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Stereographic Projections and Crystal Impurities

 A stereographic projection takes a point on a sphere and projects it onto a plane tangent to a "pole" on the sphere. Commonly, a line originating from the "north pole" passes through an arbitrary point P on the surface of the sphere and the line extends to a plane tangent to the "south pole" of the sphere. As the point P approaches the north pole of the sphere, the line from the north pole through the point P which extends to the plane tangent to the south pole increases in length. Conversely, as the point P approaches the south pole, the line passing through P which extends to the tangent plane decreases in length. 

An image taken from Wikipedia, created by Mark Howison shows lines projecting from the north pole of a sphere passing through various points on the sphere which all lie on the same height. The lines are equal in length, and their projections form a circle on the tangent plane. 


Interestingly, the projection is conformal, which is to say that the angles between the lines projected from the sphere are equal to the angles the lines form as they are projected onto the plane. However, the area of a region on the surface does not equal the area as projected on the plane! This is why the area of a country on a map can be much different than its real area on the Earth. It's a conformal projection!

Likewise, because of the sphere projects onto a 2D surface, "walking" along this surface feels the same as "walking" along a truly flat surface! Someone "standing" on the surface would not be able to distinguish the two from each other. 

The stereographic projection has many applications outside of mathematics and cartography though. In X-Ray Crystallography, the scattering of x-rays from a crystal form a sphere as the detector sweeps through space. 

As an x-ray beam hits a molecule, the electrons of the molecule scatter the x-rays. The more electron density, the more a beam will scatter, and the atoms and bonds of a molecule will be seen as clumps of electron density. However, the picture is a tad more complicated. The diffraction of the beam creates a periodic set of points called the reciprocal lattice. These points form all around the crystal. A special set of points surrounding the crystal form a sphere, known as the Ewald Sphere. They satisfy a special set of equations that make them visible to a detector, but this is far beyond the scope of this blog post.

The diffracted beams which pass through this sphere project onto the detector. This is the same as a stereographic projection! Below is a 2D schematic of a beam diffracting through a point on the Ewald Sphere and projecting itself onto a detector plate. This was produced by Dr. Sean Parkin. The blue points are the reciprocal lattice of the crystal. 


Unfortunately, not all crystals are perfect. Crystals will have impurities, residual solvent entrapped, cracks on the inside, and sometimes even multiple polymorphs! This changes how the reciprocal lattice forms as the beam diffracts from the crystal, and ultimately the projection onto the detector plate. Crystal cracking, which is known as mosaic spread is the simplest case of this distortion. In the above image, the reciprocal lattice points are perfect infinitesimal dots. Through, with cracks in the crystal, the points on the Ewald Sphere become more like smudges as they are projected onto the detector. The extent of how much of a smudge a point becomes, depends on how severe the crack is. 


For my object, I chose to roughly model how a diffraction pattern of Benzene would be distorted if there were cracks in a Benzene crystal. 


I illustrate this concept by creating a lattice of hexagons on the sphere. These hexagons represent the points which form from the uncracked portions of the crystal. Interspersed in the lattice are greatly elongated hexagons which were created from the cracks in the crystal. These hexagons became smudge-like due to the improper diffraction of the x-ray beam. Towards the bottom of the sphere, the hexagons are much smaller to represent weaker diffraction patterns. 

When light is projected through the surface, intense, large hexagons will be seen towards the middle to respresent how diffraction occurs most strongly through the center of the crystal and smaller hexagons will exist towards the edge to show how diffraction is weaker through the parameter of the crystal. The "smudges" are interspersed through the "lattice" to represent the effect of cracks. 

I also made a "perfect" crystal that does not have the elongated hexagons to show what a more ideal diffraction would look like. 


When light is projected through this, the big hexagons will be seen in the middle, and the smaller hexagons will be seen in the parameter. 

I will attempt to print both in order to illustrate both concepts. The radii of both spheres are 10 units, and will be printed to have an radius of roughly 2 inches.

The concept of x-ray crystallography is much more involved, but the concept of stereographic projection is a good segue into one of the most fascinating fields of chemical analysis. 











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