Atomic Physics

We now address the spectral lines and the information contained in the spectral lines. Based on the experiments of Bunsen and Kirchhoff, it was known that the wavelengths (energies) of the spectral lines for particular elements were the same every time they performed their experiments and the patterns of the lines were unique to each element. The lines thus served as the fingerprints of the atoms.

To understand why this is true we need to discuss the structure of atoms for a little while. An atom roughly has the following structure:

The mass of a proton is roughly 1,836 times that of an electron while neutrons and protons have roughly the same masses. So, the mass of an atom is contained in its nucleus whlie the nucleus is only

(1 fermi)**3/(1 Angstrom)**3 = (10**(-13) cm)**3/(10**(-8) cm)**3 = 10**(-15)

of the volume of the atom. Atoms are primarily empty space. Comment--note that the density of a helium nucleus is roughly

density = mass / volume = (2 protons + 2 neutrons) / volume = 5x10**14 grams/cc

(The mass of a proton is around 1.7x10**(-24) grams.) The density of lead is only around 11 grams/cc. If we had a nucleus the size of a golfball, it would weigh, 38 billion tons. That's a lot.

Now, back to atomic structure. If we make an analogy with the Solar System, we can imagine that the nucleus is equivalent to the Sun, the electrons are like the planets, and the electrical force plays the role of gravity. There is, however, a huge difference between how our Solar System works and how an atom works.

In our Solar System, the planets, in principle, can move about the Sun in orbits of arbitrary size. That is, if I give a planet a large kick, it will move around the Sun in a large orbit. If I inject a planet into the Solar System with only a small kick, the planet will settle into a small orbit around the Sun. Depending the size of the initial kick, I get a planet to settle into any size orbit (that is, I can give the planets arbitrary amounts of energy, and so they can settle into orbits of arbitrary energies).
In an atom, the above is not true. It turns out that the electrons cannot, orbit about the nuclues at all and, furthermore, they cannot move about the the nucleus in arbitrary sized orbits.
1. Recall that an accelerated (jiggled) charge radiates electromagnetic radiation. It turns that if an electron in an atom radiated electromagnetic radiation due to an orbital motion, it would lose energy so fast that the atom would collapse in less than a billion-th of a second!! This tells us that the electron cannot be in a classical orbit about the nucleus.
2. Further, there can only be very well-defined orbits which electrons can attain. We note that what this means is that electrons can exist with only very well-defined amounts of energy in an atom. They cannot have arbitrary energies and be in an atom.

These properties of atoms are not intuitive and it wasn't until the 20-th century when such things could even be modeled. They weren't understood then and they are still not understood today. The physics used to model these and other related phenomena is referred to as Quantum Mechanics. We will have more to say about Quantum Mechanics in the future. For now we consider the following representations for the structures of atoms:

From hereon, we will use the well-type picture to represent the structure of an atom. To make our ideas more concrete, we will consider the hydrogen atom and its spectrum. The hydrogen atom is particularly attractive to theorists because of its relative simplicity. It is the simplest element; its most common isotope contains only one proton and one electron. Here are some facts about hydrogen atoms

The energy required to remove an electron from the lowest level in the well is equal to 2.2x10**(-18) Joules. This is such a tiny number, that a different unit is used known as the electron volt. An electron volt is tiny in that there are around 6,000,000,000,000,000,000 electron volts in one Joule!! In terms of electron volts, the energy required to remove the lowest lying electron from a hydorgen atom is 13.6 electron volts.
Note that the process of removing an electron from an atom is known as ionization
The energy required to remove an electron from a higher lying level in the well is easily found. Bohr showed that the amount of energy needed was given by
Energy = 13.6 eV / n**2
where n tells you the level. The lowest lying level is n = 1, the next higher level is n = 2, and so on.
A further interesting note is that because the electrons can exist only in certain levels (energy states), an atom cannot absorb arbitrary amounts of energy. Due to the notion of the conservation of energy, an atom can only absorb an amount of energy which will just promote an electron to a higher lying state. (And What does this say about the production of emission lines?) The energy of a photon capable of inducing a transition between levels n = 1 and N = 2 is
Energy = 13.6 eV / 1**2 - 13.6 eV / 2**2 = 10.2 eV
Because the strength of the electrical force depends upon the charge of the nucleus (number of protons) and the number of electrons, each type of atom will have a different energy level structure, It will be harder to strip electrons off of different elements (nuclei with different numbers of protons).
Comment: An iron atom (Fe) has 26 proton while a hydrogen atom has 1 proton. It is easier to strip off one electron from a hydrogen atom than it is to strip off one electron from an iron atom. Can you figure out why this is true?

Now,let us make some remarks about the above figure. First, note that the largest transitions (longest arrows) require the highest energy photons (because the transitions have the largest changes in energy). Alternatively, this means that the largest transitions involve the shortest wavelength photons (E = hc/W). Now, let us make some observations

If an electron goes to a higher energy level, it must absorb some energy and so this corresponds to an absorption lines
If an electron drops to a lower energy level, it must get rid of some energy and so this corresponds to the production of a photon (an emission line is produced).
If the electron leaves the atom, we say that the atom was ionized
A comment about ionization. In order to ionize an atom, all that is required is that the photon which is absorbed have enough energy to lift the electron out of the well and set it free. If the photon has more than this threshold energy, it simply gives the excess energy to the electron (as kinetic energy). So, in terms of the appearance of the spectrum, what happens is that there can be a threshold where ionization begins and a broad trough which extends to shorter wavelengths (higher energies). Such ionization edges are seen in the spectra of many stars.
If a free electron is captured by the atom, we say that a recombination has taken place

Now for some definitions:

Lyman line: An electron starts from n = 1 and jumps to a higher level or an electron drops from a higher level to n = 1,
Balmer line: An electron starts from n = 2 and jumps to a higher level or an electron drops from a higher level to n = 2,
Paschen line: An electron starts from n = 3 and jumps to a higher level or an electron drops from a higher level to n = 3,
Brackett line: An electron starts from n = 4 and jumps to a higher level or an electron drops from a higher level to n = 4,
Pfund line: An electron starts from n = 5 and jumps to a higher level or an electron drops from a higher level to n = 5,

The Lyman lines fall in the ultraviolet portion of the sprecturm. The Balmer lines fall in the optical portion of the spectrum. The Paschen, Brackett, and Pfund lines fall in the infrared portion of the spectrum.