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 of magnetisation is 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 bulk (non-uniform) bound charge densities

Magnetisation described in terms of surface (uniform) or bulk (non-uniform) magnetisation current densities

Magnetism in MatterParamagnetism

Found in atoms, molecules with unpaired electron spins (magnetic moments)Examples O2, haemoglobin (Fe ion)

Paramagnetic substances become weakly magnetised in an applied fieldMagnetic moments align parallel to applied magnetic field to lower energy

Paramagnetic susceptibility is therefore positive

Moments fluctuate because system is at finite temperature

Energy of magnetic moment in B field Um = -m.BUm = -9.27.10-24 J for a moment of 1 mB aligned in a field of 1 TUthermal = kT = 4.14.10-21 J at 300K >> Um Um/kT=2.24.10-3

This implies little net magnetisation at room temperature

Magnetism in MatterDiamagnetism

Found in atoms, molecules, solids with paired electron spinsExamples H2O, N2

Induced electric currents shield interior of a body from applied magnetic fieldMagnetic field of induced current opposes the applied field (Lenz’s Law)

Diamagnetic susceptibilty is therefore negative

Generally small except for type I superconductor where interior is completely shielded from magnetic fields by surface currents in superconducting state

Strong, non-uniform magnetic fields can be used to levitate bodies via diamagnetism

Magnetism in Matter

Ferromagnetism, Ferrimagnetism, Antiferromagnetism

Found 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

Magnetic susceptibilities non-linear because of domain formation

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

p electric dipole moment of m magnetic dipole moment of

localised charge distribution localised current distribution

rrrp

rPrj

n.rjn.rP

)d(

t

)()(

dt)()(

allspace

pol

0

pol

ˆˆ

space all

)(x2

1

)(x )(

)(x2

1)(

dr rj rm

rMrj

rj r rM

M

Magnetisation

Electric polarisation Magnetisation

)(Amm

A.m

VCm

m

C.m

V1-

3

2i

i2-

3i

i

mM

pP )(

I

z

yx

xyΔx

yΔM

I

z

zI

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)

Magnetisation

Uniform magnetisation and surface current density

Symbol: aM current density (vector )Units: 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

M

m

Magnetisation

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

aM is perpendicular to both M and the surface normal n

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.

nP nM . polM c.f.

M n

aM

MagnetisationNon-uniform magnetisation and bulk current density

Rectangular 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-direction Magnetisation current density

z

x

My

zMj

I1 I2 I3

I1-I2 I2-I3

Magnetisation

Consider 3 identical element boxes, centres separated by dx

If the circulating current on the central box is , on the left and rightboxes, respectively, it is

dyMy

dx dx

dy dxx

MManddydx

x

MM y

yy

y

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

x

MMM2

1y

yy

yyy

x

Mjdxdyjdxdy

x

Mdydx

x

M22

1 yMM

yy

zz

Magnetisation

Rectangular 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

y

Mj xMz

y

M

x

Mj xyMz

Total magnetisation current || z

Similar analysis for x, y components yields MMj

Magnetic SusceptibilitySolenoid 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. e the relative permittivity

Substitute for aM

INB ovac

MNB o I

vacMo

Moenclo

BNB

LNLBLB.d

I

IIB

IL

Magnetic SusceptibilityMacroscopic 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 + moM

Define magnetic susceptibility via M = cBBMac/mo

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

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

Magnetic SusceptibilityMagnetic 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

xq2

1

d)(xq2

1

)(q)(

i

iei

ie

iiei

i

iii

i space all

iii

iiii

LL Lm

v r

v r m

r rrv r m

rrvrj

Ov1

r1

v4

v3v2

v5r5

r4

r3

r2

Magnetic SusceptibilityDiamagnetic susceptibility

Induced 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

Zeω

a4

Zeam

aBee Bv

-eB

Loe

o

3o

e2

e3

eo

2

2o

22

e

2m

eB

a

ZmB

2m

eB

am4

Ze

inquadratic aBea4

Zeam

21

wL is the Larmor frequency

Magnetic SusceptibilityPair of electrons in a pz orbital

w = wo + wL

|ℓ| = +mewLa2 m = -e/2me ℓ

w = wo - wL

|ℓ| = -mewLa2 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

Magnetic 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

2m

aeB

2m

aea

2m

eB2m

2m

em

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:aaZ2m

e

B

224-B

27-

2o

2o

e

2

Bm

Magnetic FieldRewrite BMac= B + moM as

BMac - moM = B

LHS contains only fields inside matter, RHS fields outside

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

= BMac/mo - cBBMac/mo

= BMac (1- cB) /mo

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

The two constitutive relations

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

Relative permeability Relative permittivity

Boundary conditions on B, H

21

2211

BB

0S cosBS cosB

0.d0.

S

SBB

1

2

B1

B2q2

q1

S

||2||1

freeencl2211

freeencl

HH

0L sinHL sinH

.d

I

I

H

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

222

A

B

22

111

B

A

11

sin H .d

sin H- .d

H

H

1

2 H2

H1

q2

q1dℓ1

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

tan

tanc.f.

tan

tan

cosH

sinH

cosH

sinH

cosHcosH

cosBcosBBB

sinHsinH

HH