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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
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)
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)
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
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)
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 ↑
L absorption of a right pol. photon2
∆ 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 ↑
L absorption of a right pol. photon2
∆ 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:
J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).
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:
J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).
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
+==
=
=
=
∑
∑
±=
J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).
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
J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).
Peter Fischer
David Attwood
Bending Magnet Radiation:
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)
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:
J. Stohr and H. C. Siegmann, Magnetism, From Fundamentals to Nanoscale Dynamics, (Springer-Verlag Berlin Heidelberg, 2006).
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