1. Lecture: Basics of Magnetism: Magnetic reponse Hartmut Zabel - - PowerPoint PPT Presentation

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1. Lecture: Basics of Magnetism: Magnetic reponse Hartmut Zabel Ruhr-University Bochum Germany Lecture overview 1. Lecture: Basic magnetostatic properties 2. Lecture: Paramagnetism 3. Lecture: Local magnetic moments 2 H. Zabel, RUB 1.

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  • 1. Lecture:

Basics of Magnetism:

Magnetic reponse

Hartmut Zabel Ruhr-University Bochum Germany

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Lecture overview

2

  • H. Zabel, RUB
  • 1. Lecture: Magnetic Response
  • 1. Lecture: Basic magnetostatic properties
  • 2. Lecture: Paramagnetism
  • 3. Lecture: Local magnetic moments
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Content

3

  • 1. Lecture: Magnetic Response
  • 1. Definitions
  • 2. Electron in an external field
  • 3. Diamagnetism
  • 4. Paramagnetism: classical treatment of
  • H. Zabel, RUB
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4

  • 1. Classical magnetic moments

L L m q 2 1 ω πr m m q π 2 1 A π 2 qω IA m

e 2 e e

   γ = = = = =

Magnetic dipole moment = current × enclosed area Loop current generates a magnetic field Loop current has an angular momentum

π 2 qω T q I = =

γ = gyromagnetic ratio, me= electron mass

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Torque and precession

5

Zeeman energy of magnetic moment in an external magnetic field:

B m

  • =

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:

B m T    × =

If m were just a dipole, such as the electric dipole, it would be turned into the field direction to minimize the energy. However, m is connected with an angular momentum, thus torque causes the dipole to precess:

B L γ dt L d T     × = =

Assuming B = Bz, the precessional frequency is:

z L

B γ = ω

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

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Bohr magneton

6

B e e Bohr

μ m e 2 1 m q 2 1 m

  • =

= =  L

An electron in the first Bohr orbit with a Bohr radius rBohr has the angular momentum: Then magnetic moment is:

L 

B

µ 

Because of negative charge, L and m are opposite.

  γ = =

e B

m e 2 1 μ

µB is the Bohr magneton. [µB] = 9.274 x 10-24 Am2. Magnetic moment: [m] = A m2

 = = ω

2 Bohr er

m L

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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7

LOrb S

Spin S of the electon contributes to the magnetic moment:

S m q m

e

  =

spin

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

) S 2 L γ( ) S 2 L ( m q 2 1 m

e

     + − = + =

Electron spin

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

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Magnetic field and magnetic induction

8

Oersted field H due to dc current:

πr 2 I H =

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

( )

A B dt d Uind   ⋅ − =

Therefore B is called the magnetic induction or the magnetic flux density B = Φ/A. In vacuum both quantities are connected via the permeability of the vacuum:

H μ B =

  • 7

V s μ 4 10 A m ⋅ = π ⋅

[ ] [ ]

2

V s A V s × = = T 2 A m m m I Bμ πr ⋅ ⋅   = ⋅ =   ⋅  

4 2

V s 1 1 T 10 G m ⋅ = =

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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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 i

m V 1 M  

m V N M   =

H M χ , H χ M

mag mag

∂ ∂ = =  

( )

H μ H μ μ ) χ (1 H μ M H μ B

r mag

      = = + = + =

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 Response
  • H. Zabel, RUB
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10

Potential energy (Zeeman – term):

B m EZeeman   ⋅ − = B E m

Zeeman

∂ ∂ − =

2 2

B E V N H M

Zeeman mag

∂ ∂ µ − = ∂ ∂ = χ

  • 1. Derivative → magnetic moment:
  • 2. Derivative → Susceptibility:

Potential Energy and Derivatives

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

The susceptibility is the response f

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SLIDE 11

What is more fundamental, H or B?

11

( )

[ ]

N = F × = B v q F   

Lorentz force: Vector potential:

[ ]

2

฀ m Vs T / A B = = B × =  

Zeeman energy:

[ ]

Ws VAs J = = = E

  • =

B m E  

  • Oersted field:

[ ]

m A = H = πr 2 I H

Magnetization:

[ ]

m A = M χH = M

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Classification

12

  • 1. 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 to external field, field lines are more dense in the material than in vacuum. External field is weakend by inducing screening currents according to Lenz

  • rule. Field lines are less dense than

in vacuum.

  • c. Ferromagnetism:

Spontaneous Magnetization without external field due to the interaction of magnetic moments µr attaines very high values for ferromagnets, > 104-105

  • H. Zabel, RUB
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13

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

( ) 2

2 1 H A q p me   + =

A 

  • 2. Electron in an external field

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

A B    × =฀

       

2 z z

B ~ sm diamagneti B ~

  • rbital

ism paramagnet energy kinetic 2 2 2 2

12 2 H a B m e L B m p

z e z z B e

+ µ + =

2 2 2

3 2 a y x = +

( )

z

,B , = B 

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

*Lz is here a dimensionless quantum number

*

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Hamiltonian for electron with spin

14

B S m e B S g B m E

B s s Zeeman

       ⋅ = ⋅ µ = ⋅ =

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

2 24

  • 10

27 . 9 2

  • Am

m e

B

× = = µ 

2 =

s

g

Landé factor Bohr magneton

( )

           

2 z z

B ~ sm diamagneti B ~

  • rbital

spin ism paramagnet energy kinetic 2 2 2 2

12 2 2 H a B m e S L B m p

z e z z z B e

+ + µ + =

+

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

  • 1. Lecture: Magnetic Response
  • H. 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 is determined by QED.

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Response functions

15 z

B m ∂ H ∂

  • =

( )

2 > + µ

z z B

S L

z e

B a m Ze

2 2

6

2 2 0 6

a m Ze

e

µ −

2 2

∂ H ∂

z mag

B V N µ = χ

  • 1. derivative
  • 2. derivative

Diamagnetic response for Z electrons Paramagnetic response

*

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

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

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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>

2 2 2

6

  • 6
  • ∑

a m Ze V N m e V N

e e Langevin

µ µ χ = =

i 2 i

r

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Examples for Diamagnetism

17

  • 1. 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 the

much bigger paramagnetic response.

  • H. Zabel, RUB
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18

Anisotropy of diamagnetismus for Li3N

Levitation of diamagnetic materials

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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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 x coth x L 1 − = Langevin function

Hz

m 

θ

  • 4. Paramagnetism of free local moments:

classical treatment

T k H m V N T k H m L m V N m V N M

B z z T Hohe B z z z

3 ) cos(

2 −

≈         = = θ

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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Langevin function

20 ( ) ( )

∫ ∫ ∫ ∫

π π

= =

cos

cos

sin 2 cos sin 2 cos cos dθ e θ π dθ e θ θ π dΩ e dΩ θe θ

T B k θ mB B B pot B pot

T k θ mB T k θ E T k θ E

T k B

B

µ = x θ θ π = φ θ θ = Ω d d d d sin 2 sin

L(x): Langevin-Funktion

        µ = T k B m L N M

B z

( ) ( )

x L x x ds e dx d ds e ds se

sx sx

sx

≡ − = = = θ

∫ ∫ ∫

− − −

1 coth ln cos

1 1 1 1 1 1

θ = cos s

using

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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21

Magnetization of paramagnetic moments in an external field

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

µ0Hz /kBT Magnetization

Hz

  • 1. Lecture: Magnetic Response
  • H. Zabel, RUB
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22

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

T C T k m V N H M

B z z Curie

= µ = ∂ ∂ = χ 3

2

B z

k m V N C 3

2

µ =

Paramagnetic Susceptibility

( )

z z B z

S L m 2 + = µ

Magnetic moment:

χ 1 T

Linear dependence fullfilled at high temperatures. At low T often deviations observed due to interactions.

But: However, magnetism is not a classical problem, thus Langevin function is only a rough

  • approximation. 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 be treated classically as the spin orientation of nanoparticles in the field is continuous.

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

Susceptibility of the Elements

23

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

From J.M.D. Coey

Paramagnetic Diamagnetic

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SLIDE 24

Summary

24

  • 1. Lecture: Magnetic Response

χ 1 T

2 2

6

  • a

m Ze V N

e c diamagneti Langevin

µ = χ χ T

  • 3. Paramagnetic response (HTA):
  • 2. Diamagnetic response:

( )

           

2 z z

B ~ sm diamagneti B ~

  • rbital

spin ism paramagnet energy kinetic 2 2 2 2

12 2 2 H a B m e S L B m p

z e z z z B e

+ + µ + =

+

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

T k m V N

B z ic paramagnet Curie

3

2

µ = χ

  • H. Zabel, RUB