Mathematical Models of Phase Transitions in Quartz

DMS-9704621罗杰斯 该基金资助了石英中α-β相变的数学研究计划。 在无应力的石英中，这种转变发生在大约574摄氏度，此时石英晶体在形状、体积和对称性上发生变化。 高温β相比低温α相具有更大的体积和更高的对称性。 α-β转变可以伴随着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完全不同。 该问题的一个有趣的数学特征是，微结构由标量序参量的振荡来描述。 序参量在相变的许多其他研究中也有应用，但它们通常很难测量，并且定义在相当模糊的物理术语中。 另一方面，石英中的序参量在物理上有很好的定义，并且有很好的测量技术，因此，它为使用序参量的数学模型提供了一个很好的测试案例。 提议者计划使用微分方程，变分法，科学计算和分叉理论的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，我们称之为“相变”。“明显的例子是冻结和融化，蒸发和冷凝，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很重要，但描述它们的数学理论远没有描述材料运动、变形和热性质的理论那么发达。 该基金资助了一个石英相变的数学研究项目。 这种转变是由温度变化引起的，发生在大约574摄氏度。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。 这种转变可能伴随着一个有趣的微观三角形图案，这与形状记忆合金和铁磁材料中通常发现的分层图案截然不同。 对这些微观模式（称为“微观结构”）的理解被广泛认为是理解它们所伴随的相变的关键。 有很多原因需要关注这个特殊的相变。 石英具有许多重要的工业应用，如振荡器和光波导。 此外，石英中的跃迁在数学上通常用一个称为“序参量”的量来描述。序参量已被用于许多其他类型的相变，但它们通常很难测量，并且在相当模糊的物理术语中定义。 另一方面，石英中的序参量在物理上定义得很好，并且有很好的测量技术。因此，石英为使用序参量的数学模型提供了一个很好的测试案例。 ***

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. ***