Three-dimensional coupling

Three-dimensional coupled roto-vibrators by algebraic methods... 

In the preceding sections we have discussed the algebraic treatment of onedimensional coupled oscillators. We now present the general theory of two three-dimensional coupled rovibrators (van Roosmalen, Dieperink, and... 

We consider next the case of two three-dimensional coupled problems described by U(1)(4) U<2,(4). The coset space is here composed of two com-... 

Tunneling rotation of three-dimensional coupled rotors in solid methane is dealt with in many experimental and theoretical studies. The phase transition at 20.4 K turns this species from orientationally disordered phase I to partially oriented phase II. According to neutron... 

Rigorous one-dimensionality is of course a myth in organic conductors. The Coulomb interaction is inherently long range. There is also an unavoidable finite overlap and thus hopping between chains. These lead to two- and three-dimensional couplings and to dimensionality crossovers. (a) Single-Particle Crossover... 

As mentioned in Section II, LRO in two dimensions can exist only for a real order parameter, that is, for CDW in a half-filled band. This would be the case for BOW in the polymers or the Peierls state, which would be stabilized by transverse hopping or interchain coupling. This is also the case of the CDW state of the n = 1 two-dimensional Hubbard model. All other types of instabilities, such as those treated in the RPA previously in Section V, require three-dimensional coupling to stabilize any LRO. 

We have just seen that some form of three-dimensional coupling is necessary to stabilize most types of LRO. The two-dimensional/three-dimensional crossover will occur very close to the two-dimensional RPA transition temperatures. Simply put, the three-dimensional susceptibilities Xa Cfl) are related to the two-dimensional ones XaD(Q> ) of Section IV through intersheet interactions Xa(qc) in an RPA fashion ... 

The situation in the polymers as been described in Section IV.C.l. There is a rich excitation spectrum solitons, polarons, and bi-polarons. The effect of three-dimensional coupling would be to confine soliton-antisoliton pairs... 

The theoretical problems raised by the study of CPs will be discussed below as the need arises. Many special properties of CPs are related to their quasi-one-dimensional character for instance, the large influence of disorder, the importance of residual three-dimensional coupling, and the importance of electron-phonon interactions, which, among other consequences, manifests itself in the case of a half-filled band by the occurrence of the Peierls instability. Much of the early theoretical work was concerned with PA, which, as we shall see, is peculiar among presently known CPs by having a degenerate ground state (see Section IV.B). 

Up to now, we have considered CPs (almost) exclusively as being perfectly periodic, infinite, and isolated one-dimensional chains. This is not what real materials are made of. In this and the following sections, we move closer to real materials by considering the effects of three-dimensional coupling and of disorder (both intrachain and interchain). Not surprisingly, less theoretical work has been done on these issues. On the other hand, considerable experimental effort has been devoted to characterizing the disorder, which is discussed in Chapter 12, where the structure of CPs is considered. 

Let us first discuss the effects of three-dimensional coupling of CP chains, still supposed to be perfect and in perfect three-dimensional order, a situation for which theoretical results are available. The coupling can affect the electronic structure and excitations of the chain and a new problem appears, that of three-dimensional transfer of excitation. This is crucial to understanding the dc conductivity of CPs, both undoped and doped. 

Three-dimensional coupling is characterized by the interchain transfer integrals tx. Generally, only nearest-neighbor interactions are taken into account, so only one or two t are relevant. They may differ, even in sign, depending on the relative positions of the interacting chains. In this qual-... 

The question then arises of the amount of three-dimensional coupling— that is, the value of tjt —sufficient to establish three-dimensional behavior. It has been studied in different ways, showing that three-dimensional behavior wins at least if tjt 10 2, that is, t on the order of 20 meV [91,92]. This is a stringent condition. 

However, in all these calculations, the conjugated chains bear H atoms only. As bulkier side groups are introduced, the overall interchain interaction decreases and it becomes more anisotropic. If chains form stacks, relatively strong two-dimensional coupling will persist and may still have consequences similar to those of three-dimensional coupling. The question remains open. But substituted CPs are generally even more disordered than the simple ones, and we shall see now that disorder will then have more important consequences than the details of interchain interaction. 

Brown, D.P., Rubin, S.G., and Biswas, P. (1995). Development and Demonstration of a Two/Three Dimensional Coupled Flow and Aerosol Model, In Proceedings of the I3lh AIAA Applied Aerodynamics Conference, American Institute of Aeronautics and Astronautics. 

AsFg or SbFg, it follows that Tp depends on the size of the counterion and thus on the weak interstack interactions, i.e. on the three-dimensional coupling of the conducting stacks to each other. 

Makhijani, V. B., Hang, H. Q., Dionne, P. J., and Thubrikar, M. J., Three-dimensional coupled fluid-structure simulation of pericardial bioprosthetic aortic valve function, ASAIO J., 1997 43(5) M387-M392. 

The analytical solution of these three-dimensional coupled equations can be accomplished for only very specialized states of deformation (Selvadurai 2007). For this reason, computational approaches have been developed for the solution of poroelasticity problems a finite element scheme is used here to solve this problem. 

Detemmerman and Froment [1998] carried out a three dimensional coupled CFD simulation of furnaces and reactor tubes for the millisecond thermal cracking of propane into ethylene in the presence of steam, a process already discussed in Chapters 1 and 9. 

For strictly onedimensional systems, this effect of localization due to level spacing would therefore be too large to explain the experimentally observed low-temperature conductivities with reasonable values of the conjugation lengths. We conclude that the approach of treating polyacetylene as a strictly one-dimensional system is not allowed and that three-dimensional coupling is important. For the three-dimensional case we obtain ... 

The high temperatore core without the critical heat flux criterion (i.e. the MDHFR) was designed in 1998 [12]. The two-dimensiraial R-Z model of the core cannot accurately predict bum-up of fuel rods. The three-dimensional coupled neutro-nic-thermal-hydraulic core calculation was developed in 2003 [18]. It is shown in Fig. 1.9. This calculation considered the control rod pattern and fuel loading pattern [19, 20] and was similar to the core calculation for BWRs. But the finite difference code SRAC [21] was used for the three-dimensional neutronic calculation instead of a nodal code. The core design of the Super FR also adopted the three dimensional neutronic and thermal hydraulic coupled core bum-up calculation. 

The cladding temperature that was obtained by the three-dimensional coupled core calculation is the average temperature over the assembly. The peak cladding temperature of a fuel rod is necessary for the evaluation of the fuel cladding integrity. The subchannel analysis code of the Super LWR is coupled with the fuel assembly bum-up calculation code for this purpose [25]. Fuel pin-wise power distributions are produced for various bum-ups, coolant densities, and control rod positions. The pin-wise power distributions are combined with the homogenized fuel assembly power distribution to reconstmct the pin-wise power distribution of the core fuel assembly. The power distribution over the fuel assembly is taken into account as shown in Fig. 1.11. The reconstracted pin-wise power distribution is used in the evaluation of peak cladding temperature with the subchaimel analysis. 

An improved core design procedure of the Super LWR that coupled the subchannel analysis with three-dimensional coupled core calculations is described in ref. [28]. The time-dependent subchannel analysis code for safety analysis of the Super LWR is described in ref. [123]. 

Three-dimensional coupled ice-ocean models, are under development and have been applied in ocean forecasting systems, Preller and Posey (1996). For shelf seas they are now in an active research stage and will become important tools in future forecasting systems, particularly when coupled atmosphere and ocean models are introduced into the different national forecasting institutes. 

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