MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ......
Transcript of MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ......
![Page 1: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/1.jpg)
Solid State Physics
MAGNETISM ILecture 27
A.H. HarkerPhysics and Astronomy
UCL
![Page 2: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/2.jpg)
10 Magnetic Materials
Magnet technology has made enormous advances in recent years –without the reductions in size that have come with these advancesmany modern devices would be impracticable.
2
![Page 3: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/3.jpg)
The important quantity for many purposes is the energy density ofthe magnet.
3
![Page 4: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/4.jpg)
10.1 Magnetic properties - reminder
There are two fields to consider, the magnetic fieldH which is gener-ated by currents according to Amp̀ere’s law, and the magnetic induc-tion, or magnetic flux density,B, which gives the torque experiencedby a dipole momentm asG = m×B. H is measured inA m−1 (Oer-steds in old units),B in Wb m−2 or T (Gauss in old units). In freespace,B = µ0H. In a material
B = µ0(H +M)
= µ0µrH= µ0(1 + χ)H,
whereµr is the relative permeability, χ is the magnetic susceptibility,which is a dimensionless quantity.
4
![Page 5: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/5.jpg)
Note, though, thatχ is sometimes tabulated as themolar susceptibility
χm = Vmχ,
whereVm is the volume occupied by one mole, or as themass suscep-tibility
χg =χ
ρ,
where ρ is the density.M, the magnetisation, is the dipole momentper unit volume.
M = χH.
In general, µr (and henceχ) will depend on position and will be ten-sors (so thatB is not necessarily parallel toH). Even worse, somematerials are non-linear, so thatµr and χ are field-dependent.
5
![Page 6: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/6.jpg)
The effects arehighly exaggeratedin these diagrams.6
![Page 7: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/7.jpg)
10.2 Measuring magnetic properties
10.2.1 Force method
Uses energy of induced dipole
E = −1
2mB = −1
2µ0χVH2,
so in an inhomogeneous field
F = −dE
dx=
1
2µ0V χ
dH2
dx= µ0V χHdH
dx.
Practically:
• set up large uniformH;
• superpose linear gradient with additional coils
• vary second field sinusoidally and use lock-in amplifier to measurevarying force
7
![Page 8: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/8.jpg)
10.2.2 Vibrating Sample magnetometer
• oscillate sample up and down
• measure emf induced in coils A and B
• compare with emf in C and D from known magnetic moment
• hence measured sample magnetic moment8
![Page 9: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/9.jpg)
10.3 Experimental data
In the first 60 elements in the periodic table, the majority have nega-tive susceptibility – they arediamagnetic.
9
![Page 10: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/10.jpg)
10.4 Diamagnetism
Classically, we have Lenz’s law, which states that the action of a mag-netic field on the orbital motion of an electron causes a back-emfwhich opposes the magnetic field which causes it. Frankly, this isan unsatisfactory explanation, but we cannot do better until we havestudied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector potentials. Imagine an electron in an atom asa chargee moving clockwise in the x-y plane in a circle of radiusa,areaA, with angular velocity ω. This is equivalent to a current
I = charge/time = eω/(2π),
so there is a magnetic moment
µ = IA = eωa2/2.
The electron is kept in this orbit by a central force
F = meω2a.
Now if a flux density B is applied in the z direction there will be aLorentz force giving an additional force along a radius
∆F = evB = eωaB10
![Page 11: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/11.jpg)
If we assume the charge keeps moving in a circle of the same radius (avague nod towards quantum mechanics?) it will have a new angularvelocity ω′,
meω′2a = F −∆F
someω
′2a = meω2a− eωaB,
orω′2 − ω2 = −eωB
me.
If the change in frequency is small we have
ω′2 − ω2 ≈ 2ω∆ω,
where∆ω = ω′ − ω. Thus
∆ω = − eB2me
,
the Larmor frequency. Substituting back into
µ = IA = eωa2/2.
11
![Page 12: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/12.jpg)
we find a change in magnetic moment
µ = −e2a2
4meB.
Recall that a was the radius of a ring of current perpendicular to thefield: if we average over a spherical atom
a2 = 〈x2〉 + 〈y2〉 =2
3
[〈x2〉 + 〈y2〉 + 〈z2〉
]=
2
3〈r2〉,
so
µ =e2〈r2〉6me
B,
and finally, if we have n atoms per volume, each withp electrons inthe outer shells, the magnetisation will be
M = npµ,
and
χ =MH
= µ0MB
= −µ0npe2〈r2〉6me
.
12
![Page 13: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/13.jpg)
Values of atomic radius are easily calculated: we can confirm thep〈r2〉 dependence.
13
![Page 14: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/14.jpg)
14
![Page 15: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/15.jpg)
Diamagnetic susceptibility:
• Negative
• Typically −10−6 to −10−5
• Independent of temperature
• Always present, even when there are no permanent dipole mo-ments on the atoms.
15
![Page 16: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/16.jpg)
10.5 Paramagnetism
Paramagnetism occurs when the material contains permanent mag-netic moments. If the magnetic moments do not interact with eachother, they will be randomly arranged in the absence of a magneticfield. When a field is applied, there is a balance between the internalenergy trying to arrange the moments parallel to the field and en-tropy trying to randomise them. The magnetic moments arise fromelectrons, but if we they are localised at atomic sites we can regardthem as distinguishable, and use Boltzmann statistics.
16
![Page 17: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/17.jpg)
10.5.1 Paramagnetism of spin-12 ions
The spin is either up or down relative to the field, and so the magneticmoment is either+µB or −µB, where
µB =e~
2me= 9.274× 10−24 Am2.
The corresponding energies in a flux densityB are −µBB and µBB,so the average magnetic moment per atom is
〈µ〉 =µBeµBB/kBT − µBe−µBB/kBT
eµBB/kBT + e−µBB/kBT
= µB tanh
(µBBkBT
).
For small z, tanh z ≈ z, so for small fields or high temperature
〈µ〉 ≈µ2
BBkBT
.
If there are n atoms per volume, then,
χ =nµ0µ
2B
kBT.
17
![Page 18: MAGNETISM I - UCLucapahh/teaching/3C25/Lecture27s.pdf · MAGNETISM I Lecture 27 A.H. Harker ... studied the inclusion of magnetic fields into quantum mechanics us-ing magnetic vector](https://reader033.fdocuments.us/reader033/viewer/2022060411/5f1088837e708231d4499416/html5/thumbnails/18.jpg)
Clearly, though, for low T or large B the magnetic moment per atomsaturates – as it must, as the largest magnetisation possiblesaturationmagnetisationhas all the spins aligned fully,
Ms = nµB.
18