Hamiltonian, rotating-frame

If we let (p Acu i, then eqn (9) corresponds to a calculation of the effect of a frequency offset term —Awl in the rotating frame Hamiltonian. Thus a modulation by exp (iqAcDt is caused by the offset, shifting the spectrum to Aco as seen in fig. 2. 

When S is a quadrupolar nucleus the polarization matching conditions are more complex. The reason for this can be seen by considering the rotating frame Hamiltonians of the individual I and S nuclei during the application of an on-resonance x-pulse ... 

As mentioned in Section 2.3, in the absence of an rf pulse the rotating frame Hamiltonian is the sum of H0 and HE. As these two operators commute, they may be treated independently of each other. Equation (20) shows that in the absence of an rf pulse, H0 reduces to the chemical shift evolution (CSE) Hamiltonian, i.e. 

The form of the rotating frame Hamiltonian, HE, for quadrupolar relaxation is... 

The sequence begins with a 90° pulse of width T90 and a rotating-frame phase direction assumed along the x axis. The rotating-frame Hamiltonian during this RF pulse is defined by H o=- hyBiIxy where Bi is the amplitude of the rotating flux density component of the RF field. The reduced density operator immediately after this pulse reads... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

Under MAS at frequency cor = 2nvr, the high-field Hamiltonian of a spin-1/2 system takes the following form in the rotating frame ... 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

During an rf-field irradiation, the Hamiltonian governing a quadrupolar spin system can be written in the rotating frame as... 

For a non-interacting spin-1/2 system subjected to a PIP of an arbitrary shape, the Hamiltonian in the rotating frame with a RF carrier frequency /rf... 

A spin system, initially in thermal equilibrium, can be described by a density operator high temperature approximation.3 After a 90° pulse, the system will evolve under the Hamiltonian and the density operator cr(t) in the rotating frame with a speed a)tf=2nftf can be written as3,4... 

Although the same nuclear spin interactions are present in solid-state as in solution-state NMR, the manifestations of these effects are different because, in the solid, the anisotropic contribution to the spin interactions contributes large time-independent terms to the Hamiltonian that are absent in the liquid phase. Therefore, the experimental methods employed in solids differ from the ones in the liquid state. The spin Hamiltonian for organic or biological solids can be described in the usual rotating frame as the sum of the following interactions ... 

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. 

So far, the equations in this chapter are based on the laboratory frame of reference. In Section 2.8 we saw that the description of magnetic resonance can often be simplified by using a frame rotating with angular frequency coabout the z axis, where pulse frequency (and reference frequency) used to observe the spin system. Now we want to express the density matrix in the rotating frame in order to facilitate our handling of time-dependent Hamiltonians that arise when radio frequency fields are applied. 

Bucci and his coworkers (65,66) treated double resonance by means of a second quantization of the rf fields to avoid the use of the rotating frame. The total Hamiltonian 2 - is then given by equation (1) in which is the normal Hamiltonian of the isolated spin system in the absence of radiation but in the magnetic field, is the radiation Hamiltonian, and represents the interaction between the spins and the radiation field(s). 

Consider a basis sequence that consists of N square pulses (see Fig. 3). In homonuclear experiments, where all rf pulses are applied at the same frequency the spin system is conveniently analyzed in the corresponding rotating frame, where the Hamiltonian... 

In heteronuclear Hartmann-Hahn experiments, a rf sequence is irradiated simultaneously at two frequencies vf and Vg. These experiments are conveniently analyzed in the corresponding doubly rotating frame (Ernst et al., 1987). In this frame, the free-evolution Hamiltonian contains offset terms and for / and 5 spins, isotropic homonuclear /-/ and 5-5 coupling terms and, and truncated heteronuclear coupling terms... 

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