Simplified propagation procedure

As can be seen from the discussions above, the time has not played an important role in the theory. The quantities in which we are most interested are functions of the energy. The time dependence of the wavepackets serves only as a tool for finding the experimentally determinable energy dependent quantities such as reaction cross sections in which we are interested. In the theory outined below the TDSE is mapped onto another equation whose solution is much simpler but which serves the same purpose, in that it permits the calculation of the same energy dependent quantities in which we are interested. 

When we apply this mapping, the TDSE becomes 

We now choose the mapping /(x) in such a way as to simplify the propagation equation, eq. (14.3). A suitable choice is 

With this mapping the iterative equivalent to the TDSE, eq. (14.3) becomes 

Acknowledgement. The main contributions to this work were made by Dr. S. K. Gray (Argonne National Lab.), Dr. A. R. Offer, Professor F. Gogtas, Professor R. N. Dixon, Dr. C. C. Marston, and Professor A. Lagana. 

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Eq. (14.3) is still complicated and just as difficult to use as eq. (1.2). A functional mapping is discussed below which circumvents this difficulty and dramatically simplifies the propagation procedure. 

The rate of copolymerization in a binary system depends not only on the rates of the four propagation steps but also on the rates of initiation and termination reactions. To simplify matters the rate of initiation may be made independent of the monomer composition by choosing an initiator which releases primary radicals that combine efficiently with either monomer. The spontaneous decomposition rate of the initiator should be substantially independent of the reaction medium, as otherwise the rate of initiation may vary with the monomer composition. 2-Azo-bis-isobutyronitrile meets these requirements satisfactorily. The rate Ri of initiation of chain radicals of both types Ml and M2 is then fixed and equal to 2//Cd[7], or twice the rate of decomposition of the initiator I if the efficiency / is equal to unity (see Chap. IV). The relative proportion of the two types of chain radicals created at the initiation step is of no real importance, for they wall be converted one into the other by the two cross-propagation reactions of the set (1). Melville, Noble, and Watson presented the first complete theory of copolymerization suitable for handling the problem of the rate. The theory was reduced to a more concise form by Walling, whose procedure is followed here. 

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