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Chapt. 4 Magnetic properties of
materials
Prof. Kee-Joe Lim
[email protected], 261-2424
School of Electrical and Computer Engineering
Chungbuk National University
http://imt.cbucc.net
2006/3/1
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
2
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 currentcarrying wire
dF  I  Bdl
N
wb
1[
]  1[ 2 ]  1[T ]
Am
m
magnetic fields are produced by electric currents [law of Biot-Savart]
0  r dl
dB 
Ir
2
4r
0  4 10 7 [ H / m]
1
 0 0
c
; No physical significance
r ; relative permeability – depended on materials
This quantity is the parameter which can be interpreted in terms of
the atomic properties of the medium
3
• magnetic field ; H
Ampere’s law; the line integral of H around a single closed path is equal to the
current enclosed
 Hdl  I
• relation between B and H
B  0  r H
Linear and isotropic medium only
-Nonlinear ; ferromagnetic materials
- anisotropic ; single crystal- tensor expression
4
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
z
F  ( PS ) IB  ( RQ ) IB
F
P
T  F ( PQ) cos q  ( PS ) IB ( PQ) cos q
 IBA cos q  IBA sin( 90  q )
T  IAn  B  μm  B
S
n
90oq
x
I
Q
q
R
B
F
Current loop
μm  IAn
The results can be apply for a current loop of any shape
5
y
Magnetization from a macroscopic viewpoint
I
dI
H  NI / L
dA
H
B  0 r NI / L
dl
How can we achieve a flux density inside the cavity that remains the same as it
was when the material was present ? Bi  Bo  0 r H
0 H i  0  r H
H i  H  (r  1) H
Cavity내부 자계를 (r 1) H만큼 증가시키기 위하여는 솔레노이드 전류
와 동일 방향으로 (r 1) Hdl 의 전류를 소코일에 흘리면 된다.
m  dIdA  (r  1) HdldA
-자계가 인가된 자성체는 단위체적당 M의 자기쌍극자 모멘트를 가지고 있다.
M  (r 1)H  H
0M  0 (r  1)H
B  0 (H  M)
(거시적 특성량과 원자론적 의미 관계)
6
Magnetization from a macroscopic viewpoint
• correspondence
H
B
M
E
D
P
or
E
D
P
B
H
M
?
• comparison between magnetism and electricity
q
m
qv  Il
qδ
mδ
IAn
D  0E  P
B  0 H  J
P   0 ( r  1) E
J  0 H
T  μe  E
…………
T  μm  H
…………
7
B  0 ( H  M )
M  H
T  μm  B
…………
Orbital magnetic dipole moment and angular momentum
in circular Bohr orbit model
-e
i  ef  e / 2
+e
R
1
 m  iA  R e / 2  eR 2
2
2
M a  R  mv
M a  mR 2
e
μm  
Ma
2m
1 Bohr magneton 
e h
eh

 9.27 10  24[ A  m 2 ]
2m 2 4m
h  6.62 1034[ J  sec] ( Ma has same dimension as h )
8
Orbital magnetic dipole moment and angular momentum
in spherical charge cloud model

h  2 (R2  r 2 )
di  q 2rhdrf  qrhdr
R
h +e
r
-e
d m  dir2  qhr 3dr
 m  
r R
r 0
e
q
(4 / 3) R 3
1 2
qhr dr   eR 
5
3
dM a  dmvR  [
m
2rhdr ]  r  r
3
(4 / 3) R
e
d m  
dM a
2m
결과식은 두 모델에서 동일하며, r에도 무관함
Hold for any volume element of charge distribution
9
[보충자료]
 m  
r R
1 2
qhr dr   eR  운산과정
5
3
r 0
 m  
r R
r 0
qhr 3 dr  q 
R
r 0
1
2 2
2( R 2  r ) r 3 dr
1
2 2
(R  r )  t
2
R t  r
2
R
r
0
2
1
2 2
dr  
 2tdt  2 rdr
2
0
3
2 2
t
dt
r1

2( R  r ) r dr    2t ( R  t ) ( R  t ) 2 tdt
2
3
2
2
2
t R
3
5
R
t t
4 5
  2( R t  t )dt 2( R
 )  R
t 0
3 5 t 0 15
R
2 2
4
2
e
4 5
eR 2
m  
R 
3
(4 / 3) R 15
5
10
E
4.5 Lenz’s law and induced dipole moments
d
dt
 Edl  
L

d
e
dt
di
d
 Ri  e  
dt
dt
0
t
t0
i
R0
i
L
t0 ( )
R
R0
t0
t
R
 t
e
i  (1  e L )
R
L
t
di
d

dt
dt
i  / L
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.
11
Induced dipole moment in circular Bohr orbit model(1)
1
2
 m   eR 20
E
R
B
1 d
R dB
E

2R dt
2 dt
d
Edl



dt
-e
+e
F = -eE
v0
F  eE 
B인가시도 R은 일정으로 가정
eR dB
2 dt
eR dB
dt  mRd
2 dt
d (mv)
dv
F
m
dt
dt
d 
e
dB
2m
e
  0 
B  0   L L : Larmor angular frequency
2m
2
1 2
e2 2
e
2
m   eR 0 
R B   m 0   m (ind )



R
B
m
(
ind
)
2
4m
4m
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.
12
Induced dipole moment in circular Bohr orbit model(2)
mv2
R
-e
R
+e
e2
40 R
2
mv2
e2

 evB
2
R
40 R
 evB
e2
eB
eB
2
 
   0  
3
40 mR
m
m
2
v
B
02   2 
e
  0 
B  0   L
2m
13
eB
eB 2
  ( 
)
m
2m
Induced dipole moment in homogeneous spherical charge
distribution model
0
E
F
rdr
R
+e
B
r
r
B
1 2
 m 0   eR 0
5
1 d
r dB
E

2r dt
2 dt
Fdt  mrd
m0
 m (ind )
e
Edt  rd
m
e
e

Bc 
B  0   L  0
2m
2m

e
d 
dB
2m 2
e
 m (ind )  
R2B
10m
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.
14
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
Curie law
  c /T
Curie-Weiss law
  c /(T  q ), T  TN
   r  1  c /(T  q ), T  q f
15
4.7 Diamagnetism(반자성)
M  (r 1)H  H
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
r  1  105  1
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
with an atom contains
10 electrons;
eq. (4.57)
2
2
e 2
e 2
e2 2
m(ind )   R B   R 0  r H
M  N m (ind )   N R 0  r H  H
m
m
m
19 2
(1.6 10 )
 20
7
5
  5 1028
10
4


10


10
9.11031
Superconductor is a perfect diamagnetic material; susceptibility =-1
16
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
ml  1
- Orbital angular momentum of electrons
e
m  
Ma
2m
H
ml  0
h
2
l (l  1)
ml  1
-Orbital (angular) momentum quantum number
momentum which is measured in units of h/2
l determines the orbital angular
-Magnetic quantum number ml determines the component of angular momentum along
an external field direction
17
• a completely filled shell
• l=1
ml =1(h/2), 0, 1(h/2)
orbital dipole moment -eh/4m, 0, +eh/4m
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]
18
Electron spin magnetic dipole moment
• spin angular momentum along a given direction is either +1/2(=+h/4) or
-1/2(=-h/4)
 m ( spin)
e
  M a ( spin)
m
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
19
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
20
Paramagnetic spin system
가정;
자화에는 전자spin만 기여, 원자당 dipole moment = 1 Bohr magneton( b =eh/4m)
체적당 원자수 N(개/m3) 자계와 평향한 원자수 Np, 반평행 Na
H  0 N p  Na
N P  Na  N
H  0 M  ( N p  N a ) b  H  (  r  1) H
q
T  μ m  B  0μ m  H W (q )   0μ m  Hdq  0 m H cos q
q 90
N (q )  A exp[ W (q ) / kT ] Wa  W p  20 bH
N a / N p  exp[(W p  Wa ) / kT ]  exp( 20 bH / kT )
N exp(   0 bH / kT )
N
Na 

1  exp( 2 0 bH / kT ) exp(  0 bH / kT )  exp(   0 bH / kT )
Np 
N exp( 0 bH / kT )
N

1  exp( 20 bH / kT ) exp(  0 bH / kT )  exp(  0 bH / kT )
21
M  N p  N a  Nb tanh( 0 bH / kT )
(i)
x  1, tanh( x)  x
M  N 0 b H / kT
2
( ii )
(for
x  1, tanh( x)  1
M  Nb
0 bH
kT
0 bH
kT
 1 )
Saturation, see fig. 4.18
9.27 1024 1

 1
1.38 10 23  300
( At room temperature )
   r  1  N0 b 2 / kT  C / T
Curie law
  N0 b 2 / kT  5 10 28  4 10 7  (9.27 10 24 ) 2 / 1.38 10 23T  0.3 / T
Table 4.4 susceptibility of some paramagnetic materials
22
• Diamagnetic contribution도 있으나 미미함(-10-5, 10-3)
• 대표적인 응용 분야
– To obtain very low temperature(<1oK] by adiabatic
demagnetization
s
entropy
H 0
H 0
T, temperature
– MASER ( microwave amplification through stimulated
emission by radiation )
23