Rotational product distribution prior

The other difference between the usual prior distribution and PST is that the former includes all of the product rotational degrees of freedom. Because of detailed angular momentum conservation in PST the rotational product energy distributions are no longer so simple. The details are explained in the section on PST. However, it is useful to summarize the results here. In the limit of the total J the full... 

Because the prior distribution is not imiform, the characterization of a dynamical bias is a distribution of final states that is deviant Ifom the prior distribution. This, by itself, is already an insightful point. Because the total eneigy is conserved, when we specify, say, the vibrational state of the product we at the same time implicitly specify limitations on what other states are accessible. This is particularly the case for the vibration because it can take up large chunks of the total energy, but it is equally true if we were to specify, say, a rotation. There are 2j + 1 quantum states of a given j, but in the absence of a dynamical bias the rotational state distribution will not quite go as Ij + 1 because of the implications of the conservation of energy. 

By using a time-of-flight method to distinguish between the C, C2 and C3 components of a thermal (2550 K) carbon beam, it was possible to measure the visible emission from the CN (B2S+) product of the four-centre exchange reaction C2 + NO [614], Vibrational states up to v - 4 are populated and can be fitted by a temperature of TvIb 6900 ( 700) K. The rotational excitation of the CN (B2 +) decreases as the vibrational excitation increases (7000 > Trot > 3500 K for 0 < i> < 4). An information theory analysis of the data shows agreement of the experimental distributions with the prior forms. 

The prior distribution is determined by the available phase space of the products. The fraction of the products that are formed with translational energy between E and E + dE, with total rotational energy between E and E + dE, and with vibrational... 

Figure 9.6 The effect of NO2 parent total angular momentum (/ mq ) on PST calculations of NO product rotational distributions at = 1949 cm T Two prior distributions (solid lines) are included which differ by their translational degrees of freedom (x in Eq. 9.26 is V2 or 0 for 3-D or 2-D distributions, respectively). This figure was adapted from Reisler and coworkers (Hunter et al., 1993).

Now that we have expressions for the rotational degeneracies, the PST PEDs are obtained in the same manner as those for the prior distributions. That is the PED for products having angular momenta 7, and 72 is given simply by... 

The rotational PEDs for the dissociation of state-selected NO2 have been measured and analyzed using both the prior model and PST (Robie et al., 1992 Hunter et al., 1993). A convenient approximation is to assume that the product energy distributions are independent of the NO2 M and K quantum numbers. Three product angular momenta, 7no> - o> and must be combined to add up to the total angular momentum, convenience, the NO ii value is included in the Jj q term thereby adding either 0.5 or 1.5 to the rotational quantum number. The angular momenta are combined in two steps. The intermediate J is introduced which is defined by the vector addition J = Jno + that... 

A major difference between the prior and PST product energy distributions is that in PST the rotational degeneracies of the 0 atom and NO are intimately intertwined. On the other hand, in the prior distribution, the two terms are independent of each other so that the rotational degeneracies of the O atom and NO are simply multiplied together as in Eq. (9.24). 

Our purpose is to characterize the population distribution of the final states after the colhsion. We begin with the prior distribution, a metiiod where simplicity is achieved by failing to properly conserve angnlar momentum. Next, the very same considerations are applied bnt with conservation followed by a refinement of the description, where the rotational angnlar momentum of file reactants (or products) is explicitly treated, which brings ns to the level of file phase-space theory. 

Here we examine what modifications are required in the definition of the prior distribution when we explicitly take into account the conservation of angular momentum in addition to the conservation of energy. These modifications are required when, say, high final rotational states are being populated, which requires conversion of much angular momentum of initial relative motion to internal angular momentum of the products. As discussed in Section 10.2.2, when an... 

When the heating cycle is complete the mould is either cooled in situ with water sprays or moved by block and tackle to a second rocking bed for cooling so that the first machine can immediately be re-loaded with a second mould. The operator can control the direction and speed of rotation of the mould in order to achieve optimum results, and several experimental cycles are often necessary before a satisfactory product can be made. The wall thickness of the product tends to be greatest in the hottest areas of the mould. It may even be necessary to attach an insulating material such as glass fibre to certain outer areas of the mould to achieve the desired heat distribution. Similarly, it may be necessary to preheat certain detailed areas of the mould, such as rims and valve areas, with a gas flame prior to moulding to obtain a sufficient thickness. 

Let us briefly discuss this approach. Its idea is the comparison of the experimentally obtained distribution with the so-called the prior distribution, which would be obtained under the condition of the equiprobable energy distribution over all degrees of freedom of the products. Prior distributions, which we designate as p° ( V) and / (n), have a maximum entropy and, hence, give the minimum of information about the dynamics of the reaction. The real distribution obtained in experiment has lower entropy than that of the a priori one. As model trajectory calculations and analysis of data of numerous experiments show, the distribution functions over rotational P N) and vibrational P n) states can be expressed through the corresponding a priori distributions as follows ... 

---