chapter11 Paramagnetism and Diamagnetismocw.nctu.edu.tw/upload/classbfs1209042706186759.pdf · an...

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1 Chapter Eleven Diamagnetism and Paramagnetism The story of magnetism begins with a mineral called magnetite “Fe 3 O 4 ”. The first truly scientific study of magnetism was made by William Gilbert. His book “On the magnet” was published in 1600. In 1820 Hans Christian Oersted discovered that an electric current produces a magnetic field. In 1825 the first electromagnet was made by Sturgeon. Each stores the same magnetic energy ~0.4J.

Transcript of chapter11 Paramagnetism and Diamagnetismocw.nctu.edu.tw/upload/classbfs1209042706186759.pdf · an...

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Chapter Eleven Diamagnetism and ParamagnetismThe story of magnetism begins with a mineral called magnetite “Fe3O4”.

The first truly scientific study of magnetism was made by William Gilbert.His book “On the magnet” was published in 1600.

In 1820 Hans Christian Oersted discovered that an electric current produces a magnetic field.

In 1825 the first electromagnet was made by Sturgeon.

Each stores the same magnetic energy ~0.4J.

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H (Oe)

M (e

mu/

cm3 )

H (Oe)

M (e

mu/

cm3 )

diamagnetism

paramagnetism ferromagnetism

Magnetic susceptibility per unit volume

HM

≡χ M : magnetization, magnetic moment per unit volumeB : macroscopic magnetic field intensity

In general, paramagnetic material has a positive susceptibility (χ >0).diamagnetic material has a negative susceptibility (χ<0).

CGS

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Three principal sources for the magnetic moment of a free atom : Electrons’ spin Electrons’ orbital angular momentum about the nucleus Change in the orbital moment induced by an applied magnetic field

First two sources give paramagnetic contributions.

The last source gives a diamagnetic contribution.

1895 Curie proposed the Curie law

1905 Langevin proposed the theory of diamagnetism and paramagnetism

1906 Wiess proposed the theory of ferromagnetism

1920’s The physics of magnetism was developed with theories involving electron spins and exchange interactions – the beginnings of QM

1903

1859~1906

P. Curie

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Reviews, in CGS

H M 4HB

µπ =+=

VNmM

= : magnetic dipole moment

LL

Bi

ii 2mepr

2mem µ−=−=∑ ×−=

S

oBi

ioB gsgm µµ =∑=

[emu/cm3]

Bohr magneton

Electronic g-factor go=2.0023

224-B 1Vs/m1T J/T;109.27

2me

=×==µ

Angular momentum

Spin

I vµ

L and S are quantum operators with no basis in classical physics.

electron

m

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Theory of diamagnetismwas first worked out by Paul Langevin in 1905.

Motion of the electron is equivalent to current in the loop.

Faraday’s law

Negative magnetism present in an applied fieldeven though the material is composed of atoms that have no net magnetic moment.

dtdΦB−=Emf

[volt]

Consider a circular loop with radius r, the effect of applied field H is to change µ by an amount ∆µ in negative direction.

22

2A

6mce

AN RZρχ −=

[emu/Oe-cm3]

I

H

µr

1872~1946Paul Langevin

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The change in moment of a single orbit :

dtdH

2r

dtdH

r 2r

dtdHAE

2

−=−=−==ππ

Emf

dtdH

2mcer

mceE

dtdva −=== ∫∫

−= dH

2mcerdv

H2mcerv −=∆

H4mc

rev2cer

2

22

−=∆=∆µ the plane of the orbit is perpendicular to the applied field

R

All possible orientations in the hemisphere

( )( ) 22/

0

22

2

22222

32dsin2 sin

2

AdA sinsinr

RRRR

R

RR

==

==

∫π

θθπθπ

θθθ

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A single electron with consideration of all possible oriented orbits,

H6mcev

2ce

2

22RR−=∆=∆µ

For an atom that contains Z electrons,

∑∑==

−=−=∆zz

RR1i

2i2

2

1i2

2i

2

6mcHeH

6mceµ Ri is the radius of the ith orbit

Susceptibility, 22

2A

mc6e

AN RZρχ −=

The number of atoms per unit volume : NAρ/A

Avogadro’s number

Atomic mass

Volume density of mass

H6mc

eA

N1i

2i2

2A

−=∆ ∑

=

z

Rρµ

On the volume basis,

Carbon (radius =0.7Å)

( ) ( )( )( ) ( )

36-

2821028

210323

cmemu/Oe101.5

cm107.0cm/s103g1011.966esu1080.4

12.01gg/cm22.21002.6

−×−=

×××

××−= −

χ

Experimental value : -1.1×10-6 emu/Oe-cm3

Classical Langevin result

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This classical theory of Langevin is a good example of the use of a simple atomic model to quantitatively explain the bulk properties of a material.

The agreement between calculated and experimental values for other diamagnetic materials is generally not that good, but it is at least within an order of magnitude.

The other consistency is thatχ is independent of temperaturefor most diamagnetic materials.

Quantum theory of diamagnetismresults in the same expression.

Negative susceptibility

magnitude : 10-5 ~10-6 cm3/mole

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Diamagnetic materials

Closed-shell electronic structures result in no net momentThe monatomic rare gases He, Ne, Ar, …Polyatomic gases, H2, N2, …Some ionic solids, NaCl, …Some covalent solids, C, Si, Ge, …

Superconductors : Meissner effect

The first systematic measurements of susceptibility of a large number of substances over an extended range of temperature were made by Pierre Curie and reported by him in 1895.

-- mass susceptibility is independent of temperature for diamagnetics,but that it vary inversely with temperature for paramagnetics.

TC

=χ θχ

−=

TC'

Curie law Curie-Weiss law (1907)

in general

Pierre Weiss

1865~1940

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Quantum theory of diamagnetism

Hamiltonian VAH QcQp

m21 2

+

−=

Terms due to the applied field

( ) 22

2

2mce

mc2e' AAAiH +∇•+•∇=

A

×∇=B

In the uniform magnetic field kBB =

( )k

yxj

xzi

zy

kjikz

jy

ix

xyzxxz

zyx

∂∂

−∂

∂+

∂∂

−∂∂

+

∂∂

−∂∂

=

++×

∂∂

+∂∂

+∂∂

=×∇

AAAAAA

AAAA

One solution k0j2

xBi2

yB++−=A

( )222

22

yx8mc

Bex

yy

xmc4

Be' ++

∂∂

−∂∂

=iH

zL~

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z-component of angular momentum

∂∂

−∂∂

−=x

yy

xL~ z i paramagnetism

For diamagnetic materials with all electrons in filled core shells,

the sum over all electrons vanishes∑ + zz 2SL

For a spherically symmetric system 222 r32yx =+

22

2222

2

22

B12mce

yx8mc

Be'r

E =+=the additional energy

the associated magnetic moment B6mce

B'

2

22 rE−=

∂∂

−=µ

in agreement with the classical Langevin result

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Vibrating Sample Magnetometer (VSM)

→→ measuring sample’s magnetic moment

oscillate sample up and downmeasure induced emf in coils A and Bcompare with emf in coils C and D from known magnetic moment

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Classical theory of paramagnetism

In 1905, Langevin also tried to explain paramagnetism qualitatively.

He assumed that paramagnetic materials have molecules or atoms with the same non-zero net magnetic moment µ.

In the absence of magnetic field, these atomic moments point at random and cancel one another. M=0

When a field is applied, each atomic moment tends to align with the field direction; if no opposing force acts, complete alignment would be produced and the material demonstrates a very large moment. But thermal agitation of atoms tends keep the atomic moments pointed at random. Partial alignment and small positive χ.

As temperature increases, the randomizing effect of thermal agitation increases and hence, χ decreases.

This theory leads naturally into the theory of ferromagnetism.

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A unit volume material contains n atoms with a magnetic moment µ each.

θµµ cosHH −=•−=E

TkcosH

TkE

BB dsin2dA dnθµ

θθπ eKeK ==−

∫=∫=π

θµ

θθπ0

TkcosHB sind2dnn eK

∫=∫=π

θµ

θθθπµθµ0

TkcosH

0

B sincosd2cosdnM eKn

Let a = µH/kBT,

−+

=∫

∫=

∫=

a1n

dx

dx xn

sind

sincosdnM

aa

aa

1

1

ax

1

1

ax

0

cosa

0

cosa

eeee

e

e

e

e

µµ

θθ

θθθµπ

θ

πθ

( )

−=

a1acothnµ

H

θµ

Let the direction of each moment be presented by a vector, and let all vectors be drawn thru. the center of the sphere.

# at angle θ θ+dθ

total #

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Langevin function L(a)=coth(a)-1/a

a=µH/kBT

M/M

o

L(a) can be expressed in series,

...9452a

45a

3aL(a)

53

−+−=

When a is small (<0.5),L(a) is a straight line w/. a slope 1/3.

When a is large, L(a) → 1.

Two conclusions :

(1)Saturation will occurs at large a implying that at large H and low T the alignment tendency of field is going to overcome the randomizing effect of thermal agitation.

(2)At small a, L(a) depends linearly on a implying that M varies linearly with H.

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The Langevin theory also leads to the Curie law.

For small a,Tk

H3

n3

anMB

µµµ==

Tk3n

HM

B

2µχ ==TC

= B

2

k3nµ

=C

Oxygen is paramagnetic and obeys the Curie law w/. χ=1.08×10-4emu/gOe.

χµ

85.2 moleerg/Oe102.64

1002.6)emu/gOe1008.1)(erg/K1038.1)(g/mole32(3

nTk3

20-

23

416B

=−×=

×××

==−−

Even in heavy atoms or molecules containing many electrons, each with orbital and spin moments, most of the moments cancel outand leave a magnetic moment of only a few Bohr magnetons.

Typically, ( )( )( )( ) 0064.0

300Kerg/K1038.110000Oeerg/Oe1064.2

TkHa 16

20

B

×== −

−µ

smaller enough L(a)=a/3

<0.5

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The Langevin theory of paramagnetism which leads to the Curie law, is based on the assumption that individual carrier of magnetic moment do not interact with one another, but are acted on only by applied field and thermal agitation.

However, many paramagnetic materials obey the more general law,

considering the interaction between elementary moments.θ

χ−

=T

CCurie-Weiss law (1907)

Weiss suggested that this interaction could be expressed in terms of a fictitious internal field, “molecular field HM”. In addition to H.

The total field acting on material H + HM= H + γM

TMHM

HM

tot

C=

+==

γχ ( ) H M T CC =− γ

θγχ

−=

−==

TTHM C

CC

where θ (=Cγ) is a measure of the strength of the interaction.

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Theoretical predictions

θ

CCCθθ

χ−=

−= T1T1

Intercept reveals information of θθ (FeSO4)<0, θ (MnCl2)>0

θ>0 indicates that the molecular field aids the applied field, makeselementary moments align with H, and tends to increase χ.

θ<0 indicates that the molecular field opposes the applied field, makeselementary moments disalign with H, and tends to decrease χ.

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Precession of atomic magnetic moments about H,

like gyroscope, angular momentum in the presence of gravitation.

H ×= µτ

dtLdmr

=×= gτ

If the atom was isolated, the only effect of an increase in H would be an increase in the rate of precession, but no change in θ.

For a system with many atoms, all subjected to thermal agitation, there is an exchange of energy among atoms. When H is applied, the exchange of energy disturbs the precessional motion enough so that the value of θ for each atom decreases slightly, until the distribution of θ values becomes appropriate to the existing values of field and temperature.

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Quantum theory of Paramagnetism

The central postulate of QM is that the energy of a system is discrete.quanta of energyimproving the quantitative agreement

between theory and experiment.

θ

H J=1/2 J=2H

Classical case Quantum case

θ is continuous variable. θ is restricted to certain variables.Space Quantization

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The orbital and spin magnetic moments for an electron

orbitorbit p2mc

e22mc

emc4

e −===

ππµ hh

spinspin pmce

221

mce

mc4e

−===ππ

µ hh

In general

p2mc

e g−=µ where g is spectroscopic splitting factor, or g-factor g=1 for orbital motion and g=2 for spin

In a multiple-electron atom, the resultant orbital angular momentum and the resultant spin momentum are characterized by quantum numbers L and S.

Both are combined to give the total angular momentum that is described by the quantum number J.

Bgg µµ 1)J(J1)J(Jmc2e

+=+=

In the direction of H, Bgg µµ JJH mmmc2e

==

There are 2J+1 numbers of allowed azimuthal quantum number mJ : J, J-1, … , -(J-1), -J.

θ µµH

H

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For a free atom, the g factor is given by the Landé equation

1)2J(J1)L(L1)S(S1)J(J1

++−+++

+=g

The energy levels of the system in a magnetic field

2µB

ms µs1/2 -µ

-1/2 µ

HgmHE BJ µµ =•−=

For a single spin with no orbital moment, mJ=±1/2 and g=2.

The equilibrium population for the two level system :

)TH/kexp()TH/kexp()TH/kexp(

NN

)TH/kexp()TH/kexp()TH/kexp(

NN

BB

B2

BB

B1

µµµ

µµµ

−+−

=

−+= lower level

upper level

Total number of atoms is conserved.

N = N1+N2 = constant

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The resultant magnetization for N atoms per unit volume

( )xtanh NN)N(NM xx

xx

21 µµµ =+−

=−= −

eeee where

TkHx

B

µ=

For x<<1, tanh(x)~x andTkHN xNM

B

2µµ =≅

For a system with known g and J,

Potential energy of each moment in H HgmU BJ µ−=

The probability of an atom w/. energy U TH/kgmTU/k BBJB µee =−

Therefore,

Boltzmann statistics

∑=

J

BBJ

J

BBJ

m

TH/kgmm

TH/kgmBJgm

NM µ

µµ

e

e

( )

(x)BJN 2Jxcoth

2J1

2Jx12Jcoth

2J12JJNM

JB

B

µ

µ

g

g

=

++

=

Brillouin functionwhere

TkHJx

B

Bµg=

mJ : J, J-1, … , -(J-1), -J

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The saturation magnetizationBHo JNNM µµ g==

Number of atoms per unit volume

The maximum moment of each atom in the direction of H

( )

++

==2Jxcoth

2J1

2Jx12Jcoth

2J12J(x)B

MM

Jo

Brillouin function, 1927

When J=∞, the classical distribution, the Brillouin function reduces to the

Langevin function.( )

x1xcoth

MM

o

−=

When J=1/2, the magnetic moment consists of one spin per atom.

( )xtanhMM

o

= xx<<1

TkHN

TkHJNgxNM

B

2

B

2B22

Hµµµ ==≅

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The Brillouin function, like the Langevin, is zero for x equals to zero and tends to unity as x becomes large. The course of the curve in between depends on the value of J for the atom involved.

TkHx

B

µ=classical

TkH

TkHJx

B

H

B

B µµ==

gquantum≠

µ : net magnetic moment in the atom

KCr(SO4)12H2O

Henry, Phys. Rev. 88, 559 (1952)

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When H is large enough and T low enough, a paramagnetic can be saturated.

For example, KCr(SO4)12H2O at H=5Tesla and T=4.2K

only magnetic ion Cr3+, J=3/2, L=3, and S=3/2 g=2/5the lower curve shown in the figure

→ data does not follow this predicted curve

The data is well described by the quantum model with J=S=3/2 and g=2 .

BB μ1)J(JJμ +== ggµ

L=0 no orbital component

BB μ87.3μ1)23(

232 =+=

“Spin-only” moment

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Gd(C2H3SO4)3•9H2O

metals

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What magnetic moment a metal should have?

Hund’s rule incorporate coupling of magnetic moment to determine the magnetic state

(1)Maximize S subject to the Pauli exclusion principle

ms=±1/2 put spins (in different states) parallel∑= sm

(2)Maximize L subject to the Pauli exclusion principle

m= -, -+1, …, -1, fill higher m state preferentially

∑= m

(3)

J= L-S, L-S+1, …, L+S-1, L+S allowed valuesWhether it is a minimum or maximum in the ground state depends on filling function.

+−

=full half than more is shell when SLfull half than less is shell when SL

J

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Magnetic stateJ

12S L+ L : 0, 1, 2, 3, 4S, P, D, F, G

Examples

(1)Full shells : L=0, S=0, and hence, J=0

(2) Half filled shells : L=0 and S=(2+1)(1/2)

3210-1-2-3

L=0 and S=7/2

J=L+S=7/2

Eg. Gd3+ 4f75S25P6

(n=4,=3)

7/28SState

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3210-1-2-3

L=3 and S=1/2

J=L-S=5/2

Eg. Cs3+ 4f15S25P6

(n=4, =3)

5/22 FState

3210-1-2-3

L=5 and S=1

J=L-S=4

Eg. Pr3+ 4f25S25P6

(n=4, =3)

43 HState

3210-1-2-3

L=5 and S=5/2

J=L+S=15/2

Eg. Dy3+ 4f95S25P6

(n=4, =3)

15/26 HState

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Rare earth ions

1)J(Jgp += For Sm3+ and Eu3+, there are discrepancies between experimental p and theoretically calculated value.

High states of the L-S multiplet.

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Quenching of the orbital angular momentum : Iron group ions

1)S(S2p +=Effective number of Bohr magneton 1)J(Jgp +=

In the iron group ions the 3d shell, the outermost shell, is responsible for the paramagnetism and experiences the intense inhomogeneous electric field produced by neighboring ions. called “the crystal field” breaking L-S coupling (J is meaningless)

splitting 2L+1 sublevels (diminishing the contribution of orbital motion to magnetic moment)

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An atom w/. L=1 is placed in the uniaxial crystalline electric field of two positive ions along the z-axis. Take S=0 to simplify the problem.

+

− 2yxf(r) )((r)YR

z f(r) )((r)YR2

yxf(r) )((r)YR

11

10

11

i

i

n

n

n

θφ

θφ

θφ

In a free atom, a single electron in a p-state.

Three degenerate p states

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In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential ϕ about the nucleus,

222 B)z(AByAxe +−+=ϕ

Reconstructing unperturbed ground state zf(r) Uyf(r), Uxf(r),U zyx ===

+ perturbed term eϕ

0UUUUUU xzzyyx === ϕϕϕ eee

)IA(I

]dxdydzyAx[Axf(r)

]dxdydzB)z(ABy[Axxf(r)UU

21

2242

22222xx

−=

−=

+−+=

∫∫ϕe

Non-diagonal elements

Diagonal elements

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)IB(I

]dxdydzzBy[Byf(r)

]dxdydzB)z(ABy[Axyf(r)UU

21

2242

22222yy

−=

−=

+−+=

∫∫ϕe

)IB)(I(A

]dxdydzzzB)[y(Af(r)

]dxdydzB)z(ABy[Axzf(r)UU

12

4222

22222zz

−+=

−+=

+−+=

∫∫ϕe

The orbital moment of each of the states

0ULUULUULU zzzyzyxzx ===

Crystal field splits the original degenerate states into non-magnetic states separated by energies >> µH.

quenching of the orbital angular momentum

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Paramagnetic susceptibility of conduction electrons

• Conduction electrons are not bound to atoms so that they are not included in atomic magnetization.

• Each electron has µ=±µB.

• Curie-type paramagnetic contribution (electron spins align w/. H)

• Pauli pointed out that only electrons near EF (±kBT) can flip over.

TkHnM

B

2Bµ= where n is # of conduction electrons

This requires each electron spin align w/. H, independent of all others.

Fermi-Dirac distribution

0.00.10.20.30.40.50.60.70.80.91.01.1

ε

f(ε,T

) T=0T≠0

εF

kBT

εFεF

H=0 H≠0

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Fraction of electron participating FF

B

TTTk

FB

2B

B

2B

F TkHn

TkH

TTnM µµ

=

= independent of temperature

2µH

DOSDOS

Consider the density of states D(ε) in the presence of HSpin ↑ electrons lowered in energy by µH in H ↑Spin ↓ electrons raised in energy by µH in H ↑

HεF

ε

↑↓

µH

µH

flip spins

ε ε

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( ) HT2k

3NH23NH)(DM 2

FB

2

FF µµ

εµεµµ ===−= ↓↑ NN

Pauli spin magnetization

independent of temperature

The concentration of electrons w/. magnetic moment parallel to H

The concentration of electrons w/. magnetic moment antiparallel to H

H↑, T=0K

H)(D21)(Dd

21)H(Dd

21

F0H

FF

µεεµεεε

µ

+≅+= ∫∫−

↑ εεN

H)(D21)(Dd

21)H(Dd

21

F0H

FF

µεεµεεε

µ

−≅−= ∫∫↓ εεN

valid when kBT<<εF

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References :

Kittel, Introduction to Solid State Physics, 8th Ed., Ch.11-12, 2005.

Aschroft and Mermin, Solid State Physics, Harcourt, Ch.31-33, 1976.

Mattis, Theory of Magnetism, Springer, 1985.

Cullity, Introduction to Magnetic Materials, Addison-Wesley, 1972.

Comstock, Introduction to Magnetism and Magnetic Recording, Wiley-Interscience, 1999.

Spin-dependent Transport in Magnetic Nanostructures, Edited by Maekawa and Shinjo, Taylor & Francis, 2002.

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Cooling by isentropic demagnetization

Entropy : A measure of the disorder of a system

The greater the disorder, the higher is the entropy. In the magnetic field the moments will be partly lined up (partly ordered), so that the entropy is lowered by the field.

The entropy is lowered if the temperature is lowered, as more of the moments line up.

When the material is demagnetized at constant entropy, entropy can flow from the system of lattice vibrations into the spin system.

ZlnkS B=where Z is the partition function

∑=

=

=

Im

-Im

N

B

M

TkexpZ ε

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Entropy for a spin ½ system depends only for the population distribution.

a→b : the system is magnetized isothermally.b→c : the magnetic field is turned off adiabatically.

B/T

= ∆

HHTT if where H∆ is the effective field

If we start at H=5T and Ti=10mK 5.0Tk

H

B

≈µ Tf ~10-7K

The first nuclear cooling was done on Cu, Ti=20mK Tf ~1.2×10-6K

a

bc

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=

B3.1TT if

Cu nuclei in the metal

where B∆=3.1Oe.

The present record for a spin temperature is about 280pK. PRL70, 2818 (1993).

Rd

1.2 µK