NMR Spin Hamiltonian

FIGURE 19.1 The early proton NMR spectrum of 2-bromo-5-chlorothiophene. The stick figure spectrum was fitted using vo(cts —cta) = 4.7 cps and 7 = 3.9 cps. (Reprinted with permission from Anderson, W., Phys. Rev., 102, 151, 1956. Copyright 1956, America Institute of Physics.) 

Here we have used vq = where y is the gyromagnetic ratio for the nucleus in question ( H here) 

Basis Spin States for Two Protons with (a, b) as Normalized Mixing Coefficients 

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Section 3, the main section of this paper, deals with the NMR of bulk semiconductors. Section 3.1 lists the various relevant terms in the NMR spin Hamiltonian. The NMR techniques and strategies that can be employed to obtain the individual NMR parameters of the spin Hamiltonian and theoretical calculations of NMR parameters will be discussed in Sect. 3.2. The remaining subsections will provide examples from the important classes of semiconductors that illustrate the measurement and interpretation of each of the spin Hamiltonian parameters, with an emphasis on what information about semiconductors the parameters convey. 

Anisotropy of the Spin-Spin Coupling Tensor. - In NMR experiments performed in anisotropic liquid crystal (LC) phases or in the solid state, the anisotric part of an indirect nuclear spin-spin coupling tensor J appears combined with the direct dipolar coupling D. The NMR spin Hamiltonian appropriate for spin 1/2 nuclei in molecules partially oriented in uniaxial LC solvents can be written in the high field approximation as... 

The spectral parameters appearing in the effective NMR spin Hamiltonian depend on the electronic structure around the nuclei. The NMR effective spin Hamiltonian has the following form ... 

Allouche et alP calculated chemical shifts (5 values) and SSCCs (/values) for the putrescine molecule, a polyamine present in prostate tissue, using a B3LYP-DFT/6-311-I-- -G(/,/))/PCM/GIAO (gauge-including atomic orbital) approach. From the computed and J values, the H NMR spectrum of putrescine was simulated. Comparisons between the calculated and the experimental NMR spectra at 400 MHz showed a good agreement and allowed to propose reliable values for the NMR spin Hamiltonian parameters of putrescine to be used for further development of quantitative analytical methods of metabolites in prostate tissue. 

In lieu of the reduced nucle spin-spin coupling tensor, the indirect spin-spin coupling tensor J cd = hycyo co is more usually employed in the NMR spin Hamiltonian. 

For a rotating molecule in an isotropic medium, the NMR spin Hamiltonian maybe written as (assuming the external magnetic field is directed along the Z axis) ... 

Hamiltonian O Eq. 11.208 to the observed spectrum. The NMR spin Hamiltonian is usually not expressed in terms of the reduced indirect spin-spin coupling tensors Kja, but rather in terms of the indirect spin-spin coupling tensors Jjci> which are related to the reduced tensors as ... 

However, the magnetizability tensor does not enter the effective NMR spin Hamiltonian, from which the NMR spectrum may be generated. The next term in the expansion (II) measures the coupling of the magnetic moments Mk to the external magnetic induction B ... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A wide variety of ID and wD NMR techniques are available. In many applications of ID NMR spectroscopy, the modification of the spin Hamiltonian plays an essential role. Standard techniques are double resonance for spin decoupling, multipulse techniques, pulsed-field gradients, selective pulsing, sample spinning, etc. Manipulation of the Hamiltonian requires an external perturbation of the system, which may either be time-independent or time-dependent. Time-independent... 

Electron spin resonance (ESR), or electron paramagnetic resonance (EPR) as it is sometimes known, shares many similarities with its cousin, NMR. The origin of the phenomenon is the spin of the electron (rather than the nuclear spin) coupling with the nuclear spins of the atoms in the polymer, but much of the physics of their interactions are similar. The usual spin Hamiltonian, which is used to determine the energies of the interactions, can be written as... 

Average or effective Hamiltonian theory, as introduced to NMR spectroscopy by Waugh and coworkers [55] in the late 1960s, has in all respects been the most important design tool for development of dipolar recoupling experiments (and many other important experiments). In a very simple and transparent manner, this method facilitates delineation of the impact of advanced rf irradiation schemes on the internal nuclear spin Hamiltonians. This impact is evaluated in an ordered fashion, enabling direct focus on the most important terms and, in the refinement process, the less dominant albeit still important terms in a prioritized manner. 

The preceding treatment of the spin Hamiltonian terms in bulk semiconductors, where they are relatively-well understood, will provide a basis for the discussion of the NMR of nanocrystalline semiconductors in Sect. 4, since as a group they present special considerations and many unanswered questions remain. Section 5 will provide some general conclusions and suggest future promising avenues of NMR research in semiconductors. 

Recent solid state NMR studies of liquid crystalline materials are surveyed. The review deals first with some background information in order to facilitate discussions on various NMR (13C, ll, 21 , I9F etc.) works to be followed. This includes the following spin Hamiltonians, spin relaxation theory, and a survey of recent solid state NMR methods (mainly 13C) for liquid crystals on the one hand, while on the other hand molecular ordering of mesogens and motional models for liquid crystals. NMR studies done since 1997 on both solutes and solvent molecules are discussed. For the latter, thermotropic and lyotropic liquid crystals are included with an emphasis on newly discovered liquid crystalline materials. For the solute studies, both small molecules and weakly ordered biomolecules are briefly surveyed. 

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 ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Electron spin resonance (ESR) measures the absorption spectra associated with the energy states produced from the ground state by interaction with the magnetic field. This review deals with the theory of these states, their description by a spin Hamiltonian and the transitions between these states induced by electromagnetic radiation. The dynamics of these transitions (spin-lattice relaxation times, etc.) are not considered. Also omitted are discussions of other methods of measuring spin Hamiltonian parameters such as nuclear magnetic resonance (NMR) and electron nuclear double resonance (ENDOR), although results obtained by these methods are included in Sec. VI. 

The similarity of the preceding first-order ESR treatment to the first-order NMR treatment of two coupled protons is evident. For an unpaired electron interacting with n equivalent nuclei of spin the hyperfine coupling term in the spin Hamiltonian is... 

In pulse-type NMR experiments, the super-Hamiltonian HD constitutes a superoperator representation of the commutator with the spin Hamiltonian fts which is defined by equation (14) ... 

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