Basic concepts in Magnetism; Units

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1 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:

2 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.

614 pages. Published March 2010 www.cambridge.

3 614 pages. Published March 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 15 Other topics *Appendices, conversion tables.

4 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 dφ/dt have been introduced. BUT Most literature still uses cgs units, You need to understand them too.

5 SI / cgs conversions: m M σ H B Φ dφ/dt χ SI units B = μ 0 (H + M) A m 2 A m -1 (10-3 emu cc -1 ) A m 2 kg -1 (1 emu g -1 ) A m -1 (4π/ Oe) Tesla (10 kg) Weber (Tm 2 ) (10 8 Mw) V (10 8 Mw s -1 ) - (4π cgs) cgs units B = H + 4πM emu emu cc -1 (1 k A m -1 ) emu g -1 (1 A m 2 kg -1 ) Oersted (1000/4π 80 A m -1 ) Gauss (10-4 T) Maxwell (G cm 2 ) (10-8 Wb) Mw s -1 (10 nv) - (1/4π SI)

6 Mechanical Quantity Symbol Unit m l t i θ Area A m Volume V m Velocity v ms Acceleration a ms Density d kg m Energy ε J Momentum p kg m s Angular momentum L kg m 2 s Moment of inertia I kg m Force f N Force density F Nm Power P W Pressure P Pa Stress σ Nm Elastic modulus K Nm Frequency f s Diffusion coefficient D m 2 s Viscosity (dynamic) η Nsm Viscosity ν m 2 s Planck s constant Js

7 Electrical Quantity Symbol Unit m l t i θ Current I A Current density j Am Charge q C Potential V V Electromotive force E V Capacitance C F Resistance R Resistivity ϱ m Conductivity σ Sm Dipole moment p Cm Electric polarization P Cm Electric field E Vm Electric displacement D Cm Electric flux C Permittivity ε Fm Thermopower S VK Mobility µ m 2 V 1 s

8 Magnetic Quantity Symbol Unit m l t i θ Magnetic moment m Am Magnetization M Am Specific moment σ Am 2 kg Magnetic field strength H Am Magnetic flux Wb Magnetic flux density B T Inductance L H Susceptibility (M/H) χ Permeability (B/H) µ Hm Magnetic polarization J T Magnetomotive force F A Magnetic charge q m Am Energy product (BH) Jm Anisotropy energy K Jm Exchange stiffness A Jm Hall coefficient R H m 3 C Scalar potential ϕ A Vector potential A Tm Permeance P m Tm 2 A Reluctance R m AT 1 m

9 Thermal Quantity Symbol Unit m l t i θ Enthalpy H J Entropy S JK Specific heat C JK 1 kg Heat capacity c JK Thermal conductivity κ Wm 1 K Sommerfeld coefficient γ Jmol 1 K Boltzmann s constant k B JK (1) Kinetic energy of a body: ε = 1 2 mv2 [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 = qv B [f ] = [1, 1, 2, 0, 0] [q] = [0, 0, 1, 1, 0] [v] = [0, 1, 1, 0, 0] [1, 0, 2, 1, 0] [B] = [1, 1, 2, 0, 0] (3) Domain wall energy γ w = AK (γ w is an energy per unit area) [γ w ] = [εa 1 ] [ AK] = 1/2[AK] = [1, 2, 2, 0, 0] [ A] = 1 [1, 1, 2, 0, 0] 2 [ 1,1, 2, 0, 0] [ K] = 1 [1, 1, 2, 0, 0] - [0, 0, 0, 0 ] 2 [1, 0, 2, 0, 0] = [1, 0, 2, 0, 0] (4) Magnetohydrodynamic force on a moving conductor F = B B

10 = (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 + dd/dt. [ H] = [Hr 1 ] [j] = [0, 2, 0, 1, 0] [dd/dt] = [Dt 1 ] = [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]

11 SI Units J 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 4π 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.

12 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.

13 Am -1 is a unit that some people are not entirely comfortable with, perhaps because there is a special name (Tesla) for the unit of B (flux density) or J (polarization), but there no special name for the unit of H (magnetic field intensity) or M (magnetization). In cgs there are two different named units, gauss (G) and oersted (Oe), for the two different quantities, but the issue is confused by using a dimensional constant numerically equal to 1, that is generally omitted from the equations. Hence they are numerically equal, but different quantities. In free space B and H are interchangeable, because the two fields are simply proportional. The quantities are nevertheless different, with different units. It is acceptable to label field axes µ 0 H (T) and magnetization axes µ 0 M (T). But it is nonsense to write H or M in T. The tesla is not a unit of M or H in any generallyrecognized unit system. A daft practice has grown up whereby large fields are measured in tesla, and small ones on oersteds! (see ads from Quantum Design)

14 Magnetic moment of a sample is m (Am 2 ). The magnetization M is therefore m/v, with units Am -1. (The symbol for the mass of the electron is m. Its charge is e. Mass generally is m.) The symbol for specific magnetization (magnetic moment per unit mass), often measured in bulk samples, is σ = M/ρ. Here ρ is the mass density in kgm -3. Units of σ are Am2 kg -1. This is numerically the same as the cgs unit emu g -1.

15 Susceptibility needs to be treated with care. The basic definition is χ = M/H. With this definition, here are no units; it is a pure number. This quantity, also known as the volume susceptibility, is what you can measure experimentally for a thin film if you know its volume, or deduce from the fmr frequency. With bulk samples of irregular shape, you often do not know the sample volume precisely, but the sample mass m is easy to measure. Then the mass susceptibility χ m is χ/ρ (mkg -1 ). Also useful is the molar susceptibility χ mol = Mχ m, where M is the molecular weight in gmol -1. In other words, the volume, mass and molar susceptibilities are the atomic susceptibility multiplied respectively by the number of atoms per m 3, per kg, or per mole (N A = ). Use p eff for the effective Bohr magneton number, m eff for the effective moment. The Curielaw expression for the molar susceptibility is χ mol = p 2 eff /T It is probably best to avoid definitions of susceptibility based on the B-field. Permeability, symbol µ, is B/H where B is the flux density in a material induced by a field H. It has the same units as µ 0. Relative permeability µ r = 1 + χ is dimensionless.

16 cgs Units L M Most of the primary literature on magnetism is still written using cgs units, or a muddled mixture ]where large fields are quoted in teslas and small ones in oersteds, one a unit of B, the other a unit of H! Basic cgs units are cm, g and s. The electromagnetic unit of current is equivalent to 10 A. The electromagnetic unit of potential is equivalent to 10 nv. The electromagnetic unit of magnetic dipole moment (emu) is equivalent to 10 3 Am 2. Derived cgs units include the erg (10 7 J), so that an energy density of 1 J m 3 is equivalent to 10 erg cm 3. The convention relating flux density and magnetization in cgs is B = H + 4πM (2) where the flux density or induction B is measured in gauss (G) and field H in oersteds (Oe). Magnetic moment is usually expresed as emu, and magnetization is therefore in emu cm 3, although 4πM is considerd a flux-density expression, Often quoted in kilogauss. The magnetic constant µ 0 is numerically equal to 1 G Oe 1, but its general omission from the equations makes it impossible to check their dimensions.

A Question. What is Magnetism?

A Question. What is Magnetism? A Question What is Magnetism? Basic Concepts: Magnetostatics J. M. D. Coey School of Physics and CRANN, Trinity College Dublin Ireland. 1. Introduction 2. Magnets 3. Fields 4. Forces and energy 5. Units