Magnetic properties of materials

Diamagnetic materials

Narrated video. Transcript:

Bring a strong magnet near an iron nail, a piece of aluminium, and a piece of bismuth. The nail jumps to the magnet. The aluminium is pulled, but only very weakly. The bismuth is weakly pushed away.

These are the three classes of magnetic material: ferromagnetic, paramagnetic and diamagnetic. We sort them by their susceptibility χ.

Diamagnetic: χ is small and negative, between −1 and 0, so μ_r is a little less than 1. Paramagnetic: χ is small and positive, so μ_r is a little more than 1. Ferromagnetic: χ is very large, and μ_r can be in the thousands.

Some values. Bismuth, χ = −1.66 × 10⁻⁵. Copper, −9.8 × 10⁻⁶. Aluminium, +2.3 × 10⁻⁵. These are tiny numbers: a few parts in a hundred thousand.

A diamagnetic material is magnetised opposite to the applied field. Place a sample in a uniform field: the field lines are pushed out of it, and the field inside is reduced, very slightly.

In a non-uniform field, a diamagnetic sample moves from the stronger part of the field towards the weaker part. That is why the bismuth was pushed away from the magnet.

Why does this happen? In a simple picture, electrons orbit in pairs, one each way, so their magnetic moments cancel. Switch on a field, and one electron speeds up while the other slows down. A net moment appears, opposite to the field, just as Lenz’s law would say.

Every material has some diamagnetism, but it is weak and is hidden in other materials. Bismuth, copper, lead, silicon, water, nitrogen at STP and sodium chloride are diamagnetic.

The extreme case is a superconductor, cooled to a very low temperature. It pushes the field out completely: χ = −1 and μ_r = 0. This perfect diamagnetism is the Meissner effect. It can make a small magnet float above the superconductor.

Paramagnetic materials

Narrated video. Transcript:

In a paramagnetic material, every atom has its own permanent magnetic moment. With no field, thermal motion keeps these little dipoles pointing every which way. The net moment is zero.

Switch on a field B₀. The atomic dipoles tend to turn towards it. Only a little, because the random thermal jostling keeps knocking them out of line. But the material gains a small magnetisation along the field.

Place a sample in a uniform field, and the lines are drawn slightly into it. In a non-uniform field it moves from the weaker part towards the stronger part: it is weakly attracted.

Lower the temperature, or make the field stronger, and more of the dipoles line up. Experiments give Curie’s law: M = C B₀/T. Equivalently, χ = Cμ₀/T, where C is the Curie constant of the material and T is the absolute temperature.

So the susceptibility of a paramagnet falls as it gets hotter. Double the absolute temperature, and χ halves. Cool it, and in a strong enough field almost all the dipoles line up: the magnetisation saturates.

Worked example. A paramagnetic sample has magnetisation 8 A m⁻¹ in a field of 0.6 T at 4.2 K. What is M in a field of 0.2 T at 16 K? M is proportional to B₀/T.

The field becomes 0.2/0.6, one-third as much. The temperature becomes 16/4.2 times as much. So M = 8 × (1/3) × (4.2/16) = 0.7 A m⁻¹.

Aluminium, sodium, calcium, oxygen at STP and copper chloride are paramagnetic.

Ferromagnetic materials

Narrated video. Transcript:

Iron, cobalt, nickel and gadolinium are ferromagnetic. Here, the atomic dipoles interact strongly with their neighbours, and line up together, spontaneously, over a whole region.

These regions are called domains. A domain is about a millimetre across and contains about 10¹¹ atoms. Inside one domain, all the dipoles point the same way.

In an unmagnetised piece of iron, the domains point in random directions, and their moments cancel. Look closely at the boundaries between them.

Now apply a field. Domains that point along the field grow, the others shrink, and some turn round. Soon the piece is almost one big domain, and it is strongly magnetised.

Place the iron in a uniform field, and the field lines crowd into it. The field inside is many times stronger than outside. This is why iron cores are used in electromagnets.

Remove the field, and in some materials the domains drift back to random: soft iron is a soft ferromagnet. In others, like alnico, the magnetisation stays. These hard ferromagnets make permanent magnets.

Heat a ferromagnet, and the thermal motion fights the alignment. Above a certain temperature, the domains break up and the material becomes paramagnetic. This is the Curie temperature, T_C.

For iron, T_C = 1043 K. For cobalt, 1394 K. For nickel, 631 K. For gadolinium, just 317 K, a little above room temperature.

Above the Curie temperature, the susceptibility follows χ = C/(T − T_C). It is small, and it falls as the temperature rises, like any paramagnet.

Try it: materials in a field

Interactive simulation. Things to try:

• Choose each material in turn. Lines are pushed out of diamagnets, drawn into paramagnets, and crowd into iron.
• The real χ of bismuth or aluminium is about 10⁻⁵. Untick “Exaggerate” to see how small that really is.
• Choose Superconductor: χ = −1, and no field gets inside at all.
• For platinum, raise the temperature: χ falls as 1/T (Curie’s law). For iron, go past 1043 K: it becomes paramagnetic.
• Open “B–H loop” to see how a ferromagnet remembers its magnetisation. Compare soft iron with a hard magnet.

Pushed out of the sample

A sample is placed in a uniform magnetic field. The field lines are pushed slightly out of it, and in a non-uniform field it moves towards the weaker field. Which could the sample be?

1. Aluminium
2. Bismuth
3. Iron
4. Platinum

Curie’s law

A paramagnetic sample has a magnetisation of 8 A m⁻¹ in an external field of 0.6 T at 4.2 K. What is its magnetisation in a field of 0.2 T at 16 K?

1. 2.67 A m⁻¹
2. 2.1 A m⁻¹
3. 0.7 A m⁻¹
4. 10.2 A m⁻¹

Summary

• Classes by susceptibility: dia: −1 ≤ χ < 0; para: 0 < χ ≪ 1; ferro: χ ≫ 1
• Curie’s law (paramagnets): M = C B₀/T, χ = Cμ₀/T
• Above the Curie temperature: χ = C/(T − T_C)

• Diamagnetic (bismuth, copper, water): magnetised opposite to the field, lines pushed out, move to weaker field. A superconductor is a perfect diamagnet, χ = −1 (Meissner effect).
• Paramagnetic (aluminium, sodium, oxygen): permanent atomic dipoles partly line up with the field; lines drawn in, move to stronger field. χ falls as 1/T.
• Ferromagnetic (iron, cobalt, nickel, gadolinium): domains line up, χ is very large. Soft ferromagnets lose their magnetisation when the field is removed; hard ones keep it.
• Above the Curie temperature (1043 K for iron) a ferromagnet becomes paramagnetic.