Geometric frustration in granular packings

我们提出了一个易于设置和分析的实验，以表征受挫物理系统的统计特性，并定性和定量地分析相关参数对这些特性的影响。这是通过对薄容器中单分散球体的光学研究实现的。固体颗粒的堆积问题在许多情况下都起着重要的作用，然而人们对它们的了解通常还不够充分。例如，开普勒关于相同球体密度最大的猜想早在几个世纪前就已为人所知，但直到最近几年才得到证实。在实际情况中经常遇到的一个问题是密集无序（随机）填料。这个问题仍然保留了许多悬而未决的问题，甚至对于单分散球体的简单情况也是如此。如果加上几何限制，通常情况下包装问题会变得更加复杂。我们证明了球形在扁平容器中的密集填料由于受挫表现出复杂的几何性质。这些只能在统计层面上进行分析和描述。新出现的扭曲晶体包的特征不仅在这里研究的特定系统中很有趣，而且它们也与其他物理系统中的几何挫折有关，例如三角形晶格上的反铁磁自旋系统。这里提出的实验非常适合于表征受挫几何。在我们的实验中，我们将用统计方法分析扁平容器中单分散球体填料的基本特性，并推导其一般描述的模型。该提案的两个主要目标是更好地理解颗粒物质的无序堆积，其次是在实验几何的系统变化的基础上推导关于受挫系统性质的一般陈述。特别地，我们将探讨倾斜单元中的对称性破缺。根据上面的类比，直胞对应于反铁磁体的基态，而倾斜胞提升了两个胞面的简并，对自旋系统起着与外磁场相同的作用。

We propose an easily set-up and analyzed experiment to characterize the statistical properties of a frustrated physical system and to analyze the influence of relevant parameters on these properties, both qualitatively and quantitatively. This is achieved by an optical investigation of monodisperse spheres in a thin container.Packing problems of solid particles play an important role in many situations, nevertheless they are in general not well understood. The Kepler conjecture on the densest packing of identical spheres, for example, is known since centuries but was proven only in recent years. A problem that is often encountered in practical situations is that of dense disordered (random) packings. This problem still retained a number of open questions, even for the simple case of monodisperse spheres. If geometrical restrictions are added, packing problems become more complicated in general.We show that dense packings of spheres in a flat container exhibit complex geometrical properties because of frustration.. These can be analyzed and described only on a statistical level. The characteristics of the emerging distorted crystalline packinge are not only interesting in view of the particular system investigated here, but they bear relevance for geometric frustration in other physical systems, too, for example for antiferromagnetic spin systems on a triangular lattice. The experiment proposed here is excellently suitable for a characterization of frustrated geometries.In our experiment we will analyze fundamental properties of the packings of monodisperse spheres in flat containers with statistical methods, and derive models for their general description. The two main goals of the proposal are a better understanding of disordered packings of granular matter, and second the derivation of general statements about the properties of frustrated systems, on the basis of a systematic variation of the experimental geometry. In particular, we will explore the broken symmetry in tilted cells. With respect to the analogy noted above, the straight cell corresponds to the ground state of the antiferromagnet, while the tilted cell lifts degeneracy of the two cell planes and plays the same role as an external magnetic field for the spin system.