One dimensional chain

Here, u is the displacement of the /ith molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by... 

Two approaches to the attainment of the oriented states of polymer solutions and melts can be distinguished. The first one consists in the orientational crystallization of flexible-chain polymers based on the fixation by subsequent crystallization of the chains obtained as a result of melt extension. This procedure ensures the formation of a highly oriented supramolecular structure in the crystallized material. The second approach is based on the use of solutions of rigid-chain polymers in which the transition to the liquid crystalline state occurs, due to a high anisometry of the macromolecules. This state is characterized by high one-dimensional chain orientation and, as a result, by the anisotropy of the main physical properties of the material. Only slight extensions are required to obtain highly oriented films and fibers from such solutions. 

Various Ru-oxides, YBa2Cu307, c (I), Ba Ru2/3Gdi/303 (II) as well as Ru-doped a-Fe203 (III), to probe the local chemical structure around the Ru atoms. Compound (I) has interesting properties with x < 0.2 it is a superconductor and with x 1 a semiconductor. Ru oxidation state and coordination are discussed on the basis of measured isomer shifts and quadrupole splittings Ru(IV) ions exclusively occupy Cu-1 sites which form one-dimensional chains... 

The one-dimensional chain of hydrogen atoms is merely a model. Flowever, compounds do exist to which the same kind of considerations are applicable and have been confirmed experimentally. These include polyene chains such as poly acetylene. The p orbitals of the C atoms take the place of the lx functions of the H atoms they form one bonding and one antibonding n band. Due to the Peierls distortion the polyacetylene chain is only stable with alternate short and long C-C bonds, that is, in the sense of the valence bond formula with alternate single and double bonds ... 

The silicates are a large class of solids of great importance in industry as well as science, particularly geology. The prototype silicate is quartz consisting of Si04 tetrahedra which share their comers and edges and are arrayed in various three-dimensional patterns depending on the temperature. In other crystalline minerals the tetrahedra are linked in one-dimensional chains, or two-dimensional sheets. The arrays in these latter cases are combined with various metal ions. 

In contrast, within (p-EPYNN)[Ni(dmit)2], first synthesized in 1996 [79], it has been proven that spin-ladder chains of the Ni(dmit)2 moiety coexist with the ferromagnetic one-dimensional chain of the p-EPYNN radical cation. Spin-ladders are of interest because of their potential applications in the area of quantum magnets and because it has been predicted that holes doped into even-leg ladders may pair and possibly superconduct [90-92]. 

In this compound, Ni(dmit)2 molecules are arranged in columns and Fe(qsal)2 cations are dimerized by n-n interactions and construct a one-dimensional chain. Fe(qsal)2 chains are then interwoven with the Ni(dmit)2 columns (Fig. 11). 

Significant economies of computation are possible in systems that consist of a one-dimensional chain of identical reservoirs. Chapter 7 describes such a system in which there is just one dependent variable. An illustrative example is the climate system and the calculation of zonally averaged temperature as a function of latitude in an energy balance climate model. In such a model, the surface temperature depends on the balance among solar radiation absorbed, planetary radiation emitted to space, and the transport of energy between latitudes. I present routines that calculate the absorption and reflection of incident solar radiation and the emission of long-wave planetary radiation. I show how much of the computational work can be avoided in a system like this because each reservoir is coupled only to its adjacent reservoirs. I use the simulation to explore the sensitivity of seasonally varying temperatures to such aspects of the climate system as snow and ice cover, cloud cover, amount of carbon dioxide in the atmosphere, and land distribution. 

Chapter 8 describes a similar one-dimensional chain of identical reservoirs, but one that contains several interacting species. The example illustrated here is the composition of the pore waters in carbonate sediments in which dissolution is occurring as a result of the oxidation of organic matter. I calculate the concentrations of total dissolved carbon and calcium ions and the isotope ratio as functions of depth in the sediments. I present... 

This chapter demonstrated the computational simplification that is possible in systems consisting of a one-dimensional chain of coupled reservoirs, which arise in diffusion and heat conduction problems. In such systems each equation is coupled just to its immediate neighbors, so that much of the work involved in Gaussian elimination and back substitution can be avoided. I presented here two subroutines, GAUSSD and SLOPERD, that deal efficiently with this kind of system. 

Like the climate system described in Chapter 7, this diagenetic system consists of a chain of identical reservoirs that are coupled only to adjacent reservoirs. Elements of the sleq array are nonzero close to the diagonal only. Unnecessary work can be avoided and computational speed increased by limiting the calculation to the nonzero elements. The climate system, however, has only one dependent variable, temperature, to be calculated in each reservoir. The band of nonzero elements in the sleq array is only three elements wide, corresponding to the connection between temperatures in the reservoir being calculated and in the two adjacent reservoirs. The diagenetic system here contains two dependent variables, total dissolved carbon and calcium ions, in each reservoir. The species are coupled to one another in each reservoir by carbonate dissolution and its dependence on the saturation state. They also are coupled by diffusion to their own concentrations in adjacent reservoirs. The method of solution that I shall develop in this section can be applied to any number of interacting species in a one-dimensional chain of identical reservoirs. 

Chapter 8 presented the last of the computational approaches that I find widely useful in the numerical simulation of environmental properties. The routines of Chapter 8 can be applied to systems of several interacting species in a one-dimensional chain of identical reservoirs, whereas the routines of Chapter 7 are a somewhat more efficient approach to that chain of identical reservoirs that can be used when there is only one species to be considered. Chapter 7 also presented subroutines applicable to a generally useful but simple climate model, an energy balance climate model with seasonal change in temperature. Chapter 6 described the peculiar features of equations for changes in isotope ratios that arise because isotope ratios are ratios and not conserved quantities. Calculations of isotope ratios can be based directly on calculations of concentration, with essentially the same sources and sinks, provided that extra terms are included in the equations for rates of change of isotope ratios. These extra terms were derived in Chapter 6. 

This model consists of a one-dimensional chain of elastically colliding particles with alternate masses m and M. In order to prevent total momentum conservation we confine the motion of particles of mass M (bars) inside separate cells. Schematically the model is shown in Fig.4 particles with mass m move horizontally and collide with bars of mass M which, besides suffering collisions with the particles, are elastically reflected back at the edges of their cells. In between collisions, particles and bars move freely. 

Quantum dynamics on graphs became an issue also in the context of quantum information. Aharonov et.al (1993) pointed out that a random quantum walk on one dimensional chains can be faster than the corresponding classical random walk. Since then, a whole field has emerged dealing with quantum effects on graphs with properties superiour to the corresponding classical operations. For an introductory overview and further references, see Kempe (2003). 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Figure 8.4 One-dimensional chain of atoms with interatomic distance a and force constant K.

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

The vibrational frequencies are obtained from the force-field using the second derivatives of the potential energy with respect to displacements of the atoms (in a more elaborate version of the argument used in Section 8.2 for the one-dimensional chain) the calculation is analogous to the calculation of normal mode frequencies for molecules. The resulting vibrational frequencies can be compared with those... 

The compound 4(Ph3SbI)2OI2 is obtained by the addition of I2 in acetonitrile/water to a solution of triphenyl-antimony. The crystal structure contains a one-dimensional chain 53 and (Ph3SbI)20 molecules which are not coordinated to I2. In the ID chain, the distance between the iodine of the (Ph3SbI)20 molecule and the diiodine... 

Tyutyulkov analysed cross-conjugated systems like [14] theoretically, using the Coulson-Rushbrooke theorem and Wannier transformation of the Bloch MO. His results hold multiple significance. First, it was suggested, by taking into account the interaction between the one-dimensional chains of the quasi-one-dimensional polymer, that the critical temperature may reach 10 -10 K (see p. 226). Secondly, the radical centres of hydrocarbons... 

Figure 31. Proposed structures for polymeric Cm The orthorhombic (O) phase is formed by one-dimensional chains the rhombohedral (R) and tetragonal (T) structures are two-dimensional...

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