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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 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

Magnetic field, Øersted 1820

• Oersted discovered in 1820 that a current carrying wire was able to rotate a compass needle

• Current and field are related by Ampere’s Law

• Example for I = 1A, integral around the loop is 2𝜋r, r = 2 mm H ~ 80 A/m

• Earth’s magnetic field ~ 40 A/m

H δl

r

I

I = ∫ Hdl

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.

• The parallel wires “look like” magnets in the perpendicular direction

• Weird but central to electromagnetism (E and B fields in light)

• Different from electrostatics as this is a dynamic effect from the motion of charge

Fl

=μ0

2πI1I2

rI1 I2

F

Equivalence of currents and magnetic moments

• So currents look like magnets… do magnets look like currents?

• Can express a current loop as an effective moment, ie a source of magnetic field

• What kind of currents do we need compared to typical magnetic fields?

m

Im = I⊥A

A

Comparison of current magnitudes and magnets

• Using the equivalence of current loops and magnetic moments we can compare the effective 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

• For a small magnet

• At small sizes, magnets generate much larger fields -> applications in motors

10 mm

1 Am2 1 A 10,000 turns

10,000 A 1 turns

m = I⊥A m = nI⊥A

Difference between magnetic moment and magnetisation

• Magnetic moment is specific to the sample (bigger magnet, bigger field)

• Magnetization is the moment density

• Magnetisation is a property of the material • Moment is a property of a magnet

• Magnetisation is scale independent

3 mm 10 mm25 mm

0.027 Am2 1 Am2

15.6 Am2

m = MV

Assume NdFeBMs ~ 1 MA/m

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

Weak repulsion

Strong attraction

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

• 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

<M>

4 Be 9.01 2 + 2s0

12Mg 24.21 2 + 3s0

2 He 4.00

10Ne 20.18

24Cr 52.00 3 + 3d3

312

19K 38.21 1 + 4s0

11Na 22.99 1 + 3s0

3 Li 6.94 1 + 2s0

37Rb 85.47 1 + 5s0

55Cs 132.9 1 + 6s0

38 Sr 87.62 2 + 5s0

56Ba 137.3 2 + 6s0

59Pr 140.9 3 + 4f2

1 H 1.00

5 B 10.81

9 F 19.00

17Cl 35.45

35Br 79.90

21Sc 44.96 3 + 3d0

22Ti 47.88 4 + 3d0

23V 50.94 3 + 3d2

26Fe 55.85 3 + 3d5

1043

27Co 58.93 2 + 3d7

1390

28Ni 58.69 2 + 3d8

629

29Cu 63.55 2 + 3d9

30Zn 65.39 2 + 3d10

31Ga 69.72 3 + 3d10

14Si 28.09

32Ge 72.61

33As 74.92

34Se 78.96

6 C 12.01

7 N 14.01

15P 30.97

16S 32.07

18Ar 39.95

39 Y 88.91 2 + 4d0

40 Zr 91.22 4 + 4d0

41 Nb 92.91 5 + 4d0

42 Mo 95.94 5 + 4d1

43 Tc 97.9

44 Ru 101.1 3 + 4d5

45 Rh 102.4 3 + 4d6

46 Pd 106.4 2 + 4d8

47 Ag 107.9 1 + 4d10

48 Cd 112.4 2 + 4d10

49 In 114.8 3 + 4d10

50 Sn 118.7 4 + 4d10

51 Sb 121.8

52 Te 127.6

53 I 126.9

57La 138.9 3 + 4f0

72Hf 178.5 4 + 5d0

73Ta 180.9 5 + 5d0

74W 183.8 6 + 5d0

75Re 186.2 4 + 5d3

76Os 190.2 3 + 5d5

77Ir 192.2 4 + 5d5

78Pt 195.1 2 + 5d8

79Au 197.0 1 + 5d10

61Pm 145

70Yb 173.0 3 + 4f13

71Lu 175.0 3 + 4f14

90Th 232.0 4 + 5f0

91Pa 231.0 5 + 5f0

92U 238.0 4 + 5f2

87Fr 223

88Ra 226.0 2 + 7s0

89Ac 227.0 3 + 5f0

62Sm 150.4 3 + 4f5

105

66Dy 162.5 3 + 4f9

179 85

67Ho 164.9 3 + 4f10

132 20

68Er 167.3 3 + 4f11

85 20

58Ce 140.1 4 + 4f0

13

Ferromagnet TC > 290K

Antiferromagnet with TN > 290K

8 O 16.00 35

65Tb 158.9 3 + 4f8

229 221

64Gd 157.3 3 + 4f7

292

63Eu 152.0 2 + 4f7

90

60Nd 144.2 3 + 4f3

19

66Dy 162.5 3 + 4f9

179 85

Atomic symbol Atomic Number Typical ionic change Atomic weight

Antiferromagnetic TN(K) Ferromagnetic TC(K)

Antiferromagnet/Ferromagnet with TN/TC < 290 K Metal

Radioactive

Magnetic Periodic Table

80Hg 200.6 2 + 5d10

93Np 238.0 5 + 5f2

94Pu 244

95Am 243

96Cm 247

97Bk 247

98Cf 251

99Es 252

100Fm 257

101Md 258

102No 259

103Lr 260

36Kr 83.80

54Xe 83.80

81Tl 204.4 3 + 5d10

82Pb 207.2 4 + 5d10

83Bi 209.0

84Po 209

85At 210

86Rn 222

Nonmetal Diamagnet

Paramagnet

BOLD Magnetic atom

25Mn 55.85 2 + 3d5

96

20Ca 40.08 2 + 4s0

13Al 26.98 3 + 2p6

69Tm 168.9 3 + 4f12

56

ESM Cluj 2015

Which elements are magnetic

From Coey

Bohr magneton

• Can consider an electron ‘orbiting’ an atom

• A moving charge looks like a ‘current’, generating an effective magnetic moment

• In Bohr’s quantum theory, orbital angular momentum l is quantized in units of ℏ; h is Planck’s constant, 6.6226 10-34 Js; ℏ=h/

2𝜋=1.05510-34 Js

• The orbital angular momentum is l = mer ∧v

• It is the z-component of lz that is quantized in units of ℏ, taking a value ml

ml is a quantum number, an integer with no units. Eliminating r in the expression for m

• μB is the Bohr magneton, the basic unit of atomic magnetism

Dublin January 2007 3

1.1 Orbital moment

The circulating current is I; I = -ev/2!r

The moment is m = IA m = -evr/2

In Bohr’s quantum theory, orbital angular momentum lis quantized in units of !; h is Planck’s constant, 6.622610-34 J s; ! = h/2! = 1.055 10-34 J s.

The orbital angular momentum is l = mer"v; Units are J sIt is the z-component of lz that is quantized in units of !, taking a value ml!

ml is a quantum number, an integer with no units. Eliminating r in the expression for m,

m = -(e/2me)l = (e!/2me)ml = mlµB

m = #l gyromagnetic ratio

The quantity µB = (e!/2me) is the Bohr magneton, the basic unit of atomic magnetism;

µB = 9.274 10-24 A m2

Electrons circulate indefinitely in stationary states; unquantized orbital motion radiates energy

e

e

m

m

l

m = IA = −evr2

m = −e

2meI =

eℏ2me

ml = mlμB

μB =eℏ2me

= 9.274 × 10−24Am2 |JT−1

*

* electrons travel in the opposite direction to currents

Non-integer magnetic moments

• Transition metal magnets tend to have non-integer magnetic moments, eg Fe ~ 2.2 μB, Co ~ 1.72 μB, Ni ~ 0.6 μB

• If electrons carry quanta of angular momentum, how is this possible?

• Classic explanation is itinerant magnetism: electrons are delocalised and form bands

• First principles calculations reveal a non-integer magnetic moment quite localised to the atom

• Effect due to electrons hopping between different d-orbitals

2670 K Schwarz et a1

due to a dip in the state densities of the non-d spin-up electrons at the Fermi energy (Terakura and Kanamori 1971, Terakura 1977, Malozemoff et a1 1984). When applied to Fe, equation ( 1 ) gives a magnetic moment of M = 2.6. That it is in reality less than that is due to the magnetic weakness of Fe where the Fermi energy cuts through the spin-up d band (see figure 6) . The straight line with a slope of + 45' is defined by

M = Z - 2 N J (2) where N' is the average number of spin-down electrons per atom. In this case the Fermi energy is pinned in a pronounced dip in the spin-down state density, fixing the number mJ at 2.93 (see the case of Fe in figure 6) .

Table 2 contains the ASW partial charges per atom, the average numbers per atom of majority- and minority-spin electrons, miT and i%" respectively, and the calculated lattice constants. Starting with the highest magnetic moment calculated in figure 8 (Fe, CO), we see from table 2 that both fit and 8' are close to the values assumed for equations (1) and ( 2 ) : around a concentration of 25 at.% CO we have strong ferromagnetism and the Fermi energy is pinned in the dip of the minority-spin state density (see also figure 6). For CO

Table 2. A S W partial charges (in electrons per atom) for all six structures. The total includes a small f contribution. & ? and I%" are average numbers per atom of majority- and minority- spin electrons, respectively. The corresponding equilibrium lattice constants which are derived from ASW total energy calculations are given at the bottom.

Fe,Co (Fe3Al)

Fe FeCo FeCo FeCo, CO ( B C C ) Fe, Fe2 (CsCl) (Zintl) (Fe3AI) ( B C C )

Fe s t 0.31 0.31 0.31 0.31 0.3 1 0.30 1 0.32 0.31 0.31 0.31 0.3 1 0.30

p T 0.36 0.35 0.35 0.36 0.35 0.34 i 0.40 0.38 0.40 0.40 0.38 0.38

d t 4.37 4.50 4.57 4.61 4.56 4.58 12.19 2.09 1.95 1.86 1.97

Total 5.07 5.20 5.26 5.31 5.25 1.92 5.26

~

1 2.93 2.80 2.69 2.59 2.67 2.63 ~ ~

FeCo (Fe,AI)

Fe Fe,Co FeCo FeCo CO (BCC) (Fe3AI) (CsCI) (Zintl) CO, CO, ( B C C )

C o s T 1

P T

d T

Total t 1

i

4

0.32 0.33 0.32 0.31 0.32 0.31 0.34 0.33 0.34 0.33 0.33 0.32 0.37 0.37 0.37 0.35 0.36 0.35 0.44 0.4 1 0.43 0.41 0.40 0.40 4.66 4.67 4.66 4.64 4.65 4.62

2.92 2.93 2.93 5.38 5.40 5.39 5.34 5.36 5.32 3.73 3.71 3.68 3.68 3.69 3.67

~ - - 2.93 2.94 2.90

.o 5.07 5.28 5.36 5.32 5.33 5.32 ~o 2.93 2.98 3.15 3.18 3.42 3.67

a (A) 2.8 I O 5.647 2.814 5.627 5.624 2.773

K Schwarz et al 1984 J. Phys. F: Met. Phys. 14 2659

Field from a dipole

• The magnetic induction (field) from a point dipole can be derived classically (see Jackson) and is given by

VAMPIRE: State of the art atomistic modeling of magnetic nanomaterials

R. F. L. Evans,∗

W. J. Fan, J. Barker, P. Chureemart, T. Ostler, and R. W. Chantrell

Department of Physics, The University of York, York, YO10 5DD, UK

(Dated: September 14, 2018)

I. INTRODUCTION

II. THEORETICAL METHODS

H =−∑i< j

Ji jSi ·S j −∑i

ku (Si · ei)2 −∑

iµiSi ·Bi (1)

Hbq =−∑i< j

Jbq (Si ·S j)2

(2)

∂Si

∂ t=− γi

(1+λ 2i )

[Si ×Bi +λiSi × (Si ×Bi)] (3)

Bi = ζi(t)−1

µi

∂H

∂Si(4)

ζi = 〈ζ ai (t)ζ

bj (t)〉= 2δi jδab(t − t ′)

λikBTµiγi

(5)

〈ζ ai (t)〉= 0 (6)

III. TWO TEMPERATURE MODEL

!TeCe

∂Te

∂ t

"=−Ge (Te −Tl)+P(t) (7)

Cl∂Tl

∂ t= Ge (Te −Tl) (8)

P(t) =P0√2π

exp

#−(t − t0)

t2p

$(9)

M(T ) = M0

!1− T

Tc

"β(10)

Bidp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(11)

B =µ0

4π

!3(m · r)r−m

|r|3

"(12)

Didp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(13)

%Bp

dm

&=

µ0

4π

#

∑p∕=q

Dinter · 〈mqmc〉

$+

2µ0

3

Dintra ·%mp

mc&

V pmc

.

(14)

%Bp

dm

&=

µ0

4π

#

∑p∕=q

3(mqmc · r)r−mq

mc

r3

$− µ0

3

mpmc

V pmc

(15)

JTi j =

'

(Jxx Jxy JxzJyx Jyy JyzJzx Jzy Jzz

)

* , (16)

H =−∑i< j

+Si

x,Siy,S

iz,'


)

*

'

(S j

x

S jy

S jz

)

* . (17)

T =−-

∂F

∂θ

.=

/

∑i

Si ×Bi

0(18)

T =−-

∂F

∂φ

.=

/

∑i

Si ×Bi

0(19)

∗ [email protected]

J. D. Jackson, Classical electrodynamics (2nd ed.). New York: Wiley. (1975)

m

r

B-field at any point from a point dipole

• Ignores any distribution of magnetic ‘charge’ at the dipole (need a multipole description)

Question: What is the size of the Earth’s magnetic moment?

• Assume an effective dipole at the centre of the Earth and a magnetic flux density at the North Pole of 50 µT and REarth = 6.36 x 106 m

|→B Npole

| =μ0

4πR3Earth

(3(→μ ⋅ r)r − →μ )

∴4πR3

Earth |→B Npole

|

μ0= 3 | →μ | r − →μ = | →μ | 3r − r = 2 | →μ |

| →μ | =4πR3

Earth |→B Npole

|

2μ0

| →μ | =2πR3

Earth |→B Npole

|

μ0=

2 ⋅ π ⋅ (6.38 × 106m)3 ⋅ (50 × 10−6T )(1.256 × 10−6NA−2)

∴ | →μ | ≈ 6.48 × 1022Am2

Question 2: What is the magnetization of the Earth?

MEarth =mEarth

VEarth

MEarth =mEarth

4π3 R3

Earth

MEarth =6.48 × 1022

4π3 (6.38 × 106)3

≈ 60A/m−1

Question 3: If the source of the magnetic field is an electrical current at the equator, what is its size?

m = IA

IEquator =mEarth

πR2Earth

IEquator =6.48 × 1022

π(6.38 × 106)2≈ 5 × 108A

Magnetic fields and demagnetising factors

What ranges of magnetic fields exist?

• Historically a terrestrial 1T field was considered ‘large’

• Today that is not generally true

• Recording Media coercivity ~1T

• MRI ~ 5T

1E-15 1E-12 1E-9 1E-6 1E-3 1 1000 1E6 1E9 1E12 1E15

MT

Magneta

r

Neutro

n Star

Explosiv

e Flux C

ompress

ion

Pulse M

agnet

Hybrid

Mag

net

Superco

nducting M

agnet

Perman

ent M

agnet

Human B

rain

Human H

eart

Interste

llar S

pace

Interplan

etary

Spac

e

Earth's

Field at

the S

urface

Solenoid

pT µT T

TT

The range of magnitude of B

Largest continuous laboratory field 45 T (Tallahassee)

• The tesla is a very large unit • Largest continuous field acheived in a lab was 45 T

ESM Cluj 2015

Typical values of magnetic fields

Human Brain 1 fT Earth 50 𝜇T Permanent Magnet 0.5-1T

Electromagnet 1T Magnetar 1012 TSuperconducting magnet 10 T

Magnetic fields in free space

• Two definitions of magnetic field

• When talking about generated magnetic fields in free space, they express the exact same physical phenomenon, and are related by

• 𝜇0 = 4pi 10-7 H/m is the permeability of free space

• The difference between H-field and B-field is a common point of confusion, but only when considering a magnetic medium B = 𝜇0(H+M)

• B-field component arising from applied H-field is exactly

Magnetic Field H [A/m] Magnetic flux density B [T]

B = 𝜇0H

B = 𝜇0H

Magnetic fields in media

• The actual B-field in response to media is generally more complex

• Or alternatively in terms of a relative permeability or susceptibility

• where the susceptibility gives the full magnetic response, or limit of small fields (initial susceptibility)

• Different media ave very different responses, ferromagnets highly non-linear

2

IV. MAGNETOSTATICS

B = µ0 (M+H) (22)

Btotal = Bmagnetization +Bapplied (23)

M =Bmagnetization

µ0(24)

M =Bmagnetization

µ0(25)

Happlied =Bapplied

µ0(26)

M(JT−1m−3)≡ µ0M

µ0

(T)(T2J−1m3)

(27)

µ0M (tesla)µ0H (tesla)

B = µrµ0H = µ0(1+χ)H (28)

B = µ0µrH = µ0(1+χ)H (29)

χ = M(H) =dM

dH

333H→0

(30)

2

IV. MAGNETOSTATICS

B = µ0 (M+H) (22)


M =Bmagnetization

µ0(24)

M =Bmagnetization

µ0(25)

Happlied =Bapplied

µ0(26)


µ0

(T)(T2J−1m3)

(27)


B = µrµ0H = µ0(1+χ)H (28)

B = µ0µrH = µ0(1+χ)H (29)

χ = M(H) =dM

dH

333H→0

(30)

2

IV. MAGNETOSTATICS

B = µ0 (M+H) (22)


M =Bmagnetization

µ0(24)

M =Bmagnetization

µ0(25)

Happlied =Bapplied

µ0(26)


µ0

(T)(T2J−1m3)

(27)


B = µrµ0H = µ0(1+χ)H (28)

B = µ0µrH = µ0(1+χ)H (29)

χ = M(H) =dM

dH

333H→0

(30)

Diamagnetism and Paramagnetism

• Diamagnets and paramagnets have a weak magnetic response (𝝌 << 1), ~ 10-4 - 10-6

• Response typically isotropic with respect to the field

• Diamagnets repel external magnetic fields due to Larmor precession of bound electrons that induces a moment opposite to the applied field

• Paramagnets weakly align with an external field overcoming thermal fluctuations

M

H

Paramagnetic

Diamagnetic

Ferromagnetism

Yue Cao et al, JMMM 395, 361-375 (2015)

• Complex and anisotropic behaviour of M(H)

• Definition of 𝝌 = M/H is not very sensible in most cases

• Saturated case easier to deal with!

Relation between B and H in a saturated material

• Magnetic field around a saturated magnet simple B = 𝜇0H

• What about inside the magnet?

• Why do we care?

• In general magnetization processes are anisotropic and depend on sample shape

2

IV. MAGNETOSTATICS

B = µ0 (M+H) (22)


M =Bmagnetization

µ0(24)

M =Bmagnetization

µ0(25)

Happlied =Bapplied

µ0(26)


µ0

(T)(T2J−1m3)

(27)


B = µrµ0H = µ0(1+χ)H (28)

B = µ0µrH = µ0(1+χ)H (29)

χ = M(H) =dM

dH

333H→0

(30)

Example: thin magnetic film

• Much easier to magnetise in the plane than out-of-plane

• Origin is demagnetising field - aims to minimise surface charges

M

H

H || to film

H ⟂ to film

+ + + + - - - - + + +

+ + + + + +

M

M

Demagnetizing fields

• Local effective field inside the magnet depends on surface

• Since M is uniform, first (bulk) term is zero

• For surface term, M.en determines surface charge density, larger surface leads to larger field opposing magnetisation

• Leads to concept of a demagnetising field

Consider bulk and surface magnetic charge distributions

ρm = -∇.M and ρms = M.en

H field of a small charged volume element V is

δH = (ρmr/4πr3) δV

So

For a uniform magnetic distribution the first term is zero. ∇.M = 0

ESM Cluj 2015

Field calculations – Coulombian approach

H = Happ - Hd

Demagnetization Factor

• Calculating demagnetisation field is tedious (lots of boring and complicated integrals)

• Simplify - invent a “demagnetising factor” or “shape factor” N

• Shape factor gives a constant of proportionality between the demagnetising field and shape

• Always between 0-1 and in general a tensor with trace 1

• Known for simple geometric shapes (spheres, ellipsoids, rectangular prisms)

• Is usually calculated numerically for anything complicated

Hd = -NM

Nx + Ny + Nz = 1

Demagnetization factors for different shapes

N = 0 N = 1/3

Infinite thin film

Infinitely long cylinder

Sphere

N = 1/2 N = 1

Infinitely long cylinder

Short cylinder

Beware of non-uniformities

• In general magnetization is not uniform for other shapes

Jay Shah et al, Nature Communications 9 1173 (2018)

Dipole fields and magnetostatics

• Assume a lattice of dipoles in shape of a sphere

• Total dipole field at a point in the centre summing over all other dipoles is zero

• Where does the demagnetising field come from?


R. F. L. Evans,∗




I. INTRODUCTION


H =−∑i< j


ku (Si · ei)2 −∑

iµiSi ·Bi (1)

Hbq =−∑i< j

Jbq (Si ·S j)2

(2)

∂Si

∂ t=− γi

(1+λ 2i )


Bi = ζi(t)−1

µi

∂H

∂Si(4)



λikBTµiγi

(5)

〈ζ ai (t)〉= 0 (6)


!TeCe

∂Te

∂ t

"=−Ge (Te −Tl)+P(t) (7)

Cl∂Tl

∂ t= Ge (Te −Tl) (8)

P(t) =P0√2π

exp

#−(t − t0)

t2p

$(9)

M(T ) = M0

!1− T

Tc

"β(10)

Bidp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(11)

B =µ0

4π

!3(m · r)r−m

|r|3

"(12)

Didp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(13)

%Bp

dm

&=

µ0

4π

#

∑p∕=q


$+

2µ0

3

Dintra ·%mp

mc&

V pmc

.

(14)

%Bp

dm

&=

µ0

4π

#

∑p∕=q

3(mqmc · r)r−mq

mc

r3

$− µ0

3

mpmc

V pmc

(15)

JTi j =

'


)

* , (16)

H =−∑i< j

+Si

x,Siy,S

iz,'


)

*

'

(S j

x

S jy

S jz

)

* . (17)

T =−-

∂F

∂θ

.=

/

∑i

Si ×Bi

0(18)

T =−-

∂F

∂φ

.=

/

∑i

Si ×Bi

0(19)


Classical solution: Lorentz cavity field

• Divide the problem into local and macroscopic fields a << rc << rb

• Suggests the local field at an atom is zero, despite global “demagnetising field”

Bloc = 0

Bloc = Bsurface + Bcavity

= +2M/3 - 2M/3 = 0

++

+ + + + + + ++

+

- - - --

-

----

-

- - - -

+++ +

M

Bsurface = +2M/3

Bcavity = -2M/3

a

rc

rb

What about nanoparticles and clusters?

• Small system where Lorentz approximation is not true (a << rc << rb )

• Average field for a sphere of dipoles is zero

• Where did the demagnetising field go?

Field inside a dipole

• Inside the current loop

• Second term comes from treating limiting field at origin over volume 𝜹(x)

• Field at centre of current loop looks like macroscopic field

• BUT averaged over the volume encompassed by the loop

J. D. Jackson, Classical electrodynamics (2nd ed.). New York: Wiley. (1975)


R. F. L. Evans,∗




I. INTRODUCTION


H =−∑i< j


ku (Si · ei)2 −∑

iµiSi ·Bi (1)

Hbq =−∑i< j

Jbq (Si ·S j)2

(2)

∂Si

∂ t=− γi

(1+λ 2i )


Bi = ζi(t)−1

µi

∂H

∂Si(4)



λikBTµiγi

(5)

〈ζ ai (t)〉= 0 (6)


!TeCe

∂Te

∂ t

"=−Ge (Te −Tl)+P(t) (7)

Cl∂Tl

∂ t= Ge (Te −Tl) (8)

P(t) =P0√2π

exp

#−(t − t0)

t2p

$(9)

M(T ) = M0

!1− T

Tc

"β(10)


Bidp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(11)

B =µ0

4π

!3(m · r)r−m

|r|3

"(12)

B =µ0

4π

%3(m · r)r−m

|r|3 +8π3

mδ (x)&

(13)

H =1

4π

%3(m · r)r−m

|r|3 − 4π3

mδ (x)&

(14)

Didp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(15)

'B

pdm

(=

µ0

4π

#

∑p∕=q


$+

2µ0

3

Dintra ·'m

pmc(

V pmc

.

(16)

'B

pdm

(=

µ0

4π

#

∑p∕=q

3(mqmc · r)r−m

qmc

r3

$− µ0

3

mpmc

V pmc

(17)

JTi j =

)

*Jxx Jxy JxzJyx Jyy JyzJzx Jzy Jzz

+

, , (18)

H =−∑i< j

-Si

x,Siy,S

iz.)

*Jxx Jxy JxzJyx Jyy JyzJzx Jzy Jzz

+

,

)

*S j

x

S jy

S jz

+

, . (19)

T =−/

∂F

∂θ

0=

1

∑i

Si ×Bi

2(20)

T =−/

∂F

∂φ

0=

1

∑i

Si ×Bi

2(21)

m

Reality: much more complicated

• Dipole approximation is not terrible

• But large local deviations from the average at atomic sites

• Which field is needed for spin dynamics for an atom?

• A problem for both atomistic and micromagnetic simulations

• In the end its a moot point, only sample symmetry matters since M x H = 0 for M || H

Electronic and magnetic structure of cc Fe-CO alloys 2665

2.4. Spin density

Although we cannot settle the controversy between the localised and the itinerant pictures for metallic magnetism, we nevertheless believe that a study of the spatial distribution of the spin density is quite useful. For this purpose we have chosen FeCo in the CsCl structure and have performed new band-structure calculations by the linearised augmented plane-wave (LAPW) method (Andersen 1975, Koelling and Arbman 1975), where the potentials are taken from the self-consistent ASW calculations. Since in the ASW method the potential is defined for overlapping spheres (according to the atomic-sphere approximation) but the muffin-tin form is needed for the LAPW calculation, the potential inside the smaller atomic spheres in the LAPW can be taken directly from the ASW, but for the region outside the atomic spheres a volume average of the ASW potential is used to give the constant part of the muffin-tin potential required.

Since we used two different methods to calculate the energy bands, a comparison must be made between the two sets of results. It yields good agreement between the energy

Figure 4. Spin density pT(r ) -p l ( r ) of FeCo (CsCI) in the (1 10) plane. The results are from LAPW calculations which are based on the self-consistent ASW potentials. The peak maxima near Fe and CO are slightly above 1 1 electrons A-3; zero in the contour maps is indicated by broken curves; the lowest contour has the value 0.1 electrons A-’ and adjacent contours differ by 0.3 electrons k3; the numbers are in units of 0.1 electrons A-’.

Calculated electron spin density in CoFe alloy

K Schwarz et al 1984 J. Phys. F: Met. Phys. 14 2659

Magnetic units

Magnetism units

• The older Gaussian/cgs units are still common in the literature

• (Some) conversion factors between the different systems

Quantity Symbol Gaussian & cgs emu Conversion factor SI

Magnetic flux density B gauss (G) 10-4 tesla (T)

Magnetic field strength H oersted (Oe) 103/4𝜋 A/m

Magnetization M emu/cc 103 A/m, J/T/m3

Magnetic Moment m emu 10-3 Am2, J/T

Permeability of free space 𝜇0 dimensionless 4𝜋 × 10-7 H/m, T2 J-1 m3

http://www.ieeemagnetics.org/images/stories/magnetic_units.pdf

Old units

• Redefinition of SI system in 2018 now makes the speed of light c and electronic charge e fixed constants.

• Now 𝜇0 is in principle a measurable quantity, defined from the fine structure constant ~ 1/137

• This breaks the previous convention fixing 𝜇0 as 4𝜋 10-7 H/m and thus compatibility between the SI units and old CGS units

1110003 IEEE MAGNETICS LETTERS, Volume 8 (2017)

III. THE FINE STRUCTURE CONSTANT

In fact, the experimental value of µ0 will be based on that of thedimensionless fine structure constant α, the coupling constant of theelectromagnetic force

µ0 = 2hα/ce2 (1)

where h is the newly fixed Planck constant, c is the fixed speed oflight in vacuum, and e is the newly fixed elementary charge (equalto the absolute value of the electron charge). The relative standarduncertainties in µ0, ε0, and α will be identical.

Another quantity of interest in magnetics, the magnetic flux quan-tum φ0 = h/2e, equal to the reciprocal of the Josephson constantK J , will be fixed in the revised SI. However, the Bohr magnetonµB = eh/4πme will depend on the value of the electron rest mass me,calculated from the experimentally determined Rydberg constant R∞

and α : me = 2h R∞/cα2. Because e and h will have no uncertainty,the uncertainty of µB in the revised SI will be an order of magnitudesmaller than at present [Mohr 2016, 2018].

The fine structure constant α, intriguingly almost equal to 1/137, hasmany physical interpretations in atomic physics, high-energy physics,quantum electrodynamics, and cosmology [Kragh 2003]. Because α ismeasurable by many different modalities, it is possible that its acceptedvalue, and therefore that of µ0, will evolve slightly over the years afterthe redefinition of the SI.

In the CODATA compilations under the present SI, h is a parameterthat is adjusted according to an algorithm that forces agreement amongfixed constants µ0 and c and experimental constants α, h, and e in (1)[Mills 2006, Mohr 2016]. In the revised SI, the fixed constants willbe h, c, and e and the experimental constant will be α. One way torepresent that evolution is

!h/e2"

exp = (µ0c/2)fixed · (1/α)exp#present SI

$(2)

(µ0)exp =!2h/ce2"

fixed · (α)exp [revised SI] (3)

where the subscript “exp” denotes “experimental.”If one uses the values of (h, c, e)fixed and (α)exp from the special

CODATA compilation [Mohr 2018] in (3), one obtains

µ0 = 1.256 637 0617 × 10−6 H/m (4)

with some uncertainty in the last decimal digit owing to the uncertaintyin α. This value is identical to 4π × 10−7 H/m to 9 significant figures.The point is, µ0 will be as close to 4π × 10−7 H/m as allowed by theuncertainty in α and the CODATA truncation of h and e.

The practical consequences of µ0 not being a fixed constant willbe nil [Mills 2006]; however, the philosophical implication for re-searchers still using the centimeter-gram-second (CGS) system ofelectromagnetic units (EMU) should provoke introspection. (Gaus-sian units and EMU are the same for magnetic properties.) As notedby Davis [2017], “conversion factors to CGS systems, which presentlymake use of the exact relation {µ0/4π} ≡ 10−7, will no longer bestrictly correct after the revised SI takes effect.” (The curly bracketsmean that one removes the units associated with the quantity within.)

IV. THE REJECTED ALTERNATIVE

Another possibility was under consideration, but rejected, by theWorking Group on the SI (WGSI) of CIPM’s Consultative Committee

for Electricity and Magnetism (CCEM). In addition to fixing thePlanck constant, WGSI could have recommended fixing the Planckcharge q P = (2ε0hc)1/2 = (2h/µ0c)1/2 = e/α1/2 instead of fixing theelementary charge e [Stock 2006]. Fixing q P would have kept µ0

at its familiar value of 4π × 10−7 H/m and made e dependent onmeasurements of α.

In its deliberations, the WGSI was partly influenced by the rationalefor a fixed value of e laid out by Mills et al. [2006]: By the 1990s,the ampere was being realized by the Josephson effect for voltage andthe quantum Hall effect for resistance (both functions of h and e) andOhm’s law, not by the force on currents in parallel wires. A definitionof current in terms of a fixed value of e would bring the practicalquantum electrical standards into exact agreement with the SI [BIPM2016]. WGSI’s decision informed CCEM’s [2007] RecommendationE1 to the CIPM, which was reaffirmed in 2009.

More recently, the Consultative Committee for Thermometry [CCT2017], the Consultative Committee for Mass and Related Quantities[CCM 2017], and the Consultative Committee for Units [CCU 2017]also recommended that the CIPM proceed with the planned redefini-tion of the kilogram, ampere, kelvin, and mole.

The change of µ0 from a fixed constant to an experimentally deter-mined value has implications for units of measure in magnetics.

V. A BRIEF HISTORY

Maxwell used the term “magnetic inductive capacity” for the ratioof flux density B and magnetic field strength H . The term “perme-ability” originated with Thomson [1872]. In Maxwell’s treatment (inCGS units), B and H were fundamentally different quantities [Silsbee1962]. This was codified in 1930 by the International ElectrotechnicalCommission (IEC), which assigned the unit “gauss” to B and theunit “oersted” to H, and specified that their ratio, the permeabilityof vacuum, had a numerical value of unity and physical dimensionsyet to be determined [Kennelly 1931].1 At meetings in 1931–1934,committees of the International Union of Pure and Applied Physics(IUPAP) endorsed the IEC resolutions and, while acknowledging the“practical” units, opined that “the CGS system of units is suitable forthe physicist” [Kennelly 1933].

Although the IEC abandoned the EMU system in 1935 (a decisionaffirmed in 1938) in favor of the Giorgi [1901] MKSX system [As-coli 1905, Giorgi 1905] (with the fourth fundamental unit “X” notassigned to the ampere “A” until 1950 by IEC and 1954 by CGPM)[Petley 1995], CGS has remained popular with generations of physi-cists. Indeed, when the MKSX system was adopted by CGPM [1948],it noted that “the International Union of [Pure and Applied] Physics. . . does not recommend that the CGS system be abandoned by physi-cists.”

In 1960, the 11th CGPM established the name Systeme Internationald’Unites (SI) for the system of MKSA units, the kelvin, and the candela[BIPM 2006]. In 1964, the U.S. National Bureau of Standards madeit a policy to require SI units in its reports [Chisholm 1967]. In 1966,the Institute of Electrical and Electronics Engineers, via its StandardsCoordinating Committee 14 on Quantities and Units, recommended SI

1Despite the IEC 1930 compromise statement on dimensions, it was universally acceptedby physicists that, in the EMU system, B and H have the same physical dimensions andthe permeability of vacuum is, in fact, dimensionless [Birge 1934], just as described byMaxwell [Birge 1935].

IEEE MAGNETICS LETTERS, Volume 8 (2017) 1110003

units for all published work. Notably, a specific recommendation wasthat “the various CGS units of electrical and magnetic quantities areno longer to be used. This includes . . . the gilbert, oersted, gauss, andmaxwell” [Page 1966]. IUPAP [1987] eventually recommended “thatauthors be encouraged to adapt [sic] the SI units for data in physicsjournals,” and more recently IUPAP [2008] endorsed the projectedrevised SI.

VI. RECOMMENDATIONS

In the still popular EMU system, quantities are exactly convertibleto the present SI by factors of 4π and powers of 10. The requirementthat the permeability of vacuum have a value of unity precluded itsactual experimental determination, just as it is a fixed constant inthe present SI. However, the nature of electromagnetic reality will bevery different in the revised SI. Compared to EMU, the permeabilityof vacuum not only will have dimensions (as it does in the presentSI), but its value will also, in principle, be measurable. That is, therelationship between B and H will be ontologically different in therevised SI compared to the EMU system.

Magnetics has been one of the scientific disciplines most resistantto adoption of the SI. With the revised SI, the “peaceful coexistence”of two systems of units [Silsbee 1962] is no longer feasible. Thefollowing recommendations warrant consideration.

1) Scholarly journals that publish articles in magnetics should re-quire use of the SI and disallow EMU such as oersted, gauss,and “emu per cubic centimeter.” Authors who find the expres-sion of magnetic field strength H in units of ampere per meterto be inconvenient could instead refer to µ0 H in units of tesla(or milli-, micro-, nano-, or picotesla). Similarly, magnetizationM could be expressed as µ0 M or as magnetic polarization J inunits of tesla or millitesla.

2) For the benefit of future generations of magneticians, professorsshould use SI in classroom instruction. Commercial instrumentsand magnetometers should be programmed to report measure-ment results in SI.

3) In writing equations, it is adequate to use phrases such as “whereµ0 is the permeability of vacuum” (or “the vacuum magneticpermeability” or “the permeability of free space” or “the mag-netic constant”) without giving a numerical value. This fol-lows typical usage when referring to the speed of light c, theBoltzmann constant k, or the Bohr magneton µB .

ACKNOWLEDGMENT

Richard S. Davis (Bureau International des Poids et Mesures, BIPM) offered manyinsightful comments and helpful suggestions on the manuscript.

REFERENCES

Ascoli M (1905), “On the systems of electric units,” J. Inst. Elect. Eng., vol. 34,pp. 176–180, doi: 10.1049/jiee-1.1905.0012.

BIPM (2006), The International System of Units, 8th ed. Sevres, France: Bureau Int.des Poids et Mesures. [Online]. Available: http://www.bipm.org/utils/common/pdf/si_brochure_8.pdf

BIPM (2016), The International System of Units, draft 9th ed. Sevres, France: Bureau Int.des Poids et Mesures. [Online]. Available: http://www.bipm.org/utils/common/pdf/si-brochure-draft-2016b.pdf

Birge R T (1934), “On electric and magnetic units and dimensions,” Amer. Phys. Teacher,vol. 2, pp. 41–48, doi: 10.1119/1.1992863.

Birge R T (1935), “On the establishment of fundamental and derived units, with specialreference to electric units. Part II,” Amer. Phys. Teacher, vol. 3, pp. 171–179, doi:10.1119/1.1992965.

CCEM (2007), Recommendation E1: Proposed changes to the International System ofUnits (SI), Bureau International des Poids et Mesures, Sevres, France. [Online]. Avail-able: http://www.bipm.org/cc/CCEM/Allowed/25/CCEM2007-44.pdf

CCM (2017), Recommendation G1: For a new definition of the kilo-gram in 2018 , Bureau International des Poids et Mesures, Sevres,France. [Online]. Available: http://www.bipm.org/cc/CCM/Allowed/16/06E_Final_CCM-Recommendation_G1-2017.pdf

CCT (2017), Recommendation T1: For a new definition of the kelvin in 2018 , Bureau Int.des Poids et Mesures, Sevres, France. [Online]. Available: https://www.bipm.org/cc/CCT/Allowed/28/Recommendation-CCT-T1-2017-EN.pdf

CCU (2017), Recommendation U1: On the possible redefinition of thekilogram, ampere, kelvin and mole in 2018 , Bureau Int. des Poidset Mesures, Sevres, France. [Online]. Available: http://www.bipm.org/cc/CCU/Allowed/23/CCU_Final_Recommendation_U1_2017.pdf

CGPM (1948), Resolution 6 of the 9th CGPM. [Online]. Available:http://www.bipm.org/en/CGPM/db/9/6/

CGPM (2011), Resolution 1 of the 24th CGPM on the Possible Future Re-vision of the International System of Units, the SI. [Online]. Available:http://www.bipm.org/en/CGPM/db/24/1/

CGPM (2014), Resolution 1 of the 25th CGPM on the Future Revision of the InternationalSystem of Units, the SI. [Online]. Available: http://www.bipm.org/en/CGPM/db/25/1/

Chisholm L J (1967), Units of Weight and Measure, National Bureauof Standards, Misc. Pub. 286. Washington, DC, USA: GPO. [Online].Available: https://www.govinfo.gov/app/details/GOVPUB-C13-a8ff2e478908a0307a1b097ecfe6a1cc

Davis R S (2017), “Determining the value of the fine-structure constant from a currentbalance: Getting acquainted with some upcoming changes to the SI,” Amer. J. Phys.,vol. 85, pp. 364–368, doi: 10.1119/1.4976701.

Giorgi G (1901), “Unita razionali di elettromagnetismo,” Atti dell’ Associazione Elet-trotecnica Italiana, vol. 5, pp. 402–418.

Giorgi G (1905), “Proposals concerning electrical and physical units,” J. Inst. Elect. Eng.,vol. 34, pp. 181–185, doi: 10.1049/jiee-1.1905.0013.

IUPAP (1987), Resolutions, IUPAP 19th General Assembly, Washington, USA. [Online].Available: http://iupap.org/general-assembly/19th-general-assembly

IUPAP (2008), Resolutions, IUPAP 26th General Assembly, Tsukuba, Japan, Res-olution proposal submitted to the IUPAP Assembly by Commission C2(SUNAMCO). [Online]. Available: http://iupap.org/general-assembly/26th-general-assembly/; http://archive.iupap.org/ga/ga26/file_50095.pdf

Kennelly A E (1931), “Magnetic-circuit units as adopted by the I.E.C.,” Trans. Amer.Inst. Elect. Eng., vol. 50, pp. 737–743, doi: 10.1109/T-AIEE.1931.5055863 and10.1109/T-AIEE.1931.5055864.

Kennelly A E (1933), “Recent actions taken by the symbols, units, and nomenclature(S.U.N.) committee of the International Union of Pure and Applied Physics (I.P.U.) inreference to C.G.S. magnetic units,” Trans. Amer. Inst. Elect. Eng., vol. 52, pp. 647–658, doi: 10.1109/T-AIEE.1933.5056365 and 10.1109/T-AIEE.1933.5056366.

Kragh H (2003), “Magic number: A partial history of the fine-structure constant,” Arch.Hist. Exact Sci., vol. 57, pp. 395–431, doi: 10.1007/s00407-002-0065-7.

Mills I M, Mohr P J, Quinn T J, Taylor B N, Williams E R (2006), “Redefinitionof the kilogram, ampere, kelvin and mole: A proposed approach to implement-ing CIPM recommendation 1 (CI-2005),” Metrologia, vol. 43, pp. 227–246, doi:10.1088/0026-1394/43/3/006.

Mills I M, Mohr P J, Quinn T J, Taylor B N, Williams E R (2011), “Adapting theInternational System of Units to the twenty-first century,” Philosoph. Trans. Roy. Soc.A, vol. 369, pp. 3907–3924, doi: 10.1098/rsta.2011.0180.

Mohr P J, Newell D B, Taylor B N (2016), “CODATA recommended values of thefundamental physical constants: 2014,” Rev. Mod. Phys., vol. 88, 035009, doi:10.1103/RevModPhys.88.035009.

Mohr P J, Newell D B, Taylor B N, Tiesinga E (2018), “Data and analysis for the CODATA2017 special fundamental constants adjustment,” Metrologia, to be published, doi:10.1088/1681-7575/aa99bc.

Newell D B (2014), “A more fundamental International System of Units,” Phys. Today,vol. 67, pp. 35–41, doi: 10.1063/PT.3.2448.

Page C H, Barrow B B, Kreer J G, Mason W, Wintringham W T (1966), “IEEE recom-mended practice for units in published scientific and technical work,” IEEE Spectr.,vol. 3, pp. 169–173, doi: 10.1109/MSPEC.1966.5216747.

Petley B W (1995), “A brief history of the electrical units to 1964,” Metrologia, vol. 31,pp. 481–494, doi: 10.1088/0026-1394/31/6/007.

Silsbee F B (1962), “Systems of electrical units,” J. Res. Nat. Bur. Stand., vol. 66C,pp. 137–183, doi: 10.6028/jres.066C.014.

Stock M, Witt T J (2006), “CPEM 2006 round table discussion ‘Proposed changes to theSI,’” Metrologia, vol. 43, pp. 583–587, doi: 10.1088/0026-1394/43/6/014.

Thomson W (1872), Reprint of Papers on Electrostatics and Mag-netism. London, U.K.: Macmillan, pp. 482–493. [Online]. Available:https://books.google.com/books?id=xGZDAAAAIAAJ&pg=PA482

Ronald B. Goldfarb IEEE Magn. Lett. 8 1110003 (2017)

• B, 𝜇0M and 𝜇0H are all defined in terms of magnetic field (intensity) in teslas (T)

• Started with Superconducting and Permanent magnet communities, probably due to avoidance of odd numerical conversions, dimensions and units

• Now common in the literature, theoretical and experimental

• Best way is to think about everything as current loop ‘sources’ of flux 𝜇0M and 𝜇0H

• This convention leads to oddities in hysteresis - what are the units of M.B, both in Tesla??

A recent trend to using teslas for everything


R. F. L. Evans,∗




I. INTRODUCTION


H =−∑i< j


ku (Si · ei)2 −∑

iµiSi ·Bi (1)

Hbq =−∑i< j

Jbq (Si ·S j)2

(2)

∂Si

∂ t=− γi

(1+λ 2i )


Bi = ζi(t)−1

µi

∂H

∂Si(4)



λikBTµiγi

(5)

〈ζ ai (t)〉= 0 (6)


!TeCe

∂Te

∂ t

"=−Ge (Te −Tl)+P(t) (7)

Cl∂Tl

∂ t= Ge (Te −Tl) (8)

P(t) =P0√2π

exp

#−(t − t0)

t2p

$(9)

M(T ) = M0

!1− T

Tc

"β(10)


Bidp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(11)

B =µ0

4π

!3(m · r)r−m

|r|3

"(12)

Didp =

µ0

4π

#

∑i∕= j

3(S j · r)r−S j

|r|3

$(13)

%B

pdm

&=

µ0

4π

#

∑p∕=q


$+

2µ0

3

Dintra ·%m

pmc&

V pmc

.

(14)

%B

pdm

&=

µ0

4π

#

∑p∕=q

3(mqmc · r)r−m

qmc

r3

$− µ0

3

mpmc

V pmc

(15)

JTi j =

'


)

* , (16)

H =−∑i< j

+Si

x,Siy,S

iz,'


)

*

'

(S j

x

S jy

S jz

)

* . (17)

T =−-

∂F

∂θ

.=

/

∑i

Si ×Bi

0(18)

T =−-

∂F

∂φ

.=

/

∑i

Si ×Bi

0(19)

IV. MAGNETOSTATICS

B = µ0 (M+H) (20)

µ0M (tesla)

2


M =Bmagnetization

µ0(22)

M =Bmagnetization

µ0(23)

Happlied =Bapplied

µ0(24)


• Not immediately obvious that this is useful - a single loop cycle should give units of energy (density)

• BUT - can easily extract the magnetization in sensible dimensions by dividing by 𝜇0

• In this case, a hysteresis cycle Int (M.B) has units of J/m3

• Same is true of Btot (H) loops but with inverted units

Making sense of M-B loops

𝜇0M (T)

𝜇0H (T)

2


M =Bmagnetization

µ0(22)

M =Bmagnetization

µ0(23)

Happlied =Bapplied

µ0(24)


µ0

(T)(T2J−1m3)

(25)


Summary

• Magnetic moments and current loops behave equivalently

• Quantum mechanical origin of magnetic moments not too far from a classical current loop

• Magnetic fields are different inside and outside magnetic media

• Internal magnetic fields in magnets are generally complicated

• Units in magnetism are generally horrible, but always use SI

• Remembering that 𝜇0 has units of T2 J-1 m3 will make you happy

magnetic moments, dipoles and ﬁeldsmagnetism.eu/esm/2018/slides/evans-slides1.pdf · compare the...

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