Magnetic moments, dipoles and fields Richard F L Evans ESM 2018 - PowerPoint PPT Presentation

Magnetic moments, dipoles and fields Richard F L Evans ESM 2018

Overview Origin of magnetic moments Magnetic fields and demagnetising factors Units in magnetism

Useful References • J. M. D. Coey; Magnetism and Magnetic Magnetic Materials. Cambridge University Press (2010) 614 pp • Stephen Blundell Magnetism in Condensed Matter, Oxford 2001 • D. C. Jiles An Introduction to Magnetism and Magnetic Magnetic Materials, CRC Press 480 pp • J. D. Jackson Classical Electrodynamics 3rd ed, Wiley, New York 1998

Magnetic moments

What is a magnet? “A magnet is a material or object that produces a magnetic field” Wikipedia

What is a magnet? “A magnet is a material or object that produces a magnetic field” Wikipedia

What is a magnetic field? • An invisible vector field that interacts with other magnets https://education.pasco.com/epub/PhysicsNGSS/BookInd-515.html

What is a magnetic field? • An invisible vector field that interacts with other magnets

What is a magnetic field? • An invisible vector field that interacts with other magnets

Magnetic field, Øersted 1820 • Oersted discovered in 1820 that a current carrying wire was able to rotate a compass needle I • r Current and field are related by Ampere’s Law H δ l I = ∫ H dl • Example for I = 1A, integral around the loop is 2 𝜌 r, r = 2 mm H ~ 80 A/m • Earth’s magnetic field ~ 40 A/m

Interaction of two current-carrying wires, Ampere 1825 • Two current carrying wires (one longer than the other) are attracted to each other for parallel current, and repel for anti-parallel current. I 1 I 2 F I 1 I 2 l = μ 0 2 π r • The parallel wires “look like” magnets in the F perpendicular direction • Weird but central to electromagnetism (E and B fields in light) • Di ff erent from electrostatics as this is a dynamic e ff ect from the motion of charge

Equivalence of currents and magnetic moments • So currents look like magnets… do magnets look like currents? m I m = I ⊥ A A • Can express a current loop as an e ff ective moment, ie a source of magnetic field • What kind of currents do we need compared to typical magnetic fields?

Comparison of current magnitudes and magnets • Using the equivalence of current loops and magnetic moments we can compare the e ff ective currents for a typical small magnet • Moment given by for a single loop and a solenoid respectively, where n is the number of turns of the coil m = I ⊥ A m = nI ⊥ A • For a small magnet 10 mm 1 A 10,000 A 1 Am 2 10,000 turns 1 turns • At small sizes, magnets generate much larger fields -> applications in motors

Difference between magnetic moment and magnetisation • Magnetic moment is specific to the sample (bigger magnet, bigger field) 0.027 Am 2 1 Am 2 15.6 Am 2 3 mm 10 mm 25 mm • Magnetization is the moment density Assume NdFeB M s ~ 1 MA/m m = M V • Magnetisation is a property of the material • Moment is a property of a magnet • Magnetisation is scale independent

Vectorial nature of magnetic moments • A magnetic moment generates a field around it • Interaction with non-magnets is weak • Interaction with magnets is stronger but orientation dependent Strong attraction Weak repulsion

Physical origin of magnetization and magnetic moment • At the atomic scale the magnetic moments fluctuate strongly in time and space due to the electrons ‘orbiting’ nuclei <M> • Use a continuous medium approximation to calculate an average magnetisation < M > (moment/volume) • Avoids all the horrible details of fluctuating moments and can treat magnetism on a continuum level • Good approximation for ferromagnets for volumes much larger than the atomic volume

Which elements are magnetic Magnetic Periodic Table 2 He 1 H 1.00 4.00 Atomic Number Atomic symbol 66 Dy Atomic weight Typical ionic change 3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne 162.5 Antiferromagnetic T N (K) Ferromagnetic T C (K) 3 + 4 f 9 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18 179 85 1 + 2 s 0 2 + 2 s 0 35 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 26.98 28.09 30.97 32.07 22.99 24.21 35.45 39.95 1 + 3 s 0 2 + 3 s 0 3 + 2 p 6 32 Ge 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 33 As 34 Se 35 Br 36 Kr 72.61 38.21 40.08 44.96 47.88 50.94 52.00 55.85 55.85 58.93 58.69 63.55 65.39 69.72 74.92 78.96 83.80 79.90 1 + 4 s 0 2 + 4 s 0 3 + 3 d 0 4 + 3 d 0 3 + 3 d 2 3 + 3 d 3 2 + 3 d 5 3 + 3 d 5 2 + 3 d 7 2 + 3 d 8 2 + 3 d 9 2 + 3 d 10 3 + 3 d 10 312 96 1043 1390 629 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe 87.62 88.91 91.22 92.91 95.94 97.9 101.1 102.4 106.4 107.9 112.4 114.8 118.7 121.8 127.6 126.9 83.80 85.47 1 + 5 s 0 2 + 5 s 0 2 + 4 d 0 4 + 4 d 0 5 + 4 d 0 5 + 4 d 1 3 + 4 d 5 3 + 4 d 6 2 + 4 d 8 1 + 4 d 10 2 + 4 d 10 3 + 4 d 10 4 + 4 d 10 55 Cs 56 Ba 57 La 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn 137.3 138.9 178.5 180.9 183.8 186.2 190.2 192.2 195.1 197.0 200.6 204.4 207.2 209.0 209 210 222 132.9 2 + 6 s 0 3 + 4 f 0 4 + 5 d 0 5 + 5 d 0 6 + 5 d 0 4 + 5 d 3 3 + 5 d 5 4 + 5 d 5 2 + 5 d 8 1 + 5 d 10 2 + 5 d 10 3 + 5 d 10 4 + 5 d 10 1 + 6 s 0 87 Fr 88 Ra 89 Ac 223 226.0 227.0 2 + 7 s 0 3 + 5 f 0 58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu 63 Eu 140.1 140.9 144.2 145 150.4 157.3 158.9 162.5 164.9 167.3 168.9 173.0 175.0 152.0 4 + 4 f 0 3 + 4 f 2 3 + 4 f 3 3 + 4 f 5 2 + 4 f 7 3 + 4 f 7 3 + 4 f 8 3 + 4 f 9 3 + 4 f 10 3 + 4 f 11 3 + 4 f 12 3 + 4 f 13 3 + 4 f 14 13 19 105 292 229 221 179 85 132 20 85 20 56 90 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100 Fm 101 Md 102 No 103 Lr 232.0 231.0 238.0 238.0 244 243 247 247 251 252 257 258 259 260 4 + 5 f 0 5 + 5 f 0 4 + 5 f 2 5 + 5 f 2 Diamagnet Ferromagnet T C > 290K Nonmetal Antiferromagnet with T N > 290K Paramagnet Metal Antiferromagnet/Ferromagnet with T N /T C < 290 K Radioactive BOLD Magnetic atom From Coey

Bohr magneton m • Can consider an electron ‘orbiting’ an atom e m • * A moving charge looks like a ‘current’, e generating an e ff ective magnetic moment l • In Bohr’s quantum theory, orbital angular m = IA = − evr momentum l is quantized in units of ︎ ℏ ; h is 2 Planck’s constant, 6.6226 10 -34 Js; ︎ ℏ =h/ 2 ︎ 𝜌 =1.05510 -34 Js e I = e ℏ • m = − m l = m l μ B The orbital angular momentum is l = 2 m e 2 m e m e r ︎ ∧ v • It is the z-component of l z that is quantized in units of ︎ ℏ , taking a value m l ︎ 
 μ B = e ℏ = 9.274 × 10 − 24 Am 2 | JT − 1 m l is a quantum number, an integer with 2 m e no units. Eliminating r in the expression for m • μ B is the Bohr magneton, the basic unit of atomic magnetism * electrons travel in the opposite direction to currents