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The RIS model can be treated in considerable detail. As a first step we
calculate the partition function

(1.3) 
Introducing the energy (1.2) and using a short hand notation for
the summations we get

(1.4) 

(1.5) 
We recognize a sequence of matrix products in Eq. (1.4), which
makes it possible to write

(1.6) 
T is a matrix with elements
.
In the case
of polyethylene it is given by







(1.7) 
where
and
except for
.
The zero of
energy has been chosen such that
.
Eq. (1.6) may also be written as

(1.8) 
Moreover, using

(1.9) 
we may write
Z = Y^{T} T^{N2} X

(1.10) 
with
and
.
It is useful to decompose the matrix T in terms of its eigenvectors:

(1.11) 

(1.12) 
where
is the diagonal matrix containing the eigenvalues of T.
For notational convenience we have introduced
B = A^{1}.
Eq. (1.12) may be written like

(1.13) 
where the A_{i} are the columns of A and the B_{i}^{T} the rows of B.
Then
T^{N} 
= 


T^{N} 
= 

(1.14) 
The partition function then reads

(1.15) 
In practice we are usually interested in the free energy per monomer

(1.16) 
where
is the largest eigenvalue of T. In the
limit of N going to infinity the second term goes to zero

(1.17) 
In the case of matrix (1.7) one easily calculates the
eigenvectors. From
one gets

(1.18) 
Next: Some probabilities
Up: The Rotational Isomeric State
Previous: The model
W.J. Briels