Long-range ordering transition

With regard to 1-D systems, it can be clearly stated that, in contrast to 2-D systems where a 2-D phase transition has been considered by Stanley and Kaplan among others28,78), the 1-D short-range order = 3-D long-range order transition only shows 3-D properties7). 

Using the so-called "block copolymers (a block of Na A-monomers at one end is covalently bonded to a block of Nb B-monomers) one can also realize the analogy of order-disorder phenomena in metallic alloys with polymers one observes transitions from the disordered melt to mesophases with various types of long range order (lamellar, hexagonal, cubic, etc ). We shall not consider these phenomena here further, however... 

The optical properties of a-Si H are of considerable importance, especially for solar-cell applications. Because of the absence of long-range order, the momentum k is not conserved in electronic transitions. Therefore, in contrast to crystalline silicon, a-Si H behaves as though it had a direct bandgap. Its absorption coefficient for visible light is about an order of magnitude higher than that of c-Si [74]. Consequently, the typical thickness (sub-micrometer) of an a-Si H solar cell is only a fraction of that of a c-Si cell. 

A switch to double-angle vectors given by Eq. (2.3.12) not only significantly simplifies the treatment of orientation phase transitions in planar systems of nonpolar molecules but also leads to a number of substantial inferences on the transition nature. First of all note that the long-range-order parameter t] (vanishing in a disordered phase and equal to unity at T = 0) in a -dimensional space (specified by the orientations of long molecular axes) can be defined as ... 

The defining property of a structural glass transition is an increase of the structural relaxation time by more than 14 orders in magnitude without the development of any long-range ordered structure.1 Both the static structure and the relaxation behavior of the static structure can be accessed by scattering experiments and they can be calculated from simulations. The collective structure factor of a polymer melt, where one sums over all scattering centers M in the system... 

Long-range order theory, 35 4-5 Looper s walk capping, 32 438-445 Lorentzian energy averaging, 34 217 Lorentzian function, energy-dependent, 34 243 Losod, 33 215, 224, 258 Low-coordinated transition-metal ions, 34 133 Low energy... 

In the model of the previous section, we have encountered the situation that outside the regime bounded by the disorder line the correlation function, Eq. (53), is characterized by an angle which is incommensurate with the lattice. Now it is well known that in d 2 dimensions, incommensurate long-range order (which would be described by <5,5 )x = cos[ r -Tjl(/ -F A< ] instead of Eq. (53)) is unstabie against thermal fluctuations (this is a two-component ordering - amplitude tj/, phase A0 - similar to the XY model). Nevertheless, it is possible to have a phase transition where in Eq. (53) diverges, but for T [Pg.126]

Continuous transitions are characterized by a critical behavior of a physical observable, which is the order parameter y of the transition. Above the transition temperature the thermodynamic average of the order parameter is zero, indicative of complete loss of long-range order. Below Tc the value of is nonzero indicating long-range order, and follows a power law when approaching the critical temperature ... 

Figure 13 shows the optical birefringence measured on the (OOl)c and (llO)c faces [19]. According to the indicatrix deformation approach [25], the optical birefringence, An, for the (llO)c face can detect the antiferrodistortive transition, the square of long-range order (P>, and its fluctuation term (SP) in STO by ... 

We have used here that X is either CO or a vacant site and (CO) + ( ) = . The main problem with this mean-field approximation is that at temperatures below the order-disorder transition there is a strong correlation between the occupations of sites. As will be shown in Section 4.3 (pco,co for neighboring sites is about 24 kJ/mol, which corresponds to a thermal energy at 2880 K. This means that neighboring sites will not be occupied simultaneously at any realistic temperatures, and a more sophisticated approach is needed that describes this well. For weak interactions eqn. (11) may be acceptable, but one should be aware that there is some correlation in the occupation of sites even if there is no long-range order. 

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