Pairs of steps

Fig. 4.3. Torsions angles at bonds of length that change the angle between a pair of steps of length L. The two torsions on the left determine the angle i-2, i-1, i, and the two torsions on the right determine the angle i-1, i, i+1...

Figure 2. Monte Carlo configuration of a pair of steps below the roughening transition temperature of the standard SOS model, at t = 5000 MCS. using evaporation kinetics. The initial width /,o of the center terrace is six lattice spacings.

A series or bunch of m initially straight and parallel steps, between heights 0 and m, may be expected to relax with the same asymptotics as a pair of steps. Modifications may occur, already for a pair of steps, when step-step interactions are present in addition to the entropic step repulsion. Here, we merely refer to recent reviews on experiments and theoretical analyses " on the much studied phenomenon of step bunching for vicinal surfaces, which is accompanied by interesting phase transistions. 

Above roughening, the simulations confirm the classical continuum theory. The width of the bumps spreads with time as fwhere b= Ml for evaporation-condensation, and b= 1/4 for surface diffusion. In the latter case, the profile shows an oscillatory behavior away from the foot of the bump, as for pairs of steps and wires. ... 

In particular, the equilibration of isolated and pairs of steps as well as ensembles of steps as realized in gratings, wires and bumps has been discussed, for different types of kinetics such as step diffusion, evaporation-condensation and surface diffusion. 

The computation, which results in Figure 8.5, which gives the result of a single Impulse, provides input into a calculation of the overall result. In the overall process we Kffeegin with an incoherent mixture of ND molecules of type D and NL molecules of 8i e L- In the first step the system is excited, as above, with a laser pulse sequence, jgl the second step, the system collisionally and radiatively relaxes so that all the pulation returns to the ground state to produce an incoherent mixture of IEl) and ip). This pair of steps is then repeated until the populations of EL) and ED) reach fhvergence. 

The rate equation 6.12 was derived without use of any short-cuts. With rules to be shown in Section 7.3.1, algebra could have been reduced. Specifically, the steps following the irreversible formation of the cobalt acyl (X5) could have been omitted (Rule 7.12) and the pairs of steps from X, to X3 and from X3 to X5 been consolidated each into a single step (Rule 7.24). This would have reduced the mathematics of conversion of the x-complex to the aldehyde to that of a pathway with only two instead of six steps. 

The model used is a walk on a diamond lattice, in which trans, gauche and gauche conformations have equal probabilities of 1/3. To obtain a simple result, the self-avoiding condition is relaxed, so the walks can intersect with themselves. The end-to-end vector r is the sum of the step vectors h. The mean square length of the walk is calculated from r, the scalar product of to with itself. The terms in the expansion are grouped according to the distance between the pairs of steps... 

An alternative form of positive feedback is indirect autocatalysis. Here, although / and Jjj are both negative, the combined effects of species i on species j and of j on i are such that a rise in one of these concentrations ultimately leads to an increase in how rapidly it is produced. Mathematically, we require that JyJji > 0. There are two possibilities competition, in which i inhibits y, and j inhibits i, so that both Jacobian elements are negative and symbiosis, in which the two species activate each other, resulting in positive Jy and /y,. A simple example of symbiosis arises from introducing an intermediate Y and breaking down reaction (5.19) into a pair of steps ... 

In the Frank theory, growth takes place on a substrate of length L. Each nucleation event ereates a pair of steps that travel to the left and to the right with a spreading rate g. There are on the average l x) of the former and r x) of the latter. Since no steps enter from outside the limits x = zt L... 

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