Magnetic Dipole Moment

1

Nano Materials and Application Lab

Magnetic Dipole Moment

A current loop produces magnetic (dipole) moment defined by

- the direction follows right-screw

Magnetic moment experiences torque in magnetic field (B) to align it with B

Magnetic moment (current loop) produces a magnetic field B around it : the produced B is proportional to m and inversely proportional to distance

nm uIA

A

I

m

un B

A

I

B

m

BrO P

µ m NS

µ m

3/ rB m

2


Atomic Magnetic Moment

An orbiting electron in atom is a current loop and has a magnetic moment

Since the angular momentum L of the electron is

The ratio e/2m = gyromagnetic ratio

(-) sign indicates orb is

in opposite direction of L I

A

–e

orb

L

r

Fig. 8.4: An orbitting electron is equivalent to a magneticdipole moment orb.

2)(

22

2 rerI

e

t

qI

orb

Lm

e

rmmvrL

orb 2

2

3


Atomic Magnetic Moment

Electron has an intrinsic angular momentum S (spin) and the spin has spin magnetic moment

The gyromagnetic ratio is a factor of 2 greater

In magnetic field along z-direction, the spin angular momentum is quantized

The spinning electron experiences torque that causes spin to precess about B

The average magnetic moment z along the field is

is called Bohr magneton (9.27x10-24 Am2 or J/T) = magnetic moment of the spin of a single electron along the field

21 sz mS

z µspin

B

z

S Sz

Sm

espin

m

e

m

eS

m

ezz 22

4


Magnetization

Magnetic field generated by solenoid is given byo is permeability of free space (H/m)

n=number of turns per unit length

Now put a material inside of solenoid, then each atom of the material respond to Bo and develops a magnetic moment m (the material is magnetized)

Define the magnetic vector M (magnetic dipole moment per unit volume) as to describe the extent of magnetization made by each current loop of atom normal to Bo

InIB ooo

I

B

I

M

A

I

I

Bo

atomatomi

mi nVM

/

5


Magnetization

In the cross-section of the medium, the current loop cancel each other except at the surface → surface current

Total magnetic moment is M·(volume)=MAl

Suppose the magnetization current on the surface per unit length is Im, then the total magnetic moment is (current)x(cross-sectional area) =

Therefore

Magnetization of a medium = surface current per unit length

Note that magnetization current is not due to a flow of free carriers, but due to localized electronic currents in atoms of the solid

lAIm

Surf ac e curr ents

Surf ac e c urr ents

mIM

6


Magnetic Field Intensity

The magnetic field within the medium arises from both current I’ in the solenoid and the magnetization current Im

Bo or B-oM is the contribution of the external current to magnetize the material → define magnetizing field H by

The magnetizing field is also called magnetic field intensity (A/m)

H is the total conduction current per unit length (nI)

H is the cause and B is the effect

MBIIB oomo )(

I

I

Im

B

M

Fig. 8.8: The field B in the material inside the solenoid isdue to the conduction current I through the wires and themagnetization current Im on the surface of the magnetized

medium, or B = Bo + o M.

nIBMBH ooo

11

7


Magnetic Permeability/Susceptibility

Magnetic permeability (what extent the material is permeable by the magnetic field?) of a medium is defined by

Relative permeability of a medium is the fractional increase in the magnetic field with respect to the field in free space

Therefore,

The dependence of M on H (similar to the dependence of P on E) is expressed by magnetic susceptibility of the medium

Therefore,

H

B

H

B

B

B

oor

ro

HM m

HHHMHB momooo )1()(

mr 1

8


Magnetic QuantitiesMagnetic quantity Symbol Definition Units Comment

Magnetic field; magnetic induction

B F = qv B T = tesla = webers m-2

Produced by moving charges or currents and acts on moving charges or currents.

Magnetic flux = Bnormal A Wb =

weber is flux through A and Bnormal is normal to A. Total flux through any closed surface is zero.

Magnetic dipole moment

m

m = I A A m2 Experiences a torque in B and a net force in a nonuniform B.

Bohr magneton = e/2me. A m2 or

J T-1 Magnetic moment due to the spin of the electron. = 9.2710-24 A m2

Magnetization vector

M Magnetic moment per unit volume

A m-1 Net magnetic moment in a material per unit volume

Magnetizing field; Magnetic field intensity

H H = B/o – M A m-1 H is due to external conduction currents only and is the cause of B in a material.

Magnetic susceptibility

m

M = m H None Relates the magnetization of a material to the magnetizing field H.

Absolute permeability

o c = [o

o]

-1/2 H m-1 = Wb m-1 A-1

A fundamental constant in magnetism. In free space o = B/H.

Relative permeability

r

r = B / (oH). None

Magnetic permeability

= o r H m-1 Not to be confused with magnetic moment.

Inductance L L = otal / I H (henries) Total flux threaded per unit current.

Magnetostatic energy density

Evol

dEvol = H dB J m-3 dE

vol is the energy required per unit volume in changing B by dB.

9


Magnetism

Type m

(typical values)

m

vs. T Comment and examples

Negative and small (–10 -6)

T independent Atoms of the material have closed shells. Organic materials, e.g. many polymers; covalent solids,

e.g. Si, Ge, diamond; some ionic solids, e.g. alkalihalides, some metals e.g. Cu, Ag, Au.

Diamagnetic

Negative and large ( –1)

Below a critical temperature

Supercond uctors

Positive and small, 10 -5 – 10 -4)

Independent of T Due to the alignment of spins of conduction electrons. Alkali and transition metals.

Paramagnetic

Positive and small (10 -5)

Curie or Curie - Weiss law

m = C / ( T–TC)

Materials in which the constituent atoms have a permanent magnetic mo ment, e.g. gaseous and

liquid oxygen, ferromagnets (Fe), antiferromagnets (Cr) and ferrimagnets (Fe 3 O 4 ) at high temperatures.

Ferromagnetic Positive and very large

Ferromagnetic below and paramagnetic above the Curie temperature

May possess a large permanent magnetization even in the absence of an applied field. Some transition and rare earth metals, Fe, Co, Ni,

Gd, Dy Antiferromagnetic Positive and

small Antiferromagnetic below and paramagnetic above

the Neel temperature

Mainly salts and oxides of trans ition metals e.g. MnO, NiO, MnF 2 , and some transition metals, –Cr, Mn.

Ferrimagnetic Positive and very large

Ferrimagnetic below and paramagnetic above the Curie temperature

May possess a large permanent magnetization even in the absence of an applied field. Ferrites.

10


Ferromagnetism

Fe atom (3d64s2) has 4 unpaired electrons with parallel spins by Hund rule

4 spin magnetic moment → 2.2The electrostatic energy of a paired spin-up and spin-down electrons at the same orbital (m

l) is higher due to Coulomb repulsion than the energy of the electrons with parallel spin (same ms) at different orbitals (different ml) → Exchange interaction (Pauli exclusion principle + electrostatic interaction)

Exchange energy for two atoms with outer electrons at a distance r is

S is spin angular momentum

Je is exchange integral

Antiferro : Je <0, S1S2(↑↓)<0 → Eex <0

Ferro : Je >0, S1S2 (↑↑)>0 → Eex <0 : spin spontaneously align to reduce exchange energy

3d6 4s2

m2

Lower energyHigher energy

m1

Je

0

+

Mn

FeCo

NiGd

Cr

rrd

212 SSJE eex

11


Curie Temperature

Saturation magnetization (Msat) is the max magnetization in ferromagnets when all magnetic moments align

At high temperature, lattice vibration agitates spins and ferromagnetism disappears at Curie temperature Tc

kTc ~ Eex

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Iron

Msat(T)

Msat(0)

T / TC

Fe Co Ni Gd

Crystal structure

BCC

HCP

FCC

HCP

Bohr magnetons per atom 2.22 1.72 0.60 7.1

Msat(0) (MA m-1) 1.75 1.45 0.50 2.0

Bsat = o Msat (T) 2.2 1.82 0.64 2.5

TC 770 C 1043 K

1127 C 1400 K

358 C 631 K

16 C 289 K

12


Magnetic Domain

Fe is not magnetic in general, why? → magnetic domain : a region of crystal in which all the spin magnetic moment is aligned

Magnetostatic energy = potential energy stored in a magnetic field

One domain has external magnetic field

Domain structure can remove external field or reduce magnetostatic energy

Domain wall (Bloch wall) is created

Domain has preferred direction for magnetization : easy direction Closure domain

SN

N

N S

S

NS

90° domain wall

N

S

M

Domain A Domain BBloch wall

Gradual rotationof magnetic moments

z or [001]

–z or [001]

Easy direction

Easy directionA grain with domains

13


Magnetic Anisotropy

Magnetization occurs by domain wall migration to enlarge the volume having specific spin direction

Magnetic anisotropy : difference in magnetic properties with crystallographic direction

Easy direction

Hard direction

Fe[100] easy axis

[111] hard axis

Magnetocrystalline anisotropy energy (K) : the excess energy required to magnetize in particular direction with respect to the easy direction

A B

[100]

HA B

Hard[111]

Medium[110]

Easy[100] OA

C

B

D

0

0.5

1

1.5

20 1 2 3 4

0 0.01 0.02 0.03 0.04 0.05 0.06

[100]

[111]

[110]

Applied magnetic field oH (T)

Magnetizing field H ( 104 A m-1)

Msat

P

Mag

neti z

ati o

n (

106 A

m-1

)

14


Magnetization Process

H H

a bReversible

boundarymotion

Irreversib

boundarymotion

c

H H

dRotation of M Saturation of M

M M

Msat

M

HO

ab

c

d

e

f

Mr

H

e

Hc

x

H

+xBarkhausen effect : jumps in magnetization due to jerkingin domain wall movement induced by defects

Remanent or residual magnetization

Coercivity or coercive field

15


M-H

M-H curve shows hysteresis

M-H can be converted to B-HB=oM+oH

The area of hysteresis is the energy dissipated per unit volume per cycle of the applied field

Soft magnetic material : easy to magnetizeRequires low magnetic field intensity in magnetization and demagnetization

Small hysteresis loop area (low loss) → motor, transformer, inductor

Hard magnetic materialLarge Hc → Magnetic storage

Bsat

B

d

Br

g

H H

B

Bsat

Br

B

H–H

–B

Hard

Soft

HdBE

16


Soft Magnetic Materials

Magnetic Material

oHc

(T)

Bsat

(T)

Br

(T)

ri

rmax

Wh Typical applications

IDEAL SOFT

0 Large 0 Large Large 0 Transformer cores, inductors, electrical machines, electromagnet cores, relays, magnetic recording heads etc.

Iron (commercial grade, 0.2% impurities)

<10-4 2.2 <0.1 150 104 250 Large eddy current losses. Generally not preferred in electrical machinery except in some specific applications (e.g., some electromagnets and relays)

Silicon iron (Fe: 2-4%Si)

<10-4 2.0 0.5-1 103 104 – 4105

30-100 Higher resistivity and hence lower eddy current losses. Wide range of electrical machinery (e.g., transformers etc.)

Supermalloy (79Ni-15Fe-5Mo-0.5Mn)

210-7 0.7-0.8 <0.1 105 106 <0.5 High permeability, low loss electrical devices, e.g. specialty transformers, magnetic amplifiers

78 Permalloy (78.5%Ni-21.5%Fe)

510-6 0.86 <0.1 8103 105 <0.1 Low-loss electrical devices, audio transformers, HF transformers, recording heads, filters etc.

Glassy metals Fe-Si-B

210-6 1.6 <10-6 105 20 Low-loss transformer cores

Ferrites Mn-Zn ferrite

10-5 0.4 <0.01 2103 5103 <0.01 HF low-loss applications. Low conductivity ensures negligible eddy current losses. HF transformers, inductors (pot cores, E and U cores etc.), recording heads.

17


Hard Magnetic Materials

Magnetic Material oHc

(T)

Br

(T)

(BH)max

(kJ m-3) Examples and Uses

IDEAL HARD Large Large Large Permanent magnets in various applications

Alnico (Fe-Al-Ni-Co-Cu)

0.19 0.9 50 Wide range of permanent magnet applications

Alnico (Columnar)) 0.075 1.35 60

Strontium ferrite (anisotropic)

0.3-0.4 0.36-0.43 24-34 dc motors, starter motors, loudspeakers, telephone receivers, various toys

Rare earth cobalt e.g. Sm2Co17 (sintered).

0.62-1.1 1.1 150-240 Servo motors, stepper motors, couplings, clutches, quality audio headphones etc.

NdFeB magnets

0.9-1.0 1.0-1.2 200-275 Wide range of applications, small motors (e.g. in hand tools), walkman equipment, CD motors, MRI body scanners, computer applications

Hard particles -Fe2O3

0.03 0.2 Audio and video tapes, floppy disks

18


Superconductivity

Superconductivity : no resistance to current flow below a critical temperature Tc

High Tc superconductorLa-Ba-Cu-O, 35K

Y-Ba-Cu-O, 95K

Hg-Ba-Ca-Cu-O, 130K

Meissner effect : perfect diamagnetic behavior of the superconductors or expel of magnetic field from the bulk

Develop M opposing to B (perfect diamagnetism, m= -1)

In perfect conductor, no field rejection occurs

For B =0, it disappears in SC, but the decreasing field induce current in the perfect conductor

T em p era tu re , T

S u p erc o n d u c to r (e .g . P b )

Tc

N o rm a l m eta l(e .g . A g )

r e s id u a l

00

Res

isti

vity

B

I

T > Tc T < Tc

Perfect conductor

Superconductor

B off

T < Tc

B

I

Documents

Magnetic Dipole Moment