Geometric frustration in granular packings

我们提出了一个易于设置和分析的实验来表征受挫折的物理系统的统计特性，并分析相关参数对这些属性的影响，定性和定量。这是通过对薄容器中单分散球体的光学研究来实现的。固体颗粒的堆积问题在许多情况下起着重要的作用，然而它们通常还没有得到很好的理解。例如，关于全同球体的密度填充的开普勒猜想，几个世纪以来就被人们所知，但直到最近几年才被证明。在实际情况中经常遇到的一个问题是密集无序（随机）填充。这个问题仍然保留了一些悬而未决的问题，即使是单分散球的简单情况。如果加上几何约束，填充问题一般会变得更加复杂.我们证明了，由于挫折，在一个扁平容器中的球体的密集填充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。新兴的扭曲的结晶包装的特点是有趣的，不仅在这里研究的特定系统，但他们承担相关的几何挫折在其他物理系统，例如反铁磁自旋系统的三角晶格。本文提出的实验非常适合于表征阻挫几何，在我们的实验中，我们将用统计方法分析单分散球在扁平容器中的堆积的基本性质，并导出一般描述它们的模型。该提案的两个主要目标是更好地理解无序包装的颗粒物质，第二个推导的一般陈述的性质的挫折系统，实验几何的系统变化的基础上。特别是，我们将探讨倾斜细胞的对称性破缺。相对于上面提到的类比，直的单元对应于反铁磁体的基态，而倾斜的单元提升了两个单元平面的简并度，并扮演着与自旋系统的外部磁场相同的角色。

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.