Landau diamagnetism and de Hass van Alphen effect by Enas Hamasha Stastical mechanics ph 761 Dr. Abdulla Obeidat

Landau diamagnetism its quantized charged particles orbits which affected by magnetic field

Landau, 1930 observed its electronic orbits to be quantized and don’t on the classical form and also conclude that the magnetic susceptibility for this materials is negative

Diamagnetism do not depend on the temperature but produced from external magnetic field effect on internal electron motion for atoms

The diamagnetism materials have a negative magnetic susceptibility and smaller than one such  <<1 materials appear this property clearly: Cu, Au, Hg, and H2O.

And landau arrives to quantized the electrons orbits using by the mathematical calculation The semi classical approximation by onsagar and Walfshter and assumptions (Bohr-sommerfeld) relation

landau concluded that the orbital electron quantized where the magnetic flux equal Φn =(n+γ)(2ΠЋc/q)

de Hass van Alphen effect

It’s the oscillation of the magnetic moment of Fermi gas in a metal where change magnetic field and at low temperature

When apply the magnetic field on the surface metal the orbits of electron is closed and magnetic parallel to Z-axis and have periodicity and cyclotron frequency ωc

When applying the B on the metal surface the temperature must be low such zero or around it, and the metal must to be Fermi surface and electrons move in the Fermi surface produce magnetic moment

we can studying the orbits energy when we applying variable magnetic field in zero Kelvin and seeing what happens for orbit energy when increasing field in the figure below

the presence of B lead to occurs the degeneracy in the levels means that increases the number of electrons orbits in one level and results from apply the magnetic field in low temperature to become levels” new levels called “ landau

At low temperature and when increases the field so landau levels start becomes increases and go on increases to leave the Fermi surface and this lead to oscillation effects in metals and called the result frequency from oscillation “de Hass van Alphen frequency “

D = ρ B ρ :constant. We conclude from this relation the degeneracy proportional to magnetic field means when increases the applying magnetic field the levels increases degeneracy

when apply B on the surface and increases the degeneracy increases for every level and then translate electron from higher level to lower such that there from S+1 level to S level and when be S +1 level vacated because have electron moved to lower one S so Fermi level translate an abrupt form to S level.

we can calculate the total number of electron in this case : by N = SρBs Where, N: the # of electron in the levels S : the completely filled levels ρ : constant Bs : the magnetic field where every level filled

And we can conclude the curve which the total electronic energy versus 1/B

the changing total energy with magnetic field changed and observe the oscillation in the energy U may be detected by measurement of the magnetic moment,

, the thermal and transport properties of the metals also oscillate as successive orbital levels cut through the Fermi levels when the field is increased And this is the oscillation for magnetic moment in the figure the de Hass van Alphen effect for Fermi gas in low temperature,

the oscillation occurs in equal intervals from 1/B such that ∆ (1/B) = 2πq/ЋcS where S: extremal area on the Fermi surface S predict the shape and the volume of the surface

Extremal area: for the Fermi surface of general shape the section at different values of radius with different periods the response will be the sum of contributions from all sections or all orbits,

But the dominant response of the system comes from orbits whose periods stationary with respect to small change in the radius of free electron Fermi sphere this orbits called extremal orbits seeing in figure below:

Thank you