The weaknesses of the Bohr model could be partially eliminated by the physicist Arnold Sommerfeld. In addition to the already introduced shells by Bohr, Sommerfeld further introduced subshells (also referred to as orbitals). With the introduction of these subshells, it was finally possible to explain the distribution of the electrons within the shells. The distribution of electrons in an atom is referred to as electron configuration. It will be explained in more detail below.

One imagines the main shells introduced by Bohr subdivided into subshells. The number of subshells depends on the main shell. The number of the main shell indicates the number of subshells. In the figure above, not all subshells of the higher main shells are shown, as these usually have no relevance. The lower shells are not labeled with numbers but with lowercase letters (s, p, d and f). A g-subshell exists only for theoretical elements with atomic numbers greater than 121 (called superactinoids), which is why this orbital has only theoretical meaning.

1^th main shell (K): 1 lower shell (s), identical to the main shell
2^nd main shell (L): 2 lower shells (s, p)
3^rd main shell (M): 3 lower shells (s, p, d)
4^th main shell (N): 4 lower shells (s, p, d, f)

For example, the shell designation 3p means the subshell p (“2nd subshell”) of the third main shell and the designation 4s the subshell s (“1st subshell”) of the fourth main shell. The shell designation 2d, however, does not exist because the second main shell has only one s and one p subshell! Subshells can only be occupied by a certain number of electrons:

s-subshell: 2 electrons
p-subshell: 6 electrons
d-subshell: 10 electrons
f-subshell: 14 electrons

Here, a subshell of a lower main shell number may well have a higher energy level than the subshell of a higher shell number (Sommerfeld explained this with elliptical orbits of electrons instead of circular orbits after Bohr)! For example, the subshell 3d has a higher energy state than the subshell 4s! The graphical representation by shells with their subdivisions in subshells is therefore no longer possible. Instead atomic orbitals are used, which will not be discussed further here.

The energetic distribution of the shells is shown in the figure above. The subshells are divided into white-framed blocks, each providing space for a total of two electrons.

In order to better remember the energetic order of the orbitals, you can first create a table. Therein, the line numbering corresponds to the main shell number and the column numbering corresponds to the subshell. Thus, the subshell with associated main shell is clearly defined for each field. The energetic order of the orbitals can now be obtained by going through the table diagonally line by line from top right to bottom left. This principle is also knows as the aufbau principle (“Aufbau” is a german word which means “configuration”).

The animation below shows the electron configuration with increasing atomic number of the atoms. Note that the number of electrons increases to the same extent as the number of protons and thus increases by one from element to element. The subshells belonging to a main shell are all marked in a uniform color. Also shown are the outer electrons (valence electrons) on the outermost main shell (valence shell), since these are decisive for the chemical behavior of an element. The animation also explains the order of the chemical elements in the periodic table.

The occupation of the shells with electrons always starts from the lowest energy state, only then are higher energy levels occupied. Each block of a subshell is initially filled with only one electron. This is symbolized by an ascending arrow. Only if all blocks of a subshell are solitary occupied by an electron, then the each block will be filled with one more electron. These second electrons are represented by a descending arrow. This symbolic distinction is due to the so-called Pauli exclusion principle of quantum mechanics. According to this principle no two identical electron states can exist. The different arrow directions take this principle into account (to be more precise: each arrow represents one of two spin quantum numbers).

The Pauli exclusion principle forbids that two electrons share the same state!

In the subshell 3p, it is striking that after it has been completely filled with electrons, it is energetically more favorable to start the fourth main shell and fill it with electrons (4s shell) instead of the subshell 3d! Note, that some subshells of a smaller main shell number are obviously of a higher energy level than subshells of a higher main shell number. Thus, the occupation of electrons of the main shells, which in the Bohr model at first appears a little bit strange, can now be explained.

Only after the 4s orbital is fully occupied, the third main shell is filled up with the energetically higher 3d orbital. This jump is also noticeable in the chemical behavior. It marks the transition from the so-called main group elements to the  transition group (transition metals). The transition metals are characterized by the fact that each of them have an incompletely occupied d-orbital (called d-block)!

Another jump where even one main shell is being skiped can be seen in the transition from the element barium (Ba) to cerium (Ce). After the 6s orbital of barium has been completely filled, two main shells are “jumped back” and the 4f orbital is filled up (the lanthanum in between of those elements is an exception to the Aufbau principle). This jump introduces a subgroup of transition metals, called lanthanides or actinides (f-block). The lanthanides or actinides are characterized by the fact that the f-orbital is gradually filled with electrons! Strictly speaking, the elements lanthanum and actinium do not belong to the group of lanthanides or actinides, although they are very often counted for practical reasons (after all, the suffix “ide” means similar to). Therefore, these elements also fall into the f-block.

Depending on which orbital an electron is added to, one can divide the periodic table into blocks, which correspond to s-, p-, d- or f-block.

Note, that there are other exceptions to the aufbau principle, for example, for the metals copper and chromium. There, an electron changes from the 4s subshell to the 3d subshell and thus remains occupied by only one electron. Such exceptions to the regular principle can be found especially at higher atomic numbers, as the electrons influence each other more and more. In addition, relativistic effects come into play, which are not taken into account by this model.

The Rutherford model in many cases provides a very good explanation of physical processes in matter. However, some phenomena can not be explained with this atomic model. For example, some atoms can only be excited to glow when bombarded with particles of specific energy. If the energy is only slightly lower, suddenly there is no illumination (for example Franck-Hertz experiment).

The physicist Niels Bohr suspected that this behavior must have something to do with the electron shell. Therefore, he expanded the atom model of Rutherford especially with regard to the atomic shell. He postulated that the electrons can only move in certain orbits around the atomic nucleus, comparable to the planetary motion around the sun. He called these discrete orbits shells. For this reason, the Bohr model is also referred to as shell model.

Each shell corresponds to a specific energy value of the electron (also called energy state or energy level). An electron can not assume an energy state that lies between two shells, since there can not be any electron there. The further away the shell is from the atomic nucleus, the more energetic is the state of an electron located there (higher energy level). This is the first innovation in the atomic concept that Bohr formulated as a postulate:

Electrons move only on discrete shells around the nucleus, each representing a certain energy level!

A postulate is a principle on which a theory is based!

With Bohr’s shell model can now finally be understood why atoms absorb only certain amounts of energy. Such an energy intake is also referred to as absorption. The absorption of energy can only happen if the energy supply is at least as large as an electron can be “lifted” from its current shell to the next higher one. Since there are no energy states between two shells, at lower energy levels, no electron can be brought to a next higher shell. The amount of energy supplied is not absorbed by the atom or by the electrons. The atom then remains in its lowest-energy state, which is also called the ground state. The state of an atom after one or more electrons have been brought to a higher energy level is called an excited state.

Conversely, when an electron “falls” to a lower energy level (towards an inner shell), only discrete energy packets can be released. The process of releasing energy is referred to as emission. The emitted energy corresponds exactly to the difference in the energy level of the two shells which are invovled in this process. This energy is emitted in terms of radiation (photons).

In this way, it can be explained why mercury, for example, emits a specific energy spectrum to which specific wavelengths (colors) in the light spectrum belong. The figure below shows the emitted spectrum of a mercury-vapor lamp. It can be seen that only certain wavelengths are emitted. So only discrete energy leaps occur. This corresponds to the leaps of the electrons from an energetically higher to an energetically lower shell. Because of the sharply defined lines in the spectrum, it is also called a line spectrum.

This provides another important insight into Bohr’s new nuclear concept:

When an electron “falls” from an outer shell to an inner shell, a photon is emitted. The energy of the photon corresponds to the energy difference of the two shells.

Note: The discretely portioned energy in the transition of an electron between two shells is referred to as quantum and the process of releasing a quant is called quantum leap. The Bohr atom model thus already contains basic features of quantum physics.

Although the Bohr model of the atom is a further development of Rutherford’s atomic model, it also contains some weak points. Thus, the imaginary circular motion of an electron around the nucleus is an accelerated motion. However, such an accelerated movement of charged particles would have to lead to an energy dissipation. Thus, after a short time the electrons should have no more energy to stably orbit around the nucleus. The consequence would be that the electrons fall into the nucleus and the atom decays. Since this is obviously not the case, Bohr had to postulate another postulate, which, however, contradicts everyday knowledge:

The electrons orbit the nucleus without emitting radiation!

Based on the different energy states of the shells Bohr also made a statement about the distribution of the electrons on the respective shells. Thus, on the innermost shell, which he called K-shell, there can be a maximum of two electrons. On the following shell, the L-shell, a maximum of 8 electrons can be found. The following M-shell contains a maximum of 18 electrons and the N-shell a maximum of 32 electrons, etc .:

• 1st shell (K-shell): 2 electrons
• 2nd shell (L-shell): 8 electrons
• 3rd shell (M-shell): 18 electrons
• 4th shell (N-shell): 32 electrons
• 5th shell (O -shell) Shell): 50 electrons
• 6th shell (P shell): 72 electrons
• 7th shell (Q shell): 98 electrons

The maximum number \( N_{max} \) of electrons on a particular shell can be determined by the following equation, where \( n \) is the shell number:
\begin{equation}
\boxed{N_{max} = 2 \cdot n^2 } \\[5px]
\end{equation}

The occupation of the shells with electrons always takes place from the lowest-energy state or the lowest-energy shell. The magnesium atom with its total of 12 electrons thus occupies 2 electrons on the K shell and 8 electrons on the L shell. These shells are now fully occupied so that the last two electrons can fit on the M shell. The electrons on an unfilled shell (in this case: the two electrons on the M shell) are also called valence electrons. The shell itself is called valence shell. The valence electrons on the outermost shell decisively determine the chemical properties of the atom and are also responsible for the position of the element in the periodic table.

The outermost shell is called valence shell and the electrons located in there are referred to as valence electrons. Chemical properties as mostly influenced by the number of valence electrons!

Note that the maximum number of electrons on a shell does not mean that an atom can have as many valence electrons! Because not always such a simple occupation rule shows up as in the case of the magnesium atom. In some cases a new shell is being filled (which then forms the outer electrons) although the underlying one is not yet fully occupied. This is evident, for example, in the case of the calcium atom. While the valence shell contains two electrons, the underlying M shell is filled only with 8 electrons and not with the maximum number of 18.

The order of occupation of the shells with electrons must therefore also be based on further influences that can not yet be explained by the Bohr model. In addition, experimental findings show that the classification of the electron orbits into the shells was too simple. For in some experiments, one also found energetic radiation transitions, which were also discrete, but located between the energy levels of the above-mentioned shells. So there had to be a finer division of the shells. The question of how and why elements form chemical compounds can not be explained by Bohr model as well. The physicist Sommerfeld provided an important development of Bohr’s atomic model (Bohr-Sommerfeld model).

In 1910, the physicist Ernest Rutherford found that when a thin gold foil was bombarded with α-particles (twice positively charged helium nuclei with two neutrons \( ^4_2\text{He}^{2+}  \)), only very few of these particles collided with the atomic nuclei of the gold atoms. Almost all α-particles traveled on a straight trajectory through the foil, while only a few were deflected.

Obviously, very few α-particles were close enough to the positive nucleus of the gold atoms that they could be deflected to a significant degree by the repulsive forces. In most cases, the α-particles traversed the gold foil at quite a distance from the respective atomic nuclei and were scarcely affected in their trajectory. This experiment concluded that the nucleus would have to be much smaller compared to the rest of the atom or rather to its atomic shell.

Today we know that the atomic nucleus has a diameter which is 10,000 to 100,000 times smaller than the atomic shell! If the atomic nucleus had the size of a dollar coin, the diameter of the atomic shell would amount to about 2 km!

The gold foil experiment further showed that some α-particles were reflected back to the gold foil with almost no energy loss. They obviously had to hit something very massive and heavy (analogous to a tennis ball that hits a massive concrete wall and flies back at almost the same speed). From this, Rutherford concluded that nearly the entire mass of an atom must be concentrated in the nucleus to produce such a strong reflection effect. And indeed, nearly 99.9% of the total mass of an atom is contained in its nucleus. Only 0.1% of the mass is therefore attributable to the atomic shell. Today we know that a proton (as well as a neutron) has a mass about 1800 times as large as an electron.

These findings formed the basis for Rutherford’s atomic model (Rutherford model), whose quintessences are summarized below:

• an atom consists of an atomic nucleus and an atomic shell,
• the nucleus is positively charged and the atomic shell carries a negative charge,
• in the nucleus are positively charged protons (and neutrons),
• in the atomic shell are the negatively charged electrons,
• the nucleus is much smaller than the atomic shell and
• almost the entire mass of an atom is concentrated in its nucleus.

With the Rutherford model, the results of scattering experiments (such as those of the gold foil experiment) could be correctly explained. The basic mass and size ratios as well as the corresponding division into atomic nucleus and electron shell also reflect this atomic model.

For example, the question of why atoms can only be excited with certain energies can not be answered by this model. Or why atoms emit characteristic line spectra. Likewise, Rutherford’s atomic model gives no explanation why an atom is stable, because the circular motion of the electrons around the nucleus would actually lead to an energy dissipation. Accordingly, the electrons ought to fall into the nucleus after only a short time and no atom should therefore be stable!

Some of the weaknesses of Rutherford’s atomic model could be corrected by the physicist Niels Bohr in his model (Bohr model).

Note

In principle, models (such as the atomic models or the particle model) never claim to give a complete explanation of reality. Models are always attempts to depict reality within certain limits and make it explainable.

The Rutherford model is not fundamentally “wrong” but has only limits of validity. Therefore, the Rutherford is not obsolete but it depends on the phenomena to be described and explained. To explain, for example, the gold foil experiment, the Rutherford model is completely sufficient; this does not require an unnecessarily complex quantum mechanical model.

Models are attempts to describe observable phenomena within certain validity limits.