2. Lecture: Basics of Magnetism: Paramagnetism Hartmut Zabel - - PowerPoint PPT Presentation

• 2. Lecture:

Basics of Magnetism: Paramagnetism

Hartmut Zabel Ruhr-University Bochum Germany

Content

2

• H. Zabel, RUB
• 2. Lecture: Paramagnetism
• 1. Orbital moments
• 2. Spin orbit coupling
• 3. Zeeman splitting
• 4. Thermal Properties – Brillouin function
• 5. High Temperature – low temperature

approximation

• 6. Van Vleck paramagnetism
• 7. Paramagnetism of conduction electrons

Orbital moments of the d-shell

3

• H. Zabel, RUB
• 2. Lecture: Paramagnetism

Hz

l

m

( )

1 + l l  2 − 1 − 1 2

( )

B B l B l l l z

m g m m µ − − = µ = µ = 2 , 1 , , 1 , 2

,

5 orthogonal and degenerate

• rbital wave functions of 3d shell

The upper three wave functions have maxima in the xy, xz, yz planes, the lower two have maximia along x,y and z coordinate. In a magnetic field the degenerate sublevels split.

=

l

m

1 + =

l

m

1 − =

l

m 2 + =

l

m 2 − =

l

m

4

• 1. Spin-Orbit-coupling

Coupling of spin and orbital moment yields the total angular momentum of electrons: The spin-orbit (so) interaction or LS-coupling is described by:

Hz

L  S 

J 

( ) ( )

shell f and d filled half than more for shell f and d filled half than less for < λ > λ µ − = λ ⋅ 1 ;

2

dr dU r ec m r S L r =λ E

e B SO

  

S L J    + =

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

Spin-orbit coupling is due to the Zeeman – splitting of the spin magnetic moment in the magnetic field that is produced by the

• rbital moment:

B m = E

L S SO

  ⋅ −

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism
• In rest frame of electron, E and B – fields act on electron due to positive

charge of the nucleus:

• The magnetic field is proportional to angular momentum of electrons: BL~ L:
• In magnetic field BL, S precesses with a angular velocity ωL and couples to L:
• L⋅S can be evaluated via:
• Yielding:

r E c m p r BL

2

   × =

( ) L

r r U r ec m BL   ∂ ∂ = 1 1

2

( )

S L r r U r ec m = E

e B SO

   ⋅ ∂ ∂ µ − 1

2

( ) ( )

2 2 2 2 2

2 1 ; S L J S L S L J         − − = ⋅ + =

( ) ( ) ( ) [ ]

1 1 1 2 + − + − + β s s l l j j = ESO

• 2. Fine structure

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism

Terms with same n and l – quantum numbers are energetially split according to whether the electron spin is parallel or antiparallel to the

• rbital moment. This is called the fine structure of atomic spectra.

Example hydrogen atom: The total splitting of 3/2β increases with the number of electrons in the atom and becomes in the order of 50 meV for 3d metals. LS-splitting lowers the energy for L and S antiparallel. Therefore level filling starts with lowest j-values. +1/2β

• β

Why ist λ changing sign?

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism

( )

< ⋅S L r =λ ESO  

λ

d-electrons

Less than half filled: L-S

L S

( )

> < ⋅ < ⋅ λ S L r =λ E S L

SO

if    

More than half filled: L+S

L S

( )

< < ⋅ > ⋅ λ S L r =λ E S L

SO

if    

8

LS coupling for light and heavy atoms

S L J , s S , l L

z i i Z i i

+ = = =

∑ ∑

= = 1 1

∑

=

= + =

z i i i i i

j J , s l j

1

 

Russel-Saunders coupling for light atoms (LS-coupling): This approximation assumes that the LS-coupling of individual electrons is weak compared to the coupling between electrons. Orbital moments of all electrons couple to a total angular momentum L and spin moments of all electrons couple to S. Finally L and S couple to J: jj-coupling of heavy atoms: In the limit of big LS – coupling, the spin and orbital moment of each individual electrons couples to j, and all j are added to total angular moment J.

9

Total angular moment and total magnetic moment

S L J + =

( )

) S J ( ) S L ( ) S g L g ( m

B B S L B S J

+ − = + − = + − =

+

µ µ µ 2 

Total angular moment is:

Hz L

 S  J  S J   + S 

S J

m +  : : : J L S Total spin

Total orbital moment Total angular moment Total magnetic moment is: Total angular moment J and the total magnetic moment are not collinear. However, in an external magentic field, mJ+S precesses fast about J, and J precesses much slower about Hz. Thus the time average component of the magnetic moment 〈m〉 = m|| is parallel to J .

||

m 

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

Different total angular moments J

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism

L  S  L  S  L  S 

Which one is the ground state? Next lecture: Hund‘s rule

SO Coulomb

E E E + =

< λ

S L J S L − ≥ ≥ +

max

J  J 

min

J 

2 3 , 3 = = S L

• 3. Zeeman-splitting
• In an external field the quantization axis is defined by the field axis Hz
• A state with total angular momentum J has a degeneracy of 2J+1 without
• field. These states are labled according to the magnetic quantum

number mJ : –J ≤ mJ ≤ J.

• In an external field Hz the states with different mJ have different energy

eigenstates, their degeneracy is lifted:

• The energy eigenstate are equidistant and

linearly proportional to the external field Hz.

( )

z J z SO z m J B J SO z m

H m E E H m g E E H E

J z J

,

,

µ + + = µ µ + + =     

J m

E

Hz

mJ= 3/2 mJ= 1/2 mJ=-1/2 mJ=-3/2

z B J

H g µ µ0

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• 2. Lecture: Paramagnetism
• H. Zabel, RUB

        

1 2

2 1

+

+ + − =

J J

J ,...., J , J , J m

LS and Zeemann Splitting for L=3, S=3/2, λ < 0

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism

λ < 0

Conversation: 1000 cm-1 = 0.124 eV

13

Landé factor

) 1 ( 2 ) 1 ( ) 1 ( 2 3 ) 1 ( 2 ) 1 ( ) 1 ( ) 1 ( 1 + + − + + = + + − + + + + = J J L L S S J J L L S S J J g J : factor Landé

Notice: gj=1 for J=L and 2 for J=S

B J J j z

g m m µ =

,

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

Z-component of the total magnetic moment:

Hz

j

m

( )

1 + J J  2 1 2 3 2 3 − 2 1 −

Evaluating the Landé-factor

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• H. Zabel, RUB
• 2. Lecture: Paramagnetism

From 1. Lecture we have for the paramagnetic response in an external B - field: Considering L+2S projected onto J and J project onto the B-axis: This yields: Using: We find: Which must equal: With:

( ) B

S L E

B

   ⋅ + µ = 2

( ) ( ) ( )

z z B B

B J S L S L J J B J J J S L E          + ⋅ + µ = ⋅ ⋅ + µ = 2 2

2

( )

z z B

B J S S L L J E

2 2 2

2 3     + ⋅ + µ =

( ) ( )

2 2 2 2 2

2 3 ; S L J S L S L J         − − = ⋅ + = 3

( )

z z B

B J J S L J E

2 2 2 2

2 3   + − µ =

z J B J

B m g E µ =

( ) ( ) ( )

1 2 1 1 2 3 + + − + + = J J L L S S g J

15

Hz =0, T=0

gs degenerate, all atoms in the same state

Hz >0, T= 0

Lifting of degeneracy, all atoms in the gs

Hz >0, T>0

At high temperature population

• f higer energy states

mJ

1

• 1

mJ

1

• 1

E N E N

( )

J z B J J

m H g m E µ µ0

• =

( )

J z B J J

m H g m E µ µ0

• =

J B J z

m g V N H E V N M µ µ = = ∂ ∂ 1

• 4. Thermal properties

Thermal population of the Zeeman-split levels in the ground state (gs). Example: J=1, mJ=-1,0,+1 Discrete energy levels with mJ=-J,…J Average thermal energy: Magnetization:

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

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( ) ( )

∑ ∑

α − α − =

j j

m j m j j j

m m m m exp exp

T k H g

B z B jµ

= α

j B j

m g V N M µ =

Thermal average of the magnetization

Thermal average of the magnetic moment follows from the partition function: with:

) ~ ( ) , ( α µ =

j B j z

jB g V N H T M

Bi is the Brillouin function. The Brillouin function replaces the Langevin function in case of discrete energy levels.

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

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        −         α + + = α j a j j j j j B j 2 ~ coth 2 1 ~ 2 1 2 coth 2 1 2 ) ~ (

Brillouin Function

mit

T k jH g j

B z B jµ

= α = α ~

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

18

J J J

Examples for the Brillouin-Function BJ:

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

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1 >> = T k j H g ~

B z B j µ

α

) ( T M M j g V N M

S B j

= = = = µ

1 B → ) ~ (α

• 4. Low and high temperature approximations

Low temperature approximation (LTA) for: the Brillouin function approaches 1 The thermally averaged magnetization then becomes: This corresponds to the saturation magnetization MS. The saturation magnetization can not become bigger than given by j. It corresponds to a state in which all atoms occupy the ground state.

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

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3 1 ) coth( x x x + ≈

z B z B eff

H T C T k H p V N T M = µ = 3 ) (

2 2

) 1 ( + = J J g p

j eff

B B eff

k p V N C 3

2 2 µ

=

High temperature approximation

In HTA for the Brillouin function can be approximated by

) ~ ( Bj α

1 << α ~

Then follows for the magnetization With the effective moment: And the Curie constant:

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

21

T C H M

z

= ∂ ∂ = χ

With C we can calculate peff and j. From j the valence of a chemical bond can be determined. Thus peff is important for chemistry.

Curie law of the magnetic susceptibility

100 200 300 400 500 600 5 10 15 20 25 30 35 40 45 50

χ

T

100 200 300 400 500 600

1 −

χ T

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

• 5. Van Vleck paramagnetism

22

• For J = 0 the paramagnetic susceptibility becomes zero.
• J = 0 occurs for shells, which are less than half filled by one electron
• In this case higher order terms contribute to the susceptibility, in

particular a diamagnetic term of second order, which is positiv.

• The higher order terms are due to excited states which may have a J ≠

0, even if for the ground state J = 0.

• Calculating in second order perturbation theory contributions to only

the ground state, one obtains:

• Van Vleck contribution to the susceptibility is weak, positive and

temperature independent. But it plays a decisive role for the paramagnetism of Sm and Eu ⇒ 3. Lecture.

( )

∑

2 1,2,... m

• =

+ µ µ = ∆ E E H S g L m E

m z z S z B Vleck Van

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

• 6. Paramagnetism of conduction electrons

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• 2. Lecture: Paramagnetism

For Fermi particles with spin S=1/2 we expect: Expected magnetization: which has 1/T dependence. Experiment shows: α. χ is independent of T b. has a value 1/100 of the calculated value at 300 K.

 Contradiction!

T k H V N H M

B z B z z 2

µ χ = =

( )

T k V N T k V N T k S S g V N T k p V N

B B B B B B S B eff B 2 2 2 2 2 2 2

3 2 3 2 1 2 3 1 3 µ µ µ µ χ = ⋅ ⋅ ⋅ = + = =

• H. Zabel, RUB

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Magnetization of a free electron gas

( ) ( )

V E D H V H E D V N V N N M

F z B z B F B B B 2 2 1

2 µ = µ × µ = ∆ µ = − µ =

↓ ↑

( )

E D↑

( )

E D↓

H 2

B

µ

( )

F

E D 2 1

With:

( )

F F

E N E D 2 3 =

Follows:

z

H

( )

        = = =

F B z 2 B F B z 2 B F z 2 B

T T T k H V N μ 2 3 T k H V N μ 2 3 V E D H μ M E

( )

F

E D

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

Spin split DOS for free electrons in an external field

25

Pauli Spin Suszeptibility of a free electron gas

F B B F B B Pauli

T k V N T k H H V N H M 1 2 3 2 3 ∂ ∂ ∂ ∂

2 2

µ =         µ = = χ

• Pauli spin susceptibility has the correct form.
• It is independent of temperature
• It is reduced by the factor T/TF ~ 100.
• Closed shells have no density of states at the Fermi level, thus

closed shells do not contribute to χPauli. Only s,p and d-electrons of unfilled shells contribute.

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

26

• Free electrons also contribute to the diamagnetic response in an

external field, forming Landau cylinders.

• According to Ginzburg und Landau the diamagnetic contribution to

the susceptibility of the conduction electrons is:

• Exemption superconductors, in which case χ=-1.
• Considering the paramagnetic and diamagnetic response, the total

susceptibility of a free electron gas is:

P Landau

χ − = χ 3 1 Landau diamagnetism

F B B

T k V N

2

µ = χ + χ = χ

Laudau Pauli electron free

• 2. Lecture: Paramagnetism
• H. Zabel, RUB

Experimental Pauli susceptibilities

27

• 2. Lecture: Paramagnetism

Experimental results for susceptibilities of monovalent and divalent metals:

From E.Y. Tysmbal

mono divalent

• H. Zabel, RUB

Summary of susceptibilities

28

• 2. Lecture: Paramagnetism

 

positiv induced, n interactio no ion, single negativ always induced,

+ + =

Pauli Curie Langevin tot

χ χ χ χ     

Langevin diamagnetism Curie paramagnetism T χ Pauli paramagnetism

T C

Curie =

χ

2 2 0 6

a m Ze

e Langevin

µ χ − =

• H. Zabel, RUB

F B B

T k V N

2

µ = χ

electron free