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
………… ………… …………
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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 )
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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
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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.
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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
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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
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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.
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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
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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]
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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
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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