin Hamiltonian of the form ... 

There are many other terms to the Hamiltonian but for spin-1/2 nuclei in liquids they can all be ignored. The dipole-dipole (dipolar or direct coupling) Hamiltonian is important in solids and partially oriented liquids, and the quadrupolar Hamiltonian is important for spins greater than 1/2. The dipolar interaction contains a multiplier of... 

Using time-dependent perturbation theory and taking full account of the symmetry and commutation relations for the high-order dipolar Hamiltonians, Hohwy et al.61 69 gave a systematic analysis of homonuclear decoupling under sample rotation and proposed a novel approach to the design of multiple-pulse experiments. Based on the theoretical analysis, they proposed a pulse sequence that can average dipolar interaction up to the fifth order. One example of these pulse sequences is shown at the top of Fig. 3. This sequence is sufficiently powerful that it is possible to obtain precise measurement of proton chemical shift anisotropies, as shown in Fig. 3. 

A pulse scheme recovering the zero-quantum Hamiltonian was proposed by Baldus and Meier.142 It is weakly dependent on spectral parameters and a faithful measure of internuclear distances. This sequence is based on the former rotor-synchronized R/L-driven polarization transfer experiments.143,144 It uses the LG or FS-LG, which is used to decouple the high-7 spins, and combined MAS and RF irradiation of low-7 spins to decouple the hetero-nuclear dipolar interactions. With phase-inversion and amplitude attenuation in the rotating frame and refocusing pulses in the laboratory frame part of the pulse sequence, a zero-quantum average Hamiltonian can be obtained with optimum chemical-shift/offset independence. 

This interaction leads to "fine-structure" splittings in the spectra of atoms and molecules. For atoms and molecules in the S = 1 triplet state, the electron spin-electron spin dipolar interaction leads to the "D and E" fine-structure Hamiltonian. 

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