Electron spin

Electron spin is the quantum mechanical property spin of electrons . This property was first discovered in electrons in 1925, and then in all other types of particles. The spin (from English spin 'rotation', 'twist') has all the properties of a classical mechanical angular momentum , with the exception of the fact that it is caused by the rotary motion of a mass.

For every electron the spin has an invariable amount, which is given by the spin quantum number s = 1/2 . Even when the electron is at rest with zero kinetic energy, it has its spin, which is why it is also known as its own angular momentum. How or how the spin comes about remains inexplicable in classical physics. Clear or semi-classical descriptions are therefore incomplete. An explanation for the spin was given in 1928 in the Dirac equation . This led to the development of relativistic quantum mechanics .

The spin of the electron is of fundamental importance for the physical world view. It plays a decisive role in the structure of the atomic shell and thus for matter up to the definition of its macroscopic properties.

For further basic properties of spin see main article: Spin .

Discovery and understanding of spin

Unsolved Problems of Atomic Physics in the Early 20th Century

After the movement of the electrons in the atom was recognized as decisive for the emission of light waves with well-defined frequencies ( spectral lines ) (convincing e.g. in Bohr's atomic model in 1913), the fine splitting of many lines, which had been observed for a long time, represented a problem that remained unsolved. It is true that an additional splitting caused by the application of a strong magnetic field ( Zeeman effect , found as early as 1897) could in principle be explained by a magnetic influence on the movement of electrons on their stable orbits.

The basis of the explanation is the Larmor theorem: It predicts the precession of the entire trajectory around the magnetic field axis, angular frequency ; therein e and m charge and mass of the electron, c speed of light, B magnetic field.

{\ displaystyle \ omega _ {\ text {Larmor}} = {\ tfrac {e} {mc}} \; B}

However, this explanation only fit in cases where the split was threefold (hence called the “normal Zeeman effect”). In Bohr-Sommerfeld's atomic model of 1916, higher magnetic splits, if they were odd, could be explained by the directional quantization of the orbital angular momentum of the electron levels:

For an orbital angular momentum quantum number exactly are different tilt angles of the angular momentum to the direction of the magnetic field are available, each with a to -shifted energy (in the divided by 2π Planck quantum of action , the magnetic quantum number with their different possible values of up ). Since it can only be an integer, there is always an odd number of split levels.

{\ displaystyle \, l}

{\ displaystyle \, (2l + 1)}

{\ displaystyle \, m_ {l} \, \ hbar \, \ omega _ {\ text {Larmor}}}

{\ displaystyle \, \ hbar}

{\ displaystyle \, m_ {l}}

{\ displaystyle \, (2l + 1)}

{\ displaystyle \, m = -l}

{\ displaystyle \, m = + l}

{\ displaystyle \, l}

The Bohr-Sommerfeld model could also explain fine splits that were not caused by a magnetic field, because it also makes the electron energy somewhat dependent on the same principal quantum number due to relativistic effects . However, the frequently observed even-numbered magnetic splits into two or more levels, as well as the twofold splitting of a level without a magnetic field (e.g. with the intense yellow spectral line of sodium, on which Zeeman was also able to demonstrate the magnetic effect for the first time) remained unexplained . . ${\ displaystyle \, l}$

Introduction of the electron spin

To solve this riddle, Samuel Goudsmith and George Uhlenbeck suggested in 1925 that the electron should be assigned an additional intrinsic angular momentum, spin . It had to have a half-integer angular momentum quantum number so that the magnetic spin quantum number remained limited to two possible values and thus a double or, together with an orbital angular momentum , a higher even-numbered split resulted. ${\ displaystyle \, s = {\ tfrac {1} {2}}}$ ${\ displaystyle \, m_ {s} = \ pm {\ tfrac {1} {2}}}$ ${\ displaystyle \, l \ geq 1}$

Regarding the reception of this daring idea, it should be noted that its two authors immediately backed off and tried again to prevent the publication that had already been prepared. Their head of the institute, Paul Ehrenfest, forbade them on the grounds: “They are both young enough to be able to afford something stupid.” Physically weighty counter-arguments at the time were:

In order for the electron to be able to generate its own angular momentum through a rapid rotation around its center, it would either have to have an impossibly large radius or move at the "equator" at a multiple of the speed of light.
The influence of the magnetic spin quantum number on the level splitting must be set exactly twice as large as the influence of the magnetic orbital angular momentum quantum number , incompatible with the well-founded Larmor theorem. ${\ displaystyle \, m_ {s}}$ ${\ displaystyle \, m_ {l}}$

Therefore, Wolfgang Pauli initially contradicted the idea of the intrinsic angular momentum with half-integer value, although in the previous year he had assigned an inner two-valued quantum number to the electron in addition to the three spatial quantum numbers in order to explain the systematics of the spectra and the shell structure of the atomic shell to be able to formulate Pauli's principle of exclusion . This quantum number has now been identified as. But in 1927 Pauli himself established the use of half-integer electron spin in (non-relativistic) quantum mechanics in the form of the Pauline spin matrices . ${\ displaystyle \, m_ {s} = \ pm {\ tfrac {1} {2}}}$

Anomalous magnetic moment of the electron

Anomalous spin g-factor of the electron

All particles that have electrical charge and angular momentum have a magnetic (dipole) moment , often illustrated as a small bar magnet parallel to the axis of rotation. Therefore the energy levels are influenced by a magnetic field ( Zeeman effect ). Classical physics makes a clear statement about the relationship between the magnitude of the angular momentum and the magnetic moment, which is also correct for the orbital angular momentum of the electrons in the atomic shell (Larmor theorem see above). However, an (almost, see below) exactly twice as large a magnetic moment belongs to the electron spin. This correction is taken into account by means of a number called a g-factor . For orbital angular momentum, the classic value applies to the spin of the electron . This anomalous spin g-factor was discovered by analyzing the Zeeman effect. The Dirac theory provides a theoretical explanation. After the discovery of a deviation from the Dirac value of 1.1 ‰, only this deviation is increasingly referred to as an anomaly of the g-factor. It is explained by the effects of quantum electrodynamics , which are based on otherwise unobservable processes such as vacuum polarization and self-energy . Because of the ability to determine this g-factor anomaly with extreme accuracy, experiments and theoretical calculations have been advanced to the 12th decimal place over the past 50 years without showing any significant difference. ${\ displaystyle g_ {l} {\ mathord {=}} 1}$ ${\ displaystyle g_ {s} {\ mathord {=}} 2}$ ${\ displaystyle g_ {s} {\ mathord {=}} 2}$

Electron spin and magnetic materials

The magnetic dipoles associated with the electron spin are macroscopically directly noticeable in the form of the permanent magnetism of all magnetic materials . These materials contain atoms of the elements around iron or rare earths with about half-filled inner shells (3d or 4f shell). The energetically preferred configuration of the electrons shows that all spins are parallel, while all other angular momenta add up to zero. Macroscopically noticeable (permanent) magnetism occurs in the case of materials for which it also applies that neighboring atoms also prefer the parallel alignment of their magnetic moments energetically. (This is explained by the exchange interaction of the electrons.) The ferromagnetism therefore appears with the anomalous value . This can be demonstrated by the Einstein-de-Haas effect , in which an iron rod, which is only at rest, starts rotating when its magnetization is reversed, i.e. a large number of spins flip over at the same time. In order to maintain the initial total angular momentum zero, the flipping of the spins must be compensated for by an opposite macroscopic angular momentum of the rod. The experiment is not easy and in the first few years supposedly confirmed the classical value expected at the time . Only after the discovery of the anomalous magnetic moment of the electron with the help of the Zeeman effect did the measurement results from the Einstein-de-Haas effect level off at . ${\ displaystyle g_ {s} = 2}$ ${\ displaystyle g_ {l} = 1}$ ${\ displaystyle g_ {s} = 2}$

Important experiments on the magnetic moment of the electron

The magnetic moment of the electron spin enabled the first direct proof of directional quantization in the Stern-Gerlach experiment . The effects of magnetic electron spin resonance are used for the detailed investigation of paramagnetic substances.

Consequences for the development of the theory

The spin and anomalous g-factor were predicted without further assumption by the Dirac theory of the electron in 1928 , which quickly made this theory famous. A small deviation of the electron g-factor from the value found in 1946 was theoretically predicted by the effect of vacuum polarization, which is possible in the theory of quantum electrodynamics . The deviation from value 2 is only 1 ‰, but today it has been measured with an accuracy of 10 decimal places in order to test this theory first and then the full standard model . No significant deviation has yet been found. ${\ displaystyle s = {\ tfrac {1} {2}}}$ ${\ displaystyle g_ {s} = 2}$ ${\ displaystyle g_ {s} = 2}$

literature

↑ GE Uhlenbeck, S. Goudsmit: Replacement of the hypothesis of non-mechanical compulsion by a requirement regarding the internal behavior of each individual electron . In: Natural Sciences . Vol. 13 No. 47, 1925, p. 953.
^ S. Goudsmit, GE Uhlenbeck: Spinning Electrons and the Structure of Spectra . In: Nature . Vol. 117, 1926, pp. 264-265.
^ Max Jammer: The Conceptual Development of Quantum Mechanics , McGraw-Hill, New York, 1966, p. 150
↑ source!

Web links

S. Goudsmit: The discovery of the electron spin (English)

[1] GE Uhlenbeck, S. Goudsmit: Replacement of the hypothesis of non-mechanical compulsion by a requirement regarding the internal behavior of each individual electron . In: Natural Sciences . Vol. 13 No. 47, 1925, p. 953.

[2] S. Goudsmit, GE Uhlenbeck: Spinning Electrons and the Structure of Spectra . In: Nature . Vol. 117, 1926, pp. 264-265.

[3] Max Jammer: The Conceptual Development of Quantum Mechanics , McGraw-Hill, New York, 1966, p. 150

[4] source!