1. Lecture: Basics of Magnetism: Magnetic reponse Hartmut Zabel Ruhr-University Bochum Germany

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1. Lecture: Basics of Magnetism:

Magnetic reponse

Hartmut ZabelRuhr-University Bochum

Germany

Lecture overview

2H. Zabel, RUB 1. Lecture: Magnetic Response

1. Lecture: Basic magnetostatic properties2. Lecture: Paramagnetism3. Lecture: Local magnetic moments

Content

31. Lecture: Magnetic Response

1. Definitions2. Electron in an external field3. Diamagnetism4. Paramagnetism: classical treatment of

H. Zabel, RUB

4

1. Classical magnetic moments

LLmq

21

ωπrmmq

π21

Aπ2

qωIAm

e

2e

e

γ=====

Magnetic dipole moment = current × enclosed area

Loop current generates a magnetic field

Loop current has an angular momentum

π2qω

TqI ==

γ = gyromagnetic ratio, me= electron mass

1. Lecture: Magnetic ResponseH. Zabel, RUB

Torque and precession

5

Zeeman energy of magnetic moment in an external magnetic field: Bm-=E

⋅Energy is minimized for m || B. B is the magnetic induction or the magnetic field density. Applying B, a torque is exerted on m:

BmT

×=If m were just a dipole, such as the electric dipole, it would beturned into the field direction to minimize the energy. However, m isconnected with an angular momentum, thus torque causes thedipole to precess:

BLγdtLdT

×==

Assuming B = Bz, the precessional frequency is:

zL Bγ=ωBz

ωL is called the Lamor frequency. See also EPR, FMR, MRI, etc.

1. Lecture: Magnetic ResponseH. Zabel, RUB

Bohr magneton

6

Bee

Bohr μme

21

mq

21

m -- === L

An electron in the first Bohr orbit with a Bohr radius rBohr has the angular momentum:

Then magnetic moment is:

L

Bµ

Because of negative charge, L and m are opposite.

γ==e

B me

21μ

µB is the Bohr magneton. [µB] = 9.274 x 10-24 Am2.

Magnetic moment: [m] = A m2

== ω2BohrermL

1. Lecture: Magnetic ResponseH. Zabel, RUB

7

LOrb SSpin S of the electon contributes to the magnetic moment:

Smqm

e

=spin

Including orbital and spin contributions, the magnetic moment of an electron is:

)S2Lγ()S2L(mq

21m

e

+−=+=

Electron spin

The missing factor ½ is of quantum mechanical origin and will bediscussed later.

1. Lecture: Magnetic ResponseH. Zabel, RUB

Magnetic field and magnetic induction

8

Oersted field H due to dc current: πr2IH =

Any time variation of the magnetic flux Φ = BA through the loop causes an induced voltage: ( )AB

dtdUind

⋅−=

Therefore B is called the magnetic induction or the magnetic fluxdensity B = Φ/A. In vacuum both quantities are connected via the permeability of thevacuum: HμB 0=

-70

V sμ 4 10A m

⋅= π

⋅[ ] [ ]0 2

V s A V s× = = T2 A m m m

IBμπr

⋅ ⋅ = ⋅ = ⋅ 4

2

V s1 1 T 10 Gm

⋅= =

1. Lecture: Magnetic ResponseH. Zabel, RUB

9

1. Magnetization is the sum over all magnetic moments in a volume element normalized by the volume element:

2. Thermal average of the magnetization:

3. Magnetic susceptibility:

4. Magnetic Induction: .

∑=i

imV1M

mVNM

=

HM

χ ,HχM magmag ∂∂

==

( ) HμH μμ)χ(1HμMHμB r0mag00

==+=+=

Definitions

H = magnetic field, usually externally applied by a magnet. µ0 = magnetic permeability of the vacuum. µr = relative magnetic permeability µr = (1+χ) (tensor, or a number for collinearity)

1. Lecture: Magnetic ResponseH. Zabel, RUB

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Potential energy (Zeeman – term):

BmEZeeman

⋅−=

BEm Zeeman

∂∂

−=

2

2

0 BE

VN

HM Zeeman

mag ∂∂

µ−=∂

∂=χ

1. Derivative → magnetic moment:

2. Derivative → Susceptibility:

Potential Energy and Derivatives

1. Lecture: Magnetic ResponseH. Zabel, RUB

The susceptibility is the response f

What is more fundamental, H or B?

11

( ) [ ] N=F ×= BvqF

Lorentz force:

Vector potential: [ ] 2∇ mVsT /AB ==B ×=

Zeeman energy: [ ] WsVAsJ ===E •= BmE-

Oersted field: [ ] mA=H =πr2I

H

Magnetization: [ ] mA=M χH=M

1. Lecture: Magnetic ResponseH. Zabel, RUB

Classification

121. Lecture: Magnetic Response

Application of an external field:a. Paramagnetism: χ>0 und µr >1

b. Diamagnetism: χ< 0 und µr <1

Ideal diagmagnetism, realized in superconductors with M and B antiparallel, for χ = − 1 and µr =0.

Magnetic moments align parallel toexternal field, field lines are moredense in the material than in vacuum.

External field is weakend by inducingscreening currents according to Lenz rule. Field lines are less dense thanin vacuum.

c. Ferromagnetism: Spontaneous Magnetization withoutexternal field due to the interaction of magnetic moments

µr attaines very high values forferromagnets, > 104-105

H. Zabel, RUB

13

Consider a non-relativistic Hamilton operator for electrons in an external magnetic field:

( ) 2

21H Aqpme

+=

A

2. Electron in an external field

The vector potential: is defined by the Coulomb gauge: and using

AB

×=∇

2z

z B~smdiamagnetiB~

orbitalismparamagnet

energykinetic

2222

122H aB

meLB

mp

ze

zzBe

+µ+=

222

32 ayx =+

( )z,B,=B 00

Where we assumed an average over the electron orbitperpendicular to the magnetic field:

*Lz is here a dimensionless quantum number

*

1. Lecture: Magnetic ResponseH. Zabel, RUB

Hamiltonian for electron with spin

14

BSmeBSgBmE BssZeeman

⋅=⋅µ=⋅= --

Considering the electron spin in the external field with a Zeeman energy:

224-1027.92

- Amm

eB ×==µ

2=sgLandé factor

Bohr magneton

( )

2z

z B~smdiamagnetiB~

orbital spinismparamagnet

energykinetic

2222

122

2H aB

meSLB

mp

ze

zzzBe

++µ+=

+

Hamilton operator for spin and orbital contributions of a single bondelectron then is:

1. Lecture: Magnetic ResponseH. Zabel, RUB

The gS=2 for the electron is put into the Schrödinger equation by „hand“ but would occur naturally using the Dirac equation. The exact value of 2.0023 isdetermined by QED.

Response functions

15

zBm

∂H∂-=

( ) 02 >+µ zzB SL

ze

Bam

Ze 22

62

2

0 6a

mZe

e

µ−

2

2

0 ∂H∂

zmag BV

Nµ=χ -

1. derivative 2. derivative

Diamagnetic responsefor Z electrons

Paramagnetic response *0

1. Lecture: Magnetic ResponseH. Zabel, RUB

*For single atom we can not define a paramagnetic susceptibility. This is onlypossible for an ensemble of atoms.

16

3. Properties of the Langevin diamagnetism

χLangevin is constant, independent of field strength; χLangevin is induced by external field; χLangevin < 0, according to Lenz‘ rule; χLangevin is alway present, but mostly covered by bigger and positive

paramagnetic contribution; χLangevin the only contribution to magnetism for empty or filled

electron orbits; χLangevin yiels ⟨a⟩ and the symmetry of the electron distribution; χLangevin is proportional to the area of an atom perpendicular to the

field direction, important for chemistry; χLangevin is temperature independent.

With Z electrons in an atom and an effective radius of <a>

22

0

2

0 6-

6- ∑ a

mZe

VN

me

VN

eeLangevin µµχ ==

i2

ir

1. Lecture: Magnetic ResponseH. Zabel, RUB

Examples for Diamagnetism

171. Lecture: Magnetic Response

Material χLangevin at RT

He -1.9 ⋅ 10-6cm3/mol Xe -43 ⋅ 10-6cm3/mol Bi -16 ⋅ 10-6cm3/g Cu -1.06 ⋅ 10-6cm3/g Ag -2.2 ⋅ 10-6cm3/g Au -1.8 ⋅ 10-6cm3/g

( χ is normalized to the magnetization of 1 cm3 containing one 1 Mol of gas at 1 Oe)

• All noble metals and noble gases are diamagnetic. In case of the nobel metals Ag, Au, Cu mainly the d-electrons contribute to the diamagnetism.

• In 3d transition metals the diamagnetismus is usually exceeded by themuch bigger paramagnetic response.

H. Zabel, RUB

18

Anisotropy of diamagnetismus for Li3N

Levitation of diamagnetic materials

1. Lecture: Magnetic ResponseH. Zabel, RUB

19

(free = without interactions)Orientation of permanent and isolated magentic moments in an external field Bz = µ0Hz parallel to the z-axis (orientational polarization)

( ) ( )x

xcothxL 1−= Langevin function

Hz

m

θ

4. Paramagnetism of free local moments: classical treatment

TkHm

VN

TkHmLm

VN

mVNM

B

zz

THohe

B

zzz

3

)cos(

2

−

≈

=

= θ

1. Lecture: Magnetic ResponseH. Zabel, RUB

Langevin function

20

( )

( )

∫

∫

∫

∫π

π

==

0

0

cos

cos

sin2

cossin2coscos

dθeθπ

dθeθθπ

dΩe

dΩθeθ

TBkθmB

B

B

pot

B

pot TkθmB

TkθE

TkθE

TkB

B

0µ=xθθπ=φθθ=Ω dddd sin2sin

L(x): Langevin-Funktion

µ=

TkBmLNM

B

z 00

( ) ( )xLx

xdsedxd

dse

dssesx

sx

sx

≡−===θ ∫∫

∫−

−

− 1cothlncos1

11

1

1

1

θ= cossusing

1. Lecture: Magnetic ResponseH. Zabel, RUB

21

Magnetization of paramagnetic moments in an external field

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ0Hz /kBT

Mag

netiz

atio

n

Hz

1. Lecture: Magnetic ResponseH. Zabel, RUB

22

Curie-Suszeptibilität χCurie in HTA with the Curie-constant C:

TC

Tkm

VN

HM

B

z

zCurie =

µ=

∂∂

=χ3

20

B

z

km

VNC

3

20µ

=

Paramagnetic Susceptibility

( )zzBz SLm 2+= µMagnetic moment:

χ1

T

Linear dependence fullfilledat high temperatures. At lowT often deviations observeddue to interactions.

But: However, magnetism is not a classical problem, thus Langevin function is only a roughapproximation. As quantum mechanics allows only discrete values for the z-component of the magnetic moments, a different approach has to be chosen ⇒Brillouin function replaces the Langevin function. –The susceptibility of superparamagnetic particles containing a macrospin can betreated classically as the spin orientation of nanoparticles in the field is continuous.

1. Lecture: Magnetic ResponseH. Zabel, RUB

Susceptibility of the Elements

23H. Zabel, RUB 1. Lecture: Magnetic Response

From J.M.D. Coey

Paramagnetic

Diamagnetic

Summary

241. Lecture: Magnetic Response

χ1

T

22

0 6- a

mZe

VN

e

cdiamagnetiLangevin µ=χ

χ

T

3. Paramagnetic response (HTA):

2. Diamagnetic response:

( )

2z

z B~smdiamagnetiB~

orbital spinismparamagnet

energykinetic

2222

122

2H aB

meSLB

mp

ze

zzzBe

++µ+=

+

1. Hamilton operator for an electron in an external field:

Tkm

VN

B

zicparamagnetCurie 3

20µ

=χ

H. Zabel, RUB

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