Thesus Redux
I’ve been pretty busy the past half year, and haven’t found time to work on OT. Fortunately, I recently have recovered my workflow and will update my status on OT, as well as a secret project I’ve been kicking around in the back of my head for the last several years.
My college physics thesis was a simulation and analysis of a system of close-packed spheres. Specifically, I was looking what crystalline structure arises in the system as the density of spheres increases, which is a small part of the grand question, “How do objects freeze?” Several weeks ago, I picked up my thesis to see if I could understand it better, and came to the realization that I did nothing correctly. Therefore, I’ve added to my ever-growing list of projects a complete rewrite of my thesis, to recitfy the fact that it was complete bullshit. Furthermore, after discussions with Kevin, we realized there may be a simpler method of system structural analysis than the proposed method, and I wish to expand my analysis to include the new metric, and to see if there are any correlations that may be drawn from the two structure metrics.
The system that I am analyzing consists of a quasi-2D box, packed with some density ratio of spheres; that is, the box is less than twice the diameter of a single sphere, to ensure that there can never be two layers of spheres sitting perfectly on top of one another. I am attempting a static analysis of the system, i.e. we must wait for the spheres to “settle” before feeding the system into the magic number generator black box; in order to achieve this, I used an MD simulation to calculate the final positions of the system of spheres, using the standard method; because there are no doubt a myriad of books devoted to the topic, explaining with words much more elegant than I ever can, an explanation the simulation part of this project is out of scope. Instead, I will be explaining the method used for system structure analysis.
The main underlying motivation for the analysis is a simulation of Bragg diffraction interference patterns. We can simulate and quantify this using the [static] aperture cross-correlation function
C(q, k, t=0):
q and k here are the scattering vectors originating from the pinhole [which is set at the center of the system], and ending at the scattering plane; I(q, t) is the instantaneous intensity of the scattering beam, as observed from the head of q (laying on the scattering plane):
Averaging over the particles contained in the “illuminated” area on the scattering plane (with a given aperture radius on the order of several sphere diameters), C is a metric for determining the “cross-correlational” illumination that occurs at both q and k on the scattering plane. If we hold the radial and vertical components of q and k constant, and vary only the angle at which they are oriented around the origin, we can redefine C as C(φ, t), where φ is the angle between q and k.
qr will generally be set at the first Bragg diffraction ring qDS, which falls out from the system structure factor S(q):
Here, q is the spacial frequency vector, and g is the second-order radial distribution function:
