Magnetization

Suppose a solid is composed of atomic scale circular current loops of radius r and current I so magnetic moment m=I pi r², all oriented along the same direction. Looking down on an array of such current loops, it is appears that the currents cancel in the interior but an effective surface current is circulating around the surface:

A solid cylinder of magnetic moments will generate a solenoidal magnetic field.

Apply Ampere's Law to a rectangular path of length L spanning the edge of this solenoid. The section of path in the material will link current loops within a volume Vol=pi r² L. If the density of loops is n, the linked current is (nVol) I= n m L so Ampere's Law gives the magnetic field strength

where M=nm is the magnetic moment density called the magnetization.

Orbital Currents

An electron in a atomic orbit constitutes a kind of current loop. For a circular orbit (radius r, speed v), the charge e circulates with period T=2 pi r/v.The current is I=e/T = ev/(2 pi r). The magnetic moment is

where L_orb=mvr is the orbital angular momentum. Generally, electronic orbits have random orientations so an atom has vanishing average magnetic moment!

Induced orbital magnetization

An electron in an atom is subject to a Coulomb attraction to a nucleus. The general solution to F=ma is an elliptical orbit. In the presence of a magnetic field, an orbit of any orientation is found to precess slowly about the magnetic field at the Larmor frequency (1/2 the cyclotron frequency)

omega_L=2 pi/T= eB/(2m_e)

The current associated with the Larmor precession of Z electrons in an atom gives a (dia) magnetic moment opposite to the applied field of magnitude

m = ZIA=Z(e/T )pi r²= Ze²B r²/(4m_e)

The magnetic moment is antiparallel to the magnetic field. In a region of nonuniform magnetic field, say near a pole of an external magnet, the "pole" of the diamagnetic moment nearest the external pole is of the same sign and the atom is repelled, like the Al jumping ring. Diamagnetism is relatively weak compared to "paramagnetic" effects described below.

Paramagnetism

It turns out that electrons "spin"at an unalterable rate. Think of an electron as a uniformly charged little sphere. The rotation of the charge implies a spinning electron carries a magnetic moment. The value is

m = (e/2m_e) S= m_B = 0.93x10^-23A-m²

where S=5.27x10^-35 J-s is the spin angular momentum. Electronic spin magnetic moments in a multielectron atom tend to cancel. But an atom with an odd number of electrons will have a permanent magnetic moment coming from the odd electron. If a gas of such paramagnetic atoms is subject to an applied magnetic field, the magnetic dipoles will tend to align along the field, the magnetization being limited by thermal fluctuations. A paramagnetic substance will be attracted toward a source of magnetic field.

A classical statistical argument gives an estimate for the magnetization. The potential energy of the magnetic moment in a magnetic field is

U = -m^.B

At temperature T, the probability for a given energy is given by a Boltzman factor

P = e^-U/kT ~ 1 - m^.B/kT

and the average angle between m and B is

<cos(a)> = mB/(3kT)

The average component of magnetic moment along B is

<m_z>= |m|<cos(a)> = |m|²|B|/(3kT)

Ferromagnetism

In some paramagnetic materials at low enough temperature, the electronic magnetic moments spontaneously align producing a magnetization equivalent to one spin magnetic moment per atom. Generally, the direction of the magnetization is different in different microscopic domains of material but a rather weak external field will cause complete alignment. For Fe, the saturation magnetization is about 2 Tesla. If the external field is removed, some fraction of complete alignment remains. The material is magnetized.

D. Carlsmith