The rigidity of a material is the property that enables it to preserve its structure when deformed. In a rigid body, no internal motion is possible since the degrees of freedom of the system are limited to translations and rotations only. In the macroscopic scale, the rigidity and response of a material to external load can be studied using continuum elasticity theory. But when it comes to the microscopic scale, a simple yet powerful approach is to model the structure of the material and its interparticle interactions as a ball$-$and$-$spring network. This model allows a full description of rigidity in terms of the vibrational modes and the balance between degrees of freedom and constraints in the system.

In the present work, we aim to establish a microscopic description of rigidity in \emph{disordered} networks. The studied networks can be designed to have a specific number of degrees of freedom and/or elastic properties. We first look into the rigidity transition in three types of networks including randomly diluted triangular networks, stress diluted triangular networks and jammed networks. It appears that the rigidity and linear response of these three types of systems are significantly different. In particular, jammed networks display higher levels of self-organization and a non-zero bulk modulus near the transition point. This is a unique set of properties that have not been observed in any other types of disordered networks. We incorporate these properties into a new definition of jamming that requires a network to hold one extra constraint in excess of isostaticity and have a finite non-zero bulk modulus. We then follow this definition by using a tuning by pruning algorithm to build spring networks that have both these properties and show that they behave exactly like jammed networks. We finally step into designing new disordered materials with desired elastic properties and show how disordered auxetic materials with a fully convex geometry can be produced.

Our group has constructed a ring-imaging Cherenkov (RICH) detector with the goal of testing the performance of aerogel tiles in charged particle detectors. In previous work, tiles produced by Aspen Aerogels were tested as radiators in Cherenkov threshold counters and compared to commercial-grade samples. As an extension of this work we built a counter of the RICH type, which is used in practice to extract more particle identification information than threshold counters, and we have studied the images resulting from various aerogel samples.
The detector was designed for use in table-top experiments in which our particle source would be cosmic rays. Due to the vast energy range of cosmic rays, the window in which we can discriminate velocities is relatively small. Since the particles we do detect generally have β≈1, the relativistic limit β→1 motivates imaging by the Focusing Aerogel RICH (FARICH) technique, in which photons from multiple tiles are focused together at a detection plane.
Our detection plane is an array of flat-panel, multi-anode photomultiplier tubes (PMTs). Readout consists of multiplexing the anode outputs, recording the digitized signal, and converting this into a matrix of integrated charge values. The charge distribution in that matrix should directly imply the particle's speed; however, in practice, final recorded images are the influenced by many intermediate processes, so we have attempted to make meaningful measurements by averaging over numerous events.
For a given configuration and data collection, we produce the spatial distribution of observed signals relative to the cosmic ray's point of impact. These distributions have the expected form of a ring and their characteristics compare favorably with the predictions of geometric optics. Our confidence in the images is increased by verifying that changes to the configuration are reflected by the changes in the rings. We find that FARICH improves the sharpness of our ring images, but tiles must be used individually for actual aerogel analysis. So far we have shown that the Aspen tiles behave as one would expect for the purposes of RICH. Their images do resemble those produced by commercial-grade tiles, but we do not have tiles sufficiently similar for side-by-side comparison. A method of quantifying tile performance has proven difficult and is the only remaining task for our group.