Magnetism V Consider the ground state of an ion with several electrons in an unfilled shell (→ Hund’s rule)

z

e- e-

→ Coulomb interaction between electrons. Can be described as exchange interaction J between their spins.

1 2exc J s s= − ⋅

1 20J s> ⇒ ⋅ s is at maximum! the spins should be parallel (1st Hund’s rule) ⇒

Consider 5 electrons in a d-shell

mL

+2+1

0–1–2

⇒ S = 5/2 L = 0

2nd Hund’s rule: Maximize L obeying 1st Hund’s rule This further reduces the Coulomb interaction. hand waving argument: electrons with the same sign of angular momentum can more easily avoid each other. Consider 2 electrons in a d-shell

or J = 4

S = 1 L = +3

J = 2 S = 1 L = –3

+2+1

0–1–2

+2+1

0–1–2

rd3 Hund's rule (spin-orbit interaction):shell less than half full:

more than half full:

J L S

J L S

= −

= +

;

→ this minimizes the spin-orbit (SO) coupling This is the weakest rule, it applies only for heavy elements where the SO coupling is large! → otherwise other interactions like the crystal-field-effect invalidate the 3rd Hund’s rule.

– 1 –

term symbol: 2 1 2 1 is the multiplicitySJL S+ +

where L is not written as a number but as a letter. L 0 1 2 3 4 S P D F G

exam 3 94Dy f+

S = 5/2 → 2 s + 1 = 6 L = 5 → H 6

15 / 2H

see B For thmeasuReasoit from This isouter (see B

1

2

32

2

J

p g

g

p↑

=

=

=

The cions ichargean impenviro Now worbital Recalsuch a

1 Steph

ple:mL+3+2+1

0–1–2–3

lundell1, fig. 2.13, table 2.2, fig. 2.14.

e 4 f ions we obtain very good agreement with the experimentally red values. n: the 4 f shell is “buried” underneath the shells which screen the crystal field.

5 and s 5p

not the case for the 3 transition metal series, where the shell is the one; SO coupling is rather weak → crystal field effects become important!

d 3d

lundell, table 3.1) ( )( ) (

( ))

( )

1

1 12 1

1s

J

g

J J

s s L LJ J

s s

+

+ − ++

+

+

rystal field is the electric field that originates from the neighbouring n the lattice. The neighbouring ions can be approximated as point s → ligand field theory. The symmetry of the local environment plays ortant role here. (see Blundell, fig. 3.2, octahedral and tetrahedral

nment)

e have to consider the shape (angular momentum) of the s-, p- and d-s (see Blundell, fig. 3.1)

l these are not the eigenfunctions of L but linear combinations of these in way that one gets real functions.

en Blundell, Magnetism in Condensed Matter. Oxford University Press, 2001.

– 2 –

d-wave functions: 2l = ( )

2 2

2 22 2

2

2 2

2 2

2

2

lml

im

xy

g

x y

Y e

Y YY

ieY Y

Y

φ

−

−

−

Θ

−⎧=⎪

⎪⎨ +⎪ =⎪⎩

2mlY is eigenfunction of with eigenvalue

and zL L

2, lm → L is quenched 2 0g gL t L e= = only the spin is relevant here for magnetism

1 1

1 1

2 0

2 2

2 22

2

2

2

yz

g zx

z

Y YY

iY Y

t Y

Y Y

−− −⎧

=⎪⎪⎪ − +⎨ =⎪⎪

=⎪⎩

23 gt× where the lobes point along the diagonals between , or x y z axes. 2 ge× where they point along these axes:

2

2 2

lobes along axis

lobes along and axesz

x y

d z

d x−

y

⇒ the crystal field affects the orbitals in different ways (see Blundell, fig. 3.3) → xyd has lower energy than 2 2x y

d−

octahedral environment (e.g. cuprate high Tc superconductors) → 2gt are lower than the ge orbitals octahedral

free 3d ion 3 / 5ge ∆

2 / 5∆

2gt tetrahedral free 3d ion

2

2 / 5gt

∆

3 / 5∆

ge The lowest levels will be first filled with whether the environment is octahedral

–

∆ : crystal field splitting energy

either 2gt or ge electrons depending on or tetrahedral.

3 –

low-field case: crystal field energy is smaller than the on-site Coulomb interaction → 1st Hund’s rule still applies

63 e.g. d Fe +2 octahedral

strong-field In some matefunction of tesymmetry) or Back to orbitThe crystal fiis purely ima Eigenvalues → eigenfunc

⇒ 0→ when crysmomentum. Example: Imcharged ions

+

e–

+

high-spin state

e

5 / 2S = as dictated by Hund’s first rul

case:

e break even Hund’s first rule!

low-spin stat

1/ 2S =

rials, there is a transition from a low to a high spin state as a mperature (→ the lattice shrinks or even slightly changes its pressure.

al quenching: eld hamiltonian is a real function, as opposed to L i r= − ×∇ which ginary.

always are real numbers tions 0 of must be purely real functions: crystH 0 0cryst aH E= ˆ 0L = 0 otherwise it would be purely imaginary. tal field effect dominates → 0L = : full quenching of orbital

purity state with one electron in p-orbital surrounded by 2 positively . Consider strong crystal field effect and weak -coupling. SO

single electron in p-shell → 1L =

+

+

– 4 –

:

1 3 1 31 sin2 2 2 23 20

41 312 2

l

i

m

x iyer

rx iy

r

φ

π π

π

π

++ = − Θ = −

=

−− =

The real combinations are: ( )

( )

1 31 12231 1

2230

4

x xr

i y yrz zr

π

π

π

− + − = =

+ − = =

= =

z

z

y

y

x x

The crystal field splitting will favour the z orbitals.

13∆

23∆

⎫⎪⎪⎪⎬⎪⎪⎪⎭

x y

z is the crystal field splitting∆

Now we need to consider the spin-orbit interaction: SO L sλ= ⋅ Use first order perturbation theory:

0exc.states

exc. states exc. statesSO

ex

zzE E

∆

Ψ = +−

∑

ground state wave functions: , ; ,z z+ − excited state wave functions: , ; ,

, ; ,x xy y+ −

+ −

rewrite ( )12

SOz zL S L S L S L Sλ λ + − − +

⎡ ⎤= ⋅ = + +⎢ ⎥⎣ ⎦ with ( )

( )and

x y

x y

L L i L

S S i S

± = ±

± = ±

– 5 –

raising or lowering operators

( ) ( ), 1 1 ,g g gJ J m J J m m J m± = + − ± ±1g general form

11 2 0 ; 0 2 1 ;2 2

L L S± ± ±→ = = ± = ±∓ ∓ 1 ; all others are zero.

( ),

exc.state

z ze L S L S L S zz e

λ + − − ++ + ±⎡ ⎤⎣ ⎦Ψ± = ± +∆

∑½

, 1, 1,2

SOz zx z L Sλ

− + = − − + − −1 12 2

L S L S+ − − ++ +

{ }

0,

1 1, 0,22

1 1 122 22

L Sλ+ −

⎡ ⎤+⎢ ⎥

⎣ ⎦

= − − +

= ⋅ =

analogous , 02

SOy i λ− + = −

{ }, , , ,2

, 1,2

z x i y

z

λ

λ

⇒ Ψ + = + + − − −∆

= + + −∆

, , 1,

2z λ

Ψ − = − + − +∆

the spin-orbit coupling mixes back in the states with L , of the order of 0>2λ∆

Procedure for obtaining the ground state electronic configuration of an ion. Hund’s rules → know local environment → crystal field interaction vs. spin-orbit interaction ⇒ ground state wave function → magnetic properties of the ion

– 6 –

Warning: Sometimes the magnetic properties can influence the symmetry of the local environment → e.g. an octahedron can spontaneously distort because the cost in elastic energy is balanced by electronic energy saving due to charge in crystal-field splitting = Jahn-Teller effect. Example: 3 , 3 4Mn d+ in octahedral environment

2 2x yd

−

2zd

g

e

phenomenological description:

( ) 2 212

latticeE Q M Qω=

Q

( )E Q

xyd

,xz yd d

2gt

z

Q: distortion parameter M: mass of the anion ω: eigenfrequency of the phonon mode

effective energy reduction

linear approximation: .

2 2

0 2

2

min 2

12

: 0

2

minimum

elect

tot

E AQ

E AQ M

E AQQ M

AE

ω

ω

ω

= ±

→ = ± +

∂= → =

∂

=

Q

– 7 –