Chapt. 4 Magnetic properties of materials Prof. Kee-Joe Lim [email protected]@chungbuk.ac.kr, 261-2424 School of Electrical and Computer Engineering.

Chapt. 4 Magnetic properties of materials

Chapt. 4 Magnetic properties of materials

Prof. Kee-Joe [email protected], 261-2424

School of Electrical and Computer EngineeringChungbuk National University

http://imt.cbucc.net2006/3/1

2

Scope of this chapter

• some fundamental concepts concerning magnetic fields

• essence of the atomic theory of magnetic dipoles

• atomic interpretation of dia-, para-, ferro-, antiferro- and ferri

-magnetism

• application of magnetic materials

3

Summary of concepts pertaining to magnetic fields

• Definition of magnetic flux density ; B defined in terms of the force exerted by a magnetic field on a current-carryi

ng wire

magnetic fields are produced by electric currents [law of Biot-Savart]

dld BIF ][1][1][12

Tm

wb

Am

N

rIB 2

0

4 r

dld r

]/[104 70 mH

c00

1

r ; relative permeability – depended on materials

This quantity is the parameter which can be interpreted in terms of the atomic properties of the medium

; No physical significance

4

• magnetic field ; H

Ampere’s law; the line integral of H around a single closed path is equal to the current enclosed

IdlH

• relation between B and H

HB r0

Linear and isotropic medium only

-Nonlinear ; ferromagnetic materials

- anisotropic ; single crystal- tensor expression

5

4.2 Magnetic dipole moment of a current loop• difference between electricity and magnetism

• 정자기학• magnetic dipole = motion of electric charges

• Relation between a current loop and magnetic dipole

P

Q

R

S

BF

In

F

x

y

z

Current loop

IBRQIBPSF )()(

cos)()(cos)( PQIBPSPQFT )90sin(cos IBAIBA

BμBnT m IAnμm IA

The results can be apply for a current loop of any shape

6

Magnetization from a macroscopic viewpoint

HdA

dl

LNIH / LNIB r /0

HBB roi 0How can we achieve a flux density inside the cavity that remains the same as it was when the material was present ?

HH ri 00 HHH ri )1(

HdldAdIdA rm )1(

HHM )1( r HM )1(00 r)(0 MHB

IdI

Cavity 내부 자계를 만큼 증가시키기 위하여는 솔레노이드 전류와 동일 방향으로 의 전류를 소코일에 흘리면 된다 .

Hr )1( Hdlr )1(

- 자계가 인가된 자성체는 단위체적당 M 의 자기쌍극자 모멘트를 가지고 있다 .

( 거시적 특성량과 원자론적 의미 관계 )

7

Magnetization from a macroscopic viewpoint

• correspondence

EDP

H

MB

EDP

HB

Mor ?

• comparison between magnetism and electricity

q m Ilqv δq δm nIA

EμT e HμT m BμT m HM HJ 0EP r )1(0

PED 0 JHB 0 )(0 MHB

………… ………… …………

8

Orbital magnetic dipole moment and angular momentum in circular Bohr orbit model

+e

-e

R 2/eefi

22

2

12/ ReeRiAm

vRM ma 2RmM a

am m

eMμ

2

][1027.9422

224 mAm

ehh

m

e

sec][1062.6 34 Jh

1 Bohr magneton

( Ma has same dimension as h )

9

Orbital magnetic dipole moment and angular momentum in spherical charge cloud model

-e

+eR

h

)(2 22 Rh

hdqfhdqdi 2 3)3/4( R

eq

dqhdid m32

23

0 5

1eRdqh

R

m

]2)3/4(

[3

hdR

mdmvRdM a

am dMm

ed

2

결과식은 두 모델에서 동일하며 , 에도 무관함Hold for any volume element of charge distribution

10

23

0 5

1eRdqh

R

m

[ 보충자료 ]

운산과정

dRqdqh

RR

m3

0

2

1223

0)(2

tR 2

122 )(

222 tR dtdt 22 dtt

d

02

1222

32232

1

0

22 )()(2)(2Rt

RtdttRtRtdR

5

0

532

0

422

15

4)

53(2)(2 R

ttRdtttR

R

t

R

t

515

4

)3/4(

25

3

eRR

R

em

11

4.5 Lenz’s law and induced dipole moments

dt

dE

0

t

)(0 R

Lt

t

t

i

i

0R

0R

0t

0t

edt

dEdl

dt

deRi

dt

diL

)1(t

L

R

eR

ei

dt

d

dt

diL

Li /

The current remains constant for t > t0. Thus, a permanent change has been accomplished; the current can be made equal to zero only by reducing the flux to zero.

12

Induced dipole moment in circular Bohr orbit model(1)

B

R

E

-e

+e

v0

F = -eE

02

2

1 eRm

dt

dEdl

dt

dBR

dt

d

RE

22

1

dt

dBeReEF

2

dt

dvm

dt

mvdF

)(

mRddtdt

dBeR

2dB

m

ed

2

LBm

e 00 2)(0

22

02

42

1indmmm BR

m

eeR BR

m

eindm

22

)( 4

L : Larmor angular frequency

Induced dipole moment has a direction opposite to the applied magnetic flux density. This results is independent of the initial direction of rotation. It will also keep its new angular frequency as long as B remains constant.

B 인가시도 R 은 일정으로 가정

13

Induced dipole moment in circular Bohr orbit model(2)

B

R+e

-e

v

R

mv2

evBR

e

20

2

4

evBR

e

R

mv

20

22

4

m

eB

m

eB

mR

e 2

030

22

4

2220 )

2(

m

eB

m

eB

LBm

e 00 2

14

Induced dipole moment in homogeneous spherical charge distribution model

B

B+e

R

E

dF

0

)(indm0m

02

0 5

1 eRm

dt

dB

dt

dE

22

1

dmFdt

dEdtm

e

dBm

ed

2 0022

LBm

ecB

m

e

BRm

eindm

22

)( 10

Induced dipole moment is independent of the initial angular frequency 0 of the charge distribution. Hence, a magnetic dipole moment will be induced in the atomic model, independent of whether the model has a “permanent” magnetic dipole moment or not.

15

4.6 classification of magnetic materials• 분류 기준 : 영구자기쌍극자 모멘트 유무 , 쌍극자모멘트간 상호작용

classification Permanent dipole interaction

diamagnetic No -

paramagnetic Yes Negligible

ferromagnetic Yes Parallel orientation

antiferromagnetic Yes Antiparallel orientation of equal moments

ferrimagnetic yes Antiparallel orientation of unequal moments

fr TTc ),/(1 NTTTc ),/(

Tc /Curie law

Curie-Weiss law

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4.7 Diamagnetism( 반자성 )

HHM )1( r

This expression is valid for diamagnetic and paramagnetic materials at all temperatures, but for the other classes only above a certain temperature.

Table 4.2 the susceptibility of some diamagnetic materials

1101 5 r

As long as the electronic structure of the material is independent of temperature, the magnetic susceptibility is also essentially independent of temperature.Comparing with experimental value and theoretical value (by Lenz’s law)in solid wi

th an atom contains 10 electrons; eq. (4.57)

HRm

eBR

m

erindm 0

22

22

)( HHRm

eNNM rindm 0

22

)(

572031

21928 1010410

101.9

)106.1(105

Superconductor is a perfect diamagnetic material; susceptibility =-1

17

Origin of permanent magnetic dipoles in matter

Whenever a charged particle has an angular momentum, the particle will contribute to the permanent dipole moment.

- There are three contributions to the angular momentum of a atom

(i) orbital angular momentum of electrons

(ii) electron spin angular momentum

(iii) nuclear spin angular momentum

- Orbital angular momentum of electrons

am Mm

e

2

-Orbital (angular) momentum quantum number l determines the orbital angular momentum which is measured in units of h/2

-Magnetic quantum number ml determines the component of angular momentum along an external field direction

H

1lm

1lm

0lm

)1(2

llh

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• a completely filled shell• l=1 ml =1(h/2h

orbital dipole moment -eh/4m, 0, +eh/4m

a completely filled electronic shell contributes nothing to the orbital permanent dipole moment of a atom.

• Incomplete outer shell no contribution because of “frozen in”

• Transition elements (incomplete inner shell)

•Iron group(Z=21 ~ 28; 3d)

•(Z=39~45; 4d)

•Rare earth group(Z=58~71; 4f)

•(Z=89~92; 6d)

•In case of the elements of the rare earths group, the permanent orbital dipole moments do contribute to the magnetic susceptibility, but contribution from orbital magnetic dipoles will be neglected.

1 Bohr magneton( 보아磁子 ) = eh/4 m =9.27 x 10-24 [Am2]

19

Electron spin magnetic dipole moment

• spin angular momentum along a given direction is either +1/2(=+h/4or -1/2(=-h

)()( spinaspinm Mm

e

S=1, angular momentum=h/4 dipole moment=-eh/4 m = - 1Bohr magneton

• complete electronic shell

• incomplete outer shell

• incomplete inner shell (transition element; iron group)

•Hund’s rule

•Iron group ; 1s2, 2s2, 2p6, 3s2, 3p6, 3d0~10, 4s2

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Nuclear magnetic moments

• angular momentum associated with the nuclear spin is measured in units h/2 , and is of the same order of magnitude as electron spin and orbital angular momentum of the electrons.

•Mass of the nucleus is larger than that of an electron by a factor of the order of 103.

•The magnetic dipole moment associated with nuclear spin is of the order of 10 -

3 Bohr magnetons.

•Since nuclear magnetic dipole moments are small compared to those associated with electrons, its contribution may be neglected

21

Paramagnetic spin system

가정 ;

자화에는 전자 spin 만 기여 , 원자당 dipole moment = 1 Bohr magneton( =eh/4m)

체적당 원자수 N( 개 /m3) 자계와 평향한 원자수 Np, 반평행 Na0H ap NN NNN aP 0H HHNNM rap )1()(

HμBμT mm 0

cos)( 090 0 HdW mm

Hμ

]/)(exp[)( kTWAN HWW pa 02

)/2exp(]/)exp[(/ 0 kTHkTWWNN appa

)/exp()/exp(

)/exp(

)/2exp(1 00

0

0 kTHkTH

kTHN

kTH

NNa

)/exp()/exp(

)/exp(

)/2exp(1 00

0

0 kTHkTH

kTHN

kTH

NN p

22

)/tanh( 0 kTHNNNM ap

xxx )tanh(,1

kTHNM /20

1)tanh(,1 xxNM

10 kT

H

13001038.1

11027.923

240

kT

H

TCkTNr //1 20

TTkTN /3.01038.1/)1027.9(104105/ 2322472820

( i )

( ii )

(for )

( At room temperature )

Curie law

Saturation, see fig. 4.18

Table 4.4 susceptibility of some paramagnetic materials

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• Diamagnetic contribution 도 있으나 미미함 (-10-5, 10-3)• 대표적인 응용 분야

– To obtain very low temperature(<1oK] by adiabatic demagnetization

– MASER ( microwave amplification through stimulated emission by radiation )

s

entropy

T, temperature

0H0H