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

www.tcd.ie/Physics/MagnetismComments and corrections please: jcoey@tcd.ie

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

15 Other topics

*Appendices, conversion tables.

614 pages. Published March 2010

www.cambridge.org/9780521816144

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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.

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SI / cgs conversions:

SI units

B = μ0(H + M)

A m2

A m-1 (10-3 emu cc-1)

A m2 kg-1 (1 emu g-1)

A m-1 (4π/1000 ≈ 0.0125 Oe)

Tesla (10 kG)

Weber (Tm2) (108 Mw)

V (108 Mw s-1)

- (4π cgs)

cgs units

B = H + 4πM

emu

emu cc-1 (1 k A m-1)

emu g-1 (1 A m2 kg-1)

Oersted (1000/4π ≈ 80 A m-1)

Gauss (10-4 T)

Maxwell (G cm2) (10-8 Wb)

Mw s-1 (10 nV)

- (1/4π SI)

m

M

σ

H

B

Φ

dΦ/dt

χ

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592 Appendices

B.3 Dimensions

Any quantity in the SI system has dimensions which are a combination of thedimensions of the five basic quantities, m, l, t , i and θ . In any equation relatingcombinations of physical properties, each of the dimensions must balance, andthe dimensions of all the terms in a sum have to be identical.

B3.1 Dimensions

Mechanical

Quantity Symbol Unit m l t i θ

Area A m2 0 2 0 0 0Volume V m3 0 3 0 0 0Velocity v m s−1 0 1 −1 0 0Acceleration a m s−2 0 1 −2 0 0Density d kg m−3 1 −3 0 0 0Energy ε J 1 2 −2 0 0Momentum p kg m s−1 1 1 −1 0 0Angular momentum L kg m2 s−1 1 2 −1 0 0Moment of inertia I kg m2 1 2 0 0 0Force f N 1 1 −2 0 0Force density F N m−3 1 −2 −2 0 0Power P W 1 2 −3 0 0Pressure P Pa 1 −1 −2 0 0Stress σ N m−2 1 −1 −2 0 0Elastic modulus K N m−2 1 −1 −2 0 0Frequency f s−1 0 0 −1 0 0Diffusion coefficient D m2 s−1 0 2 −1 0 0Viscosity (dynamic) η N s m−2 1 −1 −1 0 0Viscosity ν m2 s−1 0 2 −1 0 0Planck’s constant ! J s 1 2 −1 0 0

Thermal

Quantity Symbol Unit m l t i θ

Enthalpy H J 1 2 −2 0 0Entropy S J K−1 1 2 −2 0 −1Specific heat C J K−1 kg−1 0 2 −2 0 −1Heat capacity c J K−1 1 2 −2 0 −1Thermal conductivity κ W m−1 K−1 1 1 −3 0 −1Sommerfeld coefficient γ J mol−1 K−1 1 2 −2 0 −1Boltzmann’s constant kB J K−1 1 2 −2 0 −1

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593 Appendix B Units and dimensions

Electrical

Quantity Symbol Unit m l t i θ

Current I A 0 0 0 1 0Current density j A m−2 0 −2 0 1 0Charge q C 0 0 1 1 0Potential V V 1 2 −3 −1 0Electromotive force E V 1 2 −3 −1 0Capacitance C F −1 −2 4 2 0Resistance R " 1 2 −3 −2 0Resistivity ϱ " m 1 3 −3 −2 0Conductivity σ S m−1 −1 −3 3 2 0Dipole moment p C m 0 1 1 1 0Electric polarization P C m−2 0 −2 1 1 0Electric field E V m−1 1 1 −3 −1 0Electric displacement D C m−2 0 −2 1 1 0Electric flux % C 0 0 1 1 0Permittivity ε F m−1 −1 −3 4 2 0Thermopower S V K−1 1 2 −3 −1 −1Mobility µ m2 V−1 s−1 −1 0 2 1 0

Magnetic

Quantity Symbol Unit m l t i θ

Magnetic moment m A m2 0 2 0 1 0Magnetization M A m−1 0 −1 0 1 0Specific moment σ A m2 kg−1 −1 2 0 1 0Magnetic field strength H A m−1 0 −1 0 1 0Magnetic flux ' Wb 1 2 −2 −1 0Magnetic flux density B T 1 0 −2 −1 0Inductance L H 1 2 −2 −2 0Susceptibility (M/H) χ 0 0 0 0 0Permeability (B/H) µ H m−1 1 1 −2 −2 0Magnetic polarization J T 1 0 −2 −1 0Magnetomotive force F A 0 0 0 1 0Magnetic ‘charge’ qm A m 0 1 0 1 0Energy product (BH ) J m−3 1 −1 −2 0 0Anisotropy energy K J m−3 1 −1 −2 0 0Exchange stiffness A J m−1 1 1 −2 0 0Hall coefficient RH m3 C−1 0 3 −1 −1 0Scalar potential ϕ A 0 0 0 1 0Vector potential A T m 1 1 −2 −1 0Permeance Pm T m2 A−1 1 2 −2 −2 0Reluctance Rm A T−1 m−2 −1 −2 2 2 0

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593 Appendix B Units and dimensions

Electrical

Quantity Symbol Unit m l t i θ

Current I A 0 0 0 1 0Current density j A m−2 0 −2 0 1 0Charge q C 0 0 1 1 0Potential V V 1 2 −3 −1 0Electromotive force E V 1 2 −3 −1 0Capacitance C F −1 −2 4 2 0Resistance R " 1 2 −3 −2 0Resistivity ϱ " m 1 3 −3 −2 0Conductivity σ S m−1 −1 −3 3 2 0Dipole moment p C m 0 1 1 1 0Electric polarization P C m−2 0 −2 1 1 0Electric field E V m−1 1 1 −3 −1 0Electric displacement D C m−2 0 −2 1 1 0Electric flux % C 0 0 1 1 0Permittivity ε F m−1 −1 −3 4 2 0Thermopower S V K−1 1 2 −3 −1 −1Mobility µ m2 V−1 s−1 −1 0 2 1 0

Magnetic

Quantity Symbol Unit m l t i θ

Magnetic moment m A m2 0 2 0 1 0Magnetization M A m−1 0 −1 0 1 0Specific moment σ A m2 kg−1 −1 2 0 1 0Magnetic field strength H A m−1 0 −1 0 1 0Magnetic flux ' Wb 1 2 −2 −1 0Magnetic flux density B T 1 0 −2 −1 0Inductance L H 1 2 −2 −2 0Susceptibility (M/H) χ 0 0 0 0 0Permeability (B/H) µ H m−1 1 1 −2 −2 0Magnetic polarization J T 1 0 −2 −1 0Magnetomotive force F A 0 0 0 1 0Magnetic ‘charge’ qm A m 0 1 0 1 0Energy product (BH ) J m−3 1 −1 −2 0 0Anisotropy energy K J m−3 1 −1 −2 0 0Exchange stiffness A J m−1 1 1 −2 0 0Hall coefficient RH m3 C−1 0 3 −1 −1 0Scalar potential ϕ A 0 0 0 1 0Vector potential A T m 1 1 −2 −1 0Permeance Pm T m2 A−1 1 2 −2 −2 0Reluctance Rm A T−1 m−2 −1 −2 2 2 0

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592 Appendices

B.3 Dimensions

Any quantity in the SI system has dimensions which are a combination of thedimensions of the five basic quantities, m, l, t , i and θ . In any equation relatingcombinations of physical properties, each of the dimensions must balance, andthe dimensions of all the terms in a sum have to be identical.

B3.1 Dimensions

Mechanical

Quantity Symbol Unit m l t i θ

Area A m2 0 2 0 0 0Volume V m3 0 3 0 0 0Velocity v m s−1 0 1 −1 0 0Acceleration a m s−2 0 1 −2 0 0Density d kg m−3 1 −3 0 0 0Energy ε J 1 2 −2 0 0Momentum p kg m s−1 1 1 −1 0 0Angular momentum L kg m2 s−1 1 2 −1 0 0Moment of inertia I kg m2 1 2 0 0 0Force f N 1 1 −2 0 0Force density F N m−3 1 −2 −2 0 0Power P W 1 2 −3 0 0Pressure P Pa 1 −1 −2 0 0Stress σ N m−2 1 −1 −2 0 0Elastic modulus K N m−2 1 −1 −2 0 0Frequency f s−1 0 0 −1 0 0Diffusion coefficient D m2 s−1 0 2 −1 0 0Viscosity (dynamic) η N s m−2 1 −1 −1 0 0Viscosity ν m2 s−1 0 2 −1 0 0Planck’s constant ! J s 1 2 −1 0 0

Thermal

Quantity Symbol Unit m l t i θ

Enthalpy H J 1 2 −2 0 0Entropy S J K−1 1 2 −2 0 −1Specific heat C J K−1 kg−1 0 2 −2 0 −1Heat capacity c J K−1 1 2 −2 0 −1Thermal conductivity κ W m−1 K−1 1 1 −3 0 −1Sommerfeld coefficient γ J mol−1 K−1 1 2 −2 0 −1Boltzmann’s constant kB J K−1 1 2 −2 0 −1

594 Appendices

B3.2 Examples

(1) Kinetic energy of a body: ε = 12mv2

[ε] = [1, 2,−2, 0, 0] [m] = [1, 0, 0, 0, 0]

[v2] = 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]

[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] = [εA−1] [√

AK] = 1/2[AK]= [1, 2,−2, 0, 0] [

√A] = 1

2 [1, 1,−2, 0, 0]

−[ 1, 1, −2, 0, 0] [√

K] = 12

[1,−1,−2, 0, 0][1, 0,−2, 0, 0]

= [1, 0,−2, 0, 0](4) Magnetohydrodynamic force on a moving conductor F = σv × 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][1,−2,−2, 0, 0]

[B2] = 2[1, 0,−2,−1, 0][1,−2,−2, 0, 0]

(5) Flux density in a solid B = µ0(H + M) (note that quantities added orsubtracted in a bracket must have the same dimensions)[B] = [1, 0,−2,−1, 0] [µ0] = [1, 1,−2,−2, 0]

[M], [H ] = [0,−1, 0, 1, 0][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]

- [0, 2, 0, 0, 0 ]

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594 Appendices

B3.2 Examples

(1) Kinetic energy of a body: ε = 12mv2

[ε] = [1, 2,−2, 0, 0] [m] = [1, 0, 0, 0, 0]

[v2] = 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]

[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] = [εA−1] [√

AK] = 1/2[AK]= [1, 2,−2, 0, 0] [

√A] = 1

2 [1, 1,−2, 0, 0]

−[ 1, 1, −2, 0, 0] [√

K] = 12

[1,−1,−2, 0, 0][1, 0,−2, 0, 0]

= [1, 0,−2, 0, 0](4) Magnetohydrodynamic force on a moving conductor F = σv × 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][1,−2,−2, 0, 0]

[B2] = 2[1, 0,−2,−1, 0][1,−2,−2, 0, 0]

(5) Flux density in a solid B = µ0(H + M) (note that quantities added orsubtracted in a bracket must have the same dimensions)[B] = [1, 0,−2,−1, 0] [µ0] = [1, 1,−2,−2, 0]

[M], [H ] = [0,−1, 0, 1, 0][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]

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 = µ0(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 m3) are measured in Am-1. The constant µ0 in (1) is precisely 4π 10-7 TmA-1. There are other equivalent units for µ0, 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.

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The Kennelly convention is compatible with the above. 

B = µ0H + J where J = µ0M 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.

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 µ0H (T) and magnetization axes µ0M (T). But it is nonsense to write H or M in T. The tesla is not a unit of M or H in any generally-recognized 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)

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Magnetic moment of a sample is m (Am2). 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 Am2kg-1. This is numerically the same as the cgs unit emu g-1.

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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 χm is χ/ρ (mkg-1). Also useful is the molar susceptibility χmol = Mχm, 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 m3, per kg, or per mole (NA = 6.022 1023).Use ­peff­ for the effective Bohr magneton number, m eff­ for the effective moment. The Curie-law expression for the molar susceptibility is χmol = 1.591 10-6 peff­

2/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 µ0. Relative permeability µr = 1 + χ is dimensionless.

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cgs Units LM Most of the primary literature on magnetism is still written using cgs units, ora muddled mixture ]where large fields are quoted in teslas and smallones 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. Theelectromagnetic unit of potential is equivalent to 10 nV. The electromagneticunit of magnetic dipole moment (emu) is equivalent to 10−3 Am2 . Derived cgsunits include the erg (10−7 J), so that an energy density of 1 J m−3 is equivalentto 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 inoersteds (Oe). Magnetic moment is usually expresed as emu, and magnetizationis therefore in emu cm−3 , although 4πM is considerd a flux-density expression,Often quoted in kilogauss. The magnetic constant µ0 is numerically equalto 1 G Oe−1, but its general omission from the equations makes it impossibleto check their dimensions.

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