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ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2007/section-a/a3/c/solution

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Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A3 / c / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

The polymer overlap concentration corresponds to about one chain per coil volume. Approximating the occupied volume of one statistical segment by

a^{3}

gives

ϕ^{*} \sim \frac{N a ^{3}}{R _{θ}^{3}} = N^{- 1/2} = 0.0316.

(1)

Thus the site-volume model predicts overlap at a few percent volume fraction. A separately specified molecular segment volume or a coil-volume prefactor would change this numerical prefactor.

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