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The treatment of paramagnetism in the
most simple way is exactly identical to the treatment of
orientation
polarization. All you have to do is to replace the electric dipoles by magnetic dipoles, which we call
magnetic moments. |
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We have permanent dipole moments in the material,
they have no or negligible interaction between them, and they are free to point
in any direction even in solids. |
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This is a major difference to
electrical dipole moments which can only
rotate if the whole atom or molecule rotates; i.e. only in liquids. This is why
the treatment of magnetic materials focusses on ferromagnetic materials and why
the underlying symmetry of the math is not so obvious in real materials. |
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In an external magnetic field the magnetic dipole
moments have a tendency to orient themselves into the field direction, but this
tendency is opposed by the thermal energy, or better entropy of the system.
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Using
exactly the same
line of argument as in the case of orientation polarization, we have for
the potential energy W of a magnetic moment (or dipole)
m in a magnetic field H |
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| W(j) = –
µ0 · m · H =
– µ0 · m · H · cos
j |
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With j = angle
between H and m. |
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In thermal
equilibrium, the number of magnetic moments with the energy
W will be N[W(j)], and that number is once more given by the
Boltzmann factor: |
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| N[W(j)] =
c · exp –(W/kT) = c
· exp |
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m · µ0 · H · cos
j
kT |
= N(j) |
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As befoere, c is some as yet
undetermined constant. |
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As before, we have to take the
component in field direction of all the moments having the same angle with
H and integrate that over the unit sphere. The result for the
induced magnetization mind and the total magnetization
M is the same as before for the induced dielectric dipole
moment: |
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| mind |
= |
m · |
æ
è |
coth b |
– |
1
b |
ö
ø |
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| M |
= |
N · m · L(b) |
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| b |
= |
µ0 · m · H
kT |
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With L(b) =
Langevin
function = cothb – 1/b |
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The only interesting point is the
magnitude of b. In the case of the orientation polarization
it was £ 1 and we could use a
simple approximation for the Langevin function. |
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We know that m will be of the order
of magnitude of 1 Bohr
magneton. For a rather large magnetic field strength of 5 ·
106 A/m, we obtain as an estimate for an upper limit b = 1.4 · 10–2, meaning that the
range of b is even smaller as in the case of
the electrical dipoles. |
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We are thus justified to use the
simple
approximation L(b) = b/3 and obtain |
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| M = N · m · (b/3) |
= |
N · m2 · µ0 ·
H
3kT |
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The paramagnetic susceptibility c = M/H, finally, is |
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Plugging in some typical numbers (A
Bohr magneton for m and typical densities), we obtain cpara »
+10–3; i.e. an exceedingly
small effect, but with certain characteristics that will carry over
to ferromagnetic materials: |
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There is a strong temperature dependence and it
follows the "Curie law": |
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Since ferromagnets of all types turn into
paramagnets above the Curie temperature TC, we may simply
expand Curies law for this case to |
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| cferro(T > TC)
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= |
const*
T – TC |
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In summary, paramagnetism, stemming
from some (small) average alignement up of permanent magnetic dipoles
associated with the atoms of the material, is of no
(electro)technical consequence. It is, however, important for
analytical purposes called "Electron
spin resonance" (ESR) techniques. |
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There are other types of
paramagnetism, too. Most important is, e.g., the
paramagnetism of the free
electron gas. Here we have magnetic moments associated with spins of
electrons, but in a mobile way - they are
not fixed at the location of the atoms |
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But as it turns out, other kinds of paramagnetism
(or more precisely: calculations taking into account that magnetic moments of
atoms can not assume any orientation but only sone quantized ones) do not
change the general picture: Paramagnetism is a weak
effect. |
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© H. Föll (Advanced Materials B, part 1 - script)
