Celebrating Numbers

  We like to use numbers to represent things in the natural world. I believe this is because we can manipulate, modify, and understand things when put into numbers. We grow up learning to count things in order to share toys, candy and turns in games. We use numbers every day. As character Charlie Eppes in the American crime drama television series NUMB3RS so eloquently states, “Math is nature’s language: its method of communicating directly with us. Everything is numbers.”
  Many of you have probably heard of Pi day, March 14, because the date 3.14 matches the constant known as Pi, which is the ratio between the circumference and diameter of a circle. This week I thought about how other mathematical constants in our world get ignored. I figured, why not talk about some of the lesser known constants and pick one to celebrate.
  For our new math celebration, I would like to recommend Bohr day, May 29. I know, now you are all wondering what is a Bohr? Well for those who are not into nuclear physics, or computational chemistry, it’s probably a real bore. However, to me it is fascinating.

llustration by Stephen Lower, https://chem.libretexts.org

  Niels Henrik David Bohr, a Danish physicist, made foundational contributions to our understanding of the atomic structure of atoms. He received a Nobel Prize in Physics in 1992 for his work. He was best known for the development of the Bohr model of the atom. He proposed that energy levels of electrons were discrete, causing them to obtain stable orbits around the atomic nucleus, but they can jump from one orbit (or energy level) to another.
  Bohr’s model is no longer the accepted atomic model, but the principles of his model remain completely valid, even his theoretical measurement of the Bohr radius (5.29 x 10E-11). This is the average distance of the electrons in Hydrogen from the nucleus. In the Bohr model proposed in 1913, it is stated that electrons only orbit at set distances from the nucleus, depending on their energy.  In the simplest atom, hydrogen, a single electron orbits the nucleus and it smallest possible radius, with the lowest energy, has an orbital radius almost equal to the Bohr radius.
  Although Bohr’s model is no longer in use, the Bohr radius remains very useful in atomic physics calculations, mostly because of its simple relationship to other fundamental constants. It is the unit of length in atomic units, just like we use, inches, feet, and miles to measure length; at the atomic level the Bohr radius is like the inch.
  The Bohr radius is one of three units of length used in atomic physics, the other two being the Compton wavelength of the electron and the classical electron radius. The Bohr radius is calculated from the electron mass, Planck’s constant, and the electron charge. The Compton wavelength is built from the electron mass, Planck’s constant, and the speed of light. The classic electron radius is built from the electron mass, the speed of light, and the electron charge. Any of the three can be converted to the others by using the fine structure constant.  Interestingly the Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the Compton wavelength.