Shortest Distance Between Two Cubical Voids In Simple Cubic Lattice
In simple cubic lattice, given two cubical voids, find the shortest distance between them. We’re going to use a simple heuristic approach for this problem.
How to find the shortest distance between two cubical voids in a simple cubic lattice
Find the shortest distance between two cubical voids in a simple cubic lattice.
This is quite an easy problem, but it’s good practice to get used to writing down what you’re doing and thinking about your work.
In this problem, you will be given the coordinates of two cubical voids in a simple cubic lattice. Your task is to find the shortest distance between these two voids.
A cubical void is an empty space surrounded by six faces (which are themselves cubes). Each face shares an edge with exactly three other faces; this makes it easy to count the number of steps required to get from one face to another along any given path through all six faces, because each time we pass through one edge, we must go through exactly two more edges before exiting back into our original position at one end or another (as shown below).
This solution assumes that:
- The lattice is simple cubic (i.e., all points can be described by the same set of three Cartesian coordinates).
- The lattice is infinite, i.e., there are no boundaries or other obstacles that would prevent a void from moving in any direction.
- There are only two voids to consider; this may seem like an arbitrary restriction but it’s actually necessary for the solution to work out in general (see below).
The first step is to simplify the problem. We can do this by assuming that every void has a radius equal to 1 and that all of them are connected, which means we only have to consider one sphere with center at (0,0) and radius 1.
Next we use an algorithm:
- Calculate the distance from (0,0) to all points on the surface of our sphere.
- For each point whose distance from (0,0) was calculated in step 2 above: find its nearest neighbor on this list; call this point P1
Heuristic for solution
This is a heuristic for finding the shortest distance between two cubical voids in a simple cubic lattice.
The algorithm starts by finding the center of mass of one of the voids. This can be done by taking all lines connecting adjacent vertices and treating them as springs, so that they’re pushed apart by their masses (i=0..n-1). Then consider this spring system in equilibrium, where each vertex moves along its shortest path from its neighbors’ centers of mass; these are called “relaxed” positions for each vertex. Now take any other point p_2 within some radius r_2 from p_1’s relaxed position; if it lies on one side or another of this line connecting them then we know that p_1 must have been closer than r_2 to p_2 when they were relaxed!
A new record is set for the shortest distance between two cubical voids in a simple cubic lattice
The shortest distance between two cubical voids in a simple cubic lattice is 1.431 nm, achieved by Xiang Li at the University of Science and Technology Beijing (China) and colleagues. They used an atomic force microscope to measure the distance between two holes in an ultrathin silicon film. The previous record was held by Jun Zhang, who measured a distance of 2.50 nm using scanning tunnelling microscopy at his lab at Nankai University (China).
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