Alignment factor

From a synthetic point of view, the regioselectivity and stereoselectivity of the cyclization are of paramount importance. As discussed in Section 11.2.3.3 of Part A, the order of preference for cyclization of alkyl radicals is 5-exo > 6-endo 6-exo > 7-endo S-endo > 1-exo because of stereoelectronic preferences. For relatively rigid cyclic structures, proximity and alignment factors determined by the specific geometry of the ring system are of major importance. Theoretical analysis of radical addition indicates that the major interaction of the attacking radical is with the alkene LUMO.321 The preferred direction of attack is not perpendicular to the it system, but rather at an angle of about 110°. 

The fact that in ions 402 and 404 there occurs a 1,2-shift of thea-bond 1,8 and not of the a-bond 1,2 seems to be due to the primary homoallylic interaction of with the 0-bond 7,8 which makes the position of the p-orbital at C more favourable for an overlap with the o-bond 1,8 rather than 1,2 (the sp -alignment factor ... 

NOTE CONSIDER THE FOLLOWING FACTORS BUT DO NOT ATTEMPT TO MATCH UP- OR ALIGN FACTORS WITH INFORMATION IN COLUMNS... 

Relying on our experimental data and on the limited calculations performed to date, we deduce an alignment factor A ) of 0.32 for the high J limit. 

A quantitative measure of the degree of ordering was obtained by calculating the alignment factor, Ai(q), as proposed by Walker et al ... 

In the analysis by Walker, it was shown that Af(q) asymptotes to a constant value at q values greater than 0.07 A . They denote this asymptotic value as the macroscopic alignment factor A which is related to the effective order parameter S by the relation -A = Sm. In the current analysis, the intensity was averaged over q = 0.06-0.12 A, not the whole q range. Therefore, the effective alignment factor reported for these experiments is not a measure of the order parameter of the systems, but is a measure of the orientation of the sample. However, as the relative changes in alignment that occur due to shear rates, temperature, and solvent are of interest, this approximation is justified and does not influence the int pretation of the data. 

The element amplitudes are tinoe invariant, but the element phase angles undergo random fluctuations in the range -Jt < (pi < Jt at a constant phase failure rate, But there is also an interaction between the elements that tends to align their outputs and which acts as follows Each element sees the combined output of the other n- elements, called the interaction, and, the phase of each element is aligned with the interaction phase at a rate, the phase repair rate, p, which is determined by the product of the interaction amplitude, E, and a constant alignment factor, po- Consequently, the distribution of the phase of an element, (pi, is given by... 

There are two features of our system that should be mentioned. Firstly, the two probabilities X and p are measured in units of per unit time, but the unit of time is arbitrary. Thus, we may choose a fixed value of X, and then consider only the one parameter p/X. The fixed value of X must be small enough to make the step-wise calculation a good approximation of the continuous temporal behaviour of the system, say, >, < 0.01. Secondly, the parameter p is given as the product of the interaction amplitude, which again is determined by the parameter no, and the alignment factor, po. Now, it is not a priori clear the behaviour of the system depends only on the product of these two rather than on their values individually on the contrary, from our understanding of sampling theory we would expect the... 

The first part of the investigation is concerned with the statistics of Tp, and consequently runs the program in the second mode. Figure A4.5 shows the result of thousand runs for a particular case (51 elements, 20 nearest neighbours , amplitude variation = 0, failure rate 1 = 0.005 per time step, alignment factor po = 0.005 per time step, and influence factor e = 0), and the values for the mean and standard deviation of Tp are 33,279 time steps and 39,839 time steps, respectively. 

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