Interaction bilinear

Here, I denotes the lattice sites, while and Xfo are the a components of the n-dimensional momentum and displacement vectors, respectively. M denotes the mass of particle A, C is the model parameter, and h is a homogeneous external field. The bilinear interaction is restricted to the nearest neighbor only. For the anharmonic part of the interaction ... 

Recent theoretical treatments of the soft-mode behaviour include a detailed study by Onodera using classical mechanics, and a theory of hydrogen-bond mechanics, including tunnelling effects, by Stamenkovic and Novakovic. ° Onodera assumes a quartic potential function for his individual oscillators, with a bilinear interaction which reduces to c x, where x is the displacement, under the Weiss-molecular-field approximation. The model is soluble without further approximation (in series of elliptic functions), yielding the temperature variation of frequency and damping. If the quartic potential has a central hump larger than kTc,... 

Equations (13.6), (13.7), (13.9), and (13.10) provide the general stracture of our model. The relaxation process depends on details of this model mainly through the form and the magnitude of T/sb- We will consider in particular two fonns for this interaction. One, which leads to an exactly soluble model, is the bilinear interaction model in which the force F( ry ) is expanded up to first order in the deviations rj of the solvent atoms from their equilibrium positions, F( ry ) = F ( rJ )5ry,... 

This beta function is zero in the absence of dipolar interactions. It vanishes for interactions linear in the spin operator and has nonzero values only for bilinear interactions. Furthermore, it is zero for Je approaching zero, so that a nonzero signal indicates residual anisotropic interactions, and it is free of signal attenuation by relaxation. The shape of the beta function has been shown to depend strongly on the strain in rubber samples [Cal3]. 

Line broadening in solid-state NMR arises from spin interactions which can be described in first order by coupling tensors of rank two (cf. Section 3.1) [Hael, Mehl, Schl], The spin interactions are either linear or bilinear in the spin operator. Linear interactions are the Zeeman interaction, the chemical shielding, and the interaction with the rf field. Bilinear interactions are the J coupling, the dipole-dipole coupling, and the quadrupolar interaction. In isotropic materials like powders, glasses, and undrawn polymers, wide lines are observed as a result of an isotropic orientational distribution of coupling tensors. 

For a two-level systan, = ( l)(l - 0)(0 ).Hereo)(t) = a)o + (Owitho)o8S a static frequency and F(i) = F(t + 2jt/Q) denoting a periodic modulation. Assuming bilinear interactions and weak coupling, going into the Markovian limit, the probabilities to occupy the states Band 1 satisfy the master equation (12.20)with time-dependent... 

Let us take a closer look at the nature of these interactions. The achiral bilinear interactions have rather simple desires . If they are negative, they favour a parallel orientation in the interacting layers. If they are positive, they favour antiparallel orientations in the interacting layers. Chiral interactions are different. They have opposite signs in oppositely handed enantiomers. Interactions favour, in this basic linear approximation, perpendicular tilts in interacting layers. The favourable sense, i.e. the 90°... 

The term ai gives achiral bilinear interactions to nearest neighbom-ing layers expressed in tilts ... 

The final indirect bilinear interaction that is significant enough to be considered, is the interaction with the next nearest layers having a chiral character ... 

The High-Field Approximation In most NMR experiments the nuclear Zeeman interaction with the static external magnetic field is much stronger than all other interactions of the nuclear spins. As a result of these differences in the size, it is usually possible to treat these interactions in first order perturbation theory, i.e. use only those terms which commute with the Zeeman Hamiltonian, the so called secular terms. This approximation is called the high field approximation. While the single particle interactions like CSA or quadrupolar interaction have a unique form, for bilinear interactions, one has to distinguish between a homonuclear and a hetero-nuclear case. The secular parts of Hamiltonians discussed in the previous section are collected in Table 1. 

The result of a change in tunneling upon the electron exdtation of a chromophore is given by the second term in Eq. (124). This bilinear interaction allows for the processes of a simultaneous creation (annihilation) of a tunnelon and annihilation (creation) of a phonon. The probability of these processes is proportional to (1 + n)f — n(l — f) = [2sh(hv/kT)] . Therefore, this bilinear interaction yields the following expression for the homogeneous halfwidth of ZPL [46, 91, 92] ... 

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