### Abstract

A fundamental morphological measure of two-phase heterogenous materials is what we refer to as the lineal-path function L(z). This quantity gives the probability that a line segment of length z is wholly in one of the phases, say phase 1, when randomly thrown into the sample. For three-dimensional systems, we observe that L(z) is also equivalent to the area fraction of phase 1 measured from the projected image onto a plane: a problem of long-standing interest in stereology. We develop a theoretical means of representing and computing the lineal-path function L(z) for distributions of D-dimensional spheres with arbitrary degree of penetrability using statistical-mechanical concepts. In order to test our theoretical results, we determined L(z) from Monte Carlo simulations for the case of three-dimensional systems of spheres and found very good agreement between theory and the Monte Carlo calculations.

Original language | English (US) |
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Pages (from-to) | 922-929 |

Number of pages | 8 |

Journal | Physical Review A |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1992 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

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## Cite this

*Physical Review A*,

*45*(2), 922-929. https://doi.org/10.1103/PhysRevA.45.922