PHY294H - Lecture 28

The magnetic moment of a Bohr Hydrogen atom

According to Bohr theory, the angular momentum is quantized, so that,

$$L = l \hbar$$ (1)

where l = 0,1,2... is an integer. We then have (using m = -eL/2m_e),

$$m_{orbital} = -{e\over 2 m_e} \hbar l = -m_B l$$ (2)

where we have defined the Bohr magneton,

$$m_B = {e\hbar \over 2m_e} = 9.27 \times 10^{-24} Am^2$$ (3)

In addition, the electron carries an intrinsic magnetic moment due to its spin. The intrinsic magnetic moment of the electron has a value very close to m_B.

The total magnetic moment is found by a vector addition of the orbital and spin contributions. This sum is called $\vec{m}_0$. The magnetization which we used above is related to $\vec{m}_0$ by

$$\vec{M} = {N\over volume} \vec{m}_0$$ (4)

where N is the number of atoms in the volume of material that we are using. This formula holds when all of the atomic moments are aligned, as in a ferromagnet at low temperatures.

Magnetic hysteresis

Ferromagnetic materials are very important in technology. For example the hard drives in most computers are made using small domains on ferromagnetic materials. A small sensor (or read head) scans the surface of the hard drive. On the hard drive surface are small domains of ferromagnetic material. These domains are oriented in the plane of the surface and they have a prefered direction. The read head measures a resistivity which is sensitive to the local magnetic field. The technology of magnetic storage (e.g. hard drives) relies on a particular property of ferromagnetic materials. This property is called hysteresis. Hysteresis is a property which occurs when a magnetic field is applied to a ferromagnet which is below its Curie temperature.

In order to describe hysteresis we must describe the way in which we vary the temperature and the magnetic field. Let us start at high temperatures and quench to a temperature well below the Curie temperature. The magnetic material is frozen in a domain structure by this process. Now we apply a positive external field. The domains now begin to align with the magnetic field. At sufficiently high magnetic field the atomic magnetic moments are all aligned with the applied field. This is called the saturation magnetization.

Now consider reducing the applied field until it is oriented in the opposite direction to the direction of the magnetic moments. However, the magnetic moments in a ferromagnetic material prefer to have the same orientation so they do not want to follow the direction of the magnetic field at first. The magnetic moment then remains oriented opposite the applied field until a sufficiently large opposite magnetic field is applied. At this point a sudden switching of the orientation of the magnetic moment occurs. This is the switching field H_c. In magnetic storage, when we write information onto the hard drive, we are switching the orientation of the magnetic domains. The read operation does not do this, instead it just senses the direction of the local field. This magnetic memory is non-volatile as it is not necessary to have a power source continually applied to the material in order to maintain the orientation of the spins.

Magnetic materials with very large reversal fields (H_c)are called magnetically hard materials, while those with small hysteresis loops are called soft magnetic materials.

Some different types of magnetic materials

Ferromagnets

This is the material type discussed above. In ferromagnetic materials, the magnetic moments of the atoms in the material seek to align in the same direction. Examples are Fe and Permalloy (55% Fe, 45% Ni). It is actually quite difficult to find good ferromagnetic materials. There is a continuing search for ferromagnetic materials which have large local magnetic moments. A group at GM research in Detroit made a major breakthrough in this area about a decade ago. They helped develop the Niodymium, Iron, Boron magnets. The production of these magnets is now a multibillion dollar industry.

Antiferromagnetics and complex magnets

Antiferromagnetic materials have atomic magnetic moments which prefer to have their nearest neighbors in an antiparallel alignment. In the simplest case, the magnetic moments alternate between one orientation and another. This is easily possible in material structures which are bipartite (e.g the square lattice or the cubic lattice), however in other lattice structures, the magnetic order can be extremely complex. These complex ordered states are called spin glasses or frustrated magnets. Antiferromagnetic materials loose their order at a critical temperature called the Neel temperature. Complex magnets loose their order at a glass temperature or ordering temperature. Often complex materials exhibit hysteresis and time-dependent effects that make reproducible measurements more difficult. There are many antiferromagnets and complex magnets. This is the usual behavior of compounds and some elements. Examples are NiO, Cr,...

Paramagnets

Paramagnets do not exhibit spontaneous magnetic order, nevertheless they can have large magnetic susceptibilities. The magnetic moment of paramagnetic materials tries to align in the direction of the applied magnetic field. Actually all materials will magnetically order at sufficiently low temperatures, but when the ordering temperature is very low, materials are called paramagnetic. The susceptibility of paramagnetic materials obeys the Curie Law,

$$\chi_m = {\mu_0 C \over T}$$ (5)

Paramagnetic and ferromagnetic materials are attracted to magnets.

Diamagnets

If we ignore the intrinsic magnetic moment of materials, then all materials would be diamagnetic. That is, the magnetic moment of materials would be opposite the direction of the applied field. This is due to Lenz's law. Superconductors are the best diamagnets, but many pure normal conductors are too (e.g. Cu...). Magnetic fields are completely excluded from the interior of a superconductor, at low enough magnetic field. The phase in which this occurs is called the Meissner phase of a superconductor. From the expression,

$$\vec{B} = \mu_0(1+\chi_m) \vec{H}$$ (6)

it is evident that in the Meissner phase where $\vec{B} = 0$inside the superconductor, we have, $\chi_m = -1$. A measurement of $\chi_m$ is one of the first measurements that people do to determine if a material is in the superconducting state. Diamagnetic materials are repelled from magnets. This enables the possibility of magnetic levitation. Since superconductors are the best diamagnets, they are the primary candidates for possible magnetic levitation applications.