Basic concepts in Magnetism; Units J. M. D. Coey School of Physics - PowerPoint PPT Presentation

Basic concepts in Magnetism; Units J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. SI Units 2. cgs units 3. Conversions 4. Dimensions Comments and corrections please: jcoey@tcd.ie www.tcd.ie/Physics/Magnetism

Here SI units are summarized. Their advantages and differences with the old cgs system are outlined. Useful tables for conversions are provided. Dimensions are given for magnetic, electrical and other quantities.

1 Introduction 2 Magnetostatics 3 Magnetism of the electron 4 The many-electron atom 5 Ferromagnetism 6 Antiferromagnetism and other magnetic order 7 Micromagnetism 8 Nanoscale magnetism 9 Magnetic resonance 10 Experimental methods 11 Magnetic materials 12 Soft magnets 13 Hard magnets 14 Spin electronics and magnetic recording 614 pages. Published March 2010 15 Other topics * Appendices, conversion tables. www.cambridge.org/9780521816144 ESM Cluj 2015

A note on units: Magnetism is an experimental science, and it is important to be able to compare and calculate numerical values of the physical quantities involved. There is a strong case to use SI consistently Ø SI units relate to the practical units of electricity measured on the multimeter and the oscilloscope Ø It is possible to check the dimensions of any expression by inspection. Ø They are almost universally used in teaching Ø Units of B , H , Φ or d Φ /dt have been introduced. BUT Most literature still uses cgs units, You need to understand them too. ESM Cluj 2015

SI / cgs conversions: SI units cgs units B = μ 0 ( H + M ) B = H + 4 π M m A m 2 e mu M A m -1 (10 -3 emu cc -1 ) emu cc -1 (1 k A m -1 ) σ A m 2 kg -1 (1 emu g -1 ) emu g -1 (1 A m 2 kg -1 ) A m -1 (4 π / 1000 ≈ 0.0125 Oe) H Oersted (1000/4 π ≈ 80 A m -1 ) B Tesla (10 kG) Gauss (10 -4 T) Φ Weber (Tm 2 ) (10 8 Mw) Maxwell (G cm 2 ) (10 -8 Wb) d Φ /dt V (10 8 Mw s -1 ) Mw s -1 (10 nV) χ (4 π cgs) - - (1/4 π SI) ESM Cluj 2015

Mechanical Quantity Symbol Unit m l t i θ m 2 A Area 0 2 0 0 0 m 3 Volume V 0 3 0 0 0 m s − 1 Velocity 0 1 − 1 0 0 v m s − 2 Acceleration a 0 1 − 2 0 0 kg m − 3 d Density 1 − 3 0 0 0 Energy ε J 1 2 − 2 0 0 kg m s − 1 Momentum p 1 1 − 1 0 0 kg m 2 s − 1 Angular momentum L 1 2 − 1 0 0 kg m 2 Moment of inertia 1 2 0 0 0 I Force f N 1 1 − 2 0 0 N m − 3 Force density F 1 − 2 − 2 0 0 Power P W 1 2 − 3 0 0 Pressure P Pa 1 − 1 − 2 0 0 N m − 2 Stress σ 1 − 1 − 2 0 0 N m − 2 Elastic modulus K 1 − 1 − 2 0 0 s − 1 Frequency 0 0 − 1 0 0 f m 2 s − 1 Diffusion coefficient D 0 2 − 1 0 0 N s m − 2 Viscosity (dynamic) η 1 − 1 − 1 0 0 m 2 s − 1 Viscosity ν 0 2 − 1 0 0 Planck’s constant � J s 1 2 − 1 0 0 ESM Cluj 2015

Electrical Quantity Symbol Unit m l t i θ Current I A 0 0 0 1 0 A m − 2 Current density j 0 − 2 0 1 0 Charge q C 0 0 1 1 0 Potential V V 1 2 − 3 − 1 0 Electromotive force E V 1 2 − 3 − 1 0 Capacitance F − 1 − 2 4 2 0 C Resistance 1 2 − 3 − 2 0 R � Resistivity � m 1 3 − 3 − 2 0 ϱ S m − 1 Conductivity σ − 1 − 3 3 2 0 Dipole moment p C m 0 1 1 1 0 C m − 2 Electric polarization P 0 − 2 1 1 0 V m − 1 Electric field E 1 1 − 3 − 1 0 C m − 2 Electric displacement D 0 − 2 1 1 0 Electric flux � C 0 0 1 1 0 F m − 1 Permittivity − 1 − 3 4 2 0 ε V K − 1 Thermopower 1 2 − 3 − 1 − 1 S m 2 V − 1 s − 1 Mobility − 1 0 2 1 0 µ ESM Cluj 2015

Magnetic Quantity Symbol Unit m l t i θ A m 2 Magnetic moment 0 2 0 1 0 m A m − 1 Magnetization 0 − 1 0 1 0 M A m 2 kg − 1 Specific moment − 1 2 0 1 0 σ A m − 1 Magnetic field strength H 0 − 1 0 1 0 Magnetic flux � Wb 1 2 − 2 − 1 0 Magnetic flux density B T 1 0 − 2 − 1 0 Inductance L H 1 2 − 2 − 2 0 Susceptibility (M/H) 0 0 0 0 0 χ H m − 1 Permeability (B/H) 1 1 − 2 − 2 0 µ Magnetic polarization T 1 0 − 2 − 1 0 J Magnetomotive force F A 0 0 0 1 0 Magnetic ‘charge’ q m A m 0 1 0 1 0 J m − 3 Energy product ( BH ) 1 − 1 − 2 0 0 J m − 3 Anisotropy energy 1 − 1 − 2 0 0 K J m − 1 Exchange stiffness 1 1 − 2 0 0 A m 3 C − 1 Hall coefficient 0 3 − 1 − 1 0 R H Scalar potential ϕ A 0 0 0 1 0 Vector potential A T m 1 1 − 2 − 1 0 T m 2 A − 1 Permeance P m 1 2 − 2 − 2 0 A T − 1 m − 2 Reluctance R m − 1 − 2 2 2 0 ESM Cluj 2015

Thermal Quantity Symbol Unit m l t i θ Enthalpy H J 1 2 − 2 0 0 J K − 1 Entropy S 1 2 − 2 0 − 1 J K − 1 kg − 1 Specific heat C 0 2 − 2 0 − 1 J K − 1 Heat capacity c 1 2 − 2 0 − 1 W m − 1 K − 1 Thermal conductivity 1 1 − 3 0 − 1 κ J mol − 1 K − 1 Sommerfeld coefficient γ 1 2 − 2 0 − 1 J K − 1 Boltzmann’s constant k B 1 2 − 2 0 − 1 (1) Kinetic energy of a body: ε = 1 2 mv 2 [ ε ] = [1 , 2 , − 2 , 0 , 0] [ m ] = [1 , 0 , 0 , 0 , 0] [ v 2 ] = 2[0 , − 1 , − 1 , 0 , 0] [1 , − 2 , − 2 , 0 , 0] (2) Lorentz force on a moving charge; f = q v × B [ f ] = [1 , 1 , − 2 , 0 , 0] [ q ] = [0 , 0 , 1 , 1 , 0] [ v ] = [0 , 1 , − 1 , 0 , 0] [ B ] = [1 , 0 , − 2 , − 1 , 0] [1 , 1 , − 2 , 0 , 0] (3) Domain wall energy γ w = √ AK ( γ w is an energy per unit area) √ [ γ w ] = [ ε A − 1 ] [ AK ] = 1 / 2[ AK ] [ √ A ] = 1 = [1 , 2 , − 2 , 0 , 0] 2 [1 , 1 , − 2 , 0 , 0] [1 , − 1 , − 2 , 0 , 0] [ √ K ] = 1 - [0, 2, 0, 0, 0 ] − [ 1, 1, − 2, 0, 0] 2 [1 , 0 , − 2 , 0 , 0] = [1 , 0 , − 2 , 0 , 0] (4) Magnetohydrodynamic force on a moving conductor F = × B × B ESM Cluj 2015

− = (4) Magnetohydrodynamic force on a moving conductor F = σ v × B × B ( F is a force per unit volume) [ F ] = [ FV − 1 ] [ σ ] = [ − 1 , − 3 , 3 , 2 , 0] = [1 , 1 , − 2 , 0 , 0] [ v ] = [0 , 1 , − 1 , 0 , 0] [0 , 3 , 0 , 0 , 0] [ B 2 ] = 2[1 , 0 , − 2 , − 1 , 0] − [1 , − 2 , − 2 , 0 , 0] [1 , − 2 , − 2 , 0 , 0] (5) Flux density in a solid B = µ 0 ( H + M ) (note that quantities added or subtracted in a bracket must have the same dimensions) [ B ] = [1 , 0 , − 2 , − 1 , 0] [ µ 0 ] = [1 , 1 , − 2 , − 2 , 0] [0 , − 1 , 0 , 1 , 0] [ M ] , [ H ] = [1 , 0 , − 2 , − 1 , 0] (6) Maxwell’s equation ∇ × H = j + d D / d t . [ ∇ × H ] = [ Hr − 1 ] [d D / d t ] = [ Dt − 1 ] [ j ] = [0 , − 2 , 0 , 1 , 0] = [0 , − 1 , 0 , 1 , 0] = [0 , − 2 , 1 , 1 , 0] − [ 0, 1, 0, 0, 0] − [0, 0, 1, 0, 0] = [0 , − 2 , 0 , 1 , 0] = [0 , − 2 , 0 , 1 , 0] (7) Ohm’s Law V = IR = [1 , 2 , − 3 , − 1 , 0] [0, 0, 0, 1, 0] + [1 , 2 , − 3 , − 2 , 0] = [1 , 2 , − 3 , − 1 , 0] (8) Faraday’s Law E = − ∂� / ∂ t = [1 , 2 , − 3 , − 1 , 0] [1, 2, − 2, − 1, 0] − [0, 0, 1, 0, 0] = [1 , 2 , − 3 , − 1 , 0] ESM Cluj 2015

J SI Units SI units are used consistently throughout the lectures. The basis units are m, kg, s, A, K They have three compelling advantages: i) the dimensions are transparent; ii) they are directly related to the standard electrical units Volts, Amps, Ohms in which many measurements are made; iii) SI units they are almost universally used for undergraduate teaching. The Sommerfeld convention is preferred; B = µ 0 ( H + M ) (1) where the magnetic field strength (flux density) B is measured in tesla (T, distinguished from the physical variable temperature T ); the magnetizing force H and the magnetization of a material M (magnetic moment per m 3 ) are measured in Am -1 . The constant µ 0 in (1) is precisely 4 π 10 -7 TmA -1 . There are other equivalent units for µ 0 , but this one is preferred. The fields may be referred to as the ‘B-field’ and the ‘H-field’, or simply as the ‘magnetic field’, when it is clear (or unimportant) which one is meant. When appropriate, the applied field H ’ is distinguished from the field H which is actually present in the sample. ESM Cluj 2015

The Kennelly convention is compatible with the above. B = µ 0 H + J where J = µ 0 M is the magnetic polarization of a material, measured in tesla. Where possible, use the 3-order multiples of the basic units; nT, µT, mT, T; Am -1 , kAm -1 . MAm -1 etc. Hence nm or pm, rather than Å; mm or m, rather than cm. If you want to use Å, please ensure that it is used consistently for strictly comparable lengths. For example, lattice parameters in Å should not be mixed with film thicknesses in nm. Lattice parameters in pm and film thicknesses in nm is a preferred solution. ESM Cluj 2015