Topics in Magnetism I. Definitions and Atomic Sources Anne Reilly Department of Physics College of William and Mary

Topics in Magnetism

I. Definitions and Atomic Sources

Anne ReillyDepartment of Physics

College of William and Mary

After reviewing this lecture, you should be familiar with:

1. Definitions of M, H and B2. Magnetic units3. Atomic sources of magnetism4. Paramagnetic and Diamagnetic responses

Material from this lecture is taken from Physics of Magnetism by Chikazumi and Solid State Physics by Ashcroft and Mermin (Chp. 31)

Fundamental Definitions

Magnets have two poles (north and south)Poles exert a force on each other

Definition: magnetic pole m (SI units:Weber, Wb=m2kg/s2A)

NS

NS+m1

-m1+m2

-m2

20

21

4 r

mmF

Magnetic force (N):

0=4 x 10-7 H/m


Electric current in wire exerts a force on a magnetic pole

Definition: magnetic field H (SI units: A/m)

NS

+m-m

mHF Magnetic force (N):

IH

nIH Field from Solenoid: n = # turns/m, I = current


What happens to a magnet in a magnetic field?+mH

-mH

sinmlHMagnetic torque:

H

x

HmlFx

Translational force ONLY if there is non-uniform H (gradient):

N

S

+mH-mH

HNSl

Fundamental Definitions +mH

-mH

H

N

S

l

Definition: magnetic moment M = ml (SI units: Wb m)

(dipole moment)

HM

Magnetic torque:

HU

MMagnetic Energy:


Magnetic materials have a density of magnetic moments

Definition: Magnetization M=NM (SI units: Wb/m2 or Tesla T)

N=moments per unit volume

NSM


To measure magnetization, use induction (vibrating sample magnetometry)

M

H

B

V

dt

dBNAV Induced Voltage

Definition: magnetic flux density B (SI units: Tesla T)

HMB

0


Magnetization in materials is proportional to applied field H

HMB

0

HM

Definition: magnetic susceptibility (SI units: H/m)

HHB

0

Definition: magnetic permeability (SI units: H/m)

Fundamental Definitions:Review

M = magnetization (T)H = magnetic field strength (A/m)B = magnetic flux density (also called field) (T)

M

H (externally applied)

B

HMB

0

Fundamental Definitions:Review

In this case, the units of M are A/m

HMB

00

Note that sometimes magnetic flux density is defined as:

Gaussian System of Units:

HMB

4Oersted (Oe) CGS unit of magnetic field (H). The Oersted is defined to be the field strength in a vacuum at a distance 1 cm from a unit magnetic pole.

Gauss (G) CGS unit of magnetic flux density (B). A field of one Gauss exerts a force on a conductor of 0.1 dyne/A cm.

Electromagnetic Unit (emu)CGS unit of magnetic dipole moment (M) equal to 1.256637 x 10-5 Oe.

emu/cm3 or emu/cc CGS unit of magnetization (M) In SI units, one emu/cm3 can be interpreted either as 1.256637 mT as a unit of excess magnetic induction, or as 1000 A/m as a unit of magnetic dipole moment per unit volume.

Common system prior to 1980’s. Defined by magnetic poles.

Unit Conversion:

Gaussian unit

(cgs-emu)

Conversion

(SI/cgs)

SI unit

B Gauss (G) x 10-4 = T or Wb/m2

H Oersted (Oe) x 103/4 A/m

M emu/cm3 x 103 =

x 4

A/m

mT

Note: In free space (M=0), 1 G = 1 Oe

Source of Magnetic Moment: Moving Electric Charge (Current)

Atomic Magnetism arises from electron angular momentum and spin

ML

I

vLe-

S

ML= orbital magnetic moment = IA=eL/me

r


Atomic Magnetism arises from electron angular momentum and spin

ML

I

vLe-

S

Atomic magnetic moment: SgL BeB

M

Angular momentum vector Spin vector

Gyromagnetic ratio ge~ 2

ML= orbital magnetic moment = IA=eL/me

r

Bohr magneton

m

eB 2


Multi-electron atoms: total magnetic moment determined by total J, L and S

Hund’s rules: electrons fill shells such that1. Largest total S is achieved2. Largest total L is achieved3. J=|L-S| (minimum) in shells less than half full and J=|L+S| (maximum) in shells more than half full.

Example:

m = -2 -1 0 1 2 (lz)

1s

2p

2s

3s

4s

3p

3d

Iron (Fe)

26

n =1

n =4

n =3

n =2Maximum values:L=2+2+1+0-1-2=2S=4/2 = 2J=4

Source of Magnetic Moment: Quantum Derivation(for multi-atom systems)

H

HE

VHM

e

eHMTHM

H

HE

VHM

nn

n

TkEn

TkEn

Bn

Bn

)(1)(

)(),(

)(1)(

/

/

(at T=0)

(at T>0)

where

Magnetization Defined to be:

Source of Magnetic Moment: Quantum Derivation(for multi-atom systems)

2

2

1)(

H

F

V

N

H

M

H

F

VHM

In terms of Helmholtz free energy F:

To calculate magnetic properites, consider Hamiltonian in magnetic field and find energy

Source of Magnetic Moment: Quantum Derivation

Write Hamiltonian for atomic electrons in a Magnetic Field (ignore Vatom)

HrA

SHgrAc

ep

m

i

Beii

i

2

1

)(2

1 2 H

= Vector potential

Consider first term:

22

22

Ac

epAAp

c

eprA

c

ep iiiii

Source of Magnetic Moment: Quantum Derivation

Consider first term: With zHH ˆ

Hyxc

e

Hrc

e

ii

i

222

2

2

2

2

4

Hpr

c

ei

HLc

e

Using these relationships:

ii

i prL

CBACBA

ABBA

)()(

Hamiltonian in a Magnetic Field

SHgyxHmc

eHLp

m Bei

iiBi

i

222

2

2

82

1H

H0

)(8

)(' 2222

2

ii

ieB yxHmc

eHSgL

H

H’

Magnetic field dependent terms considered as perturbation:

nn EEE Magnetic field as a perturbation:

Energy (En is ground state energy)

nn nn

n EE

nnnnE

'

2

'

'H'H'

nn EEE Magnetic field as a perturbation:

Energy (En is ground state energy)

nn nn

n EE

nnnnE

'

2

'

'H'H'

nyxnHmc

e

EE

nSgLHnnSgLnHE

ii

i

nn nn

eB

eBn

)(8

'

2222

2

'

2

'

paramagnetism

diamagnetism

In ground state atoms or ions with closed (filled) shells:

nyxnHmc

e

EE

nSgLHnnSgLnHE

ii

i

nn nn

eB

eBn

)(8

'

2222

2

'

2

'

0000 SLJ 0 0

Larmor Diamagnetism is only response:

006

0012

22

2

20

2

222

2

0

ii

ii

rV

N

mc

e

H

E

V

N

rHmc

eE

3

rzyxlet

Summary of magnetic responses:

BM

BBMparamagnetic(aligns with H)

Diamagnetic(by Lenz’s Law, opposes H)

H

M=magnetization

H <<1, negative

<<1, positive

from http://www.geo.umn.edu

Summary of magnetic responses:

BM

BBMparamagnetic(aligns with H)

diamagnetic(opposes H)

H

H <<1, negative

<<1, positive

BBM

ferromagnetic(even without H!)

>1, positive

Documents

Topics in Magnetism I. Definitions and Atomic Sources Anne Reilly Department of Physics College of William and Mary