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where H is the Jaynes-Cummings Hamiltonian, (8.125) and (8.126) is the Lindblad term describing polarization decay. Before tackling our optical micrornaser, it is instructive to look at the effect of polarization decay on spontaneous emission. Without polarization decay, an initially excited two-level atom in a single-mode (perfect) cavity undergoes vacuum Rabi oscillations; that is, the expectation value of its inversion is given by

Cp2Zr(CH3)2 + B(C6F5)3

(o-z(t)



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Unlike the case for MAO, one needs only an equimolar amount of the coinitiator to activate the initiator. In some reaction systems, higher polymerization activity is observed with an excess of initiator relative to coinitiator. Maximum activity is observed in some polymerization systems at a 2 : 1 initiator : coinitiator ratio, which indicates that bimetallic species such as XXXVIII are the actual initiating species [Wang et al., 2003]. A unique feature of the use of organoboranes and organoborates is that the complexes, both monometallic such as XL and XLI and the corresponding bimetallic complexes, have been isolated and characterized in a number of systems. The high activity of these systems results from having a large, weakly coordinating anion to stabilize the metallocenium cation. The anion must be suf ciently weak in coordinating to the cationic active site so as not to compete with monomer for coordination to the active site. The order of coordinating ability for anions is [CH3 MAO] > [(CH3 )B(C6 F5 )3 ] > [B(C6 F5 )4 ] . [B(C6 H5 )4 ] is too strong a coordinating anion to allow initiation and propagation. Polymerization activity and polymer molecular weight generally increase in the opposite order. Deviations from this order are observed, indicating speci c interactions between some initiators and anions. Also, the effect of anion structure cannot be considered independent of reaction variables such as reactant concentrations and temperature. The effect of anion structure on stereoselectivity is not so clear. Stereoselectivity often decreases with increasing activity, but the trend is not universal and can be circumvented by alterations in the initiator symmetry and ligands as well as reaction variables.





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To sort your data, click the first and click the field by which you want to sort. You can add more sort fields as needed.

I(k', x)

8-5h 8-5h-1

%J :Z,8(x) {I - e- 2LI'} eik'L C(k') {COSk' L - k;L sink'L} + .!...Jnc Il+rI 8(x){1-e- I'} roo dk"IC(k")12cosk"L rrk' t Jo

The rate of polymerization is derived in a straightforward manner. The following derivation describes polymerization with a zirconocene (Zr) and MAO, but is general and applies to any metallocene coupled with any coinitiator. Reaction between the initiator zirconocene and MAO forms the active initiating species I*, which subsequently adds monomer to yield the initial propagating species M*:

k'~L sin k" L }

I* M*

8-57 8-58

+ kIf _

Propagation proceeds by successive additions of monomer to the propagating species:

"L -

8-59

L} 1 .

For the case where Reaction 8-58 is irreversible and fast (i.e., [M*] [I*]), the polymerization rate is

Note: Fields are sorted in ascending order by default. Click the Ascending button to toggle to descending order.

Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.

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Rp K1 kp Zr MAO M 8-60

There are many different modes, each of which will have a different resonant frequency corresponding to some particular complicated arrangement of the electric and magnetic fields. Each of these arrangements is called a resonant mode. The resonance frequency of each mode can be calculated by solving Maxwell's equations for the electric and magnetic fields in the cavity. -Richard Phillips Feynman [206]

which shows rst-order dependencies of Rp on zirconocene, MAO, and monomer. If Reaction 8-58 is reversible and slow, [M*] 6 [I*], the polymerization rate is

In Section 2.2 we showed that Maxwell equations involve only fields at the same point in Fourier space. Electrodynamics then becomes extremely simple. But that was all done in free space. For a cavity, the fields have to satisfy the constraints imposed by the boundary conditions on the walls. Here we show how the Fourier-

Rp K1 K2 kp Zr MAO M 2 8-61

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