Synchrotron Radiation for Materials Science Applicationsattwood/srms/2007/Lec21.pdfSynchrotron Radiation for Materials Science Applications Polarized X-rays from a Bending Magnet and

Synchrotron Radiation for

Materials Science Applications

Polarized X-rays from a Bending

Magnet and X-ray Magnetic Circular

Dichroism (XMCD)

Brooke L. Mesler

AS&T UC Berkeley

X-ray Magnetic Circular Dichroism

Magnetization dependent absorption of circularly polarized light

XMCD

tioncrossAbsorptionrayXabs

whereabs

xeIInotationAnother

sec

: 0

−−==

−=

σρσµ

µ

ρ

ρ

)(),(),,(

)3.1:(

:

0

σµωµσωµµ

ρµ

⋅+=⋅=

−=

MZMZ

where

aeqntextx

eI

I

lecturespreviousfromrecall

hh

µ−

µ+

xright circ.

left circ.

I0 I

I0 I

Circularly Polarized Light

kσ σ=Photon angular momentum:

Where σ=photon helicity =±1

Right circularly polarized

σ=+1

Left circularly polarized

σ=-1

Polarized Radiation at a Synchrotron

Source

Polarized light from an

EPU or from off-axis

bending magnet radiation

ultrarelativisticelectrons

ca. 30 m

mm

5

-5

1-1 010

0

Gain in polarization

⇔ Loss in intensity

Polarization properties of SR

e-

TOP VIEW

e- e-

SIDE VIEW

Bending Magnet Radiation On-axis

0=ψ

Power per Solid Angle

observeremitter

nv

β&v

av

Θ=××−=

=

∝⇒

Θ=

Ω

=

sin)(

:

)34.2:(16

sin:

)32.2:(16

),(:

:

2

3

0

2

222

023

0

2

22

aaandanna

onacceleratitransverseanote

aSolidAngle

power

eqntextc

ae

d

dP

SolidAngle

powerand

eqntextkrc

aetrS

area

power

lecturespreviousfromrecall

TT

T

T

T

vvvvvv

v

v

v

vv

vv

επ

επ

Flux per Solid Angle

bandwidth

electronsofbeamelectroncurrentI

dtetaA

SolidAngle

FluxA

e

I

dd

Fd

KimJeKwangfollowingNow

tti

B

=∆

→=

=

=

∆=

−

∫

ωω

απ

ωω

ωωω

αψθ

ω

)(

137/1

')'(2

)(

)()(

:,

)'(

22

v

K.-J. Kim,“Characteristics of Synchrotron Radiation,”pp565-632 in Physics of Particle Accelerators, AIP Conference Proceedings 184

(American Inst. of Physics, New York 1989)

Horizontal and Vertical Components

+

−+

=

∆=

→

)(1

)(

))(1(2

3

:asAwritecanwefunctionsBesselifiedmodthegsinunow

componentsverticalandhorizontalointupfluxbreak

radiationtheofonpolarizatitheinerestedintareWe

3/12

3/22

2

h,

2

,

2

,

2

η

η

ωω

γπ

ωω

α

ψθ

ψθ

ν

KX

iX

K

iXA

A

A

A

e

I

dd

Fd

dd

Fd

cv

h

v

h

vB

hB

γψ

ωω

η

γω

=

+=

=

X

X

c

m

eB

c

2/3)21(2

1

2

23

(We’ll see that this simplifies greatly for ψ 0)



Flux as a function of angle

)2

1()/(

)%1.0()/()()(1033.1

2

10,00

)(1

)(

)1(4

3

2

3/2

2

2

22

213

0

2

2

,

2

2

3/12

2

2

3/222

2

2

2,

2

,

2

)6.5:(

cc

c

c

B

c

vB

cvB

hB

KEEHwhere

BWmrad

photonsEEHAIGeVEe

dddd

Fd

dd

FdXLet

KX

X

K

Xe

I

dd

Fd

dd

Fd

eqntext

ωω

ωω

ωωψθ

ωω

ηψθ

ψ

η

η

ωω

ωω

γπα

ψθ

ψθ

ψ

=

⋅×=

===⇒=

+

+

∆=

=

γψ

ωω

η

γω

=

+=

=

X

X

c

m

eB

c

2/3)21(2

1

2

23



Behavior at Critical Energy

Horizontal (on axis) radiation as a

function of energy

Moving off axis

Photon Flux

Behavior at 700eV

L adsorption

edges of Fe are

~707eV, ~720eV

Bending magnet radiation is naturally polarized at small angles off-axis

ALS:

Ee=1.9GeV, I=400mA, B=1.27T

Ec=0.6650*(Ee)2[GeV]B[T]

EcALS=3.05keV

700eV/3.05keV =0.2

γ=1957Ee[GeV]

γALS=3718.3

γ*ψ=1 ψ~2.7*10-4 radians

Moving off axis

Photon Flux

X-ray Magnetic Circular Dichroism

(XMCD)

0

1

2

3

4

5

700 710 720 730

-2

-1

0

µ (

a.u.)

∆µ

(a.

u.)

energy (eV)

M↑↑σPhoton

M σPhoton↑↓

∆µ/µ ≈ 50%

L3

L2

706eV 719eVFe

element specific

huge magnetic contrast

M·σPhoton

Quantitative probe of

spin and orbital

moments

A typical XMCD result

@ Fe L3,2 absorption

edges

M=magnetization

= magnetic moment per volume

XMCD Measurement

dichroic signal µ+− =µ − ∆µ

transmission modeabsorber

µ −

µ +

xright circ.

left circ.

I0

I

I0

I

xe

I

I ±±

−=

ρµ

0

Dichroic signal

dichroic signal µ+− =µ − ∆µ

absorber

µ−

µ+

xright circ.

left circ.

I0I

I0

Iµ−

µ+

I0I

absorber

right circ. x

I0I

X-ray Absorption

Conservation laws

• energy Eph=Ef-Ei

• linear momentum (for

small e-energies) in

direction of E-vector

• orbital momentum

(symmetry)

L2

L3

1s1/2

2s1/2

2p1/2

2p3/2

d- d- continuum

k k

electron transitions

E E (L )γ≥ B 3

E E (L )γ≥ B 2

0

0

1,0

1

:RulesSelection

=∆

=∆

±==∆

±=∆

s

l

m

s

m

l

σ

Absorption of Circularly Polarized

X-rays

L2

L3

1s1/2

2s1/2

2p1/2

2p3/2

d- d- continuum

k k

electron transitions

E E (L )γ≥ B 3

E E (L )γ≥ B 2

2-step description:

Photon “polarizes” electron through the transfer of angular momentum to electron spin through spin-orbit coupling

Valence band only takes electrons of appropriate spin

| , >| , > | , > | , >

Origin of Spin and Orbital Polarisation

L L

l

z

z

2 3

σ -1/2 +1/4

+3/4 +3/4

m=1m = +1/2j

p1/2

m = -1/2j

2/3| >,l

↓

2/3| >,m=-1l↑ 1/3| >,m=0l

↓

1/3| , >m=0l↑

|m m>:l s 2 ↓ 1 ↑ 1 ↓0 ↑

L absorption of a right pol. photon2

∆ m=+1l

∆m=0s

d continuum

Not all transitions are allowed,

considering the allowed transitions,

we calculate average spin and

angular momentum of the excited

electrons

Transition Probabilities

m = +1/2j

p1/2

m = -1/2j

2/3| >,m =1l ↓

2/3| >,m =-1l ↑ 1/3| >,m =0l ↓

1/3| , >m =0l ↑

|m ,m >:l s | , >2 ↓ | , >1 ↑ | , >1 ↓| , >0 ↑


∆ m = +1l ∆m = 0s

d continuum

60% 10% 15% 15%

m = +1/2j

p1/2

m = -1/2j

2/3| >,m =1l ↓

2/3| >,m =-1l ↑ 1/3| >,m =0l ↓

1/3| , >m =0l ↑

|m ,m >:l s | , >2 ↓ | , >1 ↑ | , >1 ↓| , >0 ↑


∆ m = +1l ∆m = 0s

d continuum

60% 10% 15% 15%

Spin and Orbital Polarizations

-1/6-½15%1/3ml=0, spin

down

1/9+1/610%2/3ml=1, spin up

1/6+½15%1/3ml=0, spin up

-2/3-160%2/3ml=1, spin

down

Weighted SpinSpin up or down

(+ or -)

Relative

Transition

Probability

Transition

Probability

(Clebsh-

Gordon)2

Initial State<σz> for

p1/2 to d

<σz>=average spin

= (-2/3+1/6+ 1/9-1/6)/ (2/3+1/6+ 1/9+1/6)

=(-5/9) / (10/9)

=-1/2

Spin-orbit coupling

d-like final states

right circularly polarised light

+1/4-1/2

+3/4 +3/4

< >σz

<l >z

L2

L3

2p1/2

2p3/2

j = l - s

for theinitial state

< >σz <l >z

j = l s+

for the

initial state

< >σz <l >z

Photoelectron achieves Spin and Orbital polarization <σz>, <lz> in

photon propagation direction z.

L L

l

z

z

2 3

σ -1/2 +1/4

+3/4 +3/4

Ferromagnetic state of a 3d transition

metal

E= 0F

ρD−ρD

+

energy (eV)

Density of states (DOS)

+5

-10

Stoner model

Net magnetic

moment in

material caused

by exchange

splitting

Majority band Minority band

~5 eV

0

m / d ms B s B µ = ( − ) (Ε) Ε = − ∫ / µ ρ+ ρ−unocc. occ.

Splitting of “spin-up”

and “spin-down” bands

at the Fermi level

Formation of a magnetic

Spin moment

~ -10 eV

0

m / S µB = − ) (Ε) Ε∫ ( dρ+ ρ−

Define a “hole”-moment...

The “hole-moment”

E= 0F

ρD−ρD

+

energy (eV)

DOS

+5

-10

What does XMCD probe?

Absorption probes the

density of valence states

~5 eV

0

m / d ms B s B µ = ( − ) (Ε) Ε = − ∫ / µ ρ+ ρ−unocc. occ.

…or the “hole moment”

Electron moments

E= 0F

ρD−ρD

+

energy (eV)

DOS

+5

-10

EF

left circularly right circularly

polarised photon

L -absorption: 2

< > = -1/2σ z

2 p3/2

2 p1/2

2 s1/2

ρ+ ρ−

Spin of the

photoelectron

µ(Ε) ∝| ρM | (E)if

2

µ + ∝ ρ− unocc.

Fermi´s Golden Rule

)()(2

int EfEfEiiHfTyprobabilitTransition if ρδ −∝=

difference spectra

∆µ = µ+ − µ−

schematically

Sum rules:

Fit the experimental spectrum by a

weighted addition of both

contributions to obtain

spin and orbital moments

The sum rules of XMCDL3 L2

1.0

+1/4

-1/4

0

-3/8

∆µ

(E)

0

0

-3/4

X-ray energy Eγ

1.5

∆ES.O.

d-like unoccupied final states with...

Orbital and Spin Moments

operatorsmomentumspinandangulartheofvaluesectationexp,

1017.12

:

:

2

::

29

zz

o

B

z

Bz

sz

Bz

o

sl

smVm

eonBohrMagnet

where

smlm

MomentMagneticSpinMomentMagneticOrbital

−×==

−=

−=

h

hh

µµ

µµ

Magnetic moment from motion of electron

in orbit Intrinsic magnetic moment of the electron

J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).

Sum rules

)12(3

2

3

2

:

2

2)1(

1

2)1(

1

2

+=

=+−

=+−

⇒

−=∆ +−∑

L

LQRCwhere

rulesummomentorbitalmC

BA

rulesumspinmC

BA

RULESSUM

CCQRI

From

o

B

s

B

states

XMCD

µ

µ

A<0

B>0

Areas:


P. Gambardella et al., NATURE 416, 301 (2002)

XMCD of Co structures: 3D ⇒ 0D

XMCD

2q

q

2

2

PQ

Poperatorsdipoletheusingnow,

constantQ

Qint

)())((Q

abI

r

where

arbIenergyoveregrate

EbEaEbarb

res

res

abs

α

α ε

ε

ρωδεωσ

=

⋅=

=

⋅=⇒

−−⋅=

vv

vvh

hvv

h

Following Stöhr and Siegmann:


XMCD

),(12

4'

P

,,)()(),'(

PQ

,

)(

)1(

1,0

,

q

2

)1(

,,

,,,'

2q

ψθπ

δ

αα

α

α

ml

l

m

p

p

q

p

cpl

pmlmc

q

pcnlnss

res

Yl

torsensoroperasphericaltRacahCwhere

Cer

angularradialspin

mcCmlerRrrRmm

abI

+==

=

=

=

∑

∑

±=


Element Specificity

300 400 500 600 700 800 900 10000.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ca L2,3

O K

Mn L2,3

La M4,5

Energie (eV)

(a.u

)

2p->3d

2p->3d1s->2p

2p->3d

Abso

rpti

onco

effi

cien

t

µ(E

)

Energy (eV)

characteristic „White Lines“

La0.7Ca0.3MnO3

Element specificity caused by radial term:

Radial component of core levels is highly localized

)()( ,,' rRrrR cnln

XMCD difference signal

2)1(

1

2)1(

1

2

+−

+−

↑↑↑↓

−=∆

−=∆⇒

−≡∆

∑ CCQRI

IIIionmagnetizatfixedfor

III

states

XMCD


Peter Fischer

David Attwood

Bending Magnet Radiation:



D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation, Principles and Applications, (Cambridge University Press, New York,

1999).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999)

A. Hofmann, The Physics of Synchrotron Radiation, (Cambridge University Press, Cambridge, 2004).

P. J. Duke, Synchrotron Radiation, Production and Properties, (Oxford University Press Inc., New York, 2000).

XMCD:


S. W. Lovesey and S. P. Collins, X-Ray Scattering and Absorption by Magnetic Materials, (Oxford University Press Inc., New York,

1996).

G. Schutz, P. Fischer, K. Attenkofer, M. Knulle, D. Ahlers, S. Stahler, C. Detlefs, H. Ebert, and F. M. F. De Groot, J. Appl. Phys. 76,

6453 (1994).

Papers on the Sum rules:

P. Carra, B. T. Thole, M. Altarelli, and X. Wang, Phys. Rev. Lett. 70, 694 (1993).

J. Stohr and H. Konig, Phys. Rev. Lett. 75, 3748 (1995).

A. Ankudinov and J. J. Rehr, Phys. Rev. B 51, 1282 (1995).

References and Acknowledgements

Synchrotron Radiation for Materials Science Applicationsattwood/srms/2007/Lec21.pdfSynchrotron Radiation for Materials Science Applications Polarized X-rays from a Bending Magnet and

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