I eventually caught up, but I suspect that there are many others who need to understand the basics of Nuclear Physics and have not had anywhere to learn them. Therefore, I felt the need for this presentation.
So I will start out considering planets circling the Sun. We know that Newton determined the equation for gravitational attraction, and that Kepler had earlier discovered some useful relationships. We can describe a Kinetic Energy for a planet as being given by 1/2 * m * V2. As long as Gravitation is a Conservative Force Field, and it is, we can also describe a Potential Energy for a planet, which is actually calculated by mathematically Integrating Newton's Gravitation equation over distance.
It is common to define a ZERO Potential Energy for a planet if it is at infinite distance from the Sun or star. That results in all actual values of Potential Energy for planets to be negative values.
Newton also discovered that Energy is Conserved, as long as no outside force or action is occurring. This now becomes useful. At that infinite distance, the Kinetic Energy is also zero, because the velocity of orbiting is unmeasurably small. Therefore, at infinite distance, we can say that the TOTAL Energy, Kinetic plus Potential, is zero, and that that total must be Conserved if the planet is allowed to fall toward the Sun or star. This results in the POSITIVE amount of Kinetic Energy always being exactly equal to the NEGATIVE amount of Potential Energy, so that the total is always zero.
We can therefore say that 1/2 * m * V2 = m * M * G / R, where R is [the radius of an orbit or in other words] the distance of separation between the planet and Sun, and M is the mass of the Sun. We can take the simplest case where the planet is in a circular orbit.
μ is commonly used to represent the product of the central mass M and G.
We can divide both sides by the mass of the planet m and now get: 1/2 * V2 = μ / R.
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This is: 1/2 * ω2 * R2 = μ / R.
We can then get 2 * k = ω2 * R3.
We know that a full circle or orbit is 2 * ω radians and so the Period of the orbit (time to go completely around once) T is 2 * π / ω.
We can rewrite this as: ω = 2 * π / T.
This allows us to replace the omega in our main equation to get 2 * k = (2 * π / T)2 * R3.
This gives k / (2 * π2) = R3 / T2.
This is the derivation of one of Kepler's Laws that he had observationally found earlier. Newton actually found that Kepler's Law was not precisely correct, because of an assumption (which we included in brackets above) which slightly alters it. The correction is because the R distance BETWEEN the planet and Sun is not actually the same as the R which is the orbital radius of the planet, as BOTH the planet and Sun orbit around a common point. For all planets in the solar system, that point happens to still be inside the Sun, so Kepler's Law is quite close to the correct equation. Newton simply showed WHY it is close and what the actual, more correct form, actually is. Newton's work also showed the more complex equation that applies for elliptic orbits, which confirmed another of Kepler's Laws in the process.
None of this might seem to have anything to do with Nuclear Physics! But it does! This reasoning showed the connection between the Potential Energy and the Kinetic, and then the Period, or its inverse, the frequency of the planet going around the Sun. THESE are important!
During the previous 50 years, a lot was learned about the spectra of the Sun and various elements. Hydrogen has interesting patterns of spectral lines, and it was eventually discovered that Hydrogen can emit or absorb its shortest wavelength of around 915 Angstroms. During roughly the same time, other Physicists had determine the speed of light pretty accurately. It was quickly realized that the FREQUENCY of the light radiated away must be the speed divided by the wavelength. An Angstrom is 10-10 meter, and the speed of light is around 3 * 108 meters/second. If light and other radiation all travels at the speed of light, then we know that 915 Angstroms means the frequency of that light of must be (3 * 108 meters/second) / (9.15 * 10-8 meter) or around 3.3 * 1015 /second.
It was then reasonably concluded that the electron must be orbiting at a rate of 3.3 * 1015 full orbits /second.
Another thing that had been discovered in the same time period is that the attraction that keeps atoms together is NOT gravitational at all, as it would be far, far too weak. It was found that it was electrical, or actually electrostatic attraction between a positively charged nucleus and negatively charged electrons. Very importantly, it was found that electrostatic attraction obeys the very same inverse square distance dependence that gravitation obeys. THIS means that all the equations and logic we used above regarding a planet orbiting the Sun should equally apply within an atom.
The 915 Angstrom wavelength of radiation absorbed or emitted was learned to be the DIFFERENCE in energy between a Hydrogen electron orbiting in a (circular) orbit and that electron having been sent away to infinity. In other words, using our planet reasoning above, the energy in that radiation DIRECTLY tells us what the POTENTIAL ENERGY of the electron is in that (circular) orbit.
Max Planck and others did a lot of measuring of the energy in the energy of the radiation, and it was discovered that the energy is always exactly proportional to the frequency (or inversely to the wavelength) by the equation E = h * ν (where h is called the Planck Constant). You might notice that we now have the (differential) energy between an orbiting electron and that electron if it were at infinite distance away from the nucleus. This is EXACTLY what we were discussing for EITHER the Potential or the Kinetic Energy of the planet earlier. We are getting somewhere!
By knowing the frequency of the energy emitted or absorbed, Planck's Law tells us the Energy involved, which is the amount that we had earlier defined as being the Potential Energy, which is the same as the Kinetic Energy due to Conservation of Energy of Helmholtz, Newton and others, which is also called the First Law of Thermodynamics.
Let's look at the Potential Energy first. Planck's Constant is around 6.6 * 10-27 erg-second. Therefore, we have 3.3 * 1015 /second * 6.6 * 10-27 erg-second or 2.18 * 10-11 erg of energy. We know that this must be equal to the Potential Energy in the electrostatic Field. For a unit absolute electrostatic charge, this Potential Energy is given by Q/R, where Q is the charge of the nucleus (in absolute electrostatic charges) and R is again the radius or distance of the electron from the nucleus. Both an electron and a proton (the charge of a Hydrogen nucleus) has a charge of around 4.8 * 10-10 absolute electrostatic charge. We therefore have 2.18 * 10-11 = (4.8 * 10-10)2 / R.
We can solve this for R = 2.3 * 10-19 / 2.18 * 10-11 or R = 1.05 * 10-8 centimeter.
Therefore, it was early on determined that the orbital radius of the electron in a Hydrogen atom must be around 10-8 centimeter.
Also, it is not necessary for an electron to be completely ejected from an atom, as we have been discussing so far. an electron can instead move from one of these Principal Quantum States to another, and the result is that there are whole families of spectral lines which can be produced. One of the most famous for Hydrogen is called the Balmer Series of lines, where electrons which start out in the N = 2 state are energized to go into some higher N energy state, with each of these transitions producing a very prominent line in the spectrum of hydrogen.
By the way, the 1.05 * 10-8 centimeter orbital radius that we calculated here is today commonly described in picometers as being 105 pm. Also, many different effective diameters of atoms exist in chemical reactions (valence shell orbitals) and through many other types of experiments. All are roughly the same size, although different types of experiments result in values that may be different by a factor of two or three, usually larger.
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago