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# The mean square end-to-end vector

A rough measure of the average size of the polymer is given by the mean square end-to-end vector , which we shall calculate in this section. Related properties are the radius of gyration , and the persistence length . Both of them may be calculated using methods similar to the ones in this section.

The end-to-end vector is given by (1.27) = = = (1.28)

Again assuming the chain is infinitely long, we may put independent on i. Then = = (1.29)

where (N-n) is the number of times the distance n may occur along the chain.

We now set forth to calculate . In order to do so we need to calculate the scalar product as a function of the angles . To this end we associate with every monomer i a Cartesian coordinate system . Every vector may then be expanded like (1.30)

The precise definition of the local coordinate system is given in Appendix A. Here we only mention that =  = (1.31)

A particular example of Eq. (1.30) is (1.32)

The matrix is calculated in Appendix B.

The scalar product now reads = = = (1.33)

and in general (1.34)

from which we get (1.35)

We finally calculate the remaining average using the methods of the last section     (1.36)

where again we have omitted the subscript ''max''. We may write this in a concise form like  (1.37)

where E3 is the 3-d unit matrix. In terms of direct products of matrices this reads (1.38)

 S = = = (1.39)

Introducing everything into Eq. (1.35) we get (1.40)

For infinitely long chains we can analytically sum, obtaining (1.41)

where E9 is the 9-d unit matrix. In this derivation of Eq. (1.41) we have made use of    for large N.

Similar equations, but much more complicated, may be derived for . For these and other equations we refer to P.J. Flory, Statistical Mechanics of Chain Molecules .     Next: The radius of gyration Up: The Rotational Isomeric State Previous: Some probabilities
W.J. Briels