# Thesis 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**.

q_{r} will generally be set at the first Bragg diffraction ring q_{DS}, 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: