Average 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 ... 

In cases where the Hamiltonians (typically due to phase or amplitude switching in the rf fields) are discontinuously time-dependent, the average Hamiltonian may conveniently be set up using the semi-continuous Baker-Campbell-Hausdorff (scBCH) expansion [56] as... 

ROCS A makes use of the CN n sequences [82, 159] to generate a static CSA powder pattern under MAS condition. The average Hamiltonian for a CNvn sequence is [159]... 

All the symbols carry their usual meanings. If we want to create an average Hamiltonian similar to the static one, we just need to keep the terms associated with 720 Furthermore, the scaling factors of those terms should be identical. With the pulse sequence of Cn]n [160, 161], the symmetry allowed terms are... 

Taking the MREV-8 as a basic C element (duration = xr/5), one can generate a y -encoded [194] version of MREV-8 by concatenating the five C elements with an rf phase set equal to k / 2n/5, where k = 0-4 [107]. The averaged Hamiltonian calculated to the lowest order for this so-called T-MREV method has the following form ... 

According to the coherent averaging theory,3,4,53 the zero-order average Hamiltonian can be obtained straightforwardly... 

It can be shown that the second-order average Hamiltonian has the form of... 

Since the zero-order average Hamiltonian of each RF field j]m(m Q) and the first-order average Hamiltonians between any two RF fields (m, n O)... 

The zero-order average Hamiltonian can be obtained straightforwardly... 

Here 2tt Aft = trm is imposed to eliminate the effect by the zero-order interaction of the RF field, otherwise a term containing Ix will remain and it will distort the longitudinal magnetization, transverse magnetization, as well as the BSPS of the 13C . For the on-resonance condition, 5 = 0, all the higher-order average Hamiltonians vanish since [7f(/ ), = 0 for arbitrary l and... 

The first- and second-order average Hamiltonians can be calculated similarly... 

Similar results were observed for a pair of complementary PIPs with a condition of 2 rA/r = 3tr (m = 3), where the term including Iy is maximized and the term including Ix becomes zero in the average Hamiltonian... 

Eqs. (46)—(51) can be used in Eq. (5) to give the average Hamiltonian in the laboratory frame in terms of the order parameter tensor. The NMR spectrum arising from Eq. (5) can provide a means to determine the orientational ordering of molecules in uniaxial LC. 

The main purpose of the sequences is to obtain an averaged Hamiltonian H in which the dipolar term is very small compared with the chemical shift term. The zeroth-order of the average dipolar Hamiltonian term is given by the following equation ... 

A number of different multiple pulse sequences (8-, 24- and 52-pulse sequences) have also been introduced in order to obtain better resolution or line narrowing, i.e. to affect the first- and second-order terms in the average Hamiltonian. Since pulse imperfections are the major source of resolution limitations, these composite pulse sequences are designed with corresponding symmetry properties which allows the canceling of specific rf pulse imperfections. 

The inclusion of average Hamiltonian theory derived pulse sequence building blocks specifically designed for the measurement of RDCs is highly desirable. The homonuclear dipolar decoupling sequences used in ref. 190 lead to the less complex multiplet structures observed in isotropic samples with the chemical shift resolution reduced by a factor of 2-3, depending on the multiple pulse... 

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. 

Therefore, on the basis of the averaged Hamiltonian (5.2.4), a resonance of the standard mapping located at I = 2m7r overlaps with its next replica located at I = 2 m -I- 1)7T if Al = 2tt. This yields... 

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