Electrons in Atoms Reading 2

Electrons in Atoms

Photons and Electrons

Light in the visible and ultraviolet region of the electromagnetic spectrum as well as kiloelectron volt energy x-rays allow us to examine the electron energy levels in an atom. We use the photons as a probe and examine the energy spectra from a light or x-ray source to infer the energy levels in an atom.

A photon is created when an electron makes a transition from a high energy to a low energy state. In Fig. 1 we show in a) electrons occupying two energy levels in an atom. An incoming x-ray (it could also be a high-energy electron) collides with and ejects (Fig. 1b) an electron from energy level 1 and leaves a vacant (unoccupied) position in level 1 (Fig. 1c). The atom is in an excited state because of the empty site. To de-excite or to have the atom lose energy, the electron in level 2 makes a transition to the empty site in level 1 and gives off energy in the form of an emitted x-ray.

Figure 1. Schematic of X-ray emission from electrons which occupy two energy levels: a) energy levels occupied by electrons; b) incident high energy X-ray ejects electron from 1; c) empty state in level 1; d) electron mades transition to empty state in level 1 and emits an X-ray to conserve energy in the transition process.

Energy must be conserved in this process so the x-ray energy is

E_x-ray = E₂ - E₁

In order for an x-ray to be emitted there must be a full site with an electron occupying the site and a vacant site empty of an electron. The primary selection rule is that the x-ray energy is equal to the energy difference between the full and empty sites, E₂-E₁.

This concept holds for all photons including the infrared (IR) through the visible band to x-rays. We can extend our argument to the statement that emission and absorption of photons by atoms does not take place continuously but in finite energy intervals.

Without proof, we state one of the great findings of the early 1900's: the photon energy is given by

E = h f

where h is Planck's constant

h = 4.136 x 10^-15 eV-sec

with eV denoting the unit of energy, the electron-volt and f is the frequency, sec^-1, where

f = c (cm/sec) / (cm)

with c equal to the velocity of light (3 x 10¹⁰ cm/sec) and is the wavelength. With these equations we can relate energy and wavelength by

E (electron volts) = 1240 / (nanometers)

so that orange-yellow light of 620 nanometers has an energy of 2 electron volts.

Energies of Light: Optical Spectra

We have available in our laboratories optical spectroscopes with which we can determine the wavelength of light and hence the energy through the above equations. This allows us to recreate some of the excitement of the late 1800's where light emission from hydrogen atoms was analyzed. Hydrogen is the simplest of atoms with a central positive nucleus (atomic number Z = 1), one proton, that is surrounded by one electron that can occupy various energy states in its motion around the proton.

The light emitted from an excited hydrogen atom appears as a series of lines (Fig. 2).

Figure 2. The spectrum of light from a hydrogen atom showing the positions in lines by their wavelength in millimicrons = nanometers. The visible spectrum extends from 400 to 700 nanometers.

The names are associated with Johann Balmer, a mathematician who found in 1885 a formula for the wavelength, and with Theodore Lyman, who discovered the ultraviolet series in 1905.

The critical point is that the existence of sharp lines in the spectra dictates the existence of sharp and well-defined energy states both full and empty that the electron had access. The electrons can make transitions from an occupied to an empty state and emit light as shown in Figure 3 (predicted by Arnold Sommerfeld in 1916).

The model of the atom developed in 1913 by Niels Bohr was that of an electron confined in a potential well caused by the positive proton (Figure 4).

The ionization energy is the energy required to remove an electron from the lowest energy state E₁, where

E₁ = 13.58 eV

The Bohr atom remains a useful visual picture, but one that has been refined many times. For example, if the single electron of the hydrogen atom dropped from orbit 2 to orbit 1, it emits a quantum of fixed energy, and this is equivalent to a bit of radiation of fixed frequency. This would show up as a bright spectral line in a fixed position. (If the single electron rose from orbit 1 to orbit 2, this would be through the absorption of a quantum of the same fixed energy, and this would produce a dark line against a bright background in the same position).

If the single electron of the hydrogen atom dropped from orbit 3 to orbit 1, this would represent a greater difference in energy, and light of higher frequency would be emitted. Light of still higher frequency would result in a drop of an electron from orbit 4 to orbit 1, and higher frequency still in a drop from orbit 5 to orbit 1.

X-ray Energies

If we consider atoms that are more complicated than hydrogen and contain more electrons, we must remember that they also contain nuclei of higher positive charge. The innermost electrons are held progressively more firmly as that nuclear charge increases. It takes larger increments of energy to move such electrons away from the nucleus into excited states. Conversely, larger quanta of energy are given off when an electron drops closer to its ground state. Whereas the shortest wavelengths hydrogen can produce are those represented by the Lyman series in the ultraviolet, more complicated atoms can produce radiation in the X-ray region.

The energy levels can still be given in fashion similar to that in Figure 3 but here the binding energies are in kiloelectron volts (keV) rather than eV for arsenic atoms (atomic number Z = 33 for 33 positive charges in the nucleus) and for cadmium, Z = 48 (Figure 5).

Figure 6 shows an energetic x-ray penetrating into the atom, colliding with an electron in the K-shell (the most strongly bound electron), and ejecting the electron (photon absorption). The incident x-ray energy must be greater than the binding energy of the K-shell electron. The atom loses energy when an electron makes a transition from the L3 occupied state to the K shell vacancy.

Figure 6. Energy levels showing a photon is absorbed in transferring its energy to a K-shell electron and an x-ray (K_a x-ray) is emitted when an eldctron in the L₃ shell makes a transition to the unfilled K-shell.

Of course, when energetic x-rays or electrons are incident on an atom many vacancies are created and a number of x-rays are emitted. The x-ray lines are called characteristic x-rays. Figure 7 shows 3 examples of x-ray emission.

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