Electron shell: main energy levels in atoms — structure, history, and significance

Binding energy

The atomic shell consists of electrons that are bound to the positive atomic nucleus due to their negative electric charge. The total binding energy of the electrons of the shell in a neutral atom is about (a more accurate approximation is ). The average binding energy per electron therefore increases with increasing particle number approximately according to , increasing from at to at . This behavior contrasts with the situation in the nucleus, where the average binding energy per nucleon increases strongly only for small particle numbers up to about 16 nucleons ( ), but remains close to 8 MeV further on. These differences are explained by the properties of the prevailing interaction in each case. In the nucleus, both the strength and the effective saturation of the binding energy are based on the Strong interaction between every two nucleons, which produces a comparatively very strong bond, but is also of very short range, so that it can hardly attract the further nucleons beyond the directly neighbouring nucleons. In contrast, the shell is bound by the electrostatic attraction of the nucleus, which increases proportionally to much weaker than the nuclear forces, but reaches all electrons throughout the atom because of its long range.

In the simplest model of the atomic shell, a somewhat stronger increase of the binding energy per electron like would be expected if one starts from the Bohr atomic model and assumes, first, that each electron keeps to its quantum numbers when more electrons are added with increasing , and second, that no mutual electrostatic repulsion acts. For each of the electrons would then have a binding energy increasing with , because not only does the nuclear charge increase as but its orbit is also -fold closer to the nucleus. The weaker increase with instead of can be explained by the fact that with increasing electron numbers the more tightly bound orbits are already fully occupied according to the Pauli principle and the newly arriving electrons have to occupy the less tightly bound ones. In contrast, their mutual electrostatic repulsion is less important. An increase of the binding energy per electron with results from the treatment of the electron shell as a Fermi gas of electrons bound in an extended potential well (Thomas-Fermi model), which do not interact with each other except for a lumped electrostatic repulsion. The additional correction factor of the given more exact approximation is essentially due to the fact that additionally the bonding of the innermost electrons is treated separately. They are located close to the peaked potential minimum at the nuclear site, which is only insufficiently considered in the Thomas-Fermi model.

Shape and size

The atomic shell does not have a sharply defined surface, but shows an approximately exponential decrease in electron density in the outer region. The size and shape of the atom are usually defined by a surface area that is as small as possible and contains a large part (e.g. 90 %) of the total electron density. This surface area is approximately spherical in most cases, except for atoms chemically bound in a molecule or some crystal lattices, or after special preparation in the form of a Rydberg atom. The whole shell can oscillate against the nucleus, the frequency being around ^1017Hz (or excitation energy around 100 eV) for the xenon atom with 54 electrons, for example.

Due to the fuzzy edge of the atomic shell, the size of the atoms is not unambiguously fixed (see atomic radius). The tabulated values are obtained from the bond length, which is the most energetically favorable distance between the atomic nuclei in a chemical bond. Overall, increasing atomic number shows a roughly periodic variation in atomic size, which agrees well with the periodic variation in chemical behavior. In the periodic table of the elements, it is generally true that within a period, i.e. a line of the system, a certain shell is filled up. From left to right the size of the atoms decreases, because the nuclear charge increases and therefore all shells are more strongly attracted. When a certain shell is filled with the strongly bound electrons, the atom belongs to the noble gases. With the next electron the occupation of the shell with next bigger energy begins, which is connected with a bigger radius. Within a group, i.e. a column of the periodic table, the size therefore increases from top to bottom. Accordingly, the smallest atom is the helium atom at the end of the first period with a radius of 32 pm­, while one of the largest atoms is the cesium atom­, the first atom of the 5th period. It has a radius of 225 pm.

Density

Contrary to many popular representations, the atomic shell is by no means an essentially empty space. Rather, the average electron density of the shell varies between 0.01 and 0.1 kg/m3 depending on the element. For comparison, air has this density at a pressure between 10 and 100 mbar. The idea of the envelope as an (almost) empty space would result if at any given time the electrons were at specific locations in space as nearly perfect mass points. However, the idea of such localized electrons within the atom is inadmissible according to quantum mechanics.

Angular momentum

The atomic shell of a free atom has a certain angular momentum in each energy level. It is usually denoted by , its magnitude by the quantum number and the component to a freely chosen z-axis by the magnetic quantum number with . In electron shells with an even number of electrons, is an integer , for odd number of electrons half-integer. The magnitude of the angular momentum is given by ⟨ the z-component by ⟨ . Here the reduced Planck quantum of action.

The size of the atomic shell is determined primarily within the framework of kinetic gas theory and crystal structure analysis (see atomic radius). Methods for the elucidation of the structure of the atomic shell are summarized under the term methods of atomic physics. They are presented in detail in their own articles. Typical examples are (although the list is by no means exhaustive):

• X-ray photoelectron spectroscopy (XPS): The absorption of a quantum of high-energy X-rays in the photoelectric effect produces a free electron with a kinetic energy that is the difference between the energy of the absorbed quantum and the binding energy that the electron previously had in the shell. Electrons with the lowest kinetic energy had the highest binding energy and come from the K shell. After that, at a binding energy of about come the three closely spaced binding energies of the L shell, and so on. The energetic shell structure of the shell, starting from about including the splitting according to the structure after jj-coupling, is clearly shown.

• Atomic emission spectrometry and atomic absorption spectrometry: The spectral investigation of the electromagnetic radiation emitted or absorbed by atomic shells with respect to their wavelength, especially in the visible light, ultraviolet, infrared ranges, provides information about the energy distances of the different energy levels of the atom. In many cases these energies can be interpreted by the change of only one electron from one orbital to another (luminous electron). This has contributed significantly to the study of the atomic shell and thus to the discovery of quantum mechanics. Emission and absorption spectra are characteristic of the element in question and are used for chemical analysis. Therefore, optical spectroscopy is the oldest of the methods mentioned here. Other important results of the measurements are the intensity (especially in the ratio of different spectral lines) and polarization of the radiation.

• X-ray spectroscopy: Like optical spectroscopy above, but in the energy range of X-rays and therefore differently built spectrometers. The strength of the absorption increases abruptly with increasing energy of the X-ray quanta each time the binding energy of an orbital is exceeded (absorption edge). This was the first experimental evidence of the magnitude and quantization of the binding energies of the internal electrons in the atom. The emission of X-rays triggered by absorption shows a simple line spectrum characteristic of each element (characteristic X-ray, X-ray fluorescence analysis). The fact that it only arises when an inner electron has previously been knocked out was the first indication that a weaker bound electron can only jump to a lower level if there is a vacancy there ("hole state").

• Auger electron spectroscopy (AES): An excited atom can emit an electron instead of a photon (Auger effect) if the excitation energy makes this possible. In that case, the Auger effect is generally even the more common one. In detail it is based on the fact that in a level with high binding energy one electron is missing (hole state) and that two weaker bound electrons of the shell make a collision by means of their electrostatic repulsion, so that one of them fills the hole state and the gained energy is sufficient for the other to leave the atom. The energy and intensity of the emitted electrons is measured. These are also element-specific and are used for chemical analysis of the thinnest layers.

• Electron scattering: Investigation of the electrons emitted from atomic shells after the impact of a high-energy electron with respect to their energy and intensity.