Lecture 15

2 How to produce B-field

• Changing E-field • Electric currents • flowing electrons • nuclear and electron spins and orbital motion

3 Magnetism in a materials ▪ Paramagnetic materials (amplify the external B-field) ▪ Diamagnetic (decrease the external B-field) ▪ Ferromagnetic (amplify, remain)

▪ Strongly interacting atoms & electrons ▪ Quantum & classical effects

March 2, 2015 54 Magnetization

• Electrons and ions have magnetic moments • No B-field: randomly oriented

B

5 Magnetization

Paramagentic

B Diamagnetic

6 Magnetization

Ferromagenic B

7 Magnetic moments from

• orbital motion of electrons • spins of electrons • nuclei

8 Magnetic materials

▪ Magnetization ≡ net dipole moment per unit volume

! ▪ Usually all dipoles in matter are aligned in random directions, so the net magnetization is zero: M = 0. ▪ However, if a material is placed into an external magnetic field, the dipoles will tend to align with or against the applied field, and the total magnetic field is then:

March 2, 2015 52 Magnetic materials

! ▪ In weak fields

March 2, 2015 52 11 Atoms as Magnets (orbital momentum, “classical” description) ▪ The atoms that make up all matter contain moving electrons that form current loops that produce magnetic fields ▪ In most materials, these current loops are randomly oriented and produce no net magnetic field ▪ In magnetic materials, some of these current loops are naturally aligned ▪ Other materials can have these current loops aligned by an external magnetic field and become magnetized ▪ Simple model of an atom: Consider an electron moving at a constant speed v in a circular orbit with radius r as illustrated to the right

March 2, 2015 49 Atoms as Magnets (2)

▪ We can think of the moving charge of the electron as a current i ▪ Current is defined as the charge per unit time passing a particular point ▪ For this case the charge is the charge of the electron e and the time is related to the period of the orbit e e ve i = = = T 2πr / v 2πr ( )

March 2, 2015 50 Atoms as Magnets (3)

▪ The magnetic moment of the orbiting electron is given by ve ver ! µ = iA = π r 2 = orb 2 r 2 ! π ▪ We can define the orbital angular momentum of the electron to be ! Lorb = rp = rmv where m is the mass of the electron ▪ Solving and substituting gives us

! 2µorb " 2mµorb Lorb = rm# $ = % er & e

March 2, 2015 51 Atoms as Magnets (4)

▪ Rewriting and remembering that the magnetic dipole moment and the angular momentum are vector quantities we can write ! e ! µ = − L ! orb 2m orb • The negative sign arises because of the definition of current as the flow of positive charge ▪ This result can be applied to the hydrogen atom, and the correct result is obtained ▪ However, other predictions of the properties of atoms based on the idea that electrons exist in circular orbits in atoms disagree with experimental observations

March 2, 2015 52 Paramagnetism

▪ Materials containing certain transition elements, actinides, and rare earths exhibit paramagnetism ▪ Each atom of these elements has a permanent magnetic dipole, but these dipole moments are randomly oriented and produce no net magnetic field

▪ In the presence of an external magnetic field, some of these magnetic dipole moments align with the external field ▪ In the same direction as the external field • Resulting in an attractive force • Paramagnetic materials get pulled into magnetic fields ▪ When the external field is removed, the induced magnetic dipole moment disappears

March 2, 2015 57 Diamagnetism (1)

▪ Most materials exhibit diamagnetism ▪ Diamagnetism is weak compared with the other two types of magnetism and is thus masked by those forms if they are present in the material ▪ In diamagnetic materials, a weak magnetic dipole moment is induced by an external magnetic field (no net magnetic moment without B-field) ▪ The direction of the induced field is opposite the direction of the external field • Resulting in a repulsive force • Diamagnetic materials get pushed out of magnetic fields ▪ The induced field disappears when the external field is removed

March 2, 2015 55 Diamagnetism (2)

▪ An example of a strawberry exhibiting diamagnetism can be seen on the site for the High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands at: http://www.ru.nl/hfml/research/levitation/diamagnetic/ ▪ In this movie, diamagnetic forces induced by a non-uniform external magnetic field of 16 T are levitating the strawberry. ▪ The normally negligible diamagnetic force is large enough in this case to overcome gravity.

Levitating a live Flying Dutchfrog ▪An example of a frog exhibiting diamagnetism can also be seen on the site for the High Field Magnet Laboratory, Radboud University Nijmegen, The Netherlands at: http://www.ru.nl/hfml/research/levitation/diamagnetic/ ! March 2, 2015 56 19 Ferromagnetism (1)

▪ The elements iron, nickel, cobalt, gadolinium and dysprosium and alloys containing these elements exhibit ferromagnetism ▪ Ferromagnetic materials: • Long-range ordering at the atomic level • Dipole moments of atoms are lined up with each other • Typically in a limited region called a domain ▪ Within a domain, the magnetic field can be strong ▪ Domains are small and randomly oriented leaving no net magnetic field ▪ An external magnetic field can align these domains and magnetize the material Iron Ore Tilden mine, UP

March 2, 2015 53 Ferromagnetism (2)

▪ Spins try to align - spontaneously. ▪ Material separates into domains of different orientation ▪ Impose external B-field: domains align

! Bext ! ! ! ▪ Remove the external B-field - domains remain aligned - magnet.

March 2, 2015 54 Ferromagnetism (2)

▪ A ferromagnetic material will retain all or some of this induced magnetism when the external magnetic field is removed ▪ The induced magnetic field is in the same direction as the external magnetic field • Amplification of magnetic fields, for example inside electromagnets, solenoids or toroids ▪ Demo • Insert a ferromagnetic material in the core of a solenoid and see how much the magnetic field is increased

March 2, 2015 54 Magnetic hysteresis

Bext ▪ Start with domains aligned ▪ Impose B-field against ▪ Spins “want” align with each other and with external B-field ▪ Stronger external B-field - stronger alignment

March 2, 2015 54 Magnetization depends on temperature

• Magnetization (eg. permanent in ferromagnetics) decreases with T.

Curie temperature

24 Earth magnetism

• Currents flowing in the liquid core: Earth large scale B-field ! ! ! ! ! ! • Sometimes field reverse • Earth lava is hot during eruption - no magnetization • Cooling: ferromagnets “record” the direction of the B-field at time of eruption

25 Dynamo mechanism (in the core of Earth)

“Cut” & “glue”

26 Electron spin

▪ electron: small magnet (spin). Quantum effect ! ! ! ▪ Spin of electron: either up or down: s=+1/2, s= -1/2 ▪ Two electrons: total spin =0 or 1. (total spin 1 can have three projections: -1,0,1). Spin =0 is a bit preferred energetically (like two magnets) ▪ Even # of electrons: typically spin =0 in an atom ▪ There is a spin of the nucleus, and orbital momentum of elections ▪ The electron spin is the main contribution to ferromagnetism

March 2, 2015 54 Electron spin

▪ Electron spin is the main source of ferromagnetism ▪ Electron spins align – create large B-field. ▪ Need unfilled outer shells – extra electron with s=1/2 ▪ Interaction with external field and with each other ▪ Electrons “want” to align the spins with B-field ! ▪ BUT: parallel magnets repulse!!!!???

March 2, 2015 54 Ferromagnets

▪ Spins align with external B-field, but they should repulse from each other??? ▪ Quantum mechanics: Pauli exclusion principle: similar particles cannot occupy the same state. ▪ Parallel spins in a overlapping lattice “want” to be further away from each other, reducing their Coulomb interaction.

More magnetic energy, but less Coulomb energy

March 2, 2015 54 Conventional Magnets

▪ Magnets for industrial applications and scientific research can be constructed using ordinary resistive wire with current flowing through the wires ▪ The typical magnet is a large solenoid ▪ The current flowing through the wires of the magnet produces resistive heating ▪ The produced heat is usually taken away by flowing low conductivity water through the hollow conductors ▪ Low conductivity water is water that has been purified so that it does not conduct electricity ▪ These room temperature magnets typically can produce magnetic fields up to 1.5 T ▪ Room temperature magnets are usually relatively inexpensive to construct but expensive to operate

March 2, 2015 61 Superconducting Magnets

▪ Some applications such as magnetic resonance imaging require the highest possible magnetic field to extract the best signal to noise ratio in the measurements ▪ A magnet can be constructed using superconducting coils rather than resistive coils ▪ Such a magnet can produce a higher field than a room temperature magnet ▪ The disadvantage of a superconducting magnet is that the conductor must be kept at the temperature of liquid helium, which is approximately 4 K ▪ Thus the magnet must be enclosed in a cryostat filled with liquid helium to keep the coils cold

March 2, 2015 62 Superconductivity

▪ During the last two decades, physicists and engineers have discovered new materials that are superconducting at temperatures much above 4 K ▪ Some of these so-called high-temperature superconductors have been reported to work at temperatures up to 100 K, which means that they can be made superconducting by cooling them with liquid nitrogen ▪ Many groups around the world are working hard to find materials that are superconducting at room temperature ▪ These materials would revolutionize many parts of industry and in particular transportation. CPU in the Terminator is superconducting at room temperature

March 2, 2015 63 33 34 35 36 Astrophysical magnets

• Sun is magnetized • Regular B-field ~ 1 G • Random B-field ~ 1k G • Most planets have magnetic fields • Interstellar space is magnetized • B ~ micro-G 15 • Some stars have B-field 10 G Aurora Borealis on Jupiter

37 Neutron stars (Proposed to exist by Landau in 30s)

! R 10 km ! ⇠ M! 1.4 M ⇠ P! =1.5 msec 100ssec ! radio pulsar : P few sec !  B! = 108 1015 G ! 14 c.f.,The highest BQ =3 static B-field10 G in the Universe. ⇥ c.f., at Earth, B < 106 G

s-quark stabilization? Strange stars?

38 Discovered by Jocelyn Bell Pulsars: 48 yrs... (did not get Nobel prize, her advisor did)

• We do not know how and where in the magnetospher they produce radio emission Fermi satellite • Fermi: 100+ gamma-ray profiles of pulsar - we do not know how pulsars in gamma-rays and where they produce gamma-rays • Why are there different kinds • radio • magnetars • RRATs

39 Magnetospheres of isolated pulsars: beacon model

(D. Page) Goldreich & Julian 1969

• (nearly) Dipole B-field • Closed field lines

Rmax< c P/2π • Open field lines near magnetic pole • Radio emission is (somehow) generated near magnetic pole