Mathematical Models of Phase Transitions in Quartz

这笔拨款用于石英α - β相变的数学研究项目。在无应力的石英中，这种转变发生在大约574摄氏度，此时石英晶体的形状、体积和对称性都发生了变化。高温相比低温相具有更大的体积和更高的对称性。α - β转变可以伴随着一个有趣的三角形微观结构，这与通常在形状记忆合金和铁磁材料中发现的层状微观结构大不相同。这个问题的一个有趣的数学特征是微观结构是用标量阶参量的振荡来描述的。序参量已用于许多其他的相变研究，但它们通常难以测量，并且用相当模糊的物理术语来定义。另一方面，石英中的有序参数在物理上有很好的定义，并且有很好的测量技术。因此，它为使用顺序参数的数学模型提供了一个极好的测试用例。提议者计划使用微分方程、变分演算、科学计算和分岔理论等技术来研究石英的标准模型和他们将提出的模型。当一种材料的基本性质发生剧烈变化时，我们说它经历了“相变”。最明显的例子是冻结和融化，蒸发和凝结，但更微妙的原子重排也包括在这个术语中。尽管这种材料变化在技术上具有重要意义，但描述它们的数学理论远不如描述材料运动、变形和热性能的理论发达。这笔经费用于石英相变的数学研究项目。这种转变是由温度变化引起的，发生在574摄氏度左右。在这个温度下，石英晶体的形状、体积和对称性都会发生变化。这种转变可以伴随着一个有趣的微观三角形图案，这与通常在形状记忆合金和铁磁材料中发现的分层图案大不相同。对这些微观模式（称为“微观结构”）的理解被广泛认为是理解伴随它们的相变的关键。关注这种特殊的相变有很多原因。石英有许多重要的工业应用，如振荡器和光波导。此外，石英中的转变通常用一个称为“序参量”的量来描述。序参量已用于许多其他类型的相变，但它们通常难以测量，并且以相当模糊的物理术语定义。另一方面，石英中的有序参数在物理上有很好的定义，并且有很好的测量技术。因此，石英为使用有序参数的数学模型提供了一个极好的测试用例。＊＊＊

DMS-9704621 Rogers This grant funds a program of mathematical investigations of the alpha-beta phase transition in quartz. In unstressed quartz this transition occurs at about 574 degrees Centigrade, at which point a quartz crystal undergoes a change in shape, volume, and symmetry. The high temperature beta-phase has larger volume and higher symmetry than the low-temperature alpha-phase. The alpha-beta transition can be accompanied by an interesting triangular microstructure that is quite different from the layered microstructures that are usually found in shape memory alloys and ferromagnetic materials. An interesting mathematical feature of the problem is that the microstructure is describe by the oscillation of a scalar order parameter. Order parameters have been used in many other studies of phase transitions, but they are usually hard to measure and are defined in fairly fuzzy physical terms. On the other hand, the order parameter in quartz is well defined physically and there are good techniques for measuring it. Thus, it provides an excellent test case for mathematical models using order parameters. The proposers plan to use techniques of differential equations, the calculus of variations, scientific computation, and bifurcation theory to study both standard models for quartz and models that they will propose. %%% When a material undergoes a drastic change in its fundamental properties we say it has undergone a "phase transition." The obvious examples are freezing and melting, evaporation and condensation, but more subtle atomic rearrangements are covered by the term as well. Despite the technological importance of such material changes, the mathematical theory describing them is nowhere near as well developed as theory describing the motion, deformation, and thermal properties of materials. This grant funds a program of mathematical investigations of a phase transition in quartz. The transition is caused by changing temperature and occurs at about 574 degrees Centigrade. At this temperature, a quartz crystal undergoes a change in shape, volume, and symmetry. The transition can be accompanied by an interesting microscopic triangular pattern that is quite different from the layered patterns that are usually found in shape memory alloys and ferromagnetic materials. An understanding of these microscopic patterns (called "microstructures") is widely believed to be a key to the understanding of the phase transitions that they accompany. There are a number of reasons for focusing on this particular phase transition. Quartz has a number of important industrial applications such as oscillators and optical waveguides. In addition, the transition in quartz is usually described mathematically by a quantity called an "order parameter." Order parameters have been used in many other types of phase transitions, but they are usually hard to measure and are defined in fairly fuzzy physical terms. On the other hand, the order parameter in quartz is well defined physically and there are good techniques for measuring it. Thus, quartz provides an excellent test case for mathematical models using order parameters. ***