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19 October 2005C. Vettier Institut Laue Langevin1
Magnetic X-ray Scattering
Introduction
What does theory tell us ?
Experimental problems
Non-resonant scattering experiments
Resonant scattering experiments
Summary
19 October 2005C. Vettier Institut Laue Langevin2
introduction
J.D. Jackson Classical Electrodynamics (1962) page 681
coupling between magnetism and photons was predicted long ago
parallel developments
technology/theory
F.E. Low, Phys. Rev. 96, 1428 (1954)
M. Gell-Mann and M.L. Goldberger, Phys. Rev. 96, 1433 (1954)
P. Platzman and N. Tzoar, Phys. Rev. B2, 3556 (1970)
F. De Bergevin and M. Brunel, Acta Cryst. A37, 314 (1981)
M. Blume, J. Appl. Phys. 57, 3615 (1985)
19 October 2005C. Vettier Institut Laue Langevin3
introduction
F. de Bergevin M. Brunel Acta Cryst.1981
X-ray beam interacting with a single electron
Thomson scattering
spin-only scattering
momentum-only scattering
relativistic effects
rather weak effects :
no effect at Q = 0
mc
h= 2.59 A 1
hQ
mc2S 10 3
19 October 2005C. Vettier Institut Laue Langevin4
introduction
X-ray beam interacting with bound electrons
discrete electronic levels
possible transitions to excited states :
core hole + electron in valence band
magnetic sensitivity
photon in - photon out
selectivity :
chemical species,
electronic shells
19 October 2005C. Vettier Institut Laue Langevin5
introduction
what can be observed and measured ?
non-resonant scattering :
weak but simple scattering amplitude spin/orbital momentum
resonant scattering :
large scattered intensities
chemical and electronic selectivity
x-ray spectroscopy
imaging
importance of x-ray polarisation effects
links with magneto-optics (optical index, polarisation of light)
19 October 2005C. Vettier Institut Laue Langevin6
x-ray scattering
back to interactions photons-electrons
• write Hamiltonian to describe charges (electrons+nuclei) and photons
• exploit Fermi’s Golden rule to calculate scattering cross-section
• distinguish different regimes (resonant and non resonant)
Hamiltonian
M. Blume J.App.Phys. 57, 3615 (1985)
M. Altarelli 2004 lecture - X-ray magnetic scattering : theoretical introduction.
radiation field
kinetic energy
potential energy
spin-orbit
19 October 2005C. Vettier Institut Laue Langevin7
x-ray scattering
radiation field E,B scalar and vector potential and A
approximations
perturbation theory : first order linear in Hint
second order quadratic in Hint
scattering expts : conservation of number of photons
first order : quadratic terms in A(r,t)
second order : linear terms in A(r,t)
19 October 2005C. Vettier Institut Laue Langevin8
x-ray scattering
approximations
Hamiltonian for interacting electrons and photons
expansion of
contains terms of :
degree 0,2 in A(r,t) perturbation 1st order
degree 1 in A(r,t) perturbation 2nd order
19 October 2005C. Vettier Institut Laue Langevin9
x-ray scattering
H’1
H’2
H’3
H’4
quadratic in A
linear in A
linear in A
quadratic in AH’4
linear term in A reduced by 1/c2 w.r.t. H’2 and H’3 - to be neglected -
19 October 2005C. Vettier Institut Laue Langevin10
x-ray scattering
calculations rather tedious - follow review articles
M. Blume J. Appl. Phys. 57, 3615 (1985)
M. Altarelli , Lectures 2004
M. Blume, in Resonant Anomalous X-ray Scattering Elsevier (1994) p. 496
Eds: G. Materlik, C.J. Sparks and K. Fischer
eigenstate states defined by Hel and Hrad -
transitions induced by Hint
Fermi’s Golden Rule
intermediate states |n>
19 October 2005C. Vettier Institut Laue Langevin11
x-ray scattering
how to relate transition probabilities to scattering ?
two different contributions
terms with no denominator : valid over all photon energies
terms with denominators: resonance/non-resonance
H’1
polarisation dependenceunits :
19 October 2005C. Vettier Institut Laue Langevin12
x-ray scattering
H’4
electron spin-dependent term with different polarisation dependence
same units (r0) but reduced by and phase lag
real magnetic term !
are there other magnetic terms in a non-resonant regime ?
yes !
19 October 2005C. Vettier Institut Laue Langevin13
x-ray scattering
second order terms :
ground state with 1 photon
intermediate states
a - with either 0 photon
b - or 2 photons
tricks with denominators :
case a : possible resonance - damping
case b : no divergence
19 October 2005C. Vettier Institut Laue Langevin14
non-resonant scattering
away from resonance (edge)
denominators are simplified
case a
case b : exchange operators
finally with
19 October 2005C. Vettier Institut Laue Langevin15
non-resonant scattering
more algebra ! but at the end :
same units (r0) and reduced by and phase lag
electron spin AND momentum appear in matrix elements
19 October 2005C. Vettier Institut Laue Langevin16
non-resonant scattering
working out the x-ray cross-section in the non-resonant regime
19 October 2005C. Vettier Institut Laue Langevin17
non-resonant scattering
L 1
2Bohr magneton eh/2mc
operator relates to orbital momentum (see neutron case)
density of orbital magnetisation
19 October 2005C. Vettier Institut Laue Langevin18
non-resonant scattering
non-resonant scattering cross-section for x-rays :
Thomson scattering
magnetic scattering
orbital
magnetization
spin magnetization
differentiation through polarisation dependence
19 October 2005C. Vettier Institut Laue Langevin19
non-resonant scattering
use linear polarisation vectors as basis for matrix representation
fncharge(
r Q ) = n (
r Q ) ˆ . ˆ ' = n (
r Q )
1 0
0 cos 2
(2)
Use Didier’s expression
use scattering amplitudes
d
d= r0
2 tr '( ) ' = M M+ M = Mc ih
mc 2 Mm
19 October 2005C. Vettier Institut Laue Langevin20
non-resonant scattering
fnnon res (
r Q ,
r k ,
r k ' ) = i
hQ
mc2S
cos ˆ S 2(Q) sin cos ( ˆ S 1(Q)L1(Q)
S) + sin ˆ S 3(Q)
sin cos ( ˆ S 1(Q) +L1(Q)
S) + sin ˆ S 3(Q)
cos ( ˆ S 2(Q) + 2 sin2 L2(Q)
S)
non-resonant amplitude depends on Q !
no effect at Q=0 - no effect on absorption - no dichroism !
possibility to separate L(Q) and S(Q) polarisation effects
high energy photons : spin-only magnetic scattering
scaling factor : difficult to observe w/o synchrotron !
hQ
mc10 3
h
mc 2 sin =hQ
mc c=
2
19 October 2005C. Vettier Institut Laue Langevin21
non-resonant scattering
F. De Bergevin and M. Brunel, Acta Cryst. A37, 314 (1981)
first experiments NiO
Magnetic Bragg peak
sealed x-ray tube
0
5 0 0 0
10000
15000
20000
25000
29.2 29.3 29.4 29.5 29.6 29.7
counts/sec
Theta (deg)
NiO T=300 K (3/2 3/2 3/2)
V. Fernandez et al. Phys. Rev. B 57, 7870 (1998)
experiments NiO
ESRF with same crystal
below
and
above
Tn
19 October 2005C. Vettier Institut Laue Langevin22
non-resonant scattering experiments
applications
determination of magnetic structures ?
details of magnetic structures ?
magnetism at surfaces ?
L(Q)/S(Q) determination ?
experimental details
clean x-ray beams
use of polarised x-rays - linear polarisation
tunability (polarisation and energy - resonance)
19 October 2005C. Vettier Institut Laue Langevin23
experimental details
magnetic scattering amplitude very weak
spurious peaks arising from impurities defects
harmonics in the beam ( /2, /3, ….)
clean beams and clean samples
19 October 2005C. Vettier Institut Laue Langevin24
experimental details
ki
kf
experimental set-up
ID beamline ESRF
19 October 2005C. Vettier Institut Laue Langevin25
experimental details
x-ray polarisation
production of polarised x-rays :
insertion devices : linear, circular, elliptical
crystal optics - phase plate crystals
19 October 2005C. Vettier Institut Laue Langevin26
experimental details
x-ray polarisation analysis
Azimuthal
SxSy
Q (001+ )
linear polarisation : use charge scattering by analyser crystal
circular polarisation : phase plates but difficult !
19 October 2005C. Vettier Institut Laue Langevin27
experimental details
linear polarisation : charge scattering !
19 October 2005C. Vettier Institut Laue Langevin28
experimental details
analyzer crystals examples
• select crystal for reflectivity
• go for large mosaic
• beware of resolution effects : width of analyzer/divergences
Element edge E (keV) analyser crystal d-spacing 2 cos2(2 )
Dy L3 7.790 quartz(3 1 -4 3) 1.1802 84.80 0.00822
Al (2 2 2) 1.16913 85.79 0.00538
Graphite (0 0 6) 1.11931 90.63 0.00012
quartz(3 0 -3 3) 1.11467 91.11 0.00038
quartz(2 0 -2 0) 1.0615 97.13 0.01540
Element edge E (keV) analyser crystal d-spacing 2 cos2(2 )
Nd L3 6.208 Al (2 2 0) 1.43189 88.44 0.00074
LiF (2 2 0) 1.42376 89.08 0.00026
quartz(2 0 -2 3) 1.3745 93.19 0.00310
Al2O3 (0 3 0) 1.374 93.23 0.00318
19 October 2005C. Vettier Institut Laue Langevin29
basic experimental problems
• sample
quality mosaic spread similar to beam divergence
to reduce spread in Q of scattered intensity
typical values around 0.05-0.01 deg.
absorption
x-rays do not propagate deep in bulk samples
except at high energy but high quality crystals required
no resonance
at resonance strong effects due to varying absorption
surface orientation and preparation
19 October 2005C. Vettier Institut Laue Langevin30
basic experimental problems
sample environment
again absorption by windows
cryostats
vibrations
heat load (mW range)
cryomagnets limitations:angular access
stray fields
high pressure: high energy only (>8-10 keV)
19 October 2005C. Vettier Institut Laue Langevin31
experiments : non-resonant scattering
non resonant scattering experiments
essentially antiferromagnets: NiO Ho UAs Cr
L/S and interfacial/surface magnetism
only a few ferromagnets
more difficult to observe (flipping ratio)
19 October 2005C. Vettier Institut Laue Langevin32
non resonant scattering NiO
• NiO simple antiferromagnet
crystal structure NaCl fcc
Ni2+ ions 3d8 configuration 2 holes in 3d band
Hund’s rules maximise S and then mL
crystalline electrical field cubic symmetry t2g full (6)
orbital singlet eg (2) half full
L=O
however spin-orbit exists, g=2.2 in paramagnetic phase
magnetic anisotropy in AF phase TN=523 K
neutron data unable to measure <L>
x-rays can help
19 October 2005C. Vettier Institut Laue Langevin33
non resonant scattering NiO
• pioneering experiment by de Bergevin and Brunel
search for antiferromagnetic peaks with x-rays
AF structure known from neutron data: type-II
ferromagnetic [111] planes are piled up antiferromagnetically
propagation vector (1/2,1/2,1/2) moments along [11-2] directions
magnetic domains 4 propagation directions and 3 moment directionsfor each
important details
unpolarised beam
sealed x-ray tube
low power to high energy flux
not from x-ray lines
signal/noise ratio
19 October 2005C. Vettier Institut Laue Langevin34
non resonant scattering NiO
experiments at ESRF
samples: flat [111] face
azimuthal scans : easy for specular reflections (hhh)
explored 1 T-domain (1/2,1/2,1/2) but 3 S-domains
• experimental conditions
linear undulator
Si(111) optics and mirrors: beam size 0.3x0.2 mm2 incident power
sample at room temperature. mounted on 4-circle diffractometer
polarisation analysis PG(006) peak reflectivity 12%
measure in direct beam and reflected beam incident polarisation
(P1=0.995±0.005 P2=0.0±0.005 Punp or P3<0.01±0.01)
2.0 1012 ph/ s 2.5mW
V. Fernandez et al. Phys. Rev. B 57, 7870 (1998)
19 October 2005C. Vettier Institut Laue Langevin35
non resonant scattering NiO
sample quality rocking curve
no polarisation analysis
how to put intensities on absolute scale?
integrate charge peak intensities
to be compared with known structure factors
extinction effects to be taken into account when
comparing charge and magnetic intensities
I(Q) = I0
3r02
2µva2
F(Q)2
sin2I(Q) = I0
8
3
2r0
va
F(Q)
sin2
E=7.84 keV E=17.41 keV
Bragg Structure reflecting Structure reflecting Structure
peak factor calc. power factor obs. power factor obs.
(111) 14.2 (2.7±0.2)10-4
9.8 (1.35±0.1)10-4
15.7
(222) 14.8 (9.5±0.6)10-5
7.2 (7.6±0.3)10-5
16.1
temperature factors BNi = 0.37 BO = 0.26
0
5 0 0 0
10000
15000
20000
25000
29.2 29.3 29.4 29.5 29.6 29.7
counts/sec
Theta (deg)
NiO T=300 K (3/2 3/2 3/2)
mosaic crystal perfect crystal
19 October 2005C. Vettier Institut Laue Langevin36
non resonant scattering NiO
• magnetic domains measure 3 magnetic peaks (h/2,h/2,h/2)
h=1,3,5
not possible to control domains
population (even by applying field)
average over orientation with
respect to incident polarisation
(azimuthal scans)
magnetic peaks show rotated
polarisation component
0 100
1 10-3
2 10-3
3 10-3
4 10-3
5 10-3
6 10-3
28.6 29.0 29.4 29.8
theta (deg)
(5/2,5/2,5/2)
0 100
1 10-4
2 10-4
3 10-4
4 10-4
5 10-4
6 10-4
7 10-4
8 10-4
-17.5 -16.5 -15.5
theta (deg)
(1/2,1/2,1/2)
0 100
5 10-3
1 10-2
2 10-2
2 10-2
3 10-2
-17.5 -16.5 -15.5
theta (deg)
Intensit
y (
counts/m
onit
or)
(1/2,1/2,1/2)
0 100
1 10-3
2 10-3
3 10-3
4 10-3
5 10-3
28.6 29.0 29.4 29.8
theta (deg)
Intensit
y (
counts/m
onit
or)
(5/2,5/2,5/2)
19 October 2005C. Vettier Institut Laue Langevin37
magnetic domains in NiO
• weak reflections from domains with different propagation vectors
sample almost single T-domain near surface
• rotation about [111] shows S-domains
offset between L and S no S3
polarised intensities are shifted in
Offset=0
can obtain domain fractions in volume
probed by beam:
1=0.26±0.03 2=0.23±0.03 3=0.50±0.02
translations indicate domains size 300 m
averaged intensities
M = sin(2 ) sin S(Q)
M = sin(2 ) sin( ) cos S(Q) + cos( + 0 )L(Q)( )
3.50 10-4
4.00 10-4
4.50 10-4
5.00 10-4
5.50 10-4
6.00 10-4
Itotal
NiO T = 300 K (3/2 3/2 3/2) E=7.84 keV
2.00 10-5
2.50 10-5
3.00 10-5
3.50 10-5
4.00 10-5
4.50 10-5
5.00 10-5
I
1.00 10-5
1.50 10-5
2.00 10-5
2.50 10-5
3.00 10-5
- 3 0 0 - 2 4 0 -180 -120 -60 0 6 0I
angle (deg)
19 October 2005C. Vettier Institut Laue Langevin38
magnetic form factors in NiO
• polarised intensities corrected for width of rocking curve of analyser
note: change in resolution function (better use crystals with wide band)
required to obtain fully integrated intensities
• comparison with unpolarised intensities (no analyser) gives analyser
reflectivity
• experimental -averaged intensities
• extract L(Q)/2S(Q) ratio as a function of Q
I (sin2 )21
2S2(Q)
I (sin2 )2 (sin )21
2S(Q) + L(Q)( )2
(h,k,l) I I Itotal
(1/2,1/2,1/2) (2.8±0.1) 10-5 (1.32±0.08) 10-6 (2.35±0.2) 10-4
(3/2,3/2,3/2) (3.45±0.1) 10-5 (1.7±0.05) 10-5 (4.5±0.3) 10-4
(5/2,5/2,5/2) (3.5±0.1) 10-6 (6.2±0.2) 10-6 (7.7±0.4) 10-5
19 October 2005C. Vettier Institut Laue Langevin39
magnetic form factors in NiO
• large contribution (17±3%) from orbital moment exists
• increase at large Q indicates broader spatial extent of spin
density
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0 0.1 0.2 0.3 0.4 0.5 0.6
L(Q
) / 2
S(Q
)
Q/4 = sin /
NiO
T = 300 K
Electrons with same spins tend to be further apart !
19 October 2005C. Vettier Institut Laue Langevin40
magnetic form factors in NiO
• comparison of magnetic intensities to charge intensities and
extinction corr.
• polarised intensities on absolute scale
in electron units
extract S(Q) and L(Q) in real numbers
S(O)=0.95±0.1 and L(O)=0.32±0.05
magnetic moment at 300K 2.2±0.2 B
(h,k,l) F2 F2 F2total
(1/2,1/2,1/2) (8.4±0.9) 10-6 (4.0±0.5) 10-7 (9±1) 10-6
(3/2,3/2,3/2) (3.2±0.4) 10-5 (1.6±0.2) 10-5 (4.8±0.5) 10-5
(5/2,5/2,5/2) (4.2±0.5) 10-6 (7.5±0.9) 10-6 (1.1±0.2) 10-5
0.00
0.20
0.40
0.60
0.80
1.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
S(Q)
L(Q)
spin
and o
rbit
al m
om
entum
form
factors
Q/4 = sin /
NiO T = 300 K
19 October 2005C. Vettier Institut Laue Langevin41
magnetic form factors in NiO
neutrons :
pb: neutrons would tell L/S through a fit of data with electronic wave
functions
not really known in NiO; atomic wave functions not adequate.
comparison with other methods
g-factor =2.2 indicates spin-orbit/10Dq 0.1 which leads to L 0.3
19 October 2005C. Vettier Institut Laue Langevin42
magnetic form factors
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
L(Q
)/S
(Q)
Q/4 =sin /
CoO
NiO
CuO
MnO
through the whole series MnO, CoO, CuO
related to resonant effects
to be discussed later
existence of E2 resonance at K-edge linked to finite <L>
better knowledge of electronic properties of 3d oxides
W. Neubeck at al. J. Phys Chem Sol., 62, 2173 (2000)
19 October 2005C. Vettier Institut Laue Langevin43
magnetism at surfaces
NiOnon-resonant scatteringgrazing incidence scattering
A. Barbier et al. PRL 93, 25708 (2004)
depth sensitivesurface orderingwell-defined 2D order
melting of S-domains
19 October 2005C. Vettier Institut Laue Langevin44
non-resonant scattering from ferromagnet
Principle : measurement of flipping ratios
like in neutron diffraction
If we can flip fmagnetic by applying H or rotating polarisation
we define
ftotal = fcharge + fmagnetic with fmagnetic << fcharge
R =
I+ I
I+ + II+/- = |fcharge +/- f magnetic|
2
in the case of non-resonant scattering fmagnetic is linear in S and L.applying a field can change the sign of fmagnetic with different incidentpolarisations.
R is given by the interference between charge and magnetic scattering.Since Thomson scattering does not rotate polarisation, interferencearises in and channels only (linear polarisation)
in case of general incident polarisation, the full formalism has to be used.
19 October 2005C. Vettier Institut Laue Langevin45
non resonant scattering - ferromagnets
• density matrix
• scattered intensity
• scattered polarisation
• charge scattering back to equations
• pure magnetic terms complex in the case of general polarisation
interference between spin and orbital momentum
beyond this lecture
M. Blume et al. PRB 37, 1779 (1988)
=
1
2
1 + P1 P2 i P3
P2 + i P3 1 P1
P = tr( ) Poincare vector
d
d= r0
2 tr '( ) ' = M M+
P' =
tr '( )
tr '( )
d
d c= r0
2 (Q) 2 1
21 + cos22 + P1 (1 cos22 )[ ]
19 October 2005C. Vettier Institut Laue Langevin46
non resonant scattering - ferromagnets
• purely magnetic scattering
• linear polarisation P1 centrosymmetric structure real structure
factors
• circular polarisation P3
d
d m=
1
2r02 hQ
mc
2
(1 + P1)cos2 S
22 + sin2 (L1 + S1 )2[ ] + sin4 S
32
+ sin2 sin2 (L1 + S1)S3
(1 P1)cos2 (2 sin2 L2 + S2)2 + sin2 (L1 + S1 )2[ ] + sin4 S
32
sin2 sin2 (L1 + S1 )S3
d
d m= r0
2 hQ
mc
2
cos2 1
2(S
22 + (2 sin2 L2 + S2 )2 ) + sin2 (L1 + S1 )2
+ sin4 S
32
P3 sin2 cos(L1' + S1') sin2 L2'' + S2''[ ] (L1'' + S1'') sin2 L2' + S2'[ ]sin3 (S3'L2''- S3''L2')
19 October 2005C. Vettier Institut Laue Langevin47
non resonant scattering - ferromagnets
• interference terms
• linear polarisation P1 centrosymmetric structure
note that if crystal and magnetic structures are
centrosymmetric
interference occurs through imaginary part of atomic scattering
factors
• circular polarisation P3
d
d I= - r0
2 hQ
mc
cos (1 + P1 ) ''(Q)S2
+cos cos2 (1 P1 ) ''(Q) 2sin2 L2 + S2{ }
d
d I= - r0
2 hQ
mc
cos( '(Q)S''2 ''(Q)S'2 ) +
cos2 '(Q)(2sin2 L''2 + S''2 ) - ''(Q)(2sin2 L'2 + S'2 )[ ]
P31
2
(1 + cos2 )sin2 '(Q)( L'1 + S'1 ) - ''(Q)( L''1 + S''1 )[ ]
(1 cos2 )2 '(Q)S'3 ''(Q)S''2[ ]
19 October 2005C. Vettier Institut Laue Langevin48
non resonant scattering - ferromagnets
• assume fully linearly polarised radiation P1=±1
incident polarisation P1=1
incident polarisation P1=-1
in principle it is possible to measure L and S by changingpolarisation
R =hQ
mc
2 cos ''(Q)S2
(Q) 2 +hQ
mc
2magnetic contribution
hQ
mc
2 cos ''(Q)S2
(Q) 2
R =hQ
mc
2 cos ''(Q) 2sin2 L2 + S2{ }
cos2 (Q) 2 +hQ
mc
2magnetic contribution
hQ
mc
2 cos ''(Q) 2sin2 L2 + S2{ }cos2 (Q) 2
19 October 2005C. Vettier Institut Laue Langevin49
non resonant scattering - ferromagnets
• assume circularly polarised radiation P1=P2=0
scattering geometry: field (S and L) in scattering plane butperpendicular to Q
only S1 and L1 components
we can flip magnetisation or polarisation
if 2 =90°
if magnetic field perpendicular to scattering plane S1=L1=0
not very efficient
R =hQ
mcP3
(1 + cos2 ) sin2 '(Q)( L'1 + S'1 ) - ''(Q)( L''1 + S''1 )[ ]
(1 + cos22 ) (Q) 2
R
hQ
mcP3
( L1 + S1 )
(Q)
R =hQ
mc2
cos( '(Q)S''2 ''(Q)S'2 )+
cos2 '(Q)(2sin2 L''2 + S''2 ) - ''(Q)(2sin2 L'2 + S'2 )[ ]
P3
2(1 cos2 )2 ''(Q)S''2
(1 + cos22 ) (Q) 2
19 October 2005C. Vettier Institut Laue Langevin50
flipping ratios from ferromagnetic materials
• application to determination of polarisation of synchrotron beams
known magnetic substance ; in general L=0 and centrosymmetric
field in scattering plane: only S1 and S3 components
ideal if H// kicos +kf but S2=S3=0 is ok
assume P2=0 but with P1 and P3 finite
neglect imaginary parts f’ and f’’
Acta Cryst A39, 84-88 (1983) M.Brunel et al.
d
d I= r0
2 hQ
mc
P3
1
2(1 + cos2 )sin2 '(Q)( L'1 + S'1 ) - ''(Q)( L''1 + S''1 )[ ]{ }
R = 8hQ
mc
P3
S(Q)
(Q)
sin cos3
1 + cos2 2 + P1 (1 cos2 2 )
19 October 2005C. Vettier Institut Laue Langevin51
flipping ratios from ferromagnetic materials
• circular polarisation
2 types applications
Bragg scattering to increase sensitivity
coupling to real part of (Q)
field perpendicular to Q but in scattering plane 2 =90°
white beam technique; fixed 2 =90° value; energy sensitive
detector
D.Laundy et al., J.Phys. Condens Matter 3, 369-372 (1992)
magnetic Compton scattering
signal proportional to P3.
field in scattering plane q.cos + q’
Physica Scripta T35, 103-106 (1991)
R
hQ
mcP3
( L1 + S1 )
(Q)
19 October 2005C. Vettier Institut Laue Langevin52
non-resonant scattering - summary
what does non-resonant magnetic scattering bring about ?
possibility to use highly collimated beams for magnetism
surface magnetism
polarization analysis
separation of orbital and spin magnetism
however, “weak” intensities
“good” samples (single crystals) are needed
19 October 2005C. Vettier Institut Laue Langevin53
resonant scattering
as a reminder, probability of transition
at the resonance, the leading term is given by
absorption and creation of photon - case a
19 October 2005C. Vettier Institut Laue Langevin54
resonant scattering
for such process, the leading term is :
we can distinguish 3 terms :
and denominator !
D. Wermeille PhD Thesis Ecole Polytechnique de Lausanne (1998)
19 October 2005C. Vettier Institut Laue Langevin55
resonant scattering
the leading term !
J.P. Hannon et al. PRL 61, 1245 (1988)
involved algebra and calculation
see lecture by S di Matteo !
basic physical explanation :
electric transitions
but exclusion principle leads
magnetic sensitivity
spin polarization of core levels and
excited levels
19 October 2005C. Vettier Institut Laue Langevin56
resonant scattering
complex expression for scattering amplitude :
various approximations : L=1 dipolar, L=2 quadrupolar, …
many quantities to evaluate numerically !
with :
and :
19 October 2005C. Vettier Institut Laue Langevin57
resonant scattering
no simple link to spin or moment values !
geometrical dependence of scattering amplitude
all invariants that can be formed with the quantities involved
dipole : polarisation vectors and quantization axis
quadrupole : same plus x-ray wave-vectors, …..
dipole :
zJ direction of local magnetisation
anomalous scattering: no z, magnetic sscattering: terms in z and z2
19 October 2005C. Vettier Institut Laue Langevin58
resonant scattering
at lowest order it has no magnetic sensitivity
but contributes to anomalous scattering
first observations by K. Nanimakawa et al. J. Phys. Soc. Jpn 54, 4099 (1985)
at lowest order
no dichroism
19 October 2005C. Vettier Institut Laue Langevin59
resonant scattering
dipole :
zJ direction of local magnetisation
anomalous scattering :no z, magnetic terms in z and z2
focus on the term
19 October 2005C. Vettier Institut Laue Langevin60
resonant scattering
simple example/illustration at M edge (for dipole transition!)
assembly of (polarised) ions with simple ground state:
1 hole in the 4f band
Hund’s rule l=3, ml = -3, ms = -1/2 j = 7/2 mj=-7/2
only 1 possible transition possible at M5 edge
from 3d5/2, mj=-5/2 to 4f7/2, mj=-7/2
F11 = F10 = 0
contributions to 3 terms in scattering amplitude
no resonance at the M4 edge ! (in dipole approximation)
J.B. Goedkoop et al. PRB 37, 2086 (1988)
19 October 2005C. Vettier Institut Laue Langevin61
resonant scattering
connection to absorption/fluorescence
Q=0 experiments
linear term : non-zero if complex polarisation circular dichroism
quadratic term : linear dichroism, Cotton-Mouton effect
scattering experiments
constant term : “normal anomalous” term non-magnetic
linear term : Bragg peaks corresponding to magnetic lattice symmetry
at Qm in reciprocal lattice
quadratic term : “resonant” harmonics at 2xQm !
19 October 2005C. Vettier Institut Laue Langevin62
resonant scattering
fE1res
= F(0) ' i F(1) ( ' ) z + F(2) ( ' z)( z)
= F(0) 1 0
0 cos2
i F(1) 0 z1 cos + z3 sin
z1 cos z3 sin z2 sin2
+ F(2) z22 z2 z1 sin z3 cos( )
z2 z1 sin + z3 cos( ) z12 sin2
+ z32 cos2
geometry and polarisation dependence
no - scattering in first order
19 October 2005C. Vettier Institut Laue Langevin63
resonant scattering
quadrupolar approximations
fE2res contains terms up to 4th order in z
order (0) (k' k) ( ' )
order (1) i (k' k ) ( ' ) z + [k' ' ,k ]
order (2) (k' k) ( ' z) ( z) + [k' ' ] + [k ]
+ [k' ' , k ]+ (k' k) z ( ' ) z
order (3) i (k' z) (k z) ( ' ) z + [k' ' , k ]
+ [k' ' ] + [k ]
order (4) (k' z) (k z)( ' z) ( z)
more “resonant harmonics” ! up to 4xQm
19 October 2005C. Vettier Institut Laue Langevin64
resonant scattering
what does a resonance look like?
UAs simple antiferromagnet
(0,0,5/2) magnetic Bragg peak
diffracted intensity versus photon energy
D.B. McWhan et al. PRB 42, 6007 (1990)
huge intensities !
Interferences between resonances
M3, M4, M5 edges
essentially dipole transitions
M4,M5 from 3d to 5f
M3 from 3p to 6d
19 October 2005C. Vettier Institut Laue Langevin65
resonant scattering
applications :
• large scattered intensities
• chemical selectivity
• electronic selectivity
• probe of electronic densities of states
• Q=0 experiments - reflectometry - absorption - imaging
difficulties :
• how to interpret observed intensities ?
• are intensities magnetic in origin ?
• experimental limitations - absorption
19 October 2005C. Vettier Institut Laue Langevin66
resonant scattering
large intensities : many examples
but how to predict intensities ?
scattering amplitude : matrix elements (dipole/quadrupole)
spin polarisation of electronic levels
presence of spin-orbit coupling !
elements edge transitionintermediate
statesenergy (keV)
wavelength (Å)
3d K E1,E2 4p, 3d 7.112 1.743Fe L3 E1 3d 0.707 17.54
5d Pt L3 E2 5d 11.65 1.0724f L2,3 E1,E2 5d, 4f 7.24 1.71Gd M4,5 E1 4f 1.22 10.25f L2,3 E1,E2 6d, 5f 17.17 0.722U M4,5 E1 5f 3.74 3.32
100 r0
10 r0
0.1 r0
0.01 r0
1 r0
19 October 2005C. Vettier Institut Laue Langevin67
resonant scattering - chemical selectivity
chemical selectivity : absorption edges energies are element specific
3d series
elementK edge
(eV)L1 edge
(eV)L2 edge
(eV)L3 edge
(eV)Sc 4492 498 403 399
Ti 4966 561 461 454
V 5465 623 520 512
Cr 5989 696 584 574
Mn 6539 769 650 639
Fe 7112 845 720 707
Co 7709 925 793 778
Ni 8333 1008 870 853
Cu 8979 1097 952 932
Zn 9659 1196 1045 1022
across 3d series
selecting chemical species by tuning photon energies to proper edge
19 October 2005C. Vettier Institut Laue Langevin68
resonant scattering - chemical selectivity
alloys, compounds, intermetallics
hetero-magnetic multilayers,
U1-xNpxRu2Si2
antiferromagnetic system
incommensurate structure at x=0
(neutron scattering)
magnetic intensity at (00 4+q)
separation of U and Np response
E. Lidström et al. PRB 61, 1375 (2000)
19 October 2005C. Vettier Institut Laue Langevin69
resonant scattering - chemical selectivity
temperature dependence of
order parameters :
Np magnetism drives
magnetism of U lattice
19 October 2005C. Vettier Institut Laue Langevin70
resonant scattering - electronic selectivity
similarly, different edges couple to different electronic levels :
edge K L1 L2 L3 M1 M2 M3 M4 M5 …
core level 1s1/2 2s1/2 2p1/2 2p3/2 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 …
Holmium as an example
hexagonal crystal structure
helicoidal magnetic structure
propagating along c-axis
magnetic Bragg peaks at (h,k,l)+(0,0, )
at each edge, dipole and quadrupole transitions couple to
different bands
19 October 2005C. Vettier Institut Laue Langevin71
resonant scattering - electronic selectivity
D. Gibbs et al. PRL 61, 1241 (1988)
Holmium ordered phase at the L3 edge (8.07 keV)
L3 E1 leads to 5d states, E2 leads to 4f states
19 October 2005C. Vettier Institut Laue Langevin72
resonant scattering - electronic selectivity
unpolarised
resonance
harmonics
polarisation
analysis
- and -
separation
E1/E2
D. Gibbs et al. PRB 43, 5663 (1991)
d states
f states
19 October 2005C. Vettier Institut Laue Langevin73
resonant scattering - resonant harmonics
G. Helgesen et al., PRB 50, 2990 (1994)
temperature dependence of high-order harmonics
test for scaling theories
mean-field not
quite right !
19 October 2005C. Vettier Institut Laue Langevin74
resonant scattering - electronic selectivity
valence state selectivity
Coexistence of 2 valence states
Tm2+(4f13) and Tm3 +(4f12)
time scales ?
L3 edge dipole transitions to 5d states
D.B. McWhan et al. PRB 47, 8630 (1993)
19 October 2005C. Vettier Institut Laue Langevin75
resonant scattering - electronic selectivity
0
20
40
60
80
100
6.70 6.71 6.72 6.73 6.74
Peak I
nte
nsi
ty (
cts
/s/
20
0m
A)
Sm 5000 Å film
(0 0 16.5)
T = 50 K
Energy (keV)
E1
E2
application to the observation of induced magnetic moments
Samarium metallic rare-earth
exchange mediated by 5d electrons
0.1
1
1 0
100
2 0 4 0 6 0 8 0 100
E2 4f
E1 5d
Inte
gra
ted I
nte
nsi
ty
Temperature (K)
0.1
1.0
10.0
100 102 104 106
spin-polarisation of 5d bands follows 4f polarisation !
A. Stunault at al. PRB 65, 64436 (2002)
19 October 2005C. Vettier Institut Laue Langevin76
resonant scattering - selectivity ?
UGa3 simple antiferromagnet
strong resonance at the M4,5 edges of U
but resonance at K-edge of non-magnetic Ga
similar effects at K-edge of As in UAs !
K-edge transition from s to p bands
no large polarization expected !
not observed with neutrons nor Mössbauer
asymmetry between up/down orbitals ….
D. Mannix et al. PRL 86, 4128 (2001)
19 October 2005C. Vettier Institut Laue Langevin77
resonant scattering - spectrometry ?
observation of density of unoccupied electronic states !
simple energy line-shapes
UAs : M4,5 edges
E1 transition to 5f states
Ho : L3 edge
E1 transition to 4f states
E2 transition to 5d states
19 October 2005C. Vettier Institut Laue Langevin78
resonant scattering - spectrometry ?
0
20
40
60
80
100
6.70 6.71 6.72 6.73 6.74
Peak I
nte
nsi
ty (
cts
/s/
20
0m
A)
Sm 5000 Å film
(0 0 16.5)
T = 50 K
Energy (keV)
E1
E2Sm : L3 edge
E1 transition to 4f states
complex and broad
E2 transition to 5d states
19 October 2005C. Vettier Institut Laue Langevin79
resonant scattering - spectrometry ?
RbMnF3
K edge of Mn2+
K-edges are special
E1 transitions to p states
complex and very broad
E2 transitions to 3d states
very weak E2
no orbital moment
observed in other systems
difficult to analyse !
19 October 2005C. Vettier Institut Laue Langevin80
Q=0 experiments
XMCD
soft x-ray range (low energy)
circular x-ray polarisation
element sensitivity
XMCD & PhotoElectron Emission Microscopy PEEM
imaging of magnetic domain/walls
Co/Cu/Ni trilayer
coupling mechanism through domainwall stray fields
importance for thin film technology
W. Kuch et al. PRB 67, 214403 (2003)
19 October 2005C. Vettier Institut Laue Langevin81
resonant scattering - difficulties
• how to interpret observed intensities ?
no simple connection to magnetic moments
data analysis involves evaluating band calculations
• experimental limitations - absorption
large variation through the edge
corrections ?
example NiO K-edge of Ni. mu
raw
Int
corr
. Int
19 October 2005C. Vettier Institut Laue Langevin82
resonant scattering - difficulties
• are intensities magnetic in origin ?
RESONANCE is a very GENERAL PROCESS
the origin of the resonance can be very diverse
low local symmetry
incident photon promotes an electron (photo-electron) into anavailable electronic states (outer electronic shells).
these shells are sensitive to local symmetry of the siteprobabilities of transition (similar to scattering amplitudes)depend on the orientation of the symmetry axes with respect tothe incident polarisation (electrical field which actually inducesthe promotion of the electron)
atomic structure factors cannot be longer consider as spherical
objects; they depend on the orientation of the object with respect
to the incident polarisation (incident electrical field)
Templeton and Templeton Acta Cryst A38, 62 (1982)
19 October 2005C. Vettier Institut Laue Langevin83
resonant scattering - difficulties
difficult but classical problem: symmetry allows resonant scattering
but physical origin of the resonance to be determined by
experiments.
examples: low symmetry of crystal structures
MnF2 tetragonal structure
magnetic order (local moments break the symmetry)
orbital order (degenerate orbitals e.g. in a doublet
are occupied preferentially with LRO)
see S di Matteo
19 October 2005C. Vettier Institut Laue Langevin84
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0
0.001
0.002
0.003
-10 0 1 0 2 0 3 0 4 0 5 0 6 0
MnF2 (001) Templeton scattering E=6.55keV
Integrated i
ntensit
y (
cts/m
on)
Integrated in
tensit
y (c
ts/m
on)
Azimuthal angle (°)
resonant scattering - difficulties
MnF2
simple bct structure - Mn ions in low symmetry
environment D2h
(001) reflex forbidden
Mn K-edge at (001) strong resonance
0
0.05
0.1
0.15
0 . 2
6.52 6.53 6.54 6.55 6.56 6.57
Intensit
y
Energy (keV)
(0 0 1)
0
0.1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
6.52 6.53 6.54 6.55 6.56 6.57
Intensit
y (
cts/m
on)
Energy (keV)
polarisation and azimuth dependence
19 October 2005C. Vettier Institut Laue Langevin85
resonant scattering - summary
what does resonant magnetic scattering bring about ?
large scattering intensities
chemical selectivity
electronic selectivity
spectroscopy of available states
imaging methods
can we use x-rays to replace neutron scattering ?
NO !
inelastic scattering - and structure determinations
19 October 2005C. Vettier Institut Laue Langevin86
resonant scattering - summary
C.G. Shull et al. PR 76, 1256 (1949)
neutrons methods provide all
what we know about magnetic
structures
S.P. Collins et al. J.Phys. Cond. 7 L223 (1995)
resonant x-ray scattering from
UO2 powder at M4 edge !
only one powder line visible
19 October 2005C. Vettier Institut Laue Langevin87
magnetic x-ray scattering - summary
what do x-ray magnetic scattering bring about ?
new features - not available before
L/S separation
chemical selectivity
electronic selectivity
spectroscopy of available states
studies of micro-samples (surfaces !)
imaging methods and time-resolved
but requires a lot of care and caution