Even-site chains

A single soliton exists in the groimd state of an odd-site chain. For an even-site chain, however, a soliton is paired with an antisoliton so as to restore the 

Pairs of solitons are the natural excitations from the ground state. This is shown in Fig. 4.6, which shows the bond dimerization of the 1B state, obtained by iterating eqn (4.21). The bond dimerization fits the functional form (Brazovskii and Kirova 1981 Campbell and Bishop 1981), 

We see that there is a soliton at n = —no, which changes the dimerization from the A to B phase, and an antisoliton at n = no, which reverses the phase again. 

We can use eqn (4.31) to determine the adiabatic energy profiles of the lAg, IBu, and 2Ag states as a function of the soliton-antisoliton separation, R = 2no- 

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These geometric properties of the chain are also associated with mid-gap states (Pople and Walmsley 1962). To see this, consider the energy spectrum of an even-site chain. There are N/2 states in each of the valence and conduction bands. As a result of particle-hole symmetry, every valence band state with energy e" = e maps into a conduction band state with energy = —e. Thus, the energy spectrum is symmetric about e = 0, as shown in Fig. 3.4. Now, for an odd-site chain there are N — l)/2 states in each of the valence and conduction bands, and one localized gap state. As a consequence of particle-hole symmetry the localized state lies at e = 0. This mid-gap state is occupied by one electron, and is associated with the soliton, as shown in Fig. 4.3. 

Fig. 4.6. The normalized, staggered bond dimerization, S , of the state of an even-site chain. The wavefunction of the lower mid-gap state, is also shown. (The higher mid-gap state wavefrmction,, has the opposite spatial symmetry.)...

We saw that the soliton in the ground state of an odd-site chain is either neutral with spin-1/2 for the undoped chain (5°), or charged with spin 0 for the singly doped chain S ). We now discuss the types of solitons present in the excited states of an even-site chain. Suppose that an even-site chain is instantaneously excited from the ground state to the or 2Ag states. This is a vertical transition, with the soliton-antisoliton separation initially zero. Within a time 27r/o o a soliton-antisoliton pair is created and separates a distance Their... 

The effect of alkyl chain length on the structure of alkanethiols on Au(lll) was studied with CH3(CH2) iSH, where n = 2,4, 6, 8, 10, 11, 12, 14, 15, 16, and 18).i The results, in terms of HREEL spectra, are displayed in Figure 11. It is most interesting to note that the intensity of CH3 a-deformation mode at 1380 cm (171 meV) is profoundly dependent on the number of carbons in the alkyl chain It is present only when the number of carbon atoms is even (cf, the spectra labeled Cio, C12 and C le) it is absent when the number is odd (cf, the spectra labeled Cu andCis). This odd-even trend is caused by the fact that the orientation of the CH3 head is parallel to the surface for odd number of carbon atoms but perpendicular when the number is even (cf, the inset in Figure 11). As dictated by the dipole selection rules, only the oscillator that has a component perpendicular to the surface (as in the even number chain) would show HREELS activity. It can also be seen in the frequency region below 220 cm (27.3 meV) that more than one peak, separated by about 30 cm (3.7meV) are present this indicates the existence of multiple adsorption sites for the subject alkanethiols on Au(lll). 

This mechanism not only accounts for the substitution of the more labile chlorine atom on the polymer chain, it also results in the elimination of a new potential initiation site by moving the double bond out of conjugation with any adjacent chlorine atoms. The newly formed C—O or C—S bonds, with AH > 484 kJ/mol (100 kcal/mol), are significantly more thermally stable than even the normal C—Cl bonds in PVC at about 411 kj/mol (85 kcal/mol) (11). 

In a class of reahstic lattice models, hydrocarbon chains are placed on a diamond lattice in order to imitate the zigzag structure of the carbon backbones and the trans and gauche bonds. Such models have been used early on to study micelle structures [104], monolayers [105], and bilayers [106]. Levine and coworkers have introduced an even more sophisticated model, which allows one to consider unsaturated C=C bonds and stiffer molecules such as cholesterol a monomer occupies several lattice sites on a cubic lattice, the saturated bonds between monomers are taken from a given set of allowed bonds with length /5, and torsional potentials are introduced to distinguish between trans and "gauche conformations [107,108]. 

The thermal (or photochemical) decomposition of the azo group gives rise to a radically initiated polymerization. The reactive site F, the transformation site, however, can, depending on its chemical nature, initiate a condensation or addition type reaction. It can also start radical or ionic polymerizations. F may also terminate a polymerization or even enable the azo initiator to act as a monomer in chain polymerizations. 

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