classical theories of magnets

38
Classical Theories of Magnets 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.50 0.60 0.70 0.80 0.90 1.00 N 2 O 2 CO CH 4 Ne A Kr Xe TT c nn c 1 Ann c 1 2 1 A nn c 1 3 (B) 28th September 2003 c 2003, Michael Marder

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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

Scaling Theory 38

-2 -1 0 1 2 3 4-2

-1

0

1

2

3

4

log10 � x � x0 � 1 �

log 1

0� h

� x ���

Figure 14: Scaling function h � � H � � � M �� versus x � t � � M �

1 � � [Source: Vicentini-Missoni

(1972), p. 68.]

28th September 2003c 2003, Michael Marder