The Freely Jointed Chain in Three Dimensions

If one end of the freely jointed polymer chain is placed at the origin of a coordinate system, as represented in Fig. 76 and the chain is permitted to assume any con- y 

The Gaussian function (7) is shown graphically in Fig. 77. The most probable location of one end of the chain relative to the other is at coincidence, i.e., at r = 0. The density, or probability, decreases monotonically with r, exactly as noted above for the one-dimensional case. Equation (7) likewise is unsatisfactory for values of r not much less than the full extension length nl. The extent of this limitation will be discussed presently. 

The probability that r assumes a certain magnitude irrespective of 

The radial distribution function TF(r), which is shown in Fig. 78, exhibits a maximum at a value of r greater than zero. This maximum occurs at 

We turn now to the consideration of average values of r. The average of r, unlike that of one of its components, e.g., x exceeds zero inasmuch as r is restricted to values equal to or greater than zero. We have 

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