What is it?
A moire pattern is a superlattice(superpatern?) formed when overlaying similar patternson top of one another. The step of a new pattern is larger than or same as that of any of the original structures. This is related to the concept of least common multiple. Consider two patterns with step 2 and 3. Upon overlaying the two structures, a new pattern with a step of lcm(2,3)=6, will appear(See image 2). The easiest way to make a faitly interesting moire structure is to overlay two patterns of parallel lines.
Calculations
To find out a feature of a simple moire pattern, a few lines of math caculation are needed. I got the idea to define the new step 'P' with turned angle 'A' from Wikipedia, but couldn't understand the math perfectly, so did the calculations on my own. The original patterns are identical structures of pararrel lines, as shown in [Image 1]. Imagine a situation where one of the patterns is rotated in a degree of A. The length of the new step 'D' will be defined as the distance between two nearest rhombuses(See image 3).
Using the pythagorean theorm, D^2 = P^2 + {P/sin(a)+P/tan(a)}^2. By modifying this, the following steps can be derived.
When A<π/6, sin(A)≅A and cos(A)≅1. As a result, in angles smaller than 30°, relation D=2P/A will be true, which means that a denser moire structure will form the bigger the tilted angle is.
In real life and science uses
One representative use of moire structures in science research is to observe Hofstadter's butterfly in experiments. Hofstadter's butterfly is a is a mathematical model explaining the 'allowed' energy levels of electrons in a crystal lattice in the presence of a magnetic field. When drawn, with x and y axis for energy level and applied magnetic force, the graph is in the shape of a fractal, which resembles a butterfly(See image 5). This is why the model was named 'Hofstadter's butterfly'.
According to Hofstadter's butterfly model, an extreme amount of magnetic force is needed to observe the butterfly in experiments in a world where the distance between atoms are less than a nanometre long. In order to solve this problem, scientists used moire sturcture to increase the distance of microscopic steps. When overlaying graphene with boron nitride, a moire structure with a step larger than the originals' appear. This is because although boron nitride and graphene are similar in shape, the size of boron nitride is slightly smaller. The newly formed moire structure makes it seem that the pattern/"atom" is larger in size. This may sound like a joke, but it actually works in experiments. Scientists were able to identify Hofstadter's butterfly in real-world experimental data.
In real life, moire structures can be seen in fences, nets, or clothing.
'Math' 카테고리의 다른 글
| Riemann Zeta Function(리만 제타 함수) (0) | 2021.02.06 |
|---|---|
| The Nothing Grinder(aka. The Trammel of Archimedes) (0) | 2020.11.07 |
| Matrices - the basics (0) | 2020.04.11 |
