Bohr-Sommerfeld model of the atom

The Bohr-Sommerfeld atomic model , Sommerfeld atomic model or the Sommerfeld extension is a physical description of the electron paths in an atom . It was proposed by Arnold Sommerfeld in 1915/16 and represents a refinement of Bohr's atomic model.

overview

The Bohr-Sommerfeld atomic model from 1916 builds on the Bohr model from 1913 and is thus one of the older quantum theories before the development of quantum mechanics . It is assumed that the electrons move around the atomic nucleus on well-defined orbits , which result from the equations of motion of classical mechanics , i.e. on the ellipses known from planetary motion . Quantum theoretical principles are introduced through additional quantization conditions (Bohr-Sommerfeld quantization). These lead to the fact that only a small selection of the tracks is allowed that would be possible according to the classical mechanics. As a result, the conserved quantities associated with the orbital motion ( energy and angular momentum ) can no longer assume any arbitrary, but only certain, discrete values ; they are therefore “quantized”.

The advancement of the Sommerfeld atomic model compared to its predecessor consists primarily in the fact that it has the fine structure of the hydrogen spectrum , i.e. H. makes the small splits of the classically calculated energies calculable ( fine structure constant ) by taking into account the equation of motion of the special theory of relativity . The fine structure is justified with the increase in the inertial mass, which the special theory of relativity predicts for increasing speed. With the same principal quantum number n , the closer the electron flies past the nucleus in the perihelion , the greater the numerical eccentricity of the ellipse or the smaller the orbital angular momentum. The different orbits to a main quantum number no longer have exactly the same energy level , but the energy is also dependent on the orbital angular momentum.

Another advance of the Sommerfeld atomic model is that it can explain the normal Zeeman effect and the Stark effect .

The Bohr-Sommerfeld model of the atom has a high explanatory value because of its clarity; Instead of the only quantum number of the electron states previously based on Bohr's model , it correctly provided all three spatial quantum numbers and thus enabled an at least qualitative physical explanation of the periodic table of the chemical elements for the first time .

However, like Bohr's model, the Bohr-Sommerfeld model fails in all calculations of atoms with more than one electron. That this failure resulted from the erroneous assumption of defined, classical particle trajectories became clear from 1925, when the new quantum mechanics could explain considerably more observations and make predictions, and these were mostly quantitatively correct. In it there are no longer any defined paths, as one can e.g. B. can be recognized by the Heisenberg uncertainty relation, but only probability distributions .

Geometry of the electron orbits

While in Niels Bohr's model the possible orbits of the electron are circles around the atomic nucleus , Sommerfeld introduced more general elliptical orbits as an extension to these circular orbits. According to this model, the core is located in one of the two focal points of an orbital ellipse, resulting in a geometric configuration as in the planetary orbits according to Kepler's laws . As in Bohr's model, the electron should move on these orbits without the emission of electromagnetic radiation , as would have to occur according to classical electrodynamics .

The Bohr-Sommerfeld model thus represents a Kepler planetary system in miniature, while the Bohr model corresponds to the older Copernican idea. This analogy is obvious because the force fields of the Coulomb force of the atomic nucleus and the gravitation of the sun have the same shape:

{\ displaystyle F \ sim {1 \ over r ^ {2}}.}

The consideration of the special theory of relativity results in orbits approximately in the form of an ellipse, the main axis of which rotates slowly ( perihelion ).

Bohr-Sommerfeld quantization

An ellipse can no longer be described by a single parameter (radius) like a circle, rather two (e.g. large and small semi-axes ) are required . That is why two quantum numbers are necessary for elliptical orbits to determine the shape. A third one is needed to orient the plane of the path in space. A quantum number of them, but also only one, can contribute to the Bohr quantization of the circular motion, because in the spherically symmetric potential of the atomic nucleus all orbits have a certain angular momentum . Sommerfeld generalized Bohr's quantization to the effect that each coordinate must meet its own quantum condition: ${\ displaystyle q}$

{\ displaystyle \ oint p \, dq = nh.}

In it is

${\ displaystyle q}$ a coordinate
${\ displaystyle p}$ their canonically assigned impulse
${\ displaystyle n = 1, \, 2, \, \ dots}$ the quantum number
${\ displaystyle h}$ the Planck constant .

The curve integral is the area within the relevant path in the plane. It is referred to in mechanics as the effect associated with that movement. ${\ displaystyle pq}$

In the 1-dimensional case it can simply be the coordinate and the ordinary momentum . Then from the quantum condition z B. immediately quantize the harmonic oscillator with energy levels . However, through canonical transformation one can come to other variables which then automatically fulfill the same condition. ${\ displaystyle q}$ ${\ displaystyle x}$ ${\ displaystyle p}$ ${\ displaystyle hf}$

When moving in two or three dimensions, the coordinate z. B. choose an angle of rotation , which includes the angular momentum as a canonical pulse . The effect integral for a full cycle is then ${\ displaystyle \ alpha}$ ${\ displaystyle \, l}$

{\ displaystyle \ oint p \, dq = \ int _ {2 \ pi} l \, d \ alpha = 2 \ pi l,}

and the angular momentum quantization results as in Bohr ${\ displaystyle l = nh / 2 \ pi.}$

Quantum numbers

Sommerfeld considers the system in the three spherical coordinates (distance r and two angles) and subjects each to the new quantization condition. This gives him three quantum numbers: the radial , the azimuthal and the magnetic . ${\ displaystyle n_ {r}}$ ${\ displaystyle l}$ ${\ displaystyle m_ {l}}$

Principal quantum number

The quantum number n , which, as in Bohr's model and in the Rydberg formulas, also determines the energy here, is now called the principal quantum number and turns out to be

{\ displaystyle n = n_ {r} + l}

or actually as

{\ displaystyle n = n_ {r} + l + 1.}

Minor quantum number

The azimuthal quantum number , now called secondary quantum number, indicates the (orbital) angular momentum ( is Planck's constant divided by .) With a given n , the secondary quantum number can assume the natural numbers from 1 to : ${\ displaystyle l}$ ${\ displaystyle l \ hbar}$ ${\ displaystyle \ hbar}$ ${\ displaystyle h}$ ${\ displaystyle 2 \ pi}$ ${\ displaystyle n}$

{\ displaystyle l = 1,2, \ ldots, n,}

where the greatest possible angular momentum ( ) belongs to Bohr's circular path. The value is expressly excluded because in this case the electron swings back and forth on a straight line that goes through the nucleus. ${\ displaystyle l = n}$ ${\ displaystyle l = 0}$

Electron orbits for hydrogen in the Bohr-Sommerfeld atomic model for n = 1,2,3; Scale values in Ångström .

According to the quantum mechanical calculation , which replaced the Bohr-Sommerfeld model from 1925, the angular momentum is, however, exactly one unit lower and consequently the correct range of values ${\ displaystyle \ hbar}$

{\ displaystyle l = 0.1, \ ldots, (n-1)}

(see also the illustration opposite). A spherically symmetrical orbital belongs to it . ${\ displaystyle l = 0}$

Magnetic quantum number

The magnetic quantum number indicates the angle of inclination of the angular momentum against the z-axis, or more precisely the size of the projection of the angular momentum onto the z-axis: ${\ displaystyle m_ {l}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle l}$ ${\ displaystyle l}$

{\ displaystyle m_ {l} = l \ cdot \ cos \ alpha \ Leftrightarrow \ cos \ alpha = {\ tfrac {m_ {l}} {l}}.}

The range of values of this quantum number is

{\ displaystyle m_ {l} = \, - l, \, - (l-1), \, \ ldots \ ,, (l-1), \, l,}

different values overall . The directional quantization is thus predicted, because there is only this finite number of setting options from exactly parallel to exactly antiparallel. (Quantum mechanics gives the angular momentum vector instead of the length, which means that the two extreme setting options do not quite coincide with the z-axis.) ${\ displaystyle 2l + 1}$ ${\ displaystyle l}$ ${\ displaystyle {\ sqrt {l (l + 1)}},}$ ${\ displaystyle \ cos \ alpha = \ pm {\ tfrac {l} {\ sqrt {l (l + 1)}}}}$

The electron moving around the atomic nucleus forms a magnetic dipole , the direction of which is perpendicular to the orbital ellipse, i.e. parallel to the vector of the orbital angular momentum. If you bring the atom into an external magnetic field (which defines the z-axis), then its energy also depends on the setting angle. Because of the directional quantization, the energy splits into different values depending on the value of the magnetic quantum number (hence its name) ( Zeeman effect ). ${\ displaystyle {\ vec {\ mu}}}$ ${\ displaystyle {\ vec {L}}}$ ${\ displaystyle {\ vec {B}} _ {ext}}$ ${\ displaystyle 2l + 1}$

Spin quantum number

In addition to these spatial quantum numbers, which were introduced in Bohr-Sommerfeld's atomic model, there is also the spin quantum number for each electron for precisely two possible settings for its own angular momentum ( spin ). It is indicated with the values + ½ or −½ or the symbols ↑ or ↓. This quantum number does not result from Sommerfeld's quantization conditions, but was later inserted into the model due to otherwise inexplicable experimental findings (e.g. even-numbered splitting in the Stern-Gerlach experiment and in the anomalous Zeeman effect ). The energy of each of the above-mentioned orbits can thereby be split into two energies. ${\ displaystyle m_ {s}}$

Pauli ban

Due to the order in the understanding of the atomic structure made possible by the Sommerfeld model, Wolfgang Pauli was able to discover the Pauli prohibition in 1925 : Each of the orbits determined by the three spatial quantum numbers can hold a maximum of two electrons, which must then have opposite spin quantum numbers.

swell

Arnold Sommerfeld: On the quantum theory of the spectral lines (I + II) . In: Annals of Physics . 51, 1916, pp. 1-94. doi : 10.1002 / andp.19163561702 . (instead of vol. 51, Wiley-online applies: vol. 356)
Helmut Rechenberg Quanta and Quantum Mechanics in: Laurie M Brown et al. (Ed.) Twentieth Century Physics vol. I, IOP Publishing Ltd. AIP Press. Inc. 1995, ISBN 0750303530
Friedrich Hund: History of Quantum Theory , BI University Pocket Books Vol. 200 / 200a, Bibliographisches Institut Mannheim 1967