Solid-state Hamiltonian

Frohlich applied the unitary transformation on the Hamiltonian describing conductors, but his attempt to remove the degeneracy failed. Then he proposed the true many body treatment. Bardeen with Cooper and Schrieffer continued to fulfil Frohlich s idea, and with the full multiconfiguration method used on the Frohlich-transformed Hamiltonian, they attempted to remove the degeneracy. After 2-years of intensive work they had no positive solution. At the last moment Bardeen accepted the trial function proposed by Schrieffer, inconsistent with the particle conservation law, and leading to the concept of the Cooper pair based theory, known as BCS. Nevertheless, the solid-state Hamiltonian does not contain information of superconductivity, so that Frohlich as well as Bardeen in fact only calculated the correlation energy of conductors. 

The anisotropies that lead to line broadening in isotropic ESR spectra influence solid-state spectra more directly. Accordingly a more complex spin Hamiltonian is required to interpret such spectra ... 

This Hamiltonian leads to dephasing of the S -spin signal recorded as function of time (increasing number of rotor periods Nc in the REDOR experiment) as illustrated in Fig. lb. REDOR has been a key experiment in biological solid-state NMR, as for example used recently for determination of statherin binding to biomineral surfaces as illustrated in Fig. lc, with numerous REDOR determined intemuclear distances high-lighted in Fig. Id [79]. 

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 ... 

The dipolar-coupling Hamiltonian (TTy) describes the through-space coupling between two nuclear spins l and Ij. The dipolar coupling has an rk- dependence, and is key to the determination of internuclear distances in both solid-state and solution-state NMR. The high-field truncated form of the dipolar Hamiltonian is given by... 

Crowell discovered a variety of effects numerically, including modified Rabi flopping, which has an inverse frequency dependence similar to that observed in the solid state in reciprocal noise [73]. The latter is also explained by Crowell [17] using a non-Abelian model. A variety of other effects of RFR on the quantum electrodynamical level was also reported numerically [17]. The overall result is that the occurrence, classically, of the B V> field means that there is a quantum electrodynamical Hamiltonian generated by the classical term proportional to 3 2. This induces transitional behavior because it contributes to the dynamics of probability amplitudes [17]. The Hamiltonian is a quartic potential where the value of determines the value of the potential. The latter has two minima one where B = 0 and the other for a finite value of the B i) field, corresponding to states that are invariants of the Lagrangian but not of the vacuum. 

The dominant interactions of nuclei having spin I that are responsible for the broadening of solid-state NMR lines and for their characteristic line shapes are described by the total Hamiltonian ... 

As previously observed, the dominant terms in the Hamiltonian which describe a spin system in the solid state are the dipolar and quadrupolar terms. In the case of nuclei with 1 = 1/2 (such as H, 13C, 19F and 29Si) the quadrupolar interaction is zero. The dipolar Hamiltonian HD (for a homonuclear spin system) has the general form ... 

Proton and fluorine are the most frequently studied nuclei in the solid state NMR of polymers. However, useful information can be obtained from deuterium resonance, (2H I = 1). In the presence of nuclei with spin I > 1/2, which possess a quadrupolar moment, the most dominant interaction occurs between the quadrupolar moment and an electric field gradient tensor V, generated by the C—D bonding electrons. Consequently, the only dominant interaction, besides Hz (Eq. (1)), will be described by the quadrupolar Hamiltonian, which for a single spin has the form ... 

The broad line spectra of nuclei with spin I = 1/2 in the solid state are mainly a consequence of the dominant contribution of the dipolar Hamiltonian HD (Eq. (4)), which gives rise to a local field B)oc. Its magnitude varies as a function of the angle 0.. between the intemuclear vector r.j and the applied magnetic field B0. Depending on the nature of spin system, two general types of interactions can be distinguished ... 

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