lecture magnetism in solids-trinity college-js3015

8/18/2019 Lecture Magnetism in Solids-Trinity College-JS3015

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Magnetic Properties of SolidsMagnetic Properties of Solids

JS 3015 J.M. D. Coey 10 lectures + Tutorial

Recommended books

• Stephen Elliot: The Physics and Chemistry of Solids, Wiley 1998: Ch 7The recommended solid-state text. Ch 7 treats dielectrics first. There is a close analogy between dielectric

and magnetic materials, and between ferroelectricity and ferromagnetism.

• Stephen Blundell Magnetism in Condensed Matter , Oxford 2001A new book providing a good treatment of the basics

• J. Crangle; The Magnetic Properties of Solids, Arnold 1977: 1990A short book which treats the material at an appropriate level.

• David Jiles Introduction to Magnetism and Magnetic Materials, Chapman and

Hall 1991; 1997A more detailed introduction, written in a question and answer format.

Monday at 1700, Wednesday at 1100, 1200


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1 Introduction

2 Basic concepts

3. Magnetism of the electron

4. Magnetism of localized electrons on the atom

5 Paramagnetism

6. Ferromagnetism.

7. Miscellaneous Topics

8. Magnetic Applications


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1 Introduction

1.1 Historical background. See'Magnetism Through the Ages' on the webpage.

1.2 Magnetization and hysteresis;Overview

2 Basic concepts

2.1Moments and fields

2.2Magnetic field calculations

2.3 Units

2.4Dimensions


3.1 Orbital moment

3.2 Spin moment3.3Magnetism of electrons in solids.

3.4Localized and delocalized electrons

3.5Theory of the electronic magnetism

4. Magnetism of localized electrons on the atom

4.1 The hydrogenic atom and angularmomentum.

4.2The many-electron atom.4.3Spin-orbit coupling

4.4 Zeeman interaction

4.4Ions in solids

5 Paramagnetism

5.1 Classical Theory

5.2 Quantum theory

5.3 Paramagnetism of metals

6. Ferromagnetism.6.1 Mean field theory

6.2 Exchange Interactions.

6.3 Ferromagnetic domains

6.4 Magnetic measurements

7. Miscellaneous Topics

7.1 Antiferromagnetism

7.2 Ferrimagnets.

7.3 Other forms of magnetic order

7.4 Spin waves

7.5 Magnetic neutron scattering

8. Magnetic Applications

8.1 Hard magnets

8.2 Soft magnets

8.3 Magnetic recording

8.4 Spin electronics


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1 Introduction

1.1 Historical background. See 'Magnetism Through the Ages' on the webpage.

1.2 Magnetization and hysteresis; Course overview.

A wonder I experienced as a child of 4 or 5 years, when my father showed me a compass. That this

needle behaved in such a determined way did not at all fit into the nature of events. I can still

remember — or at least I believe I can remember — that this experience made a deep and lasting

impression on me. Albert instein


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Table 1. The seven ages of magnetism

GE Time Key Names Driver Achievements Materials Applications/

Devices

ncient

-1000

to

1500

Shen Kua

Petrus

Peregrinus

State Force field,

Induced

magnetism,

Thermoremanenc

e

Iron,

Lodestone

South-pointer,

Compass

arly

1500

to

1820

Gilbert

Descartes

D.

Bernoulli

Navy Earth’s field Iron,

Lodestone

Dip circle.

Horseshoe magnet

ectromagne

1820

to

1900

Oersted,

Ampere,

Faraday,

Maxwell

Hertz

Industry

(infrastruc-

ture)

Electromagnetic

induction,

Maxwell’s

equations

Electrical steels Motors, Generators,

Telegraph,

Wireless,

Magnetic recording

nderstandin1900

to

1935

Weiss,

Bohr,

Heisenberg,

Pauli,

Dirac,

Landau

Academy Spin,

Exchange

interactions

(Alnico)

gh

equency

1935

to

1960

Bloch,

Pound,

Purcell

Military Microwaves,

EPR, FMR,

NMR

Ferrites Radar,

Television

pplications1960to

1995

- Industry(consumers)

New Materialsminiaturisation of

magnetic circuits

Nd-Fe-B,Sm-Co

Consumer electronics

pin

ectronics

1995 -

…

- Industry

(consumers)

Thin film devices Multilayers High d ensity magnetic

recording, MRAM

…

Age

Ancient

Early

scientific

Elecro-

magnetic

Underrstanding

High

frequency

Alications

Sin

electronics


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!he Ancient Age

-1000 to 1500

Alications

South-"ointer

#omass

$ri%er

!he State

Scientific achievements

&orce field

'nduced magnetism

!hermoremanence

(ey names

Shen (un

"etrus "eregrinus


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!he Electromagnetic Age

1500 - 1)00

Alications

*otors+ ,enerators

!elegrah+ ireless

*agnetic recording

Driver

'ndustry

.'nfrastructure/

Scientific Achie%ements

Electromagnetic 'nduction

*aells Equations

(ey names

ersted+ Amere

&araday+ *aell

Hert


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!he Age of Understanding

1400 - 1435

Alications

$ri%er

Academy

Scientific Achie%ements

Sin+ *ean &ield !heory

Echange 'nteractions

(ey "layers

eiss+ ohr

Heisen6erg

$irac+ "auli

7andau

H

= 2J SiS j


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!he Age of Alications1480 - 1445

Key Players

??

Driver

Industry

(Consumer)

Scientific Achievements

New materials

Miniaturisation of Manetic Circuits

Alications

Consumer

!lectronics


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Ni-Fe/Fe-Co (heads)

Fe-Si

Fe-Si (oriented)

Ni-Fe/Fe-Co

Amo rphous

Others

Others

Alnico

Sm-CoNd-Fe-B

Hard ferrite

Co- γ Fe 2 O 3

(tapes, l opp! d iscs)CrO2 (tapes)

"ron (tapes)

Co-Cr (hard discs)

Soft ferrite

Others

"ron

Sot

#a$nets

%ard#a$nets

#a$netic&ecordin$

Magnet applications; A 30 B¤ market


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!he Age of Sin Electronics1445 - 9

Al"ert #ert

Peter $rune"er

Stuart Par%in

Driver

Industry

(Consumer)

Scientific Achievements

&hin film devices

Alications

'ihdensity

recordinMAM


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Magnetization and hysteresis

!he irre%ersi6le+ nonlinear resonse of a ferromagnet to an alied magnetic field is descri6ed 6y the

hysteresis loo+ hich reflects the arrangement of the magnetiation in ferromagnetic domains: !he loo

is nonlinear and multi%alued .history deendent/: ;ote that the magnet cannot 6e in thermodynamic

equili6rium anyhere around the oen art of the cur%e< 'mortant arameters are the remanenmagnetiation *r= and the coerci%e field Hc:

Ad%ances in magnetic materials and magnet alications ha%e 6een due essentially to mastery ocoerci%ity: e ha%e a ide range of soft .narro loo/+ hard .6road loo/ and intermediate magneti

materials: !he hard materials are used for ermanent magnets: !he soft materials are used in

electromagnetic machines .motors+ generators+ transformers/: !he intermediate materials are used fo

magnetic recording media


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* +asic Conce,ts

This section introduces the magnetization M and the two magnetic fields B and H. These are vectors which are

defined at every position r in a solid. Units and dimensions in magnetism are discussed.

2.1 Moments and fields

Magnetization•Magnetization is the basic magnetic quantity in a solid. It originates from the atomic electrons,

which are the carriers of magnetic moment.

m

All magnetism is due to circulating currents, The relation for a current loop is

m = I A

m is the magnetic dipole moment, I is the electric current, A is the area vector. Units of m are Am2.

It is a polar vector. Direction is given by the corkscrew rule.

The magnetization M of a solid is the magnetic dipole moment per unit volume.

M = δm / δVUnits of M are Am-1. For many purposes, the atomic-scale structure can be neglected. M is

regarded as a smoothly-varying vector (δV contains many atoms).•The magnetization of iron, for example, is 1.76 MA m-1 (Imagine the currents)•Estimate the magnetic moment of a ferrite ‘fridge magnet’ (M 0.2 MA m≈ -1 )

'

B fi ld


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B field

•The magnetic field B produced by the current loop is divergenceless (solenoidal).There are no magnetic

monopoles in Nature to act as sources or sinks of B. (Compare with E)

∇

B = 0 Maxwell’s Equation

The vector operator∇

means (∂ / ∂x, ∂ / ∂y, ∂ / ∂z).'∇

' is the divergence (div) of a vector. The scalar

product∇ Β

=∂Bx / ∂x + ∂By / ∂y + ∂Bz / ∂z

Units of B are Tesla. (abbreviation T)

The B field is also known as the magnetic flux density. Flux is defined as Φ = ∫ SB.dA, or for a uniform field normal to an area A, Φ = BA.

∇

B = 0 can be written in an equivalent integral form over any closed surface s as ∫ SB.dA = 0In other words the flux through any closed surface is zero,. Φ = 0.

There is a little-used SI unit for flux, the Weber. 1 T = 1 wb m-2

B


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Slide 10

•The relation between the B field and current density j (units Am-2) in the steady state is given by

another of Maxwell's equations which can also be written in point form or in integral form. We assume

no time variation. i.e. magnetostatics.

•In point form, ∇∧Β =µ o j

Here µ 0 is the magnetic constant µ 0 = 4π

10-7

T m A-1

.'∇∧' is the rotation (curl) of a vector∇∧Β is given by the expression ex(∂By / ∂z - ∂Bz / ∂y) + ey(∂Bz / ∂x - ∂Bx / ∂z) + ez(∂Bx / ∂y - ∂By / ∂x)•In integral form, around any closed loop threaded by a net current I

∫ loopB.dl = µ 0I

This equation is known as Ampere's law.

•An important result: It is possible to calculate the field due to a small current loop m = IA. The result

with components in polar coordinates is

Bdip = (µ 0m /4 rπ3)[2cosθer + sinθeθ]

This is equivalent to the field due to an electric dipole (it falls off as r -3, and is anisotropic being twice

as great in the axial position as in the broadside position). Hence the magnetic moment m is commonly

known as the magnetic dipole moment.

I

loop


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H field

•The difficulty when we are dealing with magnetic solids is that the current density j is made up of

contributions from currents in external circuits j0 (which we can measure) and contributions from the

atomic currents jM that create the magnetization of a solid (currents we cannot measure).

The relation between jM and M is ∇∧Μ = jM, and ∇∧Β =µ 0( j0 + jM). Hence it follows that∇

∧(Β

/µ 0 - M)= µ 0 j0. Define H = (Β/µ 0 - M)

or B = µ 0(H + M)

Now we are able to retain Ampere's law for the field H, which does not depend on the unmeasurablecurrents jM. In point form, ∇∧Η = j0 or in integral form ∫ loopH.dl = I0The H field is not solenoidal. It will have sources (or sinks) wherever

∇

∧Μ

≠ 0, e.g. at thesurface of a magnet. (These are the fabled North and South ‘poles’).

Note that the H field is not created by conduction currents only. Any piece of magnetised material

creates an H field both in the space around it, and within its own volume. Generally the field at apoint is H = Hd + H0, where H0 is the field created by currents in circuits, and Hd is the field created

by the magnet itself, which is known as the stray field outside the magnet, and the demagnetizing

field inside the magnet.


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2.2 Units and dimensions

•We use SI throughout with the Sommerfeld convention B = µ 0(H + M). Engineers prefer theKenelly convention B = µ 0H + J, where the polarization J is µ 0M. Both are acceptable in SI.The polarization of iron is J = 2.16 T.

•Flux density B and polarization J are measured in telsa (also mT, µ T). Magnetic moment m is measuredin Am2 so the magnetization M and magnetic field H are measured in Am-1. From the energy relation

E = - m.B, it is seen that an equivalent unit for magnetic moment is JT-1, so magnetization can also be

expressed as JT-1m-3. σ, the magnetic moment per unit mass in JT- 1kg-1, is the quantity most usually

measured in practice. µ 0 is exactly 4π

.10-7

T.mA-1

.

•The international system is based on five fundamental units kg, m, s, K, and A.

Derived units include the newton (N) = kg.m/s2, joule (J) = N.m, coulomb (C) = A.s, volt (V) = JC

tesla (T) = JA-1m -2 = Vsm-2, weber (Wb) = V.s = T.m2 and hertz (Hz) = s-1.

Recognized multiples are in steps of 10±3, but a few exceptions are admitted such as cm (10-2 m) and

Å (10-10 m). Multiples of the meter are fm (10-15), pm (10-12), nm (10-9), µ m (10-6), mm (10-3) m (10-0)and km (103).

• The SI system has two compelling advantages for magnetism:

(i) it is possible to check the dimensions of any expression by inspection and

(ii) the units are directly related to the practical units of electricity.

cgs Units


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g

•Much of the primary literature on magnetism is still written using cgs units. Fundamental cgs units are

cm, g and s. The electromagnetic unit of current is equivalent to 10 A. The electromagnetic unit of

potential is equivalent to 10 nV. The electromagnetic unit of magnetic dipole moment (emu) is

equivalent to 1 mA.m2. Derived units include the erg (10-7 J) so that an energy density such as K1 of

1 Jm-3 is equivalent to 10 erg cm-3. The convention relating flux density and magnetization isB = H+ 4πM

where the flux density or induction B is measured in gauss (G) and field H in oersted (Oe). Magnetic

moment is usually expresed as emu, and magnetization is therefore in emu/cm3, although 4πM is

frequently considered a flux-density expression and quoted in kilogauss. µ 0 is numerically equal to1 GOe-1, but it is normally omitted from the equations. The most useful conversion factors between SI

and cgs units in magnetism are

B 1 T ≡ 10 kG 1 G ≡ 0.1 mTH 1 kAm-1 ≡ 12.57 ( 12.5) Oe≈ 1 Oe ≡ 79.58 (≈80) A m-1

m 1 JT-1 ≡ 1000 emu 1 emu ≡ 1 mJT-1 M 1 kAm-1 ≡ 1 emu cm-3

σ 1 JT-1

kg-1

≡ 1 emu g-1

The dimensionless susceptibility M/H is a factor 4πlarger in SI than in cgs.

Dimensions


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In the SI system, the basic quantities are mass (m), length (l), time (t), charge (q) and temperature ( θ)Any other quantity has dimensions which are a combination of the dimensions of these five basic

quantities, m, l, t, q and θ. In any relation between a combination of physical properties, all thedimensions must balance.

Mechanical

Quantity symbol unit m l t q θ

area A m2 0 2 0 0 0

volume V m3 0 3 0 0 0

velocity v m.s-1 0 1 -1 0 0

acceleration a m.s-2 0 1 -2 0 0

density ρ kg.m-3 1 -3 0 0 0

energy E J 1 2 -2 0 0

momentum p kg.m.s-1 1 1 -1 0 0

angular momentum L kg.m2.s-1 1 2 -1 0 0

moment of inertia I kg.m2 1 2 0 0 0

force F N 1 1 -2 0 0

power p W 1 2 -3 0 0

pressure P Pa 1 -1 -2 0 0

stress S N.m-2 1 -1 -2 0 0

elastic modulus K N.m-2 1 -1 -2 0 0

frequency ν s-1 0 0 -1 0 0

diffusion coefficient D m2.s-1 0 2 -1 0 0viscosity (dynamic) η N.s.m-2 1 -1 -1 0 0

viscosity (kinematic) ν m2.s-1 0 2 -1 0 0

Planck’s constant h J.s 1 2 -1 0 0

Thermal

Quantity symbol unit m l t q θ

enthalpy H J 1 2 -2 0 0

entropy S J.K-1 1 2 -2 0 -1

specific heat C J.K-1.kg-1 0 2 -2 0 -1

heat capacity c J.K-1 1 2 -2 0 -1

thermal conductivity κ W.m-1.K-1 1 1 -3 0 -1

Sommerfeld

coefficient

γ J.mol-1.K-1 1 2 -2 0 -1

Boltzmann’s constant k J.K-1 1 2 -2 0 -1


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3) Domain wall energy γ w = √AK (γ w is an energy per unit area)


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[γ w] = [EA-1] [√AK] =1/2[ AK]

= [ 1, 2,-2, 0, 0] [√A]=1/2[ 1, 1,-2, 0, 0]

-[ 1, 1,-2, 0, 0] [√Κ]=1/2[ 1,-1,-2, 0, 0]

= [ 1, 0,-2, 0, 0] [ 1, 0,-2, 0, 0]

4) Magnetohydrodynamic force on a moving conductor f = σvxBxB (f is a force perunit volume)

[f] = [ FV-1] [σ] = [-1,-3, 1, 2, 0]

= [ 1, 1,-2, 0, 0] [v] = [ 0, 1,-1, 0, 0]

-[ 0, 3, 0, 0, 0] [B2] = 2[ 1, 0,-1,-1, 0]

[ 1,-2,-2, 0, 0] [ 1,-2,-2, 0, 0]

5) Flux density in a solid B = µ 0(H + M). (Note that quantities added or subtractedin a bracket must have the same dimensions)

[B]= [ 1, 0,-1,-1, 0] [µ 0] = [ 1, 1, 0,-2, 0]

[M],[H] = [ 0,-1,-1, 1, 0]

[ 1, 0,-1,-1, 0]

6) Maxwell’s equation ∇xH = j + dD /dt.

[∇xH] = [Hr-1

] [j] = [ 0,-2,-1, 1, 0] [dD /dt] = [Dt-1

]= [ 0,-1,-1, 1, 0] = [ 0,-2, 0, 1, 0]

-[ 0, 1, 0, 0, 0] -[ 0, 0, 1, 0, 0]

= [ 0,-2,-1, 1, 0] = [ 0,-2,-1, 1, 0]



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The origin of magnetism in solids is the quantized angular momentum of the electrons, with two distinct sources; orbital motion and

spin. Spin itself and spin-orbit coupling are relativistic effects. The description of magnetism in solids is fundamentally different

depending on whether the electrons are localized on the ion cores, or delocalized in energy bands. In this course we focus on

localised magnetism

The magnetic properties of solids derive from the magnetism of their electrons.

The electron is a particle with charge -e, mass m, possessing intrinsic angular momentum, known as

‘spin’.

e = 1.602 10-19 C m = 9.109 10-31 kg

(Nuclei also possess moments, but they are 1000 times smaller).≈

←

←

r6ital moment Sin

3.1 Orbital moment


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I-e

An electron in a circular orbit, radius r, is equivalent to a current loop

The circulating current is I; I = -ev/2 rπ

The moment is m = IA m = -evr/2

In Bohr’s quantum theory, orbital angular momentum l

is quantized in units of ђ; h is Planck’s constant, 6.622610-34 J s; ђ = h/2 = 1.055 10π -34 J s.

The orbital angular momentum is l = r∧mv; | l|= lђ; l = 1, 2, ....l is the angular momentum with units J s, l is a quantum number, an integer with no units.Eliminating

r in the expression for m,

m = -l(eђ /2m) = lµ BThe minus sign means that m and l are oppositely directed, since the electron charge is negative.

The quantity (eђ /2m) is defined to be the Bohr magneton (µ B), the basic unit of atomic magnetism;

µ B = 9.274 10-24 Am2

[Generally, for non-circular orbits, m = (1/2)∫ v r∧ jMdV; jM(r’) = -evδ(r - r’) ; m = er∧v /2 = (e/2m)l since l = r∧mv]

3.2 Spin moment


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The electron has a mysterious built-in spin angular momentum s. Spin arises from relativistic quantum

mechanics. The spin quantum number is 1/2.

|s| = sђ; s = 1/2Nonetheless,the magnetic moment associated with electron spin is also 1µ B. m = -2s(eђ /2m) = 1µ B

The magnetomechanical ratio γ, is defined as the ratio of magnetic moment to angular momentumFor orbital angular momentum m = γ l, hence

γ = −(e/2m)The g-factor is defined as the ratio of m (in units of µ B) to l (in units of ђ)

g = 1 for orbital motion

For spin angular momentum

γ = −(e/m)g = 2 for spin (after higher order corrections, 2.0023)

Spin angular momentum is twice as effective as orbital angular momentum in creating a magnetic

moment.

Generally there is both spin and orbital angular momentum for an atomic electron. They produce a

total angular momentum j, j = l + s ; j = l ± 1/2. | j| = jђ. m = -g(e/2m) j

Interaction with magnetic fields

From E = - m.B, we can see that if B = 1 T and m = 1µ B the interaction energy E(1T) is -9.274 10-24 J.Since k B = 1.381 10

-23 µ B /K, E/k B = 0.672 K/T These interactions are rather weak, compared to RT.

Einstein-de Haas experiment. This demonstrates the reality of the


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relation between magnetization and angular momentum. A nickel rod

is suspended on a torsion fibre. The field in the solenoid is reversed,

changing the direction of magnetization of the nickel. It rotates, to

conserve angular momentum as the angular momenta of the electrons

is reversed.

It turns out that g for Ni is g = 2.21, so the moment is essentially due

to spin. However, m for Ni is only 0.6 µ B /atom. There is less than oneelectron with an unpaired spin, yet the number of electrons/atom Z =

28

torsion fbre

solenoid

Ni rod3.3 Magnetism of electrons in solids.

Magnetism in free atoms is reduced by shell fillling.

Electrons in filled shells have paired spins with no net moment.

Only unpaired electrons in unfilled (usually outermost) shells have a moment.

Magnetism in solids tends to be destroyed by chemical interactions of the outer electrons:• electron transfer to form filled shells in ionic compounds• covalent bond formation in semiconductors• band formation in metals

Table 3.1 Magnetism of free atoms


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The only elements that are nonmagnetic as free atoms are those boxed, in bold type.

Radioactive atoms are shown in italics. Except for hydrogen, all moments are


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QuickTime™ and a GIFdecompressor are needed to see this picture.

Existence of magnetism depends critically on crystal structure and composition.

Table Atomic moments of iron in different compounds in Bohr magnetons/Fe.

γ -Fe2O3 α-Fe YFe2 γ -Fe YFe2Si2

ferrimagnet ferromagnet ferromagnet antiferromagnet Pauli

paramagnet

5.0 2.2 1.45 unstable 0

Formation of d and s-bands in a metal. Th

broad s-bands have no moment. The d-band

may have one if they are sufficiently narrow.

Nickel has a configuration 3d

9.4

4s

0.6

. There ar0.6 unpaired 3d electrons, m = 0.6 µ B

3.4 Localized and delocalized electrons


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LOCALIZED MAGNETISM DELOCALIZED MAGNETISM

Integral number of 3d or 4f electrons Nonintegral number of unpaired spins

on the ion core; Integral number of unpaired spins; per atom.

Discreet energy levels. Spin-polarized energy bands

with strong correlations.Ni2+ 3d8 m = 2 µ B Ni 3d9.44s0.6 m = 0.6 µ B

ψ ≈exp(-r/a0) ψ exp(-i≈ k.r)Boltzmann statistics Fermi-Dirac statistics

4f metals localized electrons

4f compounds localized electrons

3d compounds localized/delocalized electrons3d metals delocalized electrons.

Above the Curie temperature, neither localized nor delocalized moments disappear, they just

become disordered in the paramagnetic state, T > TC.

3d

3d

!

r

ψ

Ε

Susceptibility of localized electrons (Curie susceptibility)


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The electrons have s = 1/2, and g = 2.There are two states, also known as ‘spin up’( ↑) and ‘spin down’(↓), with ms = ±1/2, which correspond to two possible orientations of the moment relative to theapplied field. Since E = - m.B, the two energy levels Ei = -µ 0gµ BmsH, are mi= ±µ 0µ BH.

The population of an energy level is given by Boltzmann statistics; it is proportional toexp{-E(θ)/k ΒT}. The thermodynamic average is evaluated from these Boltzmann populations.

= [(1/2)gµ Bexp(x) - (1/2)gµ Bexp(-x)]/[exp(x) + exp(-x)] = mtanh(x)

where x = µ 0 µ BH/k BT.In small fields, tanh(x) x, hence≈

χr = µ 0Nµ B2 /k BT

This is the famous Curie law for the susceptibility, which varies as T-1.

In other terms χr = C/T, where C = µ 0Nµ B2 /k B is a constant with dimensions of temperature;

Assuming an electron density N of 6 1028

m-3

gives C 0.5 K. The Curie law susceptibility at room≈temperature is of order 1.6 10-3.

Many ‘two-level systems’ in physics are treated by assigning them a ‘pseudospin’ s = 1/2.

-1/2

1/2

MS

µ 0µ BH

z

g√[S(S+1)]µ

H 1/2

-1/2

S = 1/2g?H

ms

Susceptibility of delocalized electrons (Pauli paramagnetism of metals)


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The calculation for metals proceeds on a quite different

basis. The electrons are indistinguishable particles which

obey Fermi-Dirac statistics. They are not localized, so

Boltzmann statistics cannot be applied. The electrons have

s = 1/2, m = µ B. They partly-fill some energy band up tothe Fermi level EF.

A rough calculation gives the susceptibility as follows:

χr = (N↑ - N↓)µ B /H

≈2[ N (EF)µ 0gµ BH]µ B /H where N (EF) is the density of states at the Fermi level for one spindirection.

χ r 2µ ≈ 0 N (EF)µ B2

This is known as the Pauli susceptibility. Unlike the Curie susceptibility, it is very small, and

temperature independent, to first order.

The density of states N (EF) in a band is approximately N/2W, where W is the bandwidth (which is

typically a few eV). Comparing the expression for the Pauli susceptibility with that for the Curie

susceptibility χr = µ 0Nµ B2 /k BT, we see that the Pauli susceptibility is a factor k BT/W smaller than theCurie susceptibility . The factor is of order 100 at room temperature. χrPauli is of order 10-5.

H = 0

H

±µ 0µ BH

E

↓ ↑ ↓ ↑

E

E F

[3.6 Theory of the electronic magnetism]


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Maxwell’s equations relate magnetic and electric fields to their sources. The other fundamental

relation of electrodynamics is the expression for the force on a moving particle with charge q,

F = q(E + v∧B)The two terms are respectively the Coulomb and Lorentz forces. The latter gives the torque equation

Γ

= m∧B The corresponding Hamiltonian for the particle in a vector potential A representing themagnetic field B (B= ∇∧Α) and a scalar potential φε representing the electric field E (E = -∇φe) is

H = (1/2m)(p - qA)2 +qφeOrbital Moment

The Hamiltonian of an electron with electrostatic potential energy V(r) = -eφe is H = (1/2m)(p +eA)2 +V(r)

Now (p + eA)2 = p2 + e2A2 + 2eA.p since A and p commute when ∇.A = 0. So H = [p2 /2m +V(r)] + (e/m)A.p + (e2 /2m)A2

H = H 0 + H 1 + H 2where H 0 is the unperturbed Hamiltonian, H 1 gives the paramagnetic response of the orbital moment

and H 2 describes the small diamagnetic response. Consider a uniform field B along z. Then the vector

potential in component form is A = (1/2) (-By, Bx, 0),

so B = ∇∧Α = ez(∂Ay / ∂x - ∂Ax / ∂y) = ezB. More generallyA = (1/2)B∧r

Now (e/m)A.p = (e/2m)B∧r.p = (e/2m)B.r∧p = (e/2m)B.l since l = r∧p. The second terms in theHamiltonian is then the Zeeman interaction for the orbital moment

H 1 = (µ B / ђ)B.l

The third term is (e2 /8m) (B∧r)2 = (e2 /8m2)B2(x2+y2). If the orbital is spherically symmetric, = =/3.The corresponding energy E = (e2B2 /12m) . Since M = -∂E/ ∂B and susceptibility χ = µ 0NM/B,


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It follows that the orbital diamagnetic susceptibility is χ = µ 0Ne2 /6m.

Spin Moment

The time-dependent Schrödinger equation

-(ђ2

/2m)∇2

ψ + Vψ = iђ∂ψ / ∂tis not relativistically invariant because the operators ∂ / ∂t and ∂ / ∂x do not appear to the same power. Wneed to use a 4-vector X = (ct, x, y, z) with derivatives ∂ / ∂X.Dirac discovered the relativistic quantum mechanical theory of the electron, which involves the Paul

spin operatorsσI, with coupled equations for electrons and positrons. The nonrelativistic limit of the

theory, including the interaction with a magnetic field B represented by a vector potential A can b

written as

H = [(1/2m)(p +eA)2 +V(r)] - p4 /8m3c2 + (e/m)B.s + (1/2m2c2r)(dV/dr) - (1/4m2c2)(dV/dr) ∂ / ∂r•The second term is a higher-order correction to the kinetic energy•The third term is the interaction of the electron spin with the magnetic field, so that the complete

expression for the Zeeman interaction of the electron is

H Z = (µ B / ђ)B.(l + 2s)The factor 2 is not quite exact. The expression is 2(1 + α /2π- .....) 2.0023, where≈ α = e2 /4πε0hc 1/137≈

is the fine-structure constant.•The fourth term is the spin-orbit ineteraction., which for a central potential V(r) = -Ze2 /4πε0r with Ze athe nuclear charge becomes -Ze2µ 0l.s /8πm2r3 since µ 0ε0 = 1/c2. In an atom (0.1 nm)≈ 3 so thmagnitude of the spin-orbit coupling λ is 2.5 K for hydrogen (Z = 1), 60 K for 3d elements (Z 25), and≈160 K for actinides (Z 65).≈

In a noncentral otential the s in-orbit interaction is s∧∇V .

Magnetism and relativity.


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The classification of interactions according to their relativistic character is based on the kinetic energy

E = mc2√[1 + (v2 /c2)]

The order of magnitude of the velocity of electrons in solids is αc. Expanding the equation in powers oc gives

E = mc2 + (1/2)α2mc2 - (1/8)α4mc2

Here the rest mass of the electron, mc2= 511 keV; the second and third terms, which represent the order

of magnitude of electrostatic and magnetostatic energies are respectively 13.6 eV and 0.18 meV

Magnetic dipolar interactions are therefore of order 2 K.

4. Magnetism of localized electrons on the atomThe quantum mechanics of a single electron in a central potential leads to classification of the one-electron states in terms of four


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The quantum mechanics of a single electron in a central potential leads to classification of the one electron states in terms of four

quantum numbers. The individual electrons’ spin and orbital angular momenta are coupled in the many-electron ion, and spin-orbit

coupling operates to give a series of energy levels (multiplets), the lowest of which is specified by Hund’s rules. When placed in a

solid, the ion experiences a crystal field due to the charge environment which modifies the spin-orbit coupling and makes either S or

J the good quantum number. In the presence of a magnetic field, the atomic moment is aligned, or equivalently the magnetic

sublevels are split by the Zeeman interaction.

4.1 The hydrogenic atom and angular momentum.

A hydrogenic atom is composed of a nucleus of charge Ze at the origin and an

electron at r,θ,φ. First, consider a single electron in a central potential φe = Ze/4πe0r H = -(ђ" /2m)∇2 - Ze2 /4πe0r

In polar coordinates:

∇ 2 = ∂2 / ∂r2 +(2/r)∂ / ∂r + 1/r2{∂2 / ∂θ2 + cotθ∂ / ∂θ + (1/sin2θ)∂2 / ∂2φ}The term in parentheses is -l2. Schrödinger’s equation is H ψ = Eψ The wave function ψ means that the probability of finding the electron in a small volume dV ar r is ψ *(r)ψ (r)dV. (ψ * is the complex conjugate of ψ).Eigenfunctions of the Schrödinger equation are of the form ψ (r,θ,φ) = R(r)Θ(θ)Φ(φ).♦ The angular part Θ(θ)Φ(φ) is written as Ylml(θ,φ).The spherical harmonics Yl

ml(θ,φ) depend on two integers l, ml, where l is ≥ 0 and |ml| ≤ l.

Φ(φ) = exp(imlφ) where ml = 0, ±1, ±2 .......The z-component of orbital angular momentum, represented by the operator lz = -iђ∂ / ∂φ,has eigenvalues =mlђ.

Θ(θ) = Plml(cosθ), are the associated Legendre polynomials with l ≥ |ml|,so ml = 0, ±1, ±2,....±l.

z

θ

l

r

φZe

-e


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The three quantum numbers n, l, ml denote an orbital, a spatial distribution of electronic charge.Orbitals are denoted nx, x = s, p, d, f for l = 0, 1, 2, 3. Each orbital can accommodate


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up to two electrons with spin ms = ±1/2. No two electrons can be in a state with the same four quantumnumbers (Pauli exclusion principle). The hydrogenic orbitals are listed in the table

n l ml ms No of states

1s 1 0 0 ±1/2 2

2s 2 0 0 ±1/2 2

2p 2 1 0,±1 ±1/2 6

3s 3 0 0 ±1/2 2

3p 3 1 0,±1 ±1/2 63d 3 2 0,±1,±2 ±1/2 10

4s 4 0 0 ±1/2 2

4p 4 1 0,±1 ±1/2 6

4d 4 2 0,±1,±2 ±1/2 10

4f 4 3 0,±1,±2,±3 ±1/2 14•The Pauli principle states that no two electrons can have the same four quantum numbers. Each

orbital can be occupied by at most two electrons with opposite spin.


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4.2 The many-electron atom.

In the many electron atom terms like e2/4πε r must be added to the Hamiltonian One way of dealing


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In the many-electron atom, terms like e2 /4πε0rij, must be added to the Hamiltonian. One way of dealingwith the extra Coulomb interactions is to suppose that each electron sees a different spherical charge

distribution, which produces a different central potential for each one. The potential with many electron

is not a simple Coulomb potential well; the degeneracy of electrons with different l is raised. The 4s shell

for example, is then lower in energy than the 3d shell, which defines the shape of the periodic table. Thequantities VI(r) must be determined self-consistently (the Hartree-Foch approximation)

When several electrons are present on the same atom, at most two of them having opposite spin can

occupy the same orbital (Pauli principle). Their spin and orbital angular momenta add to give resultants

S = Σsi, MS = Σmsi, L = Σli, ML = Σmli.1s 2s 2p ML MS

↑ ↑ 2 1

↑

↓ 1 0

↓

↑ 1 0

↓ ↓ 1 -1

↑ ↑ 1 1

↑

↓ 0 0

↑↓ 0 0

↓

↑ 0 0

↓ ↓ 0 −1

↓ ↓ 0 −1

↑

↓ −1 0

↓

↑ −1 0

↑ ↑ −1 1

↑↓ −1 0

↑↓ ↑↓ ↑↓ 2 0

Consider the six-electron carbon atom;1s22s22p2. The 15 states fall into three

groups, or terms.

The notation for terms is to denote L = 1.

2. 3. ….by S, P, D, …. and to include the

spin multiplicity 2S + 1 as a superscript.The energy splitting of the terms is of

order 1 eV.

2S+1L

↑↓ * -

In spectroscopy, the energy unit cm-1 is used. Handy conversions are:1 eV ≡ 11605 K and 1 cm-1 ≡ 1.44 K


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4.4 Zeeman interaction

The magnetic moment of an ion is represented by the term m = (L + 2S)µB/ ђ


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The magnetic moment of an ion is represented by the term m (L + 2S)µ B / ђ The Zeeman Hamiltonian for the magnetic moment in a field B applied along z is – m.B

H Z = (µ B / ђ)B.(L + 2S)

L

S J

m

z

SThe vector model of the atom, including

magnetic moments. First project m onto J. J

then precesses around z.

We define the g-factor for the atom or ion as the

ratio of the component of magnetic moment

along J in units of µ B to the magnitude of theangular momentum in units of ђ.g = -( m.J / µ B)/(J2 / ђ) = m.J /J(J + 1)ђµ B.e

J2 = J(J + 1)ђ"# Jz = MJђ but m.J = (µ B / ђ){(L + 2S).(L + S)}

(µ B / ђ){(L2 + 3L.S + 2S2)}(µ B / ђ){(L2 + 2S2 + (3/2)(J2 - L2 - S2)}(µ B / ђ){((3/2)J2 – (1/2)L2 + (1/2)S2)}(µ B / ђ){((3/2)J(J + 1) – (1/2)L(L + 1) + (1/2)S(S + 1)}

hence

g = 3/2 + {S(S+1) - L(L+1)}/2J(J+1)

Also, from the vector diagram it follows that mz /Jz = m.J / J2 = gµ B / ђ.The magnetic Zeeman energy is EZ = –mz B. This is –(mZ Jz)./(Jz B) = (gµ B / ђ)JzB

( $


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Hence EZ = -gµ BMJB $ % &'"

-&'"

-3'"

-)'"

)'"

3'"&'"The effect of applying a magnetic field on an ion with J = 5/2.

Note the magnitudes of the energies involved: If g = 2. µ B = 9.27 10-24 J T-1. The splitting of two adjacentenergy levels is gµ BB. For B = 1 T, this is only 2 10≈ -23 J, equivalent to 1.4 K. [k B = 1.38 10-23 J K-1]The basis of electron spin resonance is to apply a magnetic field to split the energy levels, and then

apply radiation of frequency ν so that E = h ν is sufficient to induce transitions between the Zeemanlevels. Since h = 6.63 10-34 J s-1, ν 28 GHz for resonance in 1 T. This is in the microwave range.≈

It is possible to deduce the total moment from the susceptibility, which should give meff = g√[J(J+1)]µ B.for free ions (§ 5.2.2). The maximum value of mz is deduced from the saturation magnetization.

Generally meff > mz

For 4f ions J is the good quantum number, but for 3d ions S is the good quantum number


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Energy le%els of #o2 ion+ 3dF: ;ote that the

Geeman slitting is not to scale:

4.5 Ions in solids.

Summarizing for free ions;


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Summarizing, for free ions;

♦ Filled electronic shells are not magnetic (the spins are paired; m s = ±1/2)♦ Only unfilled shells may possess a magnetic moment♦ The magnetic moment is given by m = gµ BJ, where ђJ represents the total angular momentum. For a

given configuration the values of g and J in the ground state are given by Hund’s rulesWhen the ion is embedded in a solid, the crystal field interaction is important, and the third point i

modified

♦ Orbital angular momentum for 3d ions is quenched . The spin only moment is m gµ ≈ BS, with g = 2.♦ Magnetocrystalline anisotropy appears, making certain crystallographic axes easy directions omagnetization.

The Hamiltonian is now H = H 0 + H so+ H cf + H Z

Typical magnitudes of energy terms (in K) H 0 H so H cf H Z in 1 T

3d 1 - 5 104 102 -103 1 - 104 1

4f 1 - 6 105 1 - 5 103 ≈3 102 1

H so must be considered before H cf for 4f ions, and the converse for 3d ions. Hence J is a good quantum

number for 4f ions, but S is a good quantum number for 3d ions. The 4f electrons are generally localized

and 3d electrons are localized in oxides and other ionic compounds.

The most common coordination for 3d ions is 6-fold (octahedral) or 4-fold (tetrahedral). Both have cubicsymmetry, if undistorted


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Octahedral and tetrahedral sites.

To demonstrate quenching of orbital angular momentum, we consider the l = 1 states ψ 0, ψ 1, ψ-

corresponding to ml = 0, ±1.

ψ 0 = R(r) cos θ ψ ±1 = R(r) sin θ exp {±ιφ}The functions are eigenstates in the central potential V (r) but they are not eigenstates of H cf . Suppose th

oxygens can be represented by point charges q at their centres, then for the octahedron,

H cf = Vcf = D(x4 +y4 +z4 - 3y2z2 -3z2x2 -3x2y2)

where D eq/ ≈ 4πεoa6. But ψ ±1 are not eigenfunctions of Vcf, e.g. ?ψ i*Vcf ψ jdV≠ δij, where i,j = -1, 0, 1.

We seek linear combinations that are eigenfunctions, namelyψ 0 = R(r) cos θ = zR(r) = pz

(1/ √2)(ψ 1 + ψ -1)= R’(r)sinθcosφ = xR(r) = px(1/ √2)(ψ 1 - ψ -1)= R’(r)sinθsinφ = yR(r) = py

Note that the z-component of angular momentum; lzђ = iђ∂ / ∂φ is zero for these wavefunctions. Hencethe orbital angular momentum is quenched.


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The same applies to 3d orbitals; the eigenfunctions there are

dxy = (1/ √2)(ψ 2 - ψ -2) = R’(r)sin2θsin2φ xyR(r)≈

dyz = (1/ √2)(ψ 1

- ψ -1

) = R’(r)sinθcosθsinφ yzR(r)≈ t 2g orbitalsdzx = (1/ √2)(ψ 1 + ψ -1) = R’(r)sinθcosθcosφ zxR(r)≈dx

2-y

2 = (1/ √2)(ψ 2 + ψ -2) = R’(r)sin2θcos2φ (x≈ 2-y2)R(r) eg orbitalsd3z

2-r

2 = ψ 0 = R’(r)(3cos2θ − 1) (3z≈ 2-r2)R(r)

px,py,pz

dxy,dyz,dzx

t2g

egdx2-y2, dz2

dxy,dyz,dzxt2

edx2-y2, dz2

!he three -or6itals are degenerate in a

cu6ic crystal field+ hether octahedral

or tetrahedral+ hereas the fi%e d-or6italsslit into a grou of three t2g and a grou

of to eg or6itals

Notation; a or b denote a nondegenerate single-electron orbital, e a twofold degenerate orbital and t

threefold degenerate orbital. Capital letters refer to multi-electron states. a, A are nondegenerate and

symmetric with respect to the principal axis of symmetry (the sign of the wavefunction is unchanged), b

B are antisymmetric with respect to the principal axis (the sign of the wavefunction changes). Subscripts

g and u indicate whether the wavefunction is symmetric or antisymmetric under inversion. 1 refers to

mirror planes parallel to a symmetry axis, 2 refers to diagonal mirror planes.


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As the site symmetry is reduced, the degeneracy of the one-electron

energy levels is raised. For example, a tetragonal extension of the

octahedron along the z-axis will lower p and raise p and p The effect


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octahedron along the z axis will lower pz and raise px and py. The effect

on the d-states is shown below. The degeneracy of the d-levels in

different symmetry is shown in the table.

The effect of a tetragonal distortion of

octahedral symmetry on theone-electron energy levels.

The splitting of the 1-electron levels

in different symmetry

l Cubic Tetragonal Trigonal Rhombohedral

s 1 1 1 1 1

p 2 3 1,2 1,2 1,1,1

d 3 2,3 1,1,1,2 1,2,2 1,1,1,1,1

f 4 1,3,3 1,1,1,2,2 1,1,1,2,2 1,1,1,1,1,1,1

•A system with a single electron (or hole) in a degenerate level will tend to distort spontaneously. The

effect is particularly strong d4 and d9 ions in octahedral symmetry (Mn3+, Cu2+) which can lower theienergy by distorting the crystal environment. This is the Jahn-Teller effect. If the local strain is ε, thenergy change ∆E = -Aε +Bε2, where the first term is the crystal-field stabilization energy and the secondterm is the increased elastic energy.

Px,py

pz

d2-y2

d2

dy

dy+d

5 Paramagnetism

In small fields, the average atomic moment is proportional to applied magnetic field, and the Curie-law susceptibilit


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, g p p pp g , p

C/T is calculated first for classical moments, and then by quantum statistical mechanics. The complete magnetization

is given by the Langevin function or the Brillouin function in the two cases. In contrast, the susceptibility of a metal i

small and temprature-independent.

We study the response of a magnetic moment m to an applied magnetic field H.

5.1 Classical Theory

Langevin theory. This is the classical theory. Each atom has

a small moment m. which can adopt any orientation relative

to the applied field H = (B / µ 0

). The energy is – m.B = -µ 0

m.H

E(θ) = -µ 0mHcosθThe probability P(θ) of the moment making an angle θ with zis the product of a Boltzmann factor exp{-E(θ)/k ΒT} and a geometricfactor 2πsinθ dθ. Hence P(θ) = κ 2πsinθ exp{-E(θ)/k ΒT} dθ

where κ is determined by the normalization condition

κ = ∫ 0π P(θ) d(θ) = N, the number of moments per unit volume.

= ∫ 0πm cosθ P(θ) dθ /∫ 0πP(θ) dθ

H

mθ

2šsin θdθ

z

To evaluate the integrals, let a = cos θ; da = -sinθ dθ and define x = -µ 0mH/k BT which is thdimensionless ratio of magnetic to thermal energy. This gives

= m{cothx - 1/x}.


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The quantity in {} is the Langevin function L(x). At low fields, L(x) x/3. The susceptibility of an≈

ensemble of N moments is χr = N/H, henceχr = µ 0Nm2 /3k BT

This is the famous Curie law; it is also written as χr = C/T where C = µ 0Nm2 /3k B is the Curie constantUnits of C are kelvin. At high fields, x >> 1 the magnetization saturates; L(x) → 1; the moments araligned, = m

The Langevin function L(x).

The slope at the origin is 1/3.

Langevin theory is a fair approximation for atoms with large quantum numbers (the correspondence

principle). It is also applied to tiny ferromagnetic particles whose direction of magnetization is

randomized by thermal excitation (superparamagnetism). Ferrofluids are colloidal suspensions of

superparamagnetic particles

* " + , !

)

m'm

5.2 Quantum theory

S = 1/2

Th t t li it i th J 1/2 U ll thi i h S 1/2 L 0 Th l t


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The extreme quantum limit is the case J = 1/2. Usually this arises when S = 1/2, L = 0 There are only two

energy levels Ei = -µ 0gµ BMSH, where MS = ±1/2 and two possible orientations of the moment relative tothe applied field.Moments of the two states are mi = gµ BMS. The states are also known as ‘spin up’(↑) and

‘spin down’ (↓). A single electron has s = 1/2 and g = 2, so mi= ± µ B..The population of an energy level is given by Boltzmann statistics; it is proportional to exp{-E(θ)/k ΒT}The thermodynamic average is evaluated from the Boltzmann populations of the levels E i.

= Σimi exp{-Ei/k ΒT}/ ΣI exp{-Ei/k ΒT}

The denominator is a normalization factor known as the partitionfunction Z . In the present case there are only two terms in the sums

= [(1/2)gµ Bexp(x) - (1/2)gµ Bexp(-x)]/[exp(x) + exp(-x)] = mtanh(x)

where m = (1/2)gµ B. When S = 1/2, g = 2, this reduces to = µ0tanh(x). Here x = µ 0mH/k BT. At small fields, tanh(x) x,≈

henceχr = µ 0Nµ B2 /k BT

This is just three times as large as the classical value

Many ‘two-level systems’ in physics are treated by assigning

a pseudospin S = 1/2

z

g√ [S(S+1)]µ B

H 1/2

-1/2

-1/2

1/2

Ms

µ 0µ BHS = 1/2

General caseThe general quantum case was treated by Brillouin; m is gµ BJ, and x is defined as x = µ 0mH/k BT. Therare now 2J+1 energy levels E i = -µ 0gµ BMJH, with moment mi =gµ BMJ where MJ = J. J-1, J-2, … -J. Th


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sums over the energy levels now have 2J+1 terms.

a) Susceptibility To calculate the susceptibility, we can take x


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Ion ' 2SL* L S J g m+=$*

(B)

me =$(*(*)

(B)

me (B)

Ce. (' ) 2F/2 . /2 /2 0/1 2' 2' 2

3. (' 2) .%' ' '/ .2+ .4 .

Nd. (' .) '"5/2 0 ./2 5/2 4/ .21 .02 .'3m.(' ') "' 0 2 ' ./ 2'+ 204 -

Sm.(' ) 0%/2 /2 /2 2/1 +1 +4 1

6u. (' 0) 1F+ . . + - +++ +++ .'

7d. (' 1) 4S1/2 + 1/2 1/2 2 1++ 15' 45

89. (' 4) 1F0 . . 0 ./2 5++ 512 54

:!. (' 5) 0%/2 /2 /2 '/. +++ +0 +0

%o. (' +) "4 0 2 4 /' +++ +0 +'

6r . (' ) '"/2 0 ./2 /2 0/ 5++ 54 5

8m. (' 2) .%0 0 1/0 1++ 10 10

;9. (' .) 2F1/2 . /2 1/2 4/1 '++ '. '

2, Cr . (.d.) 'F./2 . ./2 ./2 2/ +11 .41 .4

Cr 2, #n. (.d') :+ 2 2 + - - '5+ '5

#n2, Fe.(.d) 0S/2 + /2 /2 2 50 52 5

Fe2, Co. (.d0) :' 2 2 ' ./2 01+4 '5+ 'Co2, Ni. (.d1) 'F5/2 . ./2 5/2 '/. 00.. .41 '4

Ni2 (.d4) .F' . ' /' 5+ 24. .2 Cu2 (.d5) 2:/2 2 /2 /2 0/ .+ 1. 5

b) MagnetizationTo calculate the complete magnetization curve, set y = µ 0gµ BH/k BT,then = gµ B∂ / ∂y[lnΣ-JJ exp{MJy}] [d(ln z)/dy = (1/z) dz/dy]Th h l l b l d i b i


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The sum over the energy levels must be evaluated; it can be written as

exp(Jy) {1 + r + r2 + .........r2J} where r = exp{-y}

The sum of a geometric progression (1 + r + r2+ .... + rn) = (rn+1 - 1)/(r - 1)

∴ Σ-JJ

exp{MJy} = (exp{-(2J+1)y} - 1)exp{Jy}/(exp{-y}-1)multiply top and bottom by exp{y/2}

= [sinh(2J+1)y/2]/[sinh y/2]

= gµ B(∂ / ∂y)ln{[sinh(2J+1)y/2]/[sinh y/2]} = gµ B /2 {(2J+1)coth(2J+1)y/2 - coth y/2}

setting x = Jy, we obtain = mBJ(x)

where the Brillouin function BJ(x) ={(2J+1)/2J}coth(2J+1)x/2J - (1/2J)coth(x/2J).

Again, this reduces to the previous equations

in the limits J → ∞ (m = gµ BJ) and J = 1/2, g = 2.

0 2 4 6 80

0.2

0.4

0.6 ∞

0.8 1/2

1.0

2

,

x

Comparison of the Langevin function and the

Brillouin functions for J = 1/2 and J = 2.


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Reduced magnetization curves for three paramagnetic salts, with Brillouin-theory predictions

The theory of localized magnetism gives a good account of magnetically-dilute 3d and 4f salts where the

magnetic moments do not interact with each other. Except in large fields or very low temperatures, the

M(H) response is linear. Fields > 100 T would be needed to approach saturation at room temperature.

The excellence of the theory is illustrated by the fact that data for quite different temperatures superpose

on a single Brillouin curve plotted as a function of x H/T≈


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Magnetization is given by the Brillouin function, = mBJ(x) where now x = µ 0mHi

/k BT. Thspontaneous magnetization at nonzero temperature Ms = N and M0= Nm. In zero external field, we

have

M /M B (x) (1)


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Ms /M0 = BJ(x) (1)

But also by eliminating Hi from the expressions for Hi and x,

Ms /M0 = (Nk BT/ µ 0M02nW)x

which can be rewritten in terms of the Curie constant C Ms /M0 = [T(J+1)/3C JnW]x (2)

The simultaneous solution of (1) and (2) is found graphically, or they can be solved numerically.

Graphical solution of (1) and (2) for J = 1/2 to find the spontaneous magnetization M s when T < TC. Eq. (2) is also

plotted for T = TC and T > TC.

* " + , !

)

m'm

! I !# ! > !# ! !#


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T > T C The paramagnetic susceptibility above TC is obtained from the linear term BJ(x) [(J+1)/3J]x≈

with x = µ 0m(nWM + H)/k BT. The result is the Curie-Weiss law

C/(T θ )


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χ = C/(T - θp)

where θp = TC = µ 0nWNg2µ B2J(J+1)/3k BIn this theory, the paramagnetic Curie temperature θp is equal to the Curie temperature TC, which is wherthe susceptibility diverges.

The Curie-law suceptibility of a paramagnet (left) compared with the Curie-Weiss susceptibility of a ferromagne

(right).

T

)' χ 1/χ

θp

6.2 Exchange Interactions.

What is the origin of the effective magnetic fields of 100 tesla or more which are responsible for

ferromagnetism ? They are not due to the atomic magnetic dipoles. The field at distance r due to a dipole


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m isBdip = (µ 0m /4 rπ3)[2cosθer + sinθeθ].

The order of magnitude of Bdip = µ 0Hdip is µ 0m /4πr3; taking m = 1 µ B and r = 0.1 nm gives Bdip = 4π. 10-7 x9.27. 10-24 / 4π10-30 1 tesla. Summing all the contributions of the neighbours on a lattice does no≈

change this order of magnitude; in fact the dipole sum for a cubic lattice is exactly zero!

The origin of the internal field Hi is the exchange interaction, which reflects the electrostatic Coulomb

repulsion of electrons on neighbouring atoms and the Pauli principle, which forbids two electrons from

entering the same quantum state. There is an energy difference between the ↑↓ and ↑↑ configurations fothe two atoms. Inter-atomic exchange is one or two orders of magnitude weaker than the intra-atomic

exchange which leads to Hund’s first rule.

The Pauli principle requires the total wave function of two electrons 1,2 to be antisymmetric on

exchanging two electrons

Φ(1,2) = -Φ(2,1)

The total wavefunction is the product of functions of space and spin coordinates Ψ(r1,r2) and χ(s1, s2)each of which must be either symmetric or antisymmetric. This follows because the electrons are

indistinguishable particles, and the number in a small volume dV can be written as Ψ2(1,2)dV =Ψ2(2,1)dV, hence Ψ(1,2) = ± Ψ(2,1).

The simple example of the hydrogen molecule H2 with two atoms a,b with two electrons 1,2 inhydrogenic 1s orbitals ψ i gives the idea of the physics of exchange. There are two molecular orbits, onespatially symmetric ΨS, the other spatially antisymmetric ΨA.ΨS(1 2) = (1/√2)(ψ 1ψb2+ ψ 2ψb1); ΨA(1 2) = (1/√2)(ψ 1ψb2 - ψ 2ψb1)


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ΨS(1,2) = (1/ √2)(ψ a1ψ b2+ ψ a2ψ b1); ΨA(1,2) = (1/ √2)(ψ a1ψ b2 ψ a2ψ b1)

The spatially symmetric and antisymmetric wavefunctions for H2.

The symmetric and antisymmetric spin functions are the spin triplet and spin singlet states

χS = |↑1,↑2>; (1/ √2)[|↑1,↓2> + |↓1,↑2>]; |↓1,↓2>. S = 1; MS = 1, 0, -1

χA = (1/ √2)[|↑1,↓2> - |↓1,↑2>] S = 0; MS = 0

According to Pauli, the symmetric space function must multiply the antisymmetric spin function, and

vice versa. Hence the total wavefunctions are

ΦI = ΨS(1,2)χA(1,2); ΦII = ΨA(1,2)χS(1,2)

The energy levels can be evaluated from the Hamiltonian H (r1, r2)

EI,II = ∫Ψ∗S,A(r1,r2) H (r1,r2) ΨS,A(r1,r2)dr1dr2

With no interaction of the electrons on atoms a and b, H (r1, r2) is just H 0= (-ђ2 /2m){∇12 + ∇12} + V1+V2.

S = 0 S = 1

The two energy levels EI, E ,II are degenerate, with energy E0. However, if the electons interact via a term H ’ = e2 /4πε0r122, we find that the perturbed energy levels are E I = E0+2 J , EII = E0 -2 J. The exchangintegral is

J ∫ * *( )H’( ) d d


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J = ∫ψ a1*ψ b2*(r) H’(r12) ψ a2ψ b2dr1dr"

and the separation (EII - EI) is 4 J . For the H2 molecule, EI is lies lower than EII, the bonding orbital single

state lies below the antibonding orbital triplet state J is negative. The tendency for electrons to pair off inbonds with opposite spin is everywhere evident in chemistry; these are the covalent interactions.We write

the spin-dependent energy in the form

E = -2( J ’/ ђ2)s1.s2

Energy splitting between the singlet and

triplet states for hydrogen.

Heisenberg generalized (6.14) to many-electron atomic spins S1 and S2, writing his famous Hamiltonian

H = -2 J S1.S2

Where ђ2 is absorbed into the J . J > 0 indicates a ferromagnetic interaction (favouring ↑↑ alignment). J <0 indicates an antiferromagnetic interaction (favouring ↑↓ alignment). When there is a lattice, theHamiltonian is generalized to a sum over all pairs i.j, -2Σi>j J ijSi.S j. This is simplified to a sum with asingle exchange constant J if only nearest-neighbour interactions count.

where the operator s1.s2 is 1/2[(s1 + s2)2 - s1

2 - s22]. According

to whether S = s1+ s2 is 0 or 1, the eigenvalues are -(3/4)ђ2

or -(1/4)ђ2

.The splitting betweeen the ↑↓ singlet state (I)and the ↑↑ triplet state (II) is then J ’.

Ε Ι

Ε ΙΙ

J'

The Heisenberg exchange constant J can be related to the Weiss constant nW in the molecular fieldtheory. Suppose the moment gµ BSi interacts with an effective field Hi = nWM = nWNgµ BS, and that in theHeisenberg model only the nearest neighours of Si have an appreciable interaction with it. Then the site

Hamiltonian is


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H i = -2(Σ j J S j).Si -H≈ igµ BSi

The molecular field approximation amounts to neglecting the local correlations between Si and S j. If Z ithe number of nearest neighbours in the sum, then J = nWNg

2µ B2 /2Z. Hence, from the expression for TC interms of the Weiss constant nW

TC = 2Z J J(J+1)/3k B

Taking the example of Gd again, where TC = 292 K, J = 7/2, Z = 12, we find J /k B = 2.3 K.

Exchange in metals

The Heisenberg theory describes the exchange coupling of electrons in localized orbitals. It does no

apply in metals, where there are partly-filled bands. Generally, the energy of any electronic system i

lowered as the wavefunctions spread out. This follows from the uncertainty principle ∆p∆x h. When≈many more-or-less delocalized electrons are present in different orbitals, the calculation of exchange is a

delicate matter. Energies involved are only 0.01 eV, compared with bandwidths W of order 1 - 10eV≈

There are competing exchange mechanisms with different signs of coupling.

The principal exchange mechanism in ferromagnetic and antiferromagnetic metals involves direc

overlap of the partly-localized atomic orbitals of adjacent atoms. Other exchange mechanisms involve the

interaction of localized and delocalized moments in the metal.

In 3d metals, the electrons are described by extended wave functions and a spin-polarized local density ostates, introduced in § 5.3. The bandwidth in the tight-binding model is W = 2Zt where Z is the numbe

of nearest neighbours and t is the transfer integral. In 3d metals t 0.1 eV. In a roughly half-filled band≈

the exchange is antiferromagnetic, whereas in a nearly-filled or nearly empty band it tends to be

f ti Th i f th h d d fi t b d th th i t t i


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ferromagnetic. The sign of the exchange depends first on band occupancy, then on the interatomic

spacing, with ferromagnetic exchange favoured at larger spacing.

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

↑ ↑ ↑ ↑ ↑ ↓↓↓↓↓

No

Yes

Yes

Yes

↑ ↑ ↑

↑

↑ ↑↑↓↑↓↑↓ ↑ ↑↑↓↑↓↑↓

Electron delocalization in d-bands which are half-full, or almost empty or almost full.

Stoner criterion.

Ferromagnetic exchange in metals does not necessarily lead to spontaneous ferromagnetic order. The

Pauli susceptibility must exceed a certain threshold. Ferromagnetic metals have an exceptionally large

d it f t t t th F i l l N(E ) St li d W i ’ l l fi ld id t th f


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density of states at the Fermi level N (EF). Stoner applied Weiss’s molecular field idea to the free

electron model.

Hi = nSM

Here nS is the molecular field constant; The bare

Pauli susceptibility χp = M/(H + nSM) is enhanced:

χ = M/H = χp /(1 - nSχp)

Hence the susceptibility diverges when

nSχp> 1.

The value of nS is about 10,000 in 3d metals.

The Pauli susceptibility (§3.4) is proportional

to the density of states N (EF).Only metals witha large N (EF) can order ferromagnetically. A big peak in the density of states at the Fermi level is

needed. This is why the late 3d elements Fe, Co, Ni are ferromagnetic, but the early 3d or 4d elements

Ti, V, Cr are not. Elements in the middle of the series Cr, V are antiferromagnetic because the 3d band

is approximately half-full. When the Stoner criterion is satisfied, the ↑ and ↓ bands split spontaneously.

Comparision of N(EF) with 1/ I for metallic elements.

6.3 Ferromagnetic domains

Having developed his molecular field theory, Weiss postulated that the reason why most lumps of iron

for example, do not appear ferromagnetic is because they contain many ferromagnetic domain

m ti d i diff t di ti Th h d m i f m i t d th m t t ti lf


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magnetized in different directions. The reason why domains form is to reduce the magnetostatic self

energy as far as possible. This self-energy Ems can be written in two equivalent forms

Ems = (1/2)∫ all spaceµ 0H2dV or Ems = -(1/2)∫ magnetM.BdV

The value of H inside the magnet is the demagnetizing field Hd = - N M, where N is the demagnetizing

factor. Hence

Ems = (1/2)µ 0 N M2V

Reduction of the demagnetizing energy of a ferromagnet by splitting up into domains. The values of Ems are

approximately 0.10, 0.05 and 0.02 in units of µ 0M2V

The particular domain structure adopted by a piece of ferromagnetic material is the result of minimizing

the total energy, which in the sum of four terms

Etot = Ems + Eex + Ea + EZ

The exchange energy Eex is the sum of (6.15) over all pairs of atoms in the sample. Ea is the anisotropy

ener discused next. E is the ener in an external field.

Some ways of visualizing domains:

♦ Bitter pattern (spread ferrofluid on the surface) [depends on stray field]♦ Atomic force microscope with a magnetic tip [depends on stray field]


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♦ Magnetooptic Kerr effect [depends on M in the surface region]♦Lorentz (transmission electron) microscopy [depends on B in a thin foil]

itter method .left/ and *&* .right/

Domain pattern on the surface of a nickel crystal reveled

by the Bitter method.

Anisotropy

The tendency for the ferromagnetic axis of a domain to lie along some fixed direction(s) in a sample

known as the easy axis is the phenomenon of magnetic anisotropy. Strong easy-axis anisotropy is a

prerequisite for hard magnetism Near zero anisotropy is needed for soft magnets


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prerequisite for hard magnetism. Near-zero anisotropy is needed for soft magnets.

Generally, the tendency for magnetization to lie along some easy axis is represented by an energy

expression of which the leading term is

Ea = K1sin2θ

where θ is the angle between M and the anisotropy axis.The anisotropy constant K1 depends on temperature, and goes to zero at TC. Units are J m

-3. Value

typically range from 102

to 106

J m-3

.

Most common source of anisotropy is magnetocrystalline anisotropy, where the magnetization process i

different along different crystallographic directions. This is related to spin-orbit coupling. The expression

Ea = K1sin2θ is valid for uniaxial crystal structures (hexagonal, tetragonal, trigonal). In cubic symmetry, a

different expression is necessary to reflect the symmetry of the cubic crystal. It is

Eacubic = K1c(α12α22 + α22α32 + α32α12)

where α1, α2 and α3 are the direction cosines of the magnetization direction with respect to the coordinateaxes. However, this expression reduces to K1sin

2θ in the limit of small deviations from an easy axis.

The magnetization curve and hysteresis loop


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Starting from the thermally-demagnetized state, the following magnetization processes are involved

around the hysteresis curve


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1 - 2 Reversible wall motion

3 irreversible wall motion

4 coherent rotation5 - 6 nucleation, irreversible wall motion.

In bulk material it is practically impossible

to calculate the hysteresis and coercivity.

One case where it is possible is in very

Small single-domain particles wherethe magnetization reverses coherently

The first two quadrants of the hysteresis loop

The energy loss on cycling the the M(H) loop is ∫ loopµ 0HdM. For minimum losses in ac applications thecoercivity nust be as low as possible.

The other extreme is needed for a permanent magnet, where the hysteresis loop should be broad to avoid

self demagnetization in the demagnetizing field - N M. If the magnet is to remain fully magnetized no

matter what shape it is, it should have a square loop with Hc > M

1

2

3

4

5

6

M

HHc

Mr

H

Domain walls

We now focus on the magnetic structure of the region separating two oppositely-magnetized domains

This 180° domain wall is known as a Bloch wall. The structure is obtained by minimizing the energy E tE + E + E We give an approximate treatment which illustrates the physics


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= Ems + Eex + Ea. We give an approximate treatment which illustrates the physics.

Granted we need to create a wall to reduce Ems, we seek a compromise between a wide wall, which would

minimize Eex and a narrow one which would minimize Ea. Suppose we have a ferromagnet with the siteon a cubic lattice with side a, and suppose also that the magnetization turns by an angle ϕ from one site tothe next across the wall, as shown in the figure.

A 180 Bloch wall between oppositely-magnetized domains

in a sample with uniaxial anisotropy.

The wall width δw is related to a and ϕ, δw =πa/ ϕ Exchange energy for a neighbouring pair of spins is

-2 J Si.S j. = -2 jS2cosϕ -2≈ jS2(1-ϕ2 /2 +....)

The extra energy due to misalignment is J S2ϕ2.For a line of spins across the wall, this is J S2πϕ,and per unit area of wall E’ex= J S

2πϕ/a2

If this were the only term, ϕ would be very small,δw →∞. But the spins in the wall are pointingaway from the easy axis. The anisotropy cost is

about (1/2)K1a3 per spin or per unit area ,

’ =

The wall energy γ wall = E’ex + E’a is mimimum with respect to ϕ when

∂Ew / ∂ϕ = J S2π /a2 - (1/2)K1a/ ϕ2 = 0.ϕ2 = (1/2)K1a3 / J S2


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δw =π√(2 J S2 /K1a)

γ w =π√(2K1 J S2 /a)

Taking the example of cobalt, K1 = 500 kJ m-3, J = S 1,and using T≈ C = 1390 K and Z = 12 to deduce

J from the expression TC = 2Z J J(J+1)/3k B gives J =1.2 10-21 J. a 0.2 nm; hence≈ δw 15 nm, about 75≈

atomic spacings; γ w = 8 mJ m-2.

Domain wall pinning

The domain wall acts like an elastic membrane with energy γ w J m-2. It will be strongly pinned at defects,especially planar defects, with different J or K1 to the bulk if these defects have dimension comparable

to the domain wall width δw. Weaker pinning occurs when there are many defects distributed throughout

the wall width. Generally, there is always some distribution of defects in anysample of magnetic material. Suppose the energy of the system depends just

on the wall position x, and applied field H, where f(x) represents the effects

of the pinning sites

We show how a hysteresis loop results from the energy landscape with several minima due to pinning. At

local energy minimum df(x)/dx = 2µ 0MH. As the wall jumps from points with the same df(x)/dx on

increasing field, the magnetization changes discontinuously in a Barkhausen jump. The hysteresis loopof a macroscopic sample consists of many such jumps.


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M x≈

1

2

34

5

6

H

x

f(x)

1

2

3

4

5

6

x

df/dx

1

6

2 3

4

5

µ 0MsH

a) Energy as a function of wall position. b) The equilibrium condition df(x)/dx = 2µ 0MH c) a hysteresis loop due tofield cycling.

a/

6/

c/

Single-domain particles.

When ferromagnetic particles are no bigger than a few tens of nanometers, it does not pay to form a

domain wall. The energy gain is of order (1/2)µ 0M2V µ ≈ 0M2r3, whereas the cost of forming the wall iγwπr2. When the particle radius r is sufficiently small, the latter outweighs the former. Again, when the


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γ wπr . When the particle radius r is sufficiently small, the latter outweighs the former. Again, when theparticle is very small, magnetization reversal takes place by coherent rotation of the magnetic moment m

If an external field H is applied at an angle ϕ to the easy direction, and the magnetization is aan angle θ to the easy direction, the energy is Etot = Ems + Ea + EZ the sum of magnetostatic, anisotropyand Zeeman terms. The first two are represented by a term Kusin

2θ, hence

Etot = KuVsin2θ − µ 0mHcos(ϕ-θ)

The energy can be minimized, and the hysteresis loop calculated

numerically for a general angle ϕ. This is the Stoner-Wohlfarth model.

Two special cases of great interest: are ϕ = 0 and ϕ =π /2. In the first case θ = 0 orπ, and a square loop isobserved with a flip from θ = 0 to θ =πat Hc = 2Ku /M, (M = m /V). These square loops are valuable for

θ

m

Hϕ

A single domain particle where the

magnetization rotates coherently.

M

H

M

H

6.4 Magnetic measurements

Methods of measuring magnetization, and hence susceptibility or hysteresis, of magnetic material

depend either on the force on a magnetic moment in a nonuniform field F = -∇ m.B, or on Faraday’s lawε = −dΦ /dt, where Φ is the flux threading a circuit and ε is the induced emf. We consider three examples:


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, g p

Force method (Faraday balance)

Considering one component of the force equation; ( m is constant)

Fz = -(mx∂Βx / ∂z + my∂Βy / ∂z +mz∂Βz / ∂z)

If the field is in the x-direction, Fz = -mx∂Βx / ∂z. The field gradient may be produced using shaped polepieces, or by special field gradient coils. The gradient is calibrated with a sample of known magnetization

of susceptibility. Sensitivity may be improved by applying an alternating current to the gradient coils, andusing lock-in detection.

Extraction method .

Here the sample is first located in a pickup coil in the field, and then removed to a distant point. An em

ε is induced in the coil, and

∫ε dt = Φ

The flux Φproduced by the dipole m is proportional to its magnetization. A SQUID detector can be usedto achieve great sensitivity in the measurement of Φ. Sensitivity of 10-11 A m2 in m is then achievable.

Vibrating-sample magnetometer (Foner magnetometer)

This is an ac variant of the extraction method. The sample vibrates with an amplitude of a few hundred

microns at about 100 Hz, and an alternating voltage is induced in pickup coils. A reference signal is

generated in another pickup coil or with a capacitor, and both signal and reference are fed into a lockin


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g p p p g

amplifier. Typical sensitivity is 10-8A m2.

Method of measuring magnetization or susceptibility. a) Faraday balance, b) extraction magnetometer and c) Vibrating

sample magnetometer

Data on some ferromagnets

material ρ TC µ 0M (290 K) m0 K1


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(kg/m3) (K) (T) (µ B /formula) (kJ/m3)Fe 7874 1044 2.15 2.2 50

Co 8836 1390 1.81 1.7 530

Ni 8902 628 0.62 0.6 -5

Fe65Co35* 8110 1240 2.34 2.5 40

Ni80Fe20 ¶ 8715 843 1.04 1.1


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molecular field theory to two sublatices. Various types of noncollinear structures arise from competing interactions

especially in noncrystalline soilds. The elementary excitations from the magnetic ground state are spin wave

(magnons) which are described by a dispersion relation E = Dswq2 in the ferromagnetic caase. Neutron scattering is

powerful general method for determining both magnetic structures (magnetic Bragg scattering) and for measuring

magnetic excitations (magnetic inelastic scattering)

In the last chapter, we discussed ferromagnetic order, which arises from ferromagnetic exchange

interactions J > 0. In the first two sections here we consider magnetic order which can arise from

antiferromagnetic exchange interactions J


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Sublattice magnetization of an antiferromgnet. TN is the Néel temperature

Above TN, Mα = χrHαi where χ= C ’ /T with C ’= µ 0(N/2)meff 2 /3k B. Here N/2 is the number of atoms per mof each sublattice. Hence

MA = (C ’ /T)(nW’MA + nWMB + H)

MB = (C ’ /T)(nWMA + nW’MB + H)

The condition for the appearance of spontaneous sublattoce magnetization is that these equations have a

nonzero solution in zero applied field. The determinant of the coefficients is zero, hence [(C /T)nW’- 1]2

[(C /T)nW]2 = 0, which yields

TN = C ’(nW’ - nW)

To calculate the susceptibility above TN we evaluate χr = (MA + MB)/H. Adding the equations for MA andMB, we find the Curie-Weiss law

χr = C /(T - θp)where C = 2C ’ and θp = C ’(nW’+ nW). In the two-sublattice model, we can therefore evaluate both n Wand nW’ from TN and θp. Since nw < 0, θp < TN, and it is usually negative. Normally 1/ χr is plotted versu

T

χ χ χParamagnet Ferromagnet Antiferromagnet


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T TC T TN Tθp

Comparison of the susceptibility of a

paramagnet, a ferromagnet and an

antiferromagnet.

The antiferromagnetic axis along which the sublattice magnetizations lie is determined by

magnetocrystalline anisotropy, and the response below TN depends on the direction of H relative to thi

axis.

Calculation of the susceptibility of an antiferromagnet

below TN. In a) the dashed lines show the configuration

after a spin flop.

If a small field is applied parallel to the axis , we can calculate χ|| by expanding the Brillouin functionabout x0, their arguments in zero applied field. For simplicity we take n W’= 0, and the result for χ|| =[MA(H) + MB(H)]/H is

χ|| = Nm2 BJ’(x0)/[k BT + N’nWm2 BJ’(x0)]This rises from 0 at T = 0 to C /(T - θp) at TN, where BJ’(0) = (J+1)/3J. x0 is µ 0mnWM/k BT, where M is thesublattice magnetization in zero field

/(0

(Ba) χ||

(0

(B

/

/b

2δ

δ

b) χ⊥

The perpendicular susceptibility can be calculated assuming the sublattices are canted by a small angle

δ, as in Fig 7.3. In equilibrium the torque on each one is zero, hence M αH = MαnWMβsin2δ. Since M =2Mαsinδ,

χ⊥ = 1/nWThe perpendicular susceptibility is therefore constant and independent of temperature up to TN. For a


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N

powder, the average is (1/3)χ|| + (2/3)χ⊥, or 2/3nW at low temperature.

Parallel and perpendicular susceptibility of an antiferromagnet.

Since χ⊥ > χ|| for all T < TN, we might expect that an antiferromagnet will always adopt the traansverse,flopped configuration. That it doesn’t is due to magnetocrystalline anisotropy, represented by an

effective anisotropy field HK, which acts on each sublattice along the antiferromagnetic axis. When H is

applied parallel to Mα the spin flop occurs when the energies of the parallel and perpendicular

configurations are equal;-2MaHK - (1/2)χ||Hsf 2 = -(1/2)χ⊥Hsf 2

Hsf = [4MαHK /(χ⊥ - χ||]1/2

When T


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7.3 Other forms of magnetic order

Various other forms of magnetic order can arise in solids. When

ferromagnetic and antiferromagnetic interactions compete, and cannot all

be satisfied simultaneously, the system is said to be frustrated . A

noncolinear spin structure may then arise such as a helimagnetic θ

2θ


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noncolinear spin structure may then arise, such as a helimagnetic

structure where there are ferromagnetic planes, but the ferromagnetic

axis turns by an angle θ from one plane to the next.

A helimagnet

Frustration may also arise with purely antiferromagnetic interactions,

when there are odd-membered rings. Some examples are shown below.

θ

In disordered and amorphous solids, frustrated antiferromagnetic interactions, or

competing ferromagnetic and antiferromagnetic interactions may give rise to spin

freezing in random directions. Such materials are known as spin glasses. A sin glass

Spin waves

The exchange energy in the ferromagnetic ground state is -2Z J S2 per site. Elementary excitations

from the ferromagnetic ground state are spin waves, illustrated below. These extended spin

deviations are also known as magnons by analogy with phonons, the quantized lattice waves. A

single localized spin reversal ↑↑↑↑↓↑↑↑↑ costs 8JS2 (2J) for S = 1/2, which is greater than kBTC


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single localized spin reversal ↑↑↑↑↓↑↑↑↑ costs 8 J S (2 J ) for S 1/2, which is greater than k BTCfor a chain. TC = 2ZS(S+1)/3 (for J = S = 1/2, Z = 2). Such expensive excitations cannot occur at low

temperature; instead the atoms all share the reversal, with periodic oscillation of spin orientation.

Illustration of a spin wave.

In one dimension, the relation between the frequency and wave-vector of a wave-like excitation of

the spin system can be calculated classically

hωq = 4 J S(1 - cos qa)In the limit of small wave vectors, the dispersion relation becomes

Eq D≈ swq2

where Eq = hωq, Dsw = 2 J Sa2; a is the interatomic spacing.The expression in any of the three basic cubic lattices is

the same, where a is the lattice parameter.

The spin-wave dispersion

relation for a chain of atoms.

&

'(a

Excitation of spin waves is responsible for the fall of magnetization with increasing T which is much

faster than expected from molecular-field theory, given J . They also contribute to resistivity andspecific heat, giving a T3/2 variation of specific heat at low temperature.


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Spontaneous magnetization of a ferromagnet

lecture magnetism in solids-trinity college-js3015

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