## Phase transitions in the spherical model of a lattice gas and related modelsShow full item record

Title | Phase transitions in the spherical model of a lattice gas and related models |
---|---|

Author | Carpenter, Kenneth Halsey |

Date | 1966 |

Genre | Dissertation |

Degree | Doctor of Philosophy |

Abstract | The spherical model in lattice statistics is obtained by replacement of occupation numbers n_j =0,.1 assigned to lattice sites by a continuous range -infinity < n_j < infinity but subject to the spherical constraint Summation from j=1 to K of (2n_j -1)^2 = K , where K is the total number of lattice sites. First the spherical model or a lattice gas in one dimension is derived to illustrate the methods used in obtaining the thermodynamic properties of these models. Then the derivations are generalized to n dimensions and properties are found in a form applicable to any number of dimensions with any interaction potential U(r) which the model can accomodate. A phase transition occurs in the model when in evaluation or the partition function the saddle point for a steepest descents integration "sticks" at the branch point of the integrand. This "sticking" will occur provided the Fourier coeffic1ents of the potential, [Diagram] vanish as w^q with q < n. Thus a transition occurs for most potentials in 3 dimensions, but infinite range potentials are require d in 1 and 2 dimensions. The internal energy per particle. E, and the specific heat at constant volume. C_v, are derived, and it is proved that no infinities can occur in C_v for any potential in any number dimensions. The pressure is found to be constant with specific volume in the transition region--this 1B one of the most desirable features of the spherical lattice gas. However. the pressure becomes negative for large specific volumes at all temperatures. and for small specific volumes at low temperatures. The equations of the P-V isotherms, vapor dome, and coexistence curve are obtained, and the correlation function is derived in a general form applicable to all spherical lattice gases. An investigation is made of the spherical lattice gas in one dimension using potentials U(r) = -gr^-alpha. A phase transition is found to occur provided 1<alpha<2. The critical temperature varies continuously from T_c to as alpha to 2 to T_c to infinity as alpha to 1, which suggests the interpretation that stronger attraction (i. e., alpha smaller) requires higher temperatures to prevent clustering and a transition. Similar considerations in two dimensions \ , d show that a transition occurs for potentials U(r)=-gr^-alpha with 2<alpha< 4. The BWG Imperfect Gas Model, a cell model similar to the , spherical model but having a variable cell size equal to the specific volume, is presented and the specific heat C_v derived. It is proved that C_v has no infinities for any interaction potential in any number of dimensions. The negative pressure at large specific volumes in the spherical lattice gas can be removed by a modification of the model. It is shown that the modification required is equivalent to the replacement of the spherical constraint on the occupation numbers. which gives the name "spherical model." by a properly chosen elliptical constraint. yielding an "elliptical model." This modification of the model causes no qualitative changes except for elimination of the negative pressure at large specific volumes. Consideration of the requirements for a transition in one dimension in the spherical model due to a "sticking" saddle point leads to the conclusion that the potential U(r) must be predominately attractive and have an asymptotic behavior between r^-1 and r^-2 for large r. An attempt to obtain a continuum gas model from the spherical lattice gas by letting the unit cell volume approach zero fails. The mathematical manifestation of the phase transition, the "sticking" saddle point. is retained, but the physical interpretation cannot be made since the thermodynamic properties diverge due to explicit dependence on the cell size. Finally. the spherical model is applied to the Fermi gas. No transition is obtained for a simple interaction, and the spherical solution is not a good approximation to the exact solution for the ideal Fermi gas. The calculation is of interest as one possible attempt to apply the methods of the spherical model to other problems. It is concluded that although the spherical model fails to give any infinity in C_v which can be compared to the logarithmic singularity in C_v at the critical point in the two dimensional lattice gas. or to the apparent singularity found experimentally in oxygen and argon. the spherical model is still significant and worthy of further study due to the insight gained into the mathematical description of a phase transition. In particular. the one dimensional transition is significant because it results from a potential that is not unreasonable physically. and the transition does not require a limiting process in an exponentially decaying potential as had been the case for one dimensional transitions reported previously in the literature. |

Link | https://repository.tcu.edu/handle/116099117/34158 |

Department | Physics and Astronomy |

Advisor | Strecker, Joseph L. |

##### This item appears in the following Collection(s)

- Doctoral Dissertations [1446]

© TCU Library 2015 | Contact Special Collections |

HTML Sitemap