phy 712 electrodynamics 10-10:50 am mwf olin 107 (make up lecture 2/17/2014 at 9 am)

PHY 712 Spring 2014 -- Lecture 13 102/17/2014

PHY 712 Electrodynamics10-10:50 AM MWF Olin 107(Make up lecture 2/17/2014 at 9 AM)

Plan for Lecture 13:Continue reading Chapter 5

1. Hyperfine interaction 2. Macroscopic magnetization density M3. H field and its relation to B4. Magnetic boundary values

PHY 712 Spring 2014 -- Lecture 13 3

Interactions between magnetic dipolesSources of magnetic dipoles and other sources of magnetism in an atom:

• Intrinsic magnetic moment of a nucleus• Intrinsic magnetic moment of an electron• Magnetic field due to electron current

Interaction energy between a magnetic dipole m and a magnetic field B:

02/17/2014

Ne

intE m B

In this case: (0)Nint e NE

eJB B

( )eJ r

Nr e

303

ˆ ˆ3 ( ) 8( ) ( )4 3r

N N

NNμ

r μ r μB r μ r

Hyperfine interaction energy:


Hyperfine interaction energy: -- continued

(0)Nint e NE

eJB B

Evaluation of the magnetic field at the nucleus due to the electron current density:The vector potential associated with an electron in a bound state of an atom as described by a quantum mechanical wavefunction can be written:( )

lnlm r

2

302 2

( )ˆ( )

4 | | sinl

e

nlml

e

e m d rm r

J

rz rA rr r

We want to evaluate the magnetic fieldin the vicinity of the nucleus

B A( 0).r


Hyperfine interaction energy: -- continued 2

302 20

0

( )ˆ(0)

4 | | sinl

e e

nlml

e

e m d rm r

J J r

r

rz rB Ar r

2

302 2

0

2

303 2 2

( )ˆ( ) ( )( )4 | | sin

( )ˆ( )( )4 sin

l

l

nlml

e

nlml

e

e m d rm r

e m d rm r r

o 3

r

o

rr r z rB rr r

rr z rB 0

2 22 2

3 30 03 2 2 3

ˆ ˆˆ ˆ ˆ ˆ( ) ( cos ) cos sin cos cos sin sin ).

( ) ( )ˆ sin ˆ( )4 sin 4

l lnlm nlml l

e e

e r em d r m d rm r r m r

2

o

r z r z 1 θ x y

r rzB 0 z

03

1ˆ =4 'l

e

e mm r

z


Hyperfine interaction energy: -- continued

(0)Nint HF e NE H

eJB B

Putting all of the terms together:

30HF 3 3

ˆ ˆ3( )( ) 8 ( ) .4 3 e

eHr m r

N e N e NN e

μ r μ r μ μ L μμ μ r

In this expression the brackets indicate evaluating the expectation value relative to the electronic state.


Macroscopic dipolar effects --Magnetic dipole moment

rJrm 21 3rd

Note that the intrinsic spin of elementary particles is associated with a magnetic dipole moment, but we often do not have a detailed knowledge of J(r).

30

4 rrmrA

Vector potential for magnetic dipole moment


Macroscopic magnetization

i

ii rrmrM 3

Vector potential due to “free” current Jfree(r) and macroscopic magnetization M(r). Note: the designation Jfree(r) implies that this current does not also contribute to the magnetization density.

3

30

''

' '

4 rrrrrM

rrrJ

rA ''rd free


''

'4

''

''

'1'

''

: thatNote

''

' '

4

30

3

330

rrrMrJ

rA

rrrM

rrrM

rrrM

rrrrrM

rrrrrM

rrrJ

rA

''rd

''

''

''rd

free

free

Vector potential contributions from macroscopic magnetization -- continued


rJrMrB

rMrJ

rMrJrr

rArArBA

rrrMrJ

rA

free

free

free

free

''rd

''rd

00

0

330

2

30

''4'4

:0 that case for the that Note

''

'4

Vector potential contributions from macroscopic magnetization -- continued


rJrH

rMrBrH

rJrMrB

free

free

)( )(

:densityflux magnetic theDefine

00

00

Magnetic field contributions

rJrH

rMrBrHrHrB

rJrMrB

free

free

)( )(

field magnetic the)( Definedensityflux magnetic the that Note

00

00


0

)(

)(

:ticsmagnetosta of equations ofSummary

0

0

rBrMrHrB

rJrHrJrB

free

total

For the case that 0 :

( ) 00

free

J r

H rB r

n̂1

2


00)(

: that case For the

rBrH

rJ free

n̂1

2nBnBnHnH

ˆˆˆˆ

:boundaryAt

21

21


Example magnetostatic boundary value problem

ararM

0ˆ

)( 0zrMM0

rMr

rMrHrBrMrHrB

rrHrH

)(

)(0)(

)()( 0)(

2

0

0

H

H

Φ

Φ


Example magnetostatic boundary value problem -- continued

ararM

0ˆ

)( 0zrMM0

'''

41

'1''

''' '

41

''''

41)(

)(

3

3

3

2

rrrM

rrrM

rrrM

rrrMr

rMr

rd

rd

rdΦ

Φ

H

H


ararM

0ˆ

)( 0zrMM0


20

0

2 20

0

330

0 3

For this example:

1( ) 4 ' '4

For : ( )2 6 3

For : ( )3 3

a

H

H

H

MΦ r dr

z r

M za rr a Φ Mz

M a zar a Φ Mz r r

r

r

r

'''

41)( 3

rrrMr

rdΦH


ararM

0ˆ

)( 0zrMM0


53

30

3

30

00

3ˆ3

)( )( 3

)( :For

ˆ3

)( )( 3

)( :For

rz

raMΦ

rzaMΦar

MΦzMΦar

HH

HH

rzrrHr

zrrHr

53

30

0

53

30

00

0

0

3ˆ3

)(

3ˆ3

)( :For

3ˆ2 )(

3ˆ

)( :For

)( )( )(

rz

raM

rz

raMar

MMar

rzrB

rzrH

zrBzrH

rMrHrB


3

30

53

30

00

ˆˆ3

ˆ)ˆ(

3ˆ3

)( :For

ˆˆ3

ˆ)ˆ( 3

ˆ )( :For

aaMa

rz

raMar

MaMar

rzrrH

rzrH

rzrrHzrH

Check boundary values:

31ˆˆ 3

ˆ)ˆ(

3ˆ3

)( :For

ˆˆ 3

2 ˆ)ˆ( 3

ˆ2 )( :For

5

2

3

30

0

53

30

0

00

00

aa

aaMμa

rz

raMar

MaMar

rzrrB

rzrB

rzrrBzrB


Variation; magnetic sphere plus external field B0

arar

0)( 0MrMM0 B0

00

0

00

00

000

000

23

material, ic"paramagnet" isotropican For 32

311

32

:For :ionsuperpositBy

BM

rHrBBrHrB

MBrH

MBrB

ar


0

)(

)(

:ticsmagnetosta of equations ofSummary

0

0

rBrMrHrB

rJrHrJrB

free

total

00)(

: that case For the

rBrH

rJ free

n̂1

2nBnBnHnH

ˆˆˆˆ

:boundaryAt

21

21


Magnetism in materials

)hysteresis(with materials agneticantiferrom tic,ferromagneFor

material cdiamagneti material icparamagnet

:magnetismlinear with materialsFor )(

0

0

0

HB

HB

rMrHrB

f


Example: permalloy, mumetal /0 ~ 104

0 0

a b

B0Spherical shell a < r < b :


Example: permalloy, mumetal /0 ~ 104 -- continued

0 0

ab

B0

)(

00)(

:case For this

rHrBrBrH

continuousˆcontinuousˆ

:boundariesat Continuity

nBnH



0 0

ab

B0

coscos)( For

cos)( For

cos)( 0For

0)( 0

)()( :Let

10

0

1

2

lll

lH

lll

lllH

ll

llH

H

H

Pr

rBbr

Pr

rbra

Prar

r

r

r

rrB

rrH



0 0

ab

B0

21

0

021

1

31

0

031

10

21

11

31

10

1

22 At

2 At

:)contribute terms1(only conditionsboundary Applying

bα

μBb

bγ bβ

bB

bbr

aγaβaδ

aar

l



0 0

ab

B0

0

03

0

0

02

03

00

01

/12/9

/1

1//22/1/2/9

:clearsdust When the

Bba

Bba


Energy associated with magnetic fields

rDrE

rHrB

BmBm

BmFBm

21 :analogy toIn

21 :shown that becan It

-- energies cMacroscopi

:bygiven is field externalan in dipole magnetic a aligning with associatedenergy that implies This

:is field externalan in dipole magnetic aon force the

-- proof without used previously We:Note

3

3

rdW

rdW

U

E

B

phy 712 electrodynamics 10-10:50 am mwf olin 107 (make up lecture 2/17/2014 at 9 am)

Documents

magnetic field contributions

magnetic field b

magnetic fieldin

variation magnetic sphere

r hyperfine interaction

vector potential contributions

continued m0

electronmagnetic field