Ladder models

In the secular approximation [89], we can eliminate the coherence terms [e.g., pr, (x)(u / u")] in Eq. (III.9) such that the only diagonal terms contribute to the vibrational transitions through which the vibrational populations in various states are coupled. By applying the ladder model [89] to the interaction between the vibrational and heat-bath modes, the vibrational population decay constant is expressed as... 

The first term in Eq. (III. 10) is originated from the decay from v to v 1, while the second one, with the Boltzmann factor, is due to the thermally activated transition from v to v + 1. The vibrational population transfer, on the other hand, is only allowed to undergo the v —> v 1 transitions in the ladder model. Thus, the rate constant of the transfer is given by... 

In the secular approximation and with the ladder model, the time evolution of the vibrational coherence, pw(x), is determined by... 

Klette, T. J. 8c Grihches, Z. 2000. Empirical patterns of firm growth and R8cD investment a quality ladder model interpretation. Economic Journal, 110(463) 363-387. 

In order to test this model, we measured the quantum yield of the electron transfer to methylviologen as a function of the particle radius of CdS nanoparticle.13) The dependence of the electron transfer yield on the particle size well proved the applicability of the 2D ladder model to this system. For a low excitation limit of gV< 1, the quantum yield is independent of the light intensity, as expressed by... 

Fig. 5.4 Two-dimensional (2D) ladder model for the photocatalytic reactions at small semiconductor particles. X"m represents the distribution of particles containing n electrons and m holes at some instant.

Fig. 27 Magnetic heat capacity for PhBABI for 7 < 100 K showing variation with external magnetic field (left) zero-field magnetic heat capacity showing fits (right) to ID AFM chain, 2D AFM square planar, 2D AFM square planar bilayer, singlet-triplet spin pairing (ST), and spin ladder models.

This model is usually called the step-ladder model. According to this model, the vibrational relaxation dynamics of the system depends only on Ay-,0 and vibrational frequency o>. 

Let us consider a more general spin-ladder model, for which the zigzag model (2) is a particular case. So, we consider the cyclic ladder model containing N — 2M spins s = (Fig.4). The proposed form of wave function (4) can be generalized for a ladder model as follows ... 

Figure 5 Stripe spin structure on the ladder model.

We studied previously a one-parameter ladder model (46-48) with non-degenerate singlet ground state. The exact ground state wave function of the cyclic ladder was written in the MP form (39). Now we write the wave function I>o in a form more suitable for subsequent generalization to other types of lattices [31]. 

This expression corresponds to the mechanical model shown in Figure 10.11, called the discrete ladder model, where o, and bf are the coefficients of the elastic and viscous components respectively. This model was initially proposed by Marvin and Gross. 

We notice that the elements in series in the mechanical model are transformed in parallel in the electrical analogy. The converse is true for the Kelvin-Voigt model. The electrical analog of a ladder model is thus an electrical filter. 

Juyeon Yi, Conduction of DNA molecules A charge-ladder model. Physical Review B, 68, 193103-1 (2003). 

Schiessel and Blumen (1995) describe a mechanical-ladder model based on the algebraic decay (decay1 ) profile of viscoelastic properties at gelation. The mechanical model does not relate to the underlying physics, but is based on a mechanical-ladder model that mimics fractional relaxation equations and is useful in determining viscoelastic decay properties of gelled systems. 

Figure 8.2.6 Stereochemical skewed ladder model of DNA with no gaps between base pairs. (Adapted from Calladine and Drew, 1992.)...

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