The equation for a diffraction pattern is
\[2d\sin{\theta}=\lambda\] where d is the, \(\theta\) is the angle of the image with the detector, and \(\lambda\) is the wavelength. An oversampling ratio is found by \[\frac{\lambda}{O\,a}=\frac{\Delta p}{z}\] where \(\lambda\) is again the wavelength, \(O\) is the linear oversampling ratio, \(a\) is the size of the sample, \(\Delta p\) is the difference in length in pixels along the detector, and \(z\) is the distance between the sample and the detector.
Since we are dealing with far-field diffraction patterns, we want to use Fraunhofer diffraction equation \[FN = \frac{x^{2}}{z\lambda}\] where \(FN\) is the Fresnel number, \(x\) is the distance along the detector, \(z\) is the distance between the sample and the detector, and \(\lambda\) is the wavelength. To get a far-field diffraction patter, \(FN << 1\) [4].
To find the find the focal point of ours lens and the proper placement of our sample and detector, we use the equation \[\frac{1}{f_{0}}=\frac{1}{d_{0}}+\frac{1}{d_{1}}\] where \(f_{0}\) is the focal length of the particular lens, and \(d_{0}\) and \(d_{1}\) are the distances of the sample and the detector relative to the lens [2].
In this experiment, there was also an attempt to perform a three dimensional reconstruction. This was done using Equally Sloped Tomography method (EST). This will be elaborated on in the 3D reconstruction section in the results.