Many Electron Atoms

©Fernando Caracena 2013


A previous discussion shows an approach to analyzing the hydrogen atom, The Hydrogen Atom Analyzed Using Commutator Algebra I. Hydrogen is the simplest atom because it contains only one electron. This is the track that most textbooks in quantum mechanics follow, which avoids for a time a big issue in atomic physics. The number of electrons orbiting the nucleus of an atom determine the way that it can bond to other types of atoms. This is the basis of the science of chemistry. Atoms all having the same number of orbiting electrons compose pure forms of matter called the chemical elements. It takes on the order of1023 atoms to make a noticeable amount of matter that can be worked with ordinary chemical equipment in a high school laboratory.

Chemists discovered after much work that the chemical properties of elements could be classified in groups having similar properties. The final form of the classification scheme that these efforts resulted in was the periodic table of the elements, first published independently by Dmitri Mendeleev and Julius Lothar Meyer, but updated with time.  Chemists found out that two numbers were important in classifying the elements: atomic number and atomic weight. The main number was found to be the atomic number. An element consists of atoms all characterized by the same atomic number, but which can vary as to atomic weight. This complexity is explained by nuclear physics. The atomic nucleus consists of two types of particles of nearly identical mass (nucleons): protons, which carry a positive unit of elementary electric charge and neutrons, which carry no charge. However, both types of nucleons have a magnetic moment that is, they behave like small magnets. The atomic number corresponds to the number of protons in the nucleus of the atoms of an element in question; but the atoms of a particular element can also have a variable number of neutrons, which cause the atomic weight to vary in what are called isotopes of that element.

The reason the atomic number is important is because the number of electrons orbiting the nucleus of an atom is equal to the number of protons in the nucleus. The number of negative elementary negative charges of the orbital electrons cancels the charge of an equal number of positive elementary charges of the protons in the nucleus. The big question that physicists faced when they came to understand the structure of the atom was "Why do not all the orbital electrons all fall into the lowest energy state?" The reason was formulated by Wolfgang Pauli who realized that something else was at work here that was beyond conventional forces. There are two kinds of elementary particles in physics: bosons, particles having integer spin and fermions, particles having half integer spin. The electron, having a half unit of spin, is a fermion. Bosons can simultaneously occupy the same state of motion at the same time. For example, this effect allows photons, which have a whole unit of spin, to occupy the same state of motion, such as in the pulse of a laser. What Pauli discovered and stated as a principle is that no two fermions can occupy the same state of motion at the same time. A more rigorous statement of the Pauli Exclusion Principle is that "the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles."

Atomic shell structure

What is covered here is just the general notion that there are structures in multi-electron atoms called shell structures and that these structures result from the constituent, orbital electrons' being fermions. It is quite a detailed subject. If the reader is interested in the details follow the link to electron configuration. Basically, what happens is that every allowed orbit in a multi-electron atom admits occupation by one pair of electrons that have opposite spins. The opposite spins contain the antisymmetry component of their joint wave function, the spatial part allowing a complete overlap of orbits while still complying with the Pauli Exclusion Principle. After that orbit is filled by two electrons, another electron cannot move into that space, unless one of the electrons already occupying that orbit is ejected. As more electrons are added to the atom, they must find other low-lying orbits to occupy. The complication comes from the readjustment of the electric field inside the atom by other electrons, which shifts the wave functions around. That is why we avoid details because we want to stay with the general principles. The details are available from a web and library search.

So it is the way that electrons stack in "shells" that causes their chemical properties when, in bulk, they combine in other atoms. The concept to take away from this discussion is that there are principles in physics that override forces in physics, which constitute a different kind of causality.



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