- Sommerfeld atomic model postulates
- Electrons follow circular and elliptical orbits
- Zeeman effect and Stark effect
- The atomic nucleus and electrons move around their center of mass
- Electrons can reach relativistic speeds
- Advantages and disadvantages
- Advantage
- Disadvantages
- Articles of interest
- References
The atomic model Sommerfeld was created by the German physicist Arnold Sommerfeld between 1915 and 1916, to explain the facts that the Bohr model, released earlier in 1913, could not satisfactorily explain. Sommerfeld first presented his results to the Bavarian Academy of Sciences and later published them in the journal Annalen der Physik.
The model of the atom proposed by the Danish physicist Niels Bohr describes the simplest atom of all, hydrogen, but could not explain why electrons in the same energy state could present different energy levels in the presence of electromagnetic fields.
Figure 1. In semi-classical models the orbits are Newtonian, but only those whose perimeter is an integer number of times the de Broglie wavelength are allowed. Source: F. Zapata.
In the theory proposed by Bohr, the electron orbiting the nucleus can only have certain values of its orbital angular momentum L, and therefore cannot be in any orbit.
Bohr also considered these orbits to be circular and a single quantum number called the principal quantum number n = 1, 2, 3… served to identify the allowed orbits.
Sommerfeld's first modification to the Bohr model was to assume that the orbit of the electron can also be elliptical.
A circumference is described by its radius, but for an ellipse two parameters have to be given: semi-major axis and semi-minor axis, in addition to its spatial orientation. With this he introduced two more quantum numbers.
The second major modification Sommerfeld made was to add relativistic effects to the atomic model. Nothing is faster than light, however Sommerfeld had found electrons with appreciably close speeds, therefore it was necessary to incorporate relativistic effects into any description of the atom.
Sommerfeld atomic model postulates
Electrons follow circular and elliptical orbits
The electrons in the atom follow elliptical orbits (circular orbits are a particular case) and their energy state can be characterized by 3 quantum numbers: the principal quantum number n, the secondary quantum number or azimuthal number l and the magnetic quantum number m L.
Unlike the circumference, an ellipse has a semi-major axis and a semi-minor axis.
But ellipses with the same semi-major axis can have different semi-minor axes, depending on the degree of eccentricity. An eccentricity equal to 0 corresponds to a circle, so it does not rule out circular paths. Furthermore, ellipses in space can have different inclinations.
Therefore Sommerfeld he added to its model number quantum secondary l to indicate the minor axis and the magnetic quantum number m L. Thus he indicated what are the allowed spatial orientations of the elliptical orbit.
Figure 2. The orbits corresponding to the energy level n = 5 are shown for different values of the angular momentum l that have full de Broglie wavelengths. Source: wikimedia commons.
Note that it does not add new principal quantum numbers, so the total energy of the electron in elliptical orbit is the same as in the Bohr model. Therefore there are no new energy levels, but a doubling of the levels given by the number n.
Zeeman effect and Stark effect
In this way it is possible to fully specify a given orbit, thanks to the 3 quantum numbers mentioned and thus explain the existence of two effects: the Zeeman effect and the Stark effect.
And so he explains the doubling of energy that appears in the normal Zeeman effect (there is also an anomalous Zeeman effect), in which a spectral line is divided into several components when it is in the presence of a magnetic field.
This doubling of the lines also occurs in the presence of an electric field, which is known as the Stark effect, which led Sommerfeld to think about modifying the Bohr model to explain these effects.
The atomic nucleus and electrons move around their center of mass
After Ernest Rutherford discovered the atomic nucleus and the fact that almost all the mass of the atom is concentrated there was revealed, scientists believed that the nucleus was more or less stationary.
However, Sommerfeld postulated that both the nucleus and the orbiting electrons move around the center of mass of the system, which of course is very close to the nucleus. His model uses the reduced mass of the electron-nucleus system, rather than the mass of the electron.
In elliptical orbits, as with the planets around the Sun, there are times when the electron is closer, and other times further from the nucleus. Therefore its speed is different at each point in its orbit.
Figure 3.- Arnold Sommerfeld. Source: Wikimedia Commons. GFHund.
Electrons can reach relativistic speeds
Sommerfeld introduced into his model the fine structure constant, a dimensionless constant related to the electromagnetic force:
α = 1 /137.0359895
It is defined as the quotient between the electron charge e squared, and the product of Planck's constant h and the speed of light c in a vacuum, all multiplied by 2π:
α = 2π (e 2 / hc) = 1 /137.0359895
The fine structure constant relates to three of the most important constants in atomic physics. The other is the mass of the electron, which is not listed here.
In this way electrons are linked with photons (moving at speed c in vacuum), and thus explain the deviations of some spectral lines of the hydrogen atom from those predicted by the Bohr model.
Thanks to relativistic corrections, energy levels with equal n but different l are separated, giving rise to the fine structure of the spectrum, hence the name of the constant α.
And all the characteristic lengths of the atom can be expressed in terms of this constant.
Figure 4. The quantization of angular momentum L. is shown. Unlike circular orbits, ellipticals allow more than one value of L for each energy level. Source: F. Zapata.
Advantages and disadvantages
Advantage
-Sommerfeld showed that a single quantum number was insufficient to explain the spectral lines of the hydrogen atom.
-It was the first model to propose a spatial quantization, since the projections of the orbits in the direction of the electromagnetic field are, in effect, quantized.
-The Sommerfeld model successfully explained that electrons with the same principal quantum number n differ in their energy state, because they can have different quantum numbers l and m L.
-Introduced the constant α to develop the fine structure of the atomic spectrum and explain the Zeeman effect.
-Included relativistic effects, since electrons can move with speeds quite close to that of light.
Disadvantages
-Your model was only applicable to atoms with one electron and in many respects to alkali metal atoms such as Li 2+, but it is not useful in the helium atom, which has two electrons.
-It did not explain the electronic distribution in the atom.
-The model allowed calculating the energies of the permitted states and the frequencies of the radiation emitted or absorbed in the transitions between states, without giving information about the times of these transitions.
-Now it is known that electrons do not follow trajectories with predetermined shapes such as orbits, but occupy orbitals, regions of space that correspond to solutions of the Schrodinger equation.
-The model arbitrarily combined classical aspects with quantum aspects.
-He failed to explain the anomalous Zeeman effect, for this the Dirac model is needed, which later added another quantum number.
Articles of interest
Schrödinger's atomic model.
De Broglie atomic model.
Chadwick's atomic model.
Heisenberg atomic model.
Perrin's atomic model.
Thomson's atomic model.
Dalton's atomic model.
Dirac Jordan atomic model.
Atomic model of Democritus.
Bohr's atomic model.
References
- Brainkart. Sommerfeld atom model and its drawbacks. Recovered from: brainkart.com.
- How We Came to Know the Cosmos: Light & Matter. Sommerfeld's atom. Recovered from: thestargarden.co.uk
- Parker, P. The Bohr-Sommerfeld Atom. Recovered from: physnet.org
- Educational Corner. Sommerfeld's model. Recovered from: rinconeducativo.com.
- Wikipedia. Sommerfeld atomic model. Recovered from: es.wikipedia, org.