Experimental Techniques for the Study of Magnetism

School on Pulsed Neutrons - October 2005 - ICTP Trieste

"Methods and Techniques"

Experimental Techniques for the Study of Magnetism

Prof. Dr. Thomas Brückel

Institute for Scattering Methods

Institute for Solid State Research

first compass

History: Loadstone Fe ₃ O ₄ ( ≈ 800 BC)

100 A.D.

Chinese "south pointer"

first compass

History: Loadstone Fe ₃ O ₄ ( ≈ 800 BC)

100 A.D.

Chinese "south pointer"

"perpetual motion machine"

1269

Europe: Petrus Perigrinus

"Epostolia de Magnete"

first compass

History: Loadstone Fe ₃ O ₄ ( ≈ 800 BC)

100 A.D.

Chinese "south pointer"

"perpetual motion machine"

1269

Europe: Petrus Perigrinus

"Epostolia de Magnete"

what’s new ? what’s new ?

magnetic nanostructures correlated electron systems

...

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

Magnetic Nanostructures

Thin Film Multilayer:

Fe ₅₀ Pt ₅₀ Nanoparticle Network by colloidal self organisation

Sun et al; Science 287 (2000), 1989

Magnetic Nanostructures

Thin Film Multilayer:

⇒Surfaces,

⇒Interfaces,

⇒Proximity effects

Fe ₅₀ Pt ₅₀ Nanoparticle Network by colloidal self organisation

Sun et al; Science 287 (2000), 1989

Magnetic Nanostructures

Thin Film Multilayer:

⇒Surfaces,

⇒Interfaces,

⇒Proximity effects

Fe ₅₀ Pt ₅₀ Nanoparticle Network by colloidal self organisation

Sun et al; Science 287 (2000), 1989

Interlayer Exchange Coupling

Peter Grünberg:

Interlayer Exchange Coupling in Fe/Cr Multilayers Phys. Rev. Lett. 57 (1986), 2442

Oscillatory coupling as function of interlayer thickness:

Co Cu Co

Co Cu Co

Ferromagnetic Antiferromagnetic

Giant Magnetoresistance (GMR)

P. Grünberg et al.

Phys. Rev. B 39 (1989), 4828 (and independently: A. Fert, Paris)

GMR-effect

Fe/Cr/Fe

1.5 %

Artificial Nano-Structures

→ purpose designed properties

Giant Magnetoresistance (GMR)

P. Grünberg et al.

Phys. Rev. B 39 (1989), 4828 (and independently: A. Fert, Paris)

GMR-effect

Fe/Cr/Fe

1.5 %

^Fe/Cr/Fe

Artificial Nano-Structures

→ purpose designed properties

Giant Magnetoresistance (GMR)

P. Grünberg et al.

Phys. Rev. B 39 (1989), 4828 (and independently: A. Fert, Paris)

GMR-effect

Fe/Cr/Fe

1.5 %

^Fe/Cr/Fe

1

Artificial Nano-Structures

→ purpose designed properties

Giant Magnetoresistance (GMR)

P. Grünberg et al.

Phys. Rev. B 39 (1989), 4828 (and independently: A. Fert, Paris)

GMR-effect

Fe/Cr/Fe

1.5 %

^Fe/Cr/Fe

1

Artificial Nano-Structures

→ purpose designed properties

Applications: Hard Disks

Applications: Hard Disks

Moor's Law

Applications: MRAM

MRAM MRAM

Magnetic Random Access Memory:

Applications: MRAM

MRAM MRAM

Magnetic Random Access Memory:

• 100 Million storage elements per mm ²

• 1 /100 Million gram mass per cm ²

Applications: MRAM

MRAM MRAM

Magnetic Random Access Memory:

• 100 Million storage elements per mm ²

• 1 /100 Million gram mass per cm ²

independently 1988 A. Fert

Applications: MRAM

MRAM MRAM

Magnetic Random Access Memory:

• 100 Million storage elements per mm ²

• 1 /100 Million gram mass per cm ²

independently 1988 A. Fert

"Spintronics":

Information transport, storage and processing

with the spin of the electron (not the charge!)

Complex transition metal oxides:

High T _C Superconductors; CMR-Manganates; …

New phenomena appear from the New phenomena appear from the bottom of the Fermi sea due to bottom of the Fermi sea due to electronic correlations:

electronic correlations:

• Magnetism

• Superconductivity

• Metal-insulator transition (CMR)

• Charge- & orbital order

• Multiferroica

Highly correlated electron systems

High T _C Materials (YBa ₂ Cu ₃ O _6+x ):

Magnetism ↔ Superconductivity Materials with collosal Magnetoresistance

Spin ↔ Charge ↔ Lattice ↔ Orbital order Oxides

No simple Fermi liquids; competing interactions

La Mn

O

Highly correlated electron systems

Fundamental microscopic understanding !

Fundamental microscopic understanding !

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

Susceptibility and Magnetisation

Susceptibility and Magnetisation

M

H

H M = χ ⋅

linear response theory

Susceptibility and Magnetisation

Scattering

Scattering

→ Internal structure? (atom positions, moment arrangement)

→ Microscopic dynamics? (atom movements, spin dynamics)

⇒ Macroscopic properties (conductivity, susceptibility, ...)

Scattering

Scattering:

interaction sample ↔ radiation weak

⇒ non-invasive, non destructive probe

for structure & dynamics

v N

µ _N

Generalised Susceptibility

linear response theory:

perturbation of magnetic system described by spacial and temporal varying magnetic field H (r, t)

system reaction:

local magnetisation M (r,t)

linear response theory → susceptibility )

, ( r t χ

( ) ( )

,

0

, =

=

ld H

ld

t M R t

R

M

^{β β}

( ^{, '} ) (

_{' '}

^{, '} ) ^'

' '

'

'

t R R t t dt

R

t

H

d l ld d

l

d

∫ ∑∑

−∞ l

− −

+

^αβ

α

α

χ

( ^{, '} ) ^{( )} (

_{0 '}

^{, '} )

1

'

Q t t e

^{iQ R R0 '}

R

_ld

R

_d

t t

dd

d

ld

− −

=

− ∑

^{⋅ −}

^αβ

αβ χ

χ

( ) ^{Q e}

dd

( ) ^{Q t dt}

t i

dd

,

_'

,

' 0

αβ ω

αβ

ω χ

χ ⁼ ∫

^{∞ −}

v N

µ _N

Generalised Susceptibility

linear response theory:

perturbation of magnetic system described by spacial and temporal varying magnetic field H (r, t)

system reaction:

local magnetisation M (r,t)

linear response theory → susceptibility )

, ( r t χ

Fourier transform:

( ^, ) ⁼ ( ^, )

₌₀

ld H

ld

t M R t

R

M

^{β β}

( ^{, '} ) (

_{' '}

^{, '} ) ^'

' '

'

'

t R R t t dt

R

t

H

d l ld d

l

d

∫ ∑∑

−∞ l

− −

+

^αβ

α

α

χ

( ^{, '} ) ^{( )} (

_{0 '}

^{, '} )

1

'

Q t t e

^{iQ R R0 '}

R

_ld

R

_d

t t

dd

d

ld

− −

=

− ∑

^{⋅ −}

^αβ

αβ χ

χ

( ) ^{Q e}

dd

( ) ^{Q t dt}

t i

dd

,

_'

,

' 0

αβ ω

αβ

ω χ

χ ⁼ ∫

^{∞ −}

Neutrons: Length and Time Scales

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

Magnetic Structures

Mn

²⁺

b=a a=4.873 Å

c= 3. 31 Å

Collinear Antiferromagnets:

F -

MnF

₂

:

Modulated Structures:

Cr:

MnO:

Complex Structures:

Er

₆

Mn

₂₃

:

Rare Earth:

General description in Fourier representation:

∑ ^{⋅ − ⋅}

=

k k l

e

m i k R

m

ij ij

exp( )

Neutron-Matter-Interaction

First Born Approximation: 2 2

|

|

|'

2 ⎟ | < >

⎠

⎜ ⎞

⎝

= ⎛

Ω m k V k

d d

π h

σ

Neutron-Matter-Interaction

First Born Approximation: 2 2

|

|

|'

2 ⎟ | < >

⎠

⎜ ⎞

⎝

= ⎛

Ω m k V k

d d

π h σ

r d e

r V

r d e r V e

r Q i

r k i r

k i

3 3 '

) (

) (

⋅

−

−

∫

∫

⋅

=

Neutron-Matter-Interaction

First Born Approximation: 2 2

|

|

|'

2 ⎟ | < >

⎠

⎜ ⎞

⎝

= ⎛

Ω m k V k

d d

π h σ

r d e

r V

r d e r V e

r Q i

r k i r

k i

3 3 '

) (

) (

⋅

−

−

∫

∫

⋅

? =

Neutron-Matter-Interaction

First Born Approximation: 2 2

|

|

|'

2 ⎟ | < >

⎠

⎜ ⎞

⎝

= ⎛

Ω m k V k

d d

π h σ

r d e

r V

r d e r V e

r Q i

r k i r

k i

3 3 '

) (

) (

⋅

−

−

∫

∫

⋅

? =

• strong interaction n ↔ nucleus

• magnetic dipole-interaction with B-field of unpaired e

^-

major

Magnetic Interaction Potential

e ^-

v _e

µ _e

R

B µ _n

n

magnetic moment of the neutron:

⋅ σ γµ

−

=

µ _{n N}

magnetic field of the electron:

L

S B

B

B = +

dipolar field of the spin moment: ; 2 S

R R

B x _{e B}

e 3

S ⎟⎟ ⎠ µ = − µ ⋅

⎜⎜ ⎞

⎝

× ⎛ µ

∇

=

field due to the movement of the electron (Biot-Savart): _L ^e ₃ R

R v

c

B = − e ×

n B

m = − µ ⋅ V

Zeeman energy:

Magnetic Scattering Cross Section

σz Vm k

σz‘ k‘

z 2 m

z 2

n 2 k ' ' k

2 m d

d ⎟⎟ ⎠ σ σ

⎜⎜ ⎞

⎝

⎛

= π Ω

σ V

h

( ) _z ( ) _z ²

B

0 2 ' M Q

2 r 1

d

d σ ⋅ σ

− µ γ

Ω = σ

σ ⊥

cm 10

539 . 0

r ₀ = ⋅ ^{− 12} γ

→"equivalent scattering length" for 1 µ

_B

(S=

2

1 ): 2.696 fm ≈ b

_co

( ) ^{Q Qˆ M} ( ) ^{Q Qˆ}

M _⊥ = × ×

( ) Q ⁼ _∫ M ( ) r e ^⋅ d r

M ^{i Q r 3}

( ) r M ( ) r M ( ) r

M = _S + _L

( ) = − µ ⋅ ( ) = − µ ∑ δ ( − )

i i i

B B

S r 2 S r 2 r r S

M

1. Born approximation

Directional Dependence

Q k‘

M

k

M_⊥

( ) ^{Q Qˆ M Qˆ}

M _⊥ = × ×

Illustration: scattering from the dipolar field

Only the component of the magnetisation perpendicular to the scattering vector gives rise to magnetic scattering!

M || Q M

Q

Planes with equal phase factor

M ⊥ Q M

Q

Pure Spin Scattering

Ri rik

tik Sik

Si

Atom i

Separation of intra-atomic quantities for localised moments:

( ) = − µ ∑ δ ( − ) ⋅ +

=

ik ik ik

B S

ik i

ik R t ; M r 2 r r s

r

( ) Q = _∫ M ( ) r e ^⋅ d r

M _S ^{i Q r 3}

∑ ∑ ⋅

∑ =

= ^{⋅ ⋅ ⋅}

i i Q R k i Q t ik ik ik

r Q

i s e e s

e ^{i i ik}

Expectation value of the operator for the thermodynamic state of the sample:

( ) Q = − 2 µ _B ⋅ f _m ( ) Q ⋅ ∑ e ^{i Q ⋅ R} ⋅ S _i

M ⁱ

( ) = _{∫ ρ} ( ) ^⋅

Atom

r 3 Q s i

m Q r e d r

f

( ) ( ) ²

i

R Q i i

2 m

0 f Q S e ⁱ

d r

d = γ ∑

Ω

σ ⊥

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

normalized form factor

sin(Θ/λ)

nuclear scattering

orbital

x-ray

spin

Chromium

0.0 0.2 0.4 0.6 0.8 1.0

sin(Θ/λ)

Q || b Q || a

a b

form factor

Form Factor: Spin, Orbit, Anisotropy

M(r)

λ

F

i. a. anisotrop:

in general anisotropic:

Magnetic Bragg Diffraction from a Type I Antiferromagnet on a tetragonal body-centered lattice

amplitude

→ intensity

nuclear structure factor

squared

square of structure

factor

Magnetic Neutron Scattering

Neutron Powder Diffraction

10 K RT

E. Gorelik (2004)

Distorted

Perovskite

structure

Magnetic Neutron Scattering

Neutron Powder Diffraction

10 K RT

E. Gorelik (2004)

Distorted

Perovskite

structure

Magnetic Neutron Scattering

Neutron Powder Diffraction

10 K RT

E. Gorelik (2004)

Spin Structure:

Distorted

Perovskite

structure

v N

µ _N Interaction:

Magnetic Dipole-Dipole

) ' ,' ( )

' , ,' , ( )

,

( r t r r t t H r t

M = χ ⋅

(

₀

)

²

1 ' ( )

²

z B z

mag n

Q M d r

d ω σ γ µ σ σ σ

⋅

⊥

−

=

Elastic scattering:

Magnetic Neutron Scattering

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

X-Ray Powder Diffraction

Chemical structure, but not

Magnetic Structure

La _7/8 Sr _1/8 MnO ₃ -Kristall

Perßon, Li, Mattauch, Kaiser, Roth, Heger (2004)

La _7/8 Sr _1/8 MnO ₃ -Kristall

Perßon, Li, Mattauch, Kaiser, Roth, Heger (2004)

< 112>

T = 120 K

ESRF @ Grenoble, France 6 GeV

APS @ Argonne/Chicago, USA 7 GeV

SPRING8, Japan, 8 GeV

Synchrotron Sources

X-Ray Probes of Magnetism

- Kerr-microscopy - Faraday effect

- Linear x-ray magnetic dichroism - Circular x-ray magnetic dichroism

- Spin resolved x-ray absorption fine structure SEXAFS - Magnetic x-ray diffraction (non-resonant scattering) - Resonant magnetic x-ray scattering (X-ray resonance

exchange scattering XRES) - Nuclear resonant scattering - Magnetic x-ray reflectivity - Magnetic Compton scattering

- Angular- and spin resolved photoemission

X-Ray Probes of Magnetism

- Kerr-microscopy - Faraday effect

- Linear x-ray magnetic dichroism - Circular x-ray magnetic dichroism

- Spin resolved x-ray absorption fine structure SEXAFS - Magnetic x-ray diffraction (non-resonant scattering) - Resonant magnetic x-ray scattering (X-ray resonance

exchange scattering XRES) - Nuclear resonant scattering - Magnetic x-ray reflectivity - Magnetic Compton scattering

- Angular- and spin resolved photoemission

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

E

H E

H

H H

E

E

interaction re-radiation

-e

-e

-e µ

µ force

-eE

-eE

grad(µH)

torque Hxµ

E-dipole

H-quadr.

E-dipole

H-dipole

σ

σ

σ

σ

σ

π,σ

π

π

µ

De Bergevin & Brunel 1981

Nonresonant Scattering: Classical

Thomson scattering from charges

⇒ Structure

E

H E

H

H H

E

E

interaction re-radiation

-e

-e

-e µ

µ force

-eE

-eE

grad(µH)

torque Hxµ

E-dipole

H-quadr.

E-dipole

H-dipole

σ

σ

σ

σ

σ

π,σ

π

π

µ

De Bergevin & Brunel 1981

Nonresonant Scattering: Classical

Thomson scattering from charges

⇒ Structure

But: X-rays are electromagnetic

radiation ⇒ non resonant magnetic x-ray

scattering

⇒ Magnetism

Cross Section for Magnetic X-Ray Scattering

Non-relativistic treatment in second order perturbation theory ( Blume 1985, Blume & Gibbs 1988 )

• Hamiltonian for e ^- in e-m field:

)) 2 (

2 (

1 A r j

c j e j m P

H = ∑ −

+ ∑

ji V ( r ij )

∑ ⋅ ∇ ×

− j s j A r j mc

e h ( )

)) (

( ) 2 (

) (

2 A r j

c j e j s j E r j P

mc

e ∑ ⋅ × −

− h

2 ) ) 1 ( ) (

∑ ( + +

+ λ ω λ λ

k h k c k c k

kinetic energy

Coulomb interaction Zeeman energy -µ · H

spin-orbit coupling -µ·H~s·(E×v)

self energy of e-m-field

• Vector potential in plane wave expansion:

2 1 q 2 V c q 2 )

r (

A = ∑σ ω π

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

h × [ ε ( q σ ) c ( q σ ) e i q ⋅ r + ε * ( q σ ) c + ( q σ ) e − i q ⋅ r ]

H = H _o + H _r + H _int

e

^-

-system e-m-wave interaction

→ perturbation theory (Fermi's "golden rule")

first order for terms quadratic in A second order for terms linear in A

2

int

, ,

,'

,' f H k i d k

d σ _∝ ε ε

Ω

Cross Section for Magnetic X-Ray Scattering

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ²

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ² magnetic~ |f _M | ²

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ² magnetic~ |f _M | ² interference~ f _C · f _M

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ² magnetic~ |f _M | ² interference~ f _C · f _M

π/2 phase shift

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ² magnetic~ |f _M | ² interference~ f _C · f _M

π/2 phase shift h/mc = 2.426 pm

Intensity ratio: I I

2 ' 2

2 2

' εε

ε ε

σ

f C

mc e d

d ⎥ ⋅

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

non-resonant elastic scattering cross section:

r _e = 2.818 fm

incident and final polarization

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

charge ~ |f _C | ² magnetic~ |f _M | ² interference~ f _C · f _M

π/2 phase shift h/mc = 2.426 pm

10 6

~ 2 f S

N f M N M

d c

~ I M C

I −

⋅

⋅ ⋅

Intensity ratio: I I λ

cross section:

scattering geometry:

Cross Section: Nonresonant

2 ' '

2 2 2

' ε ε ε ε

ε ε

λ σ

M C

C f

i d mc f

e d

d ⎥ ⋅ +

⎦

⎢ ⎤

⎣

= ⎡ Ω _→

Q=k’-k

Amplitude-matrices:

to \ ^from σ π

σ ' ρ ( Q ) 0

π ' 0 ρ (Q) cos2 ( θ )

<f

_C

> for charge scattering:

⇒ charge density ρ(Q)

Amplitude-matrices:

to \ ^from σ π

σ ' ρ ( Q ) 0

π ' 0 ρ (Q) cos2 ( θ )

<f

_C

> for charge scattering:

⇒ charge density ρ(Q)

e

^-

E Hertz

Dipole

Radiation

Amplitude-matrices:

to \ ^from σ π

σ ' ρ ( Q ) 0

π ' 0 ρ (Q) cos2 ( θ )

<f

_C

> for charge scattering:

⇒ charge density ρ(Q)

e

^-

E Hertz

Dipole Radiation

to \ ^from σ π

σ ' S ₂ ⋅ cos θ [ ( L ₁ + S ₁ ) ⋅ cos θ + S ₃ ⋅ sin θ ] ^{⋅ sin θ}

π ' [ − ( L ₁ + S ₁ ) ⋅ cos θ + S ₃ ⋅ sin θ ] ^{⋅ sin θ} [ ^{2 L} 2 ⋅ sin ² θ + S ₂ ] ^{⋅cos θ}

<f

_M

> for the magnetic part:

⇒ spin density S(Q) and orbital angular momentum density L(Q)

Amplitude-matrices:

to \ ^from σ π

σ ' ρ ( Q ) 0

π ' 0 ρ (Q) cos2 ( θ )

<f

_C

> for charge scattering:

⇒ charge density ρ(Q)

e

^-

E Hertz

Dipole Radiation

to \ ^from σ π

σ ' S ₂ ⋅ cos θ [ ( L ₁ + S ₁ ) ⋅ cos θ + S ₃ ⋅ sin θ ] ^{⋅ sin θ}

π ' [ − ( L ₁ + S ₁ ) ⋅ cos θ + S ₃ ⋅ sin θ ] ^{⋅ sin θ} [ ^{2 L} 2 ⋅ sin ² θ + S ₂ ] ^{⋅cos θ}

<f

_M

> for the magnetic part:

⇒ spin density S(Q) and orbital angular momentum density L(Q)

Charge scattering: "NSF"

Magnetic scattering: "NSF" (S

₂

, L

₂

) – _┴ scattering plane + "SF" (S

₁

, S

₃

, L

₁

) – in scattering plane

⇒ Separation S ↔L

50 m

5 mm properties calculable

small source

size wiggler clean ultra-high

vacuum source

time structure

intense continuous spectrum highly collimated

undulators

.

. polarised

Synchrotron X-Ray Source

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

resonance exchange scattering

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

resonance exchange scattering

( )

2 0

M

mag E E i / 2

E / d

d

Γ

−

−

∝ α Ω

σ

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

resonance exchange scattering

( )

2 0

M

mag E E i / 2

E / d

d

Γ

−

−

∝ α Ω

σ

neutron scattering

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

Resonant Magnetic X-Ray Scattering

ε_F

2s 2p 2p

1s

1/2 3/2

4f ↑ 4f ↓

s-p-d

up E down

exchange splitting

resonance exchange scattering

( )

2 0

M

mag E E i / 2

E / d

d

Γ

−

−

∝ α Ω

σ

neutron scattering

resonant x-ray scattering

γ

L_III

E1: 2p_3/2→5d_5/2 E2: 2p_3/2→4f_7/2

Hannon, Trammell, Blume & Gibbs PRL 61 (1988), 1245

7942 eV 7938 eV 7935 eV 7933 eV

ω 0

1000 2000 3000

5 5.2 5.4 5.6 5.8

co un ts / se c

energy

7924 eV 7930 eV

0 20 40 60 80 100 120 140

0 50 100 150 200 250

7920 7925 7930 7935 7940 7945 7950

peak intensity [a.u.] flourescence yield [a.u.]

energy [eV]

GdS 9/2 1/2 1/2

L_II edge

GdS: L _II Edge Resonance

Brückel, Hupfeld, Strempfer, Caliebe, Mattenberger, Stunault, Bernhoeft, McIntyre; Eur. Phys. J B19 (2001); 475

) ( )

( )

( )

1 (

lin E f circ E

f o E

f E E

f res = + +

Dipole Approximation:

⎥⎦

⎢ ⎤

⎣

⎟⎡

⎠⎞

⎜⎝

⎛

ε ε⋅ + + −

= 1

F 1 1 1 F ' ) E 0 ( f

⎥⎦

⎢ ⎤

⎣

⎟ ⎡

⎠⎞

⎜⎝

⎛

ε × ε ⋅ − − +

= 1

F 1 1 1 F m '

i ) E circ ( f

( )

_⎥

⎦

⎢ ⎤

⎣

⎟ ⎡

⎠⎞

⎜⎝

⎛

ε ⋅ ε ⋅ − + − −

= 1

F 1 1 1 1 F

F 0 2 m m

' ) E lin ( f

Amplitudes:

Oscillator Strengths:

2 h res i

1 M

F M ω − ω − Γ

= α

⎟⎠

⎜ ⎞

⎝⎛

( ) ^{... 2}

' 1

' '

2 2 2

' = ⋅ + + +

Ω → ⎟

⎠

⎜ ⎞

⎝

⎛

ε ε ε

λ ε ε

ε ε ε

σ E _E

f res f M

d i c f c

mc e d

d

Anomalous Scattering: Cross Section

XRES: Resonance Enhancements

elements edge transition energy range [keV]

resonance strength

comment

3d

^{K 1s → 4p 5 - 9 weak} small overlap

3d

^{LI 2s → 3d 0.5 - 1.2 weak} small overlap

3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{K 1s → 5p 40 - 63 weak} small overlap

4f

^{LI 2s → 5d 6.5 - 11 weak} small overlap

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

4f

^{MI 3s → 5p 1.4 - 2.5 weak} small overlap

4f

^{MII, MIII 3p → 5d}

3p → 4f

1.3 - 2.2 medium to strong

dipolar quadrupolar

4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

XRES: Resonance Enhancements

elements edge transition energy range [keV]

resonance strength

comment

3d

^{K 1s → 4p 5 - 9 weak} small overlap

3d

^{LI 2s → 3d 0.5 - 1.2 weak} small overlap

3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{K 1s → 5p 40 - 63 weak} small overlap

4f

^{LI 2s → 5d 6.5 - 11 weak} small overlap

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

4f

^{MI 3s → 5p 1.4 - 2.5 weak} small overlap

4f

^{MII, MIII 3p → 5d}

3p → 4f

1.3 - 2.2 medium to strong

dipolar quadrupolar

4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

thin films ^3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

XRES: Resonance Enhancements

elements edge transition energy range [keV]

resonance strength

comment

3d

^{K 1s → 4p 5 - 9 weak} small overlap

3d

^{LI 2s → 3d 0.5 - 1.2 weak} small overlap

3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{K 1s → 5p 40 - 63 weak} small overlap

4f

^{LI 2s → 5d 6.5 - 11 weak} small overlap

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

4f

^{MI 3s → 5p 1.4 - 2.5 weak} small overlap

4f

^{MII, MIII 3p → 5d}

3p → 4f

1.3 - 2.2 medium to strong

dipolar quadrupolar

4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

thin films ^3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

XRES: Resonance Enhancements

elements edge transition energy range [keV]

resonance strength

comment

3d

^{K 1s → 4p 5 - 9 weak} small overlap

3d

^{LI 2s → 3d 0.5 - 1.2 weak} small overlap

3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{K 1s → 5p 40 - 63 weak} small overlap

4f

^{LI 2s → 5d 6.5 - 11 weak} small overlap

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

4f

^{MI 3s → 5p 1.4 - 2.5 weak} small overlap

4f

^{MII, MIII 3p → 5d}

3p → 4f

1.3 - 2.2 medium to strong

dipolar quadrupolar

4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

thin films ^3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

thin films ^4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

XRES: Resonance Enhancements

elements edge transition energy range [keV]

resonance strength

comment

3d

^{K 1s → 4p 5 - 9 weak} small overlap

3d

^{LI 2s → 3d 0.5 - 1.2 weak} small overlap

3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{K 1s → 5p 40 - 63 weak} small overlap

4f

^{LI 2s → 5d 6.5 - 11 weak} small overlap

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

4f

^{MI 3s → 5p 1.4 - 2.5 weak} small overlap

4f

^{MII, MIII 3p → 5d}

3p → 4f

1.3 - 2.2 medium to strong

dipolar quadrupolar

4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

thin films ^3d

^{LII, LIII 2p → 3d 0.4 - 1.0 strong} dipolar, large overlap, high spin polarisation of 3d

4f

^{LII, LIII 2p → 5d}

2p → 4f

6 - 10 medium dipolar

quadrupolar

thin films ^4f

^{MIV, MV 3d → 4f 0.9 - 1.6 strong} dipolar, large overlap, high spin polarisation of 4f

5f

^{MIV, MII 3d → 5f 3.3 - 3.9 strong} dipolar, large overlap, high spin polarisation of 5f

Outline

• What's new in magnetism ?

• Experimental techniques

• Elastic magnetic neutron scattering

• X-ray techniques for magnetism

• Nonresonant magnetic x-ray scattering

• Resonant magnetic x-ray scattering

• Example: Non-resonant scattering from transition metal di-flourides

• Example: Resonance exchange scattering from mixed crystals

• Summary

High Energy X-Ray Scattering

Sample Ø 5 mm

W. Schweika:

ico-AlPdMn: phason modes

High Energy X-Ray Scattering

Sample Ø 5 mm

Bragg geometry 10 keV

W. Schweika:

ico-AlPdMn: phason modes

High Energy X-Ray Scattering

Sample Ø 5 mm

100 keV Sample

Ø 5 mm

W. Schweika:

ico-AlPdMn: phason modes