Larmor diamagnetism

The Larmor diamagnetism (named after the physicist Joseph Larmor ) is a special form of diamagnetism , in atoms with completely filled electron shells occurs. It is therefore also called atomic diamagnetism , in contrast to the diamagnetism of the free electron gas (Landau diamagnetism).

In the classical conception , the external magnetic field induces atomic circular currents with which a magnetic moment opposite to the magnetic field is connected ( Lenz's rule ). However, this idea serves only for illustration, since according to the Bohr-van-Leeuwen theorem a classical system cannot show magnetism. However, the exact quantum mechanical description is very complex.

description

The Larmor susceptibility (also called diamagnetic Langevin susceptibility ) is calculated as follows:

${\displaystyle {\begin{aligned}\chi _{\mathrm {Larmor} }&=-{\frac {\mu _{0}Ne^{2}}{6mV}}\sum _{i=1}^{Z}{\langle r_{i}^{2}\rangle }\\&\approx -{\frac {\mu _{0}Ne^{2}}{6mV}}Z_{\mathrm {a} }r_{\mathrm {a} }^{2}\end{aligned}}}$

Here referred to

• ${\textstyle \mu _{0}}$the magnetic permeability of the vacuum
• ${\displaystyle N}$ the number of atoms
• ${\displaystyle e}$the elementary charge
• ${\displaystyle m}$ the crowd
• ${\displaystyle V}$ the volume
• ${\textstyle Z_{\mathrm {a} }}$ the number of electrons in the outermost shell of the atom or ion
• ${\textstyle r_{\mathrm {a} }}$the atomic or ionic radius .

Overall, the susceptibility is largely independent of temperature. The use of the electron count of the outermost shell is justified by the fact that these electrons dominate because of the largest radius.

Classic derivation

The circular current that the electrons generate due to the Larmor precession with circular frequency is proportional to the magnetic flux density of the external magnetic field: ${\displaystyle I}$ ${\textstyle Z}$ ${\displaystyle \omega _{\mathrm {Lamor} }}$ ${\displaystyle B_{\mathrm {ext} }}$

${\displaystyle I=-Ze{\frac {\omega _{\mathrm {Lamor} }}{2\pi }}=-{\frac {Ze^{2}}{4\pi m}}B_{\mathrm {ext} }}$

Thus one obtains for the magnetic moment:

${\displaystyle \mu =IA=-{\frac {Ze^{2}}{4\pi m}}B_{\mathrm {ext} }\pi \left(\langle x^{2}\rangle +\langle y^{2}\rangle \right)=-{\frac {Ze^{2}}{6m}}B_{\mathrm {ext} }\langle r_{\mathrm {a} }^{2}\rangle }$

In the last step, the spherical symmetry of the charge distribution was used. ${\textstyle \langle x^{2}\rangle +\langle y^{2}\rangle ={\frac {2}{3}}\langle r_{\mathrm {a} }^{2}\rangle }$

This results in the energy shift of the states:

${\displaystyle \Delta E=-\mu B_{\mathrm {ext} }={\frac {Ze^{2}}{6m}}B_{\mathrm {ext} }^{2}\langle r_{\mathrm {a} }^{2}\rangle }$

This energy shift results in the magnetization, from which one can ultimately calculate the susceptibility:

${\displaystyle \chi _{\mathrm {Lamor} }=-\mu _{0}{\frac {N}{V}}{\frac {\partial ^{2}\Delta E}{\partial B_{\mathrm {ext} }^{2}}}=-{\frac {\mu _{0}Ne^{2}}{6mV}}Z_{\mathrm {a} }r_{\mathrm {a} }^{2}}$

Individual evidence

1. ↑ ^{a b c} Rudolf Gross, Achim Marx: Solid State Physics . De Gruyter, Oldenbourg 2012, p. 670 .