- This is a 100 year story of how physicists were able to figure out the mathematics that defines the behavior of an atom. They are still figuring, but, we have come a long way. One of the biggest issues was infinity, which is a very long way. It kept showing up in the equations. Physicists ignored it, hope it cancelled out, but, kept making discoveries to understand how infinites could be avoided.
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- Early attempts to calculate the strength of the electromagnetic interactions that hold electrons in orbits around the nucleus of atoms required the term 1/137. It was a dimensionless constant called ALPHA. The problem was 1/137 was in an infinite series of growing exponentials,( 1/137)^2 + ( 1/137)^3 + (1/137)^4 + ………. to infinity.
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- ( 1/137) became known as the Fine Structure Constant, called FSC or ALPHA. It is a constant in nature equal to 0.0072973531.… which is approximately 1/137. FSC is calculated from 4 basic physical constants:
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------------------------ electronic charge = e
------------------------ speed of light in a vacuum = c
------------------------ Planck’s Constant of action = h
------------------------ Permittivity of free space = Pe
------------------------ or, Permeability of a vacuum = Po , where Pe * Po = 1 / c
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- FSC measures the strength of an electromagnetic interaction.
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- FSC is the ratio of the velocity of the electron in orbit to the speed of light.
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-------------------------- FSC = v / c = 1 / 137 = 0.73%
-------------------------- v = 0.73% * c where c = 186,000 miles per second.
-------------------------- v = 4,888,080 miles per hour.
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- The electron is traveling almost 5 million mile per hour around the nucleus of the atom. But, the image needs to be a wave not just a single particle in orbit traveling at that speed.
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- FCS defines the interactions of all electrically charged particles, electrons, muons, and photons. The Fine Structure Constant that shows up in these equations is directly proportional to the velocity of light and the electric charge.
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-------------------- FSC = Po /2 ( velocity of light ) * ( electric charge squared ) / Plank’s Constant
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--------------------- FSC = Po * c * e^2 / 2 * h
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- FSC often showed up in math equations as a series of exponential sums to higher powers that proceeded to infinty : (0.00728)^2 + (0.00728^3 + ( 0.00728)^4 + ……………. off to infinity with smaller and smaller numbers. In the 1920”s the higher order factors were ignored , the infinities were ignored. Ignoring infinities is disturbing to pure mathematicians.
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- There are several other problems understanding the atom using classical theories. A hydrogen atom is composed of a proton nucleus and a single electron in orbit. The proton is a positive electric charge and the electron a negative electric charge. Opposite charges attract so that force pulls the electron inward. . Inside a hydrogen atom 10 electron volts separates the proton and the electron over a gap of 0.1 nanometers. Angular momentum in a circular orbit pushes the electron outward. The two forces are in balance for a neutral hydrogen atom. BUT, the electron is an accelerating charge constantly changing direction in orbit it should be radiating energy. It should spiral into the proton nucleus in 10^-10 seconds.
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- The next improvement in the math recognized the electron was not just a particle it was a wave. AND, only certain orbits were allowed for the wave to be is resonance in one complete orbit. Therefore, the electron could only exist at certain energy levels in the atom. When the electron jumped from one allowed level to another allowed level it emitted ( or absorbed) a photon of that exact difference in energy level. No two electrons can share the same energy level, called the Wolfgang Pauli Exclusion Principle.
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- The other problem was apparent for heavier atoms. Say, helium with 2 protons in the nucleus and 2 electrons in orbit. The 2 electrons can have opposite spins so they would not be sharing the same energy level. HOWEVER, The 2 protons in the nucleus have the same positive electric charge. Like charges repel. What could be holding the protons together in the nucleus.? They should be flying apart?
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- This next step required the discovery of the Strong Nuclear Force which was 100 times stronger than the Electromagnetic Force. Theories developed force carriers for the Strong Nuclear Force called “ Gluons”. The force carriers for the Electromagnetic Force were the photons.
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- An infinite sum can be a series that has a finite answer: 1 + ½ + ¼ + 1/8 + …… = 2
BUT, another infinite series can sun to INFINITY: 1 + ½ + 1/3 + ¼ + ……. infinity
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- The math used to explore the atom, particularly the nucleus, was called Quantum Electrodynamics, QED. This math gave physicists infinite series of 1 / FSC growing at an exponential rate .
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- The Strong Nuclear Force is the binding force that holds the atomic nucleus together. The Weak Nuclear Force destabilizes the nucleus causing radioactive decay. Math had the same problems with The Weak Force. The formulas all had infinity in the calculations.
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- Another mystery in studying the radioactive decay of atoms was that these subatomic particles seemed to have a left-handed screw. If you cup your left hand , point your thumb up and your fingers curl clockwise. If you point your thumb in the direction of motion the particles spins in the same direction as the fingers curl. Do the same with your right hand with the thumb pointing in the direction of motion and the fingers curl counter-clockwise. Photons can be polarized with left handed spin or right handed spin.
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- The energy of the photons is proportional to its frequency of oscillation. A photon at the high-frequency violet end of the rainbow has twice the energy of the photons at the low frequency red end of the rainbow, 350 nanometers versus 700 nanometers wavelengths.
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- Hot sodium and mercury vapor lamps glow yellow or turquoise hue because the heat is shaking photons lose from the atom’s electromagnetic force. They have characteristic colors because photons emerge with energies, or frequencies, unique to each element’s atom. The colors represent the pattern of energy levels available to the electrons within those atoms.
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- The hot Sun and all the stars emit electromagnetic radiation across the entire spectrum. There is a lot of hot gases in their outer atmospheres that contain a smorgasbord of elements. The result is white light. However, the elements can be individually identified because they also absorb photons at the same unique energy levels. Each element can be identified by these absorption lines, black lines, found in the spectrum.
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- Each atom has its own unique energy ladder for the electrons to orbit and electrons can only orbit in whole resonant wavelengths. In 1928 Paul Dirac came up with the equations to define these levels. Today the math is know as Quantum Electrodynamics, QED. In QED the electromagnetic force is transmitted by force carriers which are the photons. The scale of the force carrier interactions is controlled by the constant dimensionless number, the Fine Structure Constant, 0.00728.
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- FSC often shows up in QED equations as a series of exponential sums to higher powers that proceed off to infinity. (0.00728)^2 = 0.0000530, (0.00728)^3 = 0.000000386, off to infinity with each number being smaller and smaller. That is why in the early calculations in the 1920’s the infinities were ignored. However, ignoring infinities is disturbing to pure mathematicians.
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- In Quantum Mechanics individual particles have an infinite number of possible paths. However, a group of particles together at molecular size mutually cancel individual probability amplitudes for all the paths except those that are near the classical one’s we see. Particles will always take the path of least time. In Quantum Mechanics calculations allow particles to travel backwards and forwards in time.
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- In the 1930’s QED calculations worked as a first approximation but beyond that it got into a meaningless series of infinities. When dealing with electric and magnetic fields you need 4 numbers. The field force vectors have 3 directions and one direction of time ( put could be plus or minus). Drawing the axes horizontally and vertically in plotting a field vector is a measure of “ gauge”. Gauge invariance means the field symmetry is the same no matter how you orient the axes. The photon is known as a massless vector boson , or, a gauge boson. The photons carry energy and momentum from one electric charge to another transferring motion across space.
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- For the Strong Nuclear Force there are 3 massless gauge bosons. One is electrically neutral. One is charged positive and the third is charged negative. W+ and W- and Z bosons.
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- The electron and the neutrino are particles that do not feel the Strong Nuclear Force. They do feel the Weak Nuclear Force. These particles are called “ Leptons”.
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- The Weak Nuclear Force changes one form of matter into another form of matter through beta radioactivity. When radioactivity occurs a neutron converts into a proton and releases an electron and an anti-neutrino.
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- The Strong Nuclear Force can squeeze 4 protons into a tight clump but it is the Weak Nuclear Force the converts them into a nuclei of helium.
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- In 1964 the math was developed introducing quarks using Group Theory. The particular math groups was SU3. Each quark carries any of three varieties of a property known as “ Color”. Color has nothing to do with light rather it is similar to electric charge, where electric charge is the source of electromagnetic force , color is the source of the Strong Force interactions. The three fold nature of color allows up to 3 quarks to occupy the same quantum state in the nucleus as long as each carries a different color charge. This triplet property is the basis of Quantum Chromo dynamics which explains the behavior of Quarks and hadrons.
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- Neutrinos are like electrons but without an electric charge Highly elusive, they pass through the Universe almost as if they were mere spectators, so shy it is remarkable that we even know they exist If the Z boson did not exist they would be unable to bounce off anything.
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- Physicists in the 1970’s assumed that gravity behaved like the other 3 forces in nature and sought the theories that would bring gravity into the QED, QCD and the Standard Model. However, the theories in Quantum Mechanics produced nonsensical results when applied to gravity. Physicist tried super symmetry and string theory as new math to try to merge gravity into the Standard Model for the other particles
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- What physicists were searching for were ”Gravitons” that could carry the force of gravity similar to how “ Gluons” are the force carriers for the Strong Nuclear Force. In the 1980”s a new method proposed that a Graviton would be two Gluons stitched together. When first developed the Feynman Diagrams used in Quantum Electrodynamics, QED, math described accurately the interactions between electrons and photons.
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- In the 1990’ the Feynman diagram techniques were extended to the Gluons and the Strong Nuclear Force, the math became Quantum Chromo dynamics, QCD. But, the math got much more complex. When dealing with QCD physicists had to deal with “virtual particles” that pop in and out of existence in the space between other particles. The larger the mass of the virtual particle the shorter its lifetime. They defied the Conservation of Energy because they existed in such a short time. QCD started with Feynman diagrams for the first approximation and then added “perturbation” of virtual particle loops and virtual electron and anti-electron annihilations that produce photons. As more loops are added more precision in the calculations was attained.
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- A new method of calculating in the 1990’s was called “unitarily” in which the math required the square roots of probabilities of all the possible outcomes of an interaction sum to 100%.
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- If the Large Hadron Collider at CERN discovers the Higgs Particle it will be evidenced in the collision producing a single electron, a pair of jets, and a neutrino. The neutrino being detected as a particular energy gone missing. The math is still the method of defining the particles behavior and discovering new particles. The struggle to get rid of infinities as brought us a long way but many discoveries remain to be found. An announcement will be made shortly, stay tuned.
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(1) To learn more: The Infinity Puzzle by Frank Close, 2011.
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