Atoms and Nuclei

Atoms

Contents

Atoms are one of the most stable forms of matter. They consist of a nucleus made up of protons and neutrons, which are surrounded by a ‘cloud’ of electrons. A normal atom will have the same number of protons and electrons, making it electrically neutral. It will then have a set number of neutrons than are needed to keep the nucleus stable.

If the atom loses or gains 1 or more electrons then it become an ion, a charged atom, if it gains or loses neutron it becomes an isotope. Because the neutrons control the stability of the nucleus it is common for isotopes to be unstable, and hence radioactive.

Building Hydrogen?

If you want to build hydrogen you need one proton and and one electron. The negative electron is bound around the positive proton and you have an atom. The problem is that everyday experience tells us that the positive-negative attraction would just pull them together until they were touching, leaving no stable structure to the universe. But the experience we have turn out to be just a large scale effect, and at the quantum level things work differently.

In the quantum mechanics of atoms the energies that the bound electrons can have can only take certain values in an analogous way to the strings in a piano only being able to play certain notes. Theses energies in a basic atom are given roughly by

    \[E_{n} = \frac{-Z^{2}e^{4}m_{e}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}n^{2}} = -13.6\frac{Z^{2}}{n^{2}} \text{ eV},\]

where Z is the charge of the nucleus of the atom and n ranges from 1 to infinity and at a basic level counts the different energy levels. Each of these energies will have a corresponding orbital radius given by

    \[r_{n} = \frac{4\pi\epsilon_{0}\hbar^{2}n^{2}}{Ze^{2}m_{e}} =5.29\times 10^{-11}\frac{n^{2}}{Z} \text{ meters}\]

So if we take our proton and add an electron to make hydrogen the lowest level it can go into is n=1, which means the electron will have to have an energy of -13.6 eV and orbit at a radius of 52.9 picometers. It can’t get any closer as there are no lower energy levels or radii for it to be at.

This gives us our first step to structure, one hydrogen atom has to have a volume of around 620000 cubic picometers, you can’t squash it any smaller. But what about other elements that have more than one electron? What about carbon? For carbon we have Z=6 and 6 electrons, so do they all go in the n=1 level and have energies of -489 eV and orbit at 8.8 picometers? The answer is no.

Exclusion

The thing that gives atoms and all matter additional structure is the Pauli Exclusion Principle. If we have a system of two particles, A and B, in two possible states, 1 and 2, then the description of the system will be of the form

    \[\Psi = \phi_{1}(A)\phi_{2}(B)\]

We now need to outline two important features of particles;

  1. All particles of the same type are indistinguishable from each other.
  2. Matter particles are antisymmetric.

From point 1 we don’t know which particle is in state 1 and which is in states 2, so the proper description of the system should take into account both possible combinations, this is done in the following way:

    \[\Psi = \phi_{1}(A)\phi_{2}(B) \pm \phi_{2}(A)\phi_{1}(B)\]

From point 2 we have to pick the minus condition. If we were dealing with force particles, which are symmetric it would be the plus but for matter the antisymmetry means we have a minus sign. So now we have

    \[\Psi = \phi_{1}(A)\phi_{2}(B) - \phi_{2}(A)\phi_{1}(B)\]

This is an expression for a system containing two identical particles. What happens if we try to put them in the same state, say \phi_1?

    \[\Psi = \phi_{1}(A)\phi_{1}(B) - \phi_{1}(A)\phi_{1}(B) = 0\]

Therefore you cannot have two identical particles in the same state.

So if we go back to our case of carbon, all 6 electrons are not allowed to just orbit at the same distance with the same energy. We can add the fist electron to the n=1, -489 eV/8.8 pm state, but to add the second electron something will have to be different. In this case it will be spin. You may or may not know that electrons have a quantum property called spin, it can either be up or down and provides a way to get two electron at the same energy and radius,. One will be spinning up and the other down meaning the states will be different. As soon as you’ve got the first two electron on that state it is then filled and so you move on to n=2, with a new energy and radius.

At this point things get a lot more complicated and you get more and more ways of getting electrons into the same n level with slightly different states. You have to include things like orbital angular momentum, magnetic effects and Hund’s rules but electron by electron you can build up the entire periodic table.

Neutron Glue

For small atoms you tend to find that the number of protons and neutrons in the nucleus are roughly the same, once you start to get past Calcium you get more and more neutrons for each proton. This is because the neutrons feel the strong force but not the electromagnetic force. As the number of protons increases the electromagnetic force that’s trying to rip apart the nucleus increases, and so in order to keep the nucleus stable more neutrons are added so that the strong force is increased in line with the electromagnetic replusion.

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