classical theories of magnets
TRANSCRIPT
Classical Theories of Magnets 1
0.0 0.5 1.0 1.5 2.0 2.5 3.00.50
0.60
0.70
0.80
0.90
1.00
� � � �� � � �
��
�N2
O2
COCH4�
Ne
A
Kr
Xe
T�T c
n � nc
1� A � n � nc� 1 � 2
1� A� � n � nc� 1 � 3
(B)
28th September 2003c 2003, Michael Marder
Definitions 2
☞ Phenomenology of Magnets
☞ Dipole Moments
☞ Ferromagnets, Ferrimagnets,and Antiferromagnets
☞ Mean Field Theory
☞ The Lenz–Ising Model
☞ Domains
☞ Hysteresis
☞ Order–Disorder Transitions
☞ Critical Phenomena
☞ Landau Free Energy
☞ Scaling and Universality
28th September 2003c 2003, Michael Marder
Magnetic Moments 3
�
jmag � c
�� � �
M � (L1)
�
H � �
B� 4 � �
M (L2)
� � �B � 4 � �
jmag
c
� 4 � �
jext
c
� 1c
�
D
t(L3)
� 4 � �� � �
M � 4 � �
jext
c
� 1c
�
D
t(L4)
� � �
H � 4 � �jext
c� 1
c �
D
t
� (L5)�
B � ��
H � (L6)
� M
H
� (L7)
28th September 2003c 2003, Michael Marder
Conductivity 4
�
EL ��q �
�
E � �q �
q2 ��
ET � �
E� �
EL � (L8)
�
j � c2q2
4 � i � � 1� 1
����
ET (L9) �
j t
� � c2
4 � � 1� 1
� ��� � �� � �
E (L10)� c
4 � � � 1� 1
����� � �
B
t
� (L11)
�
j � c4 � � � 1� 1
����� � �
�
B� �
B0 � � (L12)
�� � �
B � � � 1� 1
� ��� � �
B � 1c
�
D
t
� (L13)
�� ��
B
�� �� � �
H � 1c
�
D
t
� (L14)
28th September 2003c 2003, Michael Marder
Free Energy 5
�
�
�
B ��r � �
(L15)
�
jext
Figure 1: Sample influenced only by cur-
rents
�
jext .
d �
dt
� � d �r �E �
�r � � �jext �
�r � � (L16)
�
H ��r � � 4 � � �
�
�
B
�
�
�
B ��r �
� (L17)
28th September 2003c 2003, Michael Marder
Free Energy 6
� � � 14 � d �r �
H ��r � � �
�
B ��r � � (L18)
�
t� 1
4 � d �r �
H ��r � � �
B ��r �
t(L19)
� � c4 � d �r �
H � �� � �
E (L20)
� � c4 � d �r
�
�
E � �� � �
H� �� � ��
H � �
E ��
(L21)
� � c4 � d �r �
E � �� � �
H � (L22)
�� � �
H ��r � � 4 �
c�
jext � (L23)
�
M ��r � � 1
4 � ��
B ��r �� �
H ��r �� �
(L24)
28th September 2003c 2003, Michael Marder
Free Energy 7
� � T ��
B � � � ��
B �� TS � (L25)
� � � � S � T � d �r �
H ��r � � �
�
M ��r � � 1
8 � d �r � H2
��r � � (L26)
�� � � � 1
4 � d �r �
B ��r � � �
H ��r � � (L27)
�
�� � � 1
4 � d �r �B �
�r � � �
�
H ��r � (L28)
� � d �r �
M ��r � � �
�H �
�r �� 14 � d �r �
H ��r � � �
�
H ��r � � (L29)
� � �� � 1
8 � d �r H2
��r � (L30)
� � � � S � T� d �r �
M � �
�H � (L31)
28th September 2003c 2003, Michael Marder
Magnetic Dipole Moments 8
Element Element
(10 � 6 cm3 mole � 1) (10 � 6 cm3 mole � 1)
Ar � 19.18 N2 � 12.04
As � 5.24 Ne � 7.02
B � 6.70 P � 26.63
C � 5.88 S � 15.39
Cl � 20.18 Se � 23.69
Ge � 7.99 Si � 3.09
H2 � 4.00 Te � 37.00
He � 1.88 Tl � 43.42
I � 45.68 Xe � 43.33
Kr � 28.49
28th September 2003c 2003, Michael Marder
Magnetic Dipole Moments 9
�m � d �r 12c
�r � �
j ��r � � (L32)
�F � 1
cd �r �
j ��r � � �
B ��r � � (L33)
�
F � 1c
d �r �
j ��r � �
�
�
B � 0 � � ��r � �� �
�
B � 0 � � � � ��
(L34)� 0 � �
�m � �� � � �
B (L35)
� �� ��m � �
B � (L36)
U � � �m � �
B � (L37)
28th September 2003c 2003, Michael Marder
Magnetic Dipole Moments 10
Bz
�B
0�
zBz
�
M�
Figure 2: Schematic view of Fara-
day balance.
Bz � z � � B0 � zB1 � (L38)
28th September 2003c 2003, Michael Marder
Spontaneous Magnetization of Ferromagnets11
0.0 0.5 1.00.0
0.5
1.0
NMRMossbauer
Temperature T � Tc
Mag
netiz
atio
nM
� M0
Figure 3: Internal mag-
netic fields in iron
[Source:Preston et al.
(1962).]� B � e
�
h � 2mc � (L39)
� B � 9 � 27 � 10
� 21cm esu � 9 � 27 � 10� 21erg G
� 1 � (L40)
�� 1T� � �
(L41)
28th September 2003c 2003, Michael Marder
Spontaneous Magnetization of Ferromagnets12
���
1
0 500 1000 1500 20000
5
10
Temperature T (K)
Spec
ific
heat
c P(c
alK�
1m
ole�
1)
(A) (B)
0 50 100 150 200 250 3000
10
20
30
Temperature T (K)
Figure 4: (A) Specific heat of iron. [Source: Hofmann et al. (1956) p. 53.] (B) Magnetic
susceptibility of EuO. Source: Matthias et al. (1961), p. 160.]
28th September 2003c 2003, Michael Marder
Spontaneous Magnetization of Ferromagnets13
Compound Tc � mI Compound Tc mI
(K) (K) �� B � (K) �� B �
Cr a 312 0.59 FeFe2O4 fi 858 4.1CoO a 291 � 330 3.8 (magnetite)
CuO a 230 � 745 0.5 FeNiFeO4 fi 858 2.3Mn a 100 0.5 FeLiFeO4 fi 943 2.6MnO a 122 � 610 5 FeCuFeO4 fi 728 1.3NiO a 523 � 2470 2 FeCoFeO4 fi 793 3.7O2 a 23.9 2
Co f 1394 1415 1.72Dy f 85 157 10.65Eu f 289 108 7.12Fe f 1043 1100 2.2Gd f 302 289 7.97Ho f 20 87 10.9Ni f 628 650 0.6Tb f 20 87 10.9
28th September 2003c 2003, Michael Marder
Ferrimagnets 14
0 100 200 300 400 500 600 700-5
0
5
10
15
20
25
30
35
TmErHoDy
TbGd
Temperature T (K)
Mag
neti
zati
onM���
Bpe
run
itce
ll
Figure 5: Spontaneous magnetization of rare earth iron garnets 5Fe2O3 � R2O2 [Source:
Bertaut and Pauthenet (1957).]
� 1T � � � �
� (L42)
28th September 2003c 2003, Michael Marder
Antiferromagnets 15
Figure 6: Spin structure of transi-
tion metal oxides such as
CoO or NiO.
28th September 2003c 2003, Michael Marder
Mean Field Theory and the Ising Model 16
� � ��
�
R
�
R � �
J � �
R � �
R ��
�
R
H � B � �
R � (L43)
� � � �R �� exp
��
�
��
R
�
R � �
J � �
R � �
R � � �
R
H � B � �
R
�
�� (L44)
� �R
� � � � � � �
R
� � � � � (L45)
� �
R � �
R � ��
� � � � � �
R
� � � � � �
� � � � � �R �� � � � ��
� � � � �
R � � �
R � �� � � 2 � (L46)
�
��
R
�
R � �
J � �
R � �
R ��
�
R
H � B � �
R NzJ
� � 2 � 2�
�R
� H � �
H � � B � �
R (L47)
28th September 2003c 2003, Michael Marder
Mean Field Theory and the Ising Model 17
�
H � zJ
� �� B
� (L48)
Z
� 1� � � � N
exp
�
� � NzJ
� � 2 � 2�
�
R
� H � �
H � � B � �
R ��
(L49)� e
� � NzJ �� 2
� 2
�
exp
� � H ��
H � � B
�
� exp
�
� � H ��
H � � B
�
�
N(L50)
� � � kBT lnZ � NzJ
� � 2 � 2� NkBT ln
�
2cosh � B � H � �
H � �
� (L51)
� � � 1Z
� 1� � � � N
1N �
R �� �
R � exp
�
� �
�� �
R � �
(L52)
� 1Z
� 1� � � � N
1
N � B
H
exp�
� �
�� �
R � �
(L53)
� � 1N
1
� B
�
H(L54)
� � tanh � B � H � �
H � � (L55)28th September 2003
c 2003, Michael Marder
Mean Field Theory and the Ising Model 18
� � � tanh �
zJ
� � � � BH
�
� (L56)
� zJ � 1 � 2
��
-1 0 1-1.0
-0.5
0.0
0.5
1.0
� zJ � 1
��-1 0 1
� zJ � 1� 5
��
-1 0 1
Figure 7: Graphical solution of Eq. (L56).
28th September 2003c 2003, Michael Marder
Domains 19
� � ��
�R
�R � �
J �� �
R
� �� �
R � ��
R
��� �
�� �
R
��� x �
2� � B
�
B � �� �
R
�
� 18 � d �r �
B � �
B � (L57)
� � JLla
� � la2
(L58)
�
l
� �a2
� JLl2a
(L59)
l � aJL
� a(L60)
� min
L
� 2�� J
a2
aL
� (L61)
28th September 2003c 2003, Michael Marder
Domains 20
[100]
[010]
[011][001]
L
(A)
(B)
l
x
y
Figure 8: (A) Domain formation in a rectangular bar magnet. (B) In an anisotropic crystal
28th September 2003c 2003, Michael Marder
Hysteresis 21
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
H (Gauss)
B(T
esla
)
� 2
� B� 0� 1 T� B� 0� 05 T� 4
(schematic)
Figure 9: Hysteresis in the magnetization curve of Permalloy. [Source: Bozorth (1951)]28th September 2003
c 2003, Michael Marder
Alloy Superlattices 22
f �� 1 �� 1 � � � AA � f � 1 �� 1 � � f �� 1 � 1 � � � AB � and f � 1 � 1 � � � BB � (L62)
f � � �
R � � �
R � � � C1 � C2 � � �
R � � �
R � � � C3 � �
R � �
R � � (L63)
f � � �
R � � �R � � � � BB � � AB
2 � � �
R � � �
R � ��� AB � �
R � �
R � � (L64)
� � � �
R � � exp
��
��
� �
�
R
� �R
� �
�R
�
R � �
f � � �
R � � � �
R �
��
��
(L65)
� exp
��
��
� � BH
�
R
� �
R � J�
�
R
�R � �
� �
R � �
R �
��
��
(L66)
where28th September 2003
c 2003, Michael Marder
Alloy Superlattices 23
� BH � �� � BB � � AB
2z and J � � AB � (L67)
� �
RA
� � A � � � �
RA
� � A � � � �
RB
� � B � � � �
RB
� � B � (L68)
� � exp
� � BH
�
R
� �
R � J�
�RA
�RB �
� � A � �
RB
� � B � �
RA
� � A � B ��
� (L69)
�
�
RA
exp
� � BH � �
RA
� Jz � � B � �RA
� � A � B � 2 ��
�
RB
exp
� � BH � �
RB
� Jz � � A � �
RB
� � A � B � 2 ��
� (L70)
28th September 2003c 2003, Michael Marder
Alloy Superlattices 24
� A ��
� �
RA �� e� � � BH � � Jz � B �� e�
� � � BH � � Jz � B �
e� � � BH � � Jz � B � � e�� � � BH � � Jz � B �
� (L71)� A � tanh
� � BH � z � BJ
�
(L72a)
� B � tanh
� � BH � z � AJ
�
� (L72b)
� A � � B � 0 � (L73)
� A � � tanh � Jz � A � � tanh � � J � z � A � � (L74)
28th September 2003c 2003, Michael Marder
Critical Phenomena 25
Temperature T
Mag
netic
field
H
M ��
M ��
M � 0
Temperature T
Pres
sure
P
Liquid
Gas
Critical Points
(A) (B)
Figure 10: (A) Schematic phase diagram for a ferromagnet. (B) Schematic phase diagram
of liquid–gas system. .
28th September 2003c 2003, Michael Marder
Landau Free Energy 26
� � M � T � � A0 � T � � A2 � T � M2 � A4 � T � M
4 � HM � (L75)
0
0
�(A
rbitr
ary
units
)
M (Arbitrary units)
A2� 0
A2 � 0
A2� 0
Figure 11: Landau free energy, Eq. (L75), for A2� 0, A2 � 0, and A2� 0.
t � T� Tc
Tc
� (L76)
28th September 2003c 2003, Michael Marder
Landau Free Energy 27
� � a2tM2 � a4M4 � HM � (L77)
H � 2ta2M � 4a4M3 � 0 � (L78)
M ��
��
��
2 � t � a2
4a4
� for t� 0
0 for t� 0.
(L79)
C � � �
T
� T
�
(L80)
� � 1Tc
t � 1 � t �
2 t
�
1 � t(L81)
� 1Tc
2
�
t2 (L82)
28th September 2003c 2003, Michael Marder
Landau Free Energy 28
��
��
��
�
1Tc
a22
2a4
� for t� 0
0 for t� 0.
(L83)
M � 2 � t � a2
4a4
� qH � (L84)
q � � 14a2 � t �
� (L85)
M
H
��
��
�� 1
4 � t � a2for t� 0
� 12ta2
for t� 0.(L86)
H � 4a4M3 � 0 M� H1 � 3 � (L87)
28th September 2003c 2003, Michael Marder
Critical Phenomena 29
0.0 0.5 1.0 1.5 2.00.0
5.0
10.0
15.0
CuK2Cl4� 2H2OCu � NH4 � 2Cl4� 2H2OCuRb2Cl4� 2H2OCu � NH4 � 2Br4� 2H2O
T � Tc
c P� J���
mol
e�K�
Figure 12: Molar heat capacities of four ferromagnetic copper salts versus scaled temper-
ature T � Tc. [Source Jongh and Miedema (1974).]
28th September 2003c 2003, Michael Marder
Critical Phenomena 30
NMR Frequency � M (MHz)
0.0 20.0 40.0 60.00.970
0.980
0.990
1.000
T� T c
1� AM3
(A)
0.0 0.5 1.0 1.5 2.0 2.5 3.00.50
0.60
0.70
0.80
0.90
1.00
� � � �� � � �
��
�N2
O2
COCH4�
Ne
A
Kr
Xe
T� T c
n � nc
1� A � n � nc� 1 � 2
1� A� � n � nc� 1 � 3
(B)
Figure 13: (A) Temperature versus magnetization, antiferromagnet Source:
Heller and Benedek (1962) (B) Coexistence curve for eight fluids. Source:
Guggenheim (1945).
28th September 2003c 2003, Michael Marder
Critical Phenomena 31
dP � sdT � nd � � (L88)
C � � t ��� � t �� �
�
(L89)
M� � t � � and � n� � t � � � (L90)
KT � 1n
n P� 1
nc
� n
P
� � t �� � � (L91)
M
H
� � � t �� � � (L92)
P� � � n ��� � (L93)
� M �� � H �
1 � � � (L94)28th September 2003
c 2003, Michael Marder
Critical Phenomena 32
g � r �� 1� e
� r �� (L95)
S ��q �� 1 � n d �r ei �q � �r
�
g � r �� 1
�
(L96)� d �r e
� r � � � i �q � �r� 11 � �
2q2
� (L97)
� � � t �� � � (L98)
g � r � � r
� 1 � � � (L99)
28th September 2003c 2003, Michael Marder
Critical Phenomena 33
Exponent Fluid Magnet Mean Field Theory Experiment 3d Ising
� C � � � t � � � C � � � t � � � discontinuity 0.11–0.12 0.110
� � n � � t � � M � � t � � 12 0.35–0.37 0.325
� KT � � t � � �
� � � t � � � 1 1.21–1.35 1.241
� P � � � n �� � H � � � M �� 3 4.0–4.6 4.82
� � � � t � � �
� � � t � � � 0.61–0.64 0.63
g � r � � r
� 1 � � g � r � � r
� 1 � � 0.02–0.06 0.032
28th September 2003c 2003, Michael Marder
Scaling Theory 34
�
� kBT
� � t �
x1G � t � H � � (L100)
C � � T
�
� t
� � (L101)
x1 � 2� � � (L102)
G � t � H � � GH
H0 � t � � � (L103)
limy �� G � y � � yx2 � (L104)
�
� kBT
� � t �
2 � � HH0 � t � �
x2� � t �
2 � � � � x2 � (L105)
x2 � 2� ��
� (L106)
28th September 2003c 2003, Michael Marder
Scaling Theory 35
� M � �
H
� � t �
2 � � 1H0 � t � � G �
HH0 � t � � � (L107)
� t �
2 � � � � � � t � � (L108)
� � 2� � � � (L109)
M
H
��
� H� 0
� � � t �
2 � �
H20 � t � 2 � G � �
�
HH0 � t � � �
��
� H� 0(L110)
� t �
2 � � � 2 � � � t �� � (L111)
�
� � � 2 � � 2 � (L112)
2 � � � 2 � � � (L113)
28th September 2003c 2003, Michael Marder
Scaling Theory 36
M � 1H0 � t � � � t �
2 � � HH0 � t � �
x2 � 1
(L114)
� Hx2 � 1 � H �
2 � � � �� � � (L115)
1
�
� 2� � ��
2� � � �
(L116) � � 1 � �
� (L117)
�
� N2
�
� � kBT N2
� 2
� P
� kBTn2 � KT (L118)
��
d �rd �r �
� n ��r � n ��r �
� ��
� � N �
2 (L119)
� � n 1 � n d �r � g � r �� 1 � � (L120)
g � r � � e � r ��
r1 � � � (L121)
28th September 2003c 2003, Michael Marder
Scaling Theory 37
one hasKT � d �r g � r � � (L122)
KT � �
3
�� 1 � � d �s e � s
s1 � �
(L123)� �
2 � � � � t �� �
�
2 � � � � (L124)
� 2�� ��� �� � (L125)
�
kBT �� � t �
2 � � � �� 3 (L126)
2� � � 3� � (L127)
28th September 2003c 2003, Michael Marder