2. lecture: basics of magnetism:...

2. Lecture: Basics of Magnetism:

Paramagnetism

Hartmut ZabelRuhr-University Bochum

Germany

Content

2H. Zabel, RUB 2. Lecture: Paramagnetism

1. Orbital moments2. Spin orbit coupling3. Zeeman splitting4. Thermal Properties – Brillouin function5. High Temperature – low temperature

approximation6. Van Vleck paramagnetism7. Paramagnetism of conduction electrons

Orbital moments of the d-shell


Hz

lm ( )1+ll

2−1−01

2

( ) B

BlBlllz mgmmµ−−=

µ=µ=

2,1,0,1,2,

5 orthogonal and degenerateorbital wave functions of 3d shell

The upper three wave functions havemaxima 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.

0=lm

1+=lm 1−=lm2+=lm

2−=lm

4

1. Spin-Orbit-couplingCoupling of spin and orbital moment yields the total angular momentum of electrons:

The spin-orbit (so) interaction or LS-coupling isdescribed by:

Hz

L

S

J

( ) ( )

shell f and d filled half than more for 0shell f and d filled half than less for

<λ>λ

µ−=λ⋅

0

1; 2 drdU

recmrSL r=λE

e

BSO

SLJ

+=

2. Lecture: ParamagnetismH. Zabel, RUB

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

Bm=E LSSO

⋅−


• 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:

rE

cmpr BL 2

0

×=

( ) LrrU

recm BL

∂∂

=11

20

( ) SLrrU

recm=E

e

BSO

⋅

∂∂µ

−1

2

( ) ( )22222

21; SLJSLSLJ

−−=⋅+=

( ) ( ) ( )[ ]1112

+−+−+β sslljj=ESO

2. Fine structure


Terms with same n and l – quantum numbers are energetially splitaccording to whether the electron spin is parallel or antiparallel to the orbital 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 levelfilling starts with lowest j-values.

+1/2β

-β

Why ist λ changing sign?


( ) 0<⋅ SL r=λESO

λd-electrons

Less than half filled: L-S

L S

( )0

0

0

><⋅

<⋅

λSL r=λE

SL

SO

if

More than half filled: L+S

L S

( )0

0

0

<<⋅

>⋅

λSL r=λE

SL

SO

if

8

LS coupling for light and heavy atoms

SLJ,sS,lLz

ii

Z

ii +=== ∑∑

==

11

∑=

=+=z

iiiii jJ,slj

1

Russel-Saunders coupling for light atoms (LS-coupling): This approximation assumes that the LS-coupling of individual electronsis 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 eachindividual electrons couples to j, and all j are added to total angular moment J.

9

Total angular moment and total magnetic moment

SLJ +=

( ) )SJ()SL()SgLg(m BBSLBSJ +−=+−=+−=+ µµµ 2

Total angular moment is:

HzL

S

J

SJ

+

S

SJm +

:

:

:

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 Jprecesses much slower about Hz. Thus the time average component of the magnetic moment ⟨m⟩ = m|| is parallel to J .

||m


Different total angular moments J


LS

L

S

L

S

Which one is the ground state?

Next lecture: Hund‘s rule

SOCoulomb EEE +=

0<λ

SLJSL −≥≥+

maxJ

J

minJ

23,3 == SL

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 withoutfield. These states are labled according to the magnetic quantumnumber mJ : –J ≤ mJ ≤ J.

• In an external field Hz the states with different mJ have different energyeigenstates, their degeneracy is lifted:

• The energy eigenstate are equidistant and linearly proportional to the external field Hz.

( )

zJzSO

z

m

JBJSOzm

HmEE

HmgEEHEJz

J

,00

00

,

µ++=

µµ++=

JmE

Hz

mJ= 3/2

mJ= 1/2

mJ=-1/2

mJ=-3/2zBJ Hg µµ0


12

21+

++−=J

J J,....,J,J,Jm

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


λ < 0Conversation:1000 cm-1 = 0.124 eV

13

Landé factor

)1(2)1()1(

23

)1(2)1()1()1(1

++−+

+=

++−+++

+=

JJLLSSJJ

LLSSJJgJ

:factor Landé

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

BJJjz gmm µ=,


Z-component of the total magnetic moment:

Hz

jm ( )1+JJ

2123

23−21−

Evaluating the Landé-factor


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:

( ) BSLE B

⋅+µ= 2

( ) ( ) ( ) zzB

B BJSLSLJJ

BJJ

JSLE

+⋅+µ

=⋅⋅+

µ= 222

( ) zzB BJSSLL

JE 22

2 23

+⋅+µ

=

( ) ( )22222

23; SLJSLSLJ

−−=⋅+= 3

( )zzB BJ

JSLJE 2

222

23

+−

µ=

zJBJ BmgE µ=

( ) ( )( )12

1123

++−+

+=JJ

LLSSgJ

15

Hz =0, T=0gs degenerate, all atoms in the same state

Hz >0, T= 0Lifting of degeneracy, all atoms in the gs

Hz >0, T>0At high temperature population of higer energy states

mJ10-1

mJ10-1

E

N

E

N

( ) JzBJJ mHgmE µµ0-=

( ) JzBJJ mHgmE µµ0-=

JBJz

mgVN

HE

VN

M µµ

==∂∂1

-0

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:


16

( )( )∑

∑α−

α−

=

j

j

mj

mjj

j m

mmm

exp

exp

TkHg

B

zBjµ=α

jBj mgVNM µ=

Thermal average of the magnetization

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

with:

)~(),( αµ= jBjz jBgVNHTM

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


17

−

α

++=α

ja

jjj

jjBj 2

~coth

21~

212coth

212)~(

Brillouin Function

mitTkjHg

jB

zBjµ=α=α~


18

J

J

J

Examples for the Brillouin-Function BJ:


19

1>>=Tk

jHg~B

zBj µα

)( 0TMMjgVNM SBj ==== µ

1B →)~(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.


20

31)coth( xx

x +≈

zB

zBeff HTC

TkHp

VNTM =

µ=

3)(

22

)1( += JJgp jeff

B

Beffk

pVNC

3

22 µ=

High temperature approximation

In HTA for the Brillouin function can be approximated by )~(Bj α1<<α~

Then follows for the magnetization

With the effective moment:

And the Curie constant:


21

TC

HM

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

0 100 200 300 400 500 6000

5

10

15

20

25

30

35

40

45

50χ

T0 100 200 300 400 500 600

0

1−χ

T2. Lecture: ParamagnetismH. 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 onlythe 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.

( )∑

0

2

00

0

1,2,...m --

=

+µµ=∆

EE

HSgLmE

m

zzSzBVleckVan


6. Paramagnetism of conduction electrons

232. 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!

TkH

VNHM

B

zBzz

2µχ ==

( )TkV

NTkV

NTk

SSgVN

Tkp

VN

B

B

B

B

B

BS

B

effB22

22

222

323

212

31

3µµµµχ =⋅⋅⋅=+==

H. Zabel, RUB

24

Magnetization of a free electron gas

( ) ( )V

EDHV

HEDVN

VNN

M FzB

zBFBBB

221 2

µ=µ×

µ=∆

µ=−

µ= ↓↑

( )ED↑

( )ED↓H2 Bµ

( )FED21

With: ( )F

F ENED

23

=

Follows:

zH

( )

===

FB

z2B

FB

z2B

Fz2B T

TTk

HVNμ

23

TkH

VNμ

23

VEDHμM

E

( )FED


Spin split DOS for free electronsin an externalfield

25

Pauli Spin Suszeptibility of a free electron gas

FBB

FBBPauli TkV

NTk

HHV

NHM 1

23

23

∂∂

∂∂ 22 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, thusclosed shells do not contribute to χPauli. Only s,p and d-electrons of unfilled shells contribute.


26

• Free electrons also contribute to the diamagnetic response in an external field, forming Landau cylinders.

• According to Ginzburg und Landau the diamagnetic contribution tothe 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:

PLandau χ−=χ31

Landau diamagnetism

FB

B

TkVN 2µ

=χ+χ=χ LaudauPaulielectron free


Experimental Pauli susceptibilities


Experimental results forsusceptibilities of monovalent and divalent metals:

From E.Y. Tysmbal

mono

divalent

H. Zabel, RUB

Summary of susceptibilities


positiv induced,ninteractio no ion, singlenegativ always induced,

++= PauliCurieLangevintot χχχχ

Langevin diamagnetism

Curie paramagnetism

T

χ

0

Pauli paramagnetism

TC

Curie =χ

22

0 6a

mZe

eLangevin µχ −=

H. Zabel, RUB

FB

B

TkVN 2µ

=χ electron free

2. lecture: basics of magnetism:...

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