Magnetism in Matter

Electric polarisation (P) - electric dipole moment per unit vol.Magnetic ‘polarisation’ (M) - magnetic dipole moment per unit vol.M magnetisation Am-1 c.f. P polarisation Cm-2

Element magnetic dipole moment mWhen all moments have same magnitude & direction M=NmN number density of magnetic moments

Dielectric polarisation described in terms of surface (uniform) or volume (non-uniform) bound charge densities

By analogy, expect description in terms of surface (uniform) or volume (non-uniform) magnetisation current densities

Magnetism in MatterElectric polarisation P(r) Magnetisation M(r)

p electric dipole moment of m magnetic dipole moment oflocalised charge distribution localised current distribution

rrrp

rPrj

n.rjn.rP

)d(

t)()(

dt)()(

allspace

pol

0pol

ˆˆ

space all

)(x21

)(x )(

)(x21)(

dr rj rm

rMrj

rj r rM

M

Magnetic moment and angular momentum• Magnetic moment of a group of electrons m• Charge –e mass me

momentum angular total 2m

e-2m

e- momentum angularxm

xq21

d)(xq21

)(q)(

i

iei

ie

iiei

i

iii

i space alliii

iiii

LL Lm

v r

v r m

r rrv r m

rrvrj

Ov1

r1

v4

v3 v2

v5r5

r4

r3

r2

Force and torque on magnetic moment

)(x d )(x)(x Torque

)(-U c.f. )(.-U U

)(.-U suggests )(

...d )(B.x )(0d ...)(B.)(Bx)(

expansion Taylor ...)(B.)(B)(B

)( )()( d )(x)(

ondistributi current Continuous d )(x)( )(

charges point on force Lorentz x q

space all

pm

m

space allk

space allkk

kkk

space all

space all

iii

0B mrrB rj rT

0p.E0Bm F

0Bm0m.B F

r0r rjr0r0 rj F0r0r

rvrrjrrB rj

rrB rvr F BvF

Diamagnetic susceptibilityInduced magnetic dipole moment when B field appliedApplied field causes small change in electron orbit, inducing L,m

Consider force balance equation when B = 0(mass) x (accel) = (electric force)

21

3eo

2

o2o

22oe am4

Zeωa4

Zeam

aBee Bv

-eB

Loe

o

3o

e2

e3

eo

2

2o

22

e

2meB

aZmB

2meB

am4Ze

inquadratic aBea4

Zeam

21

L is the Larmor frequency

Diamagnetic susceptibilityPair of electrons in a pz orbital

w = o + L

|ℓ| = +meLa2 m = -e/2me ℓ

w = o - L

|ℓ| = -meLa2 m = -e/2me ℓ

a v-e

m

-e v x B

v-e

m

-e v x B

B

Electron pair acquires a net angular momentum/magnetic moment

Diamagnetic susceptibilityIncrease in ang freq increase in ang mom (ℓ)Increase in magnetic dipole moment:

Include all Z electrons to get effective total induced magneticdipole moment with sense opposite to that of B

Bme

22

e

222

ee

e

2Le

e

2maeB

2maea

2meB2m

2mem

a2m2m

em

-eB

m

electron one for momentmagnetic spin''Intrinsic 1 Am9.274.10 1 c.f. 1T B 12Z for 10~

orbit electron of radiussquaremean:aaZ2me

B

224-B

27-

2o

2o

e

2

Bm

ParamagnetismFound in atoms, molecules with unpaired electron spinsExamples O2, haemoglobin (Fe ion)

Paramagnetic substances become weakly magnetised in an applied field

Energy of magnetic moment in B field Um = -m.B

Um = -9.27.10-24 J for a moment of 1 B aligned in a field of 1 TUthermal = kT = 4.14.10-21 J at 300K >> Um

Um/kT=2.24.10-3

Boltzmann factors e-Um/kT for moment parallel/anti-parallel to B differ little at room temperatureThis implies little net magnetisation at room temperature

Ferro, Ferri, Anti-ferromagnetismFound in solids with magnetic ions (with unpaired electron spins)Examples Fe, Fe3O4 (magnetite), La2CuO4

When interactions H = -J mi.mj between magnetic ions are (J) >= kTThermal energy required to flip moment is Nm.B >> m.BN is number of ions in a cluster to be flipped and Um/kT > 1

Ferromagnet has J > 0 (moments align parallel)Anti-ferromagnet has J < 0 (moments align anti-parallel)Ferrimagnet has J < 0 but moments of different sizes giving net magnetisation

Uniform magnetisation Electric polarisation Magnetisation

)(Amm

A.mV

CmmC.m

V1-

3

2i

i2-

3i

i

mM

pP )(

I

z

yx

xyΔxyΔM

I

zzI

Magnetisation is a current per unit length

For uniform magnetisation, all current localised on surface of magnetised body(c.f. induced charge in uniform polarisation)

Surface Magnetisation Current Density

Symbol: aM a vector current densityUnits: A m-1

Consider a cylinder of radius r and uniform magnetisation Mwhere M is parallel to cylinder axis

Since M arises from individual m,(which in turn arise in current loops) draw these loops on the end face

Current loops cancel in interior,leaving only net (macroscopic) surface current

Mm

Surface Magnetisation Current Density

magnitude aM = M but for a vector must also determine its direction

aM is perpendicular to both M and the surface normal

Normally, current density is “current per unit area” in this case it is “current per unit length”, length along the Cylinder - analogous to current in a solenoid.

SP nM dd c.f. polM .ˆ a

M n̂

aM

Surface Magnetisation Current DensitySolenoid in vacuum

With magnetic core (red), Ampere’s Law integration contour encloses two types of current, “conduction current” in the coils and “magnetisation current” on the surface of the core

> 1: aM and I in same direction (paramagnetic) < 1: aM and I in opposite directions (diamagnetic)

is the relative permeability, c.f. the relative permittivity

Substitute for aM

INB ovac

MNB o I

vacMo

Moenclo

BNB

LNLBLB.d

a

a

I

IIB

IL

MagnetisationMacroscopic electric field EMac= EApplied + EDep = E - P/o

Macroscopic magnetic field BMac= BApplied + BMagnetisation

BMagnetisation is the contribution to BMac from the magnetisation

BMac= BApplied + BMagnetisation = B + oM

Define magnetic susceptibility via M = cBBMac/o

BMac= B + cBBMac EMac= E - P/o = E - EMac

BMac(1-cB) = B EMac(1+c) = E

Diamagnets BMagnetisation opposes BApplied cB < 0Para, Ferromagnets BMagnetisation enhances BApplied cB > 0

cB Au -3.6.10-5 0.99996Quartz -6.2.10-5 0.99994O2 STP +1.9.10-6 1.000002

MagnetisationRewrite BMac= B + oM as

BMac - oM = B

LHS contains only fields inside matter, RHS fields outside

Magnetic field intensity, H = BMac/o - M = B/o

= BMac/o - cBBMac/o

= BMac (1- cB) /o

= BMac/o c.f. D = oEMac + P = o EMac

= 1/(1- cB) = 1 + c

Relative permeability Relative permittivity

Non-uniform MagnetisationRectangular slab of material with M directed along y-axisM increases in magnitude along x-axis

Individual loop currents increase from left to right There is a net current along the z-axis Magnetisation current density

z

x

My

zMj

I1 I2 I3

I1-I2 I2-I3

Non-uniform Magnetisation

Consider 3 identical element boxes, centres separated by dx

If the circulating current on the central box is

Then on the left and right boxes, respectively, it is

dyMy

dx dx

dy dxx

MManddydx

xM

M yy

yy

Non-uniform Magnetisation

Magnetisation current is the difference in neighbouring circulating currents, where the half takes care of the fact thateach box is used twice! This simplifies to

dyMdxx

MMdx

xM

MM21

yy

yy

yy

xM

jdxdyjdxdyx

Mdydx

xM

221 y

MMyy

zz

Non-uniform MagnetisationRectangular slab of material with M directed along x-axisM increases in magnitude along y-axis

z

x

My

I1 I2 I3

I1-I2 I2-I3z

y

-Mx

xx

Mj yMz

yMj x

Mz

yM

xM

j xyMz

Total magnetisation current || z

Similar analysis for x, y components yields MMj

Types of Current j

Polarisation current density from oscillation of charges as electric dipolesMagnetisation current density from space/time variation of magnetic dipoles

PMf jjjj

tPjP

tooo EjB

M = sin(ay) k

k

ij

jM = curl M = a cos(ay) i

Total current

MjM x

Magnetic Field Intensity HRecall Ampere’s Law

Recognise two types of current, free and bound

jBB oenclo or.d I

f

oo

oo

ffo

foMfoo

Magnetic Electric

orwhere

jH.D

MBH PED

MHBMBH

jHjMBMjjjjB

f

Magnetic Field Intensity H

D/t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor

In vacuum D = oE and displacement current exists throughout space

tt

tt

t1

t

ff

f

PMf

DjHPEjMB

EPMj

EjjjB

EjB

oo

o

oo

ooo

Boundary conditions on B, H

21

2211

BB0S cosBS cosB

0.d0.

S

SBB

1

2

B1

B22

1

S

||2||1

freeencl2211

freeencl

HH

0L sinHL sinH

.d

I

I

H

For LIH magnetic media B = oH(diamagnets, paramagnets, not ferromagnets for which B = B(H))

222

A

B22

111

B

A11

sin H .d

sin H- .d

H

H

1

2 H2

H1

2

1dℓ1

dℓ2

C ABI enclfree

Boundary conditions on B, H

||||

2

1

2

1

21

21

21

21

r

r

2

1

r

r

2

1

22or

22

11or

11

22or11or

2211

2211

tantanc.f.

tantan

cosHsinH

cosHsinH

cosHcosH

cosBcosBBBsinHsinH

HH