Amélie Juhin
Sorbonne Université-CNRS (Paris)
1 « Dichroism » (« two colors ») describes the dependence of the absorption measured with two orthogonal polarization states of the incoming light:
Circular left Circular right
k k
ε ε Linear horizontal Linear vertical
k k
ε ε
2 By extension, « dichroism » also includes similar dependence phenomena, such as:
• Low symmetry crystals show a trichroic dependence with linear light
• Magneto-chiral dichroism (MχD) is measured with unpolarized light
• Magnetic Linear Dichroism (MLD) is measured by changing the direction of magnetic ield and keeping the linear polarization ixed …
Dichroism describes an angular and /or polarization behaviour of the absorption
3 Linear dichroism (LD) : difference measured with linearly polarized light
Circular dichroism (CD) : difference measured with left / right circularly polarized light.
Natural dichroism (ND) : time-reversal symmetry is conserved
Non-Reciprocal (NR): time-reversal symmetry is not conserved
Magnetic dichroism (MD) : measured in (ferro, ferri or antiferro) magnetic materials
Dichroism Time reversal Parity symmetry symmetry Natural Linear (NLD) + + Magne c Linear (MLD) + + Non Reciprocal Linear (NRLD) - - Natural Circular (NCD) + - Magne c Circular (MCD) - + Magneto-op cal (MχD) - -
4 The measurement of dichroism is often challenging…
… but provides access to properties that cannot be measured in another way
isotropic spectra polarized spectra 9 Al K edge Al K edge [6] beryl Al
6
6 corundum -Al O α 2 3 ε //c
3 beryl 3
(BeO)3(Al2O3)(SiO2)6 Normalized absorption (arbit. units) Normalized absorption (arbit. units) ε ⊥ c
0 0 1560 1565 1570 1575 1580 1585 1590 1560 1565 1570 1575 1580 1585 1590 incident X-ray energy (eV) incident X-ray energy (eV)
Average signature Access to polarization dependent states 5 The corresponding « sum rules » relate dichroism to a ground state moment:
Widely applied in XMCD, less applied in other types of dichroisms
XMCD : average value of
6 ProblemXMCD : The parametrization in LFM
Can be measured in ferri / ferromagne c materials (XMCD = 0 in an ferromagnets) XAS measured with circularly polarized x-rays on a sample magne cally polarized by external magne c field
) XMCD = XAS right – XAS left → 1.0 XAS left
� units . 0.2
┬ ) arb L
XAS (arb. unit) (arb. XAS 0.5 2
0.0 units . XAS (
0.0
-0.2arb 710 715 720 725 730 735 Energy (eV) Energy (eV) Fe L edges Absorptionunits) (arb. 2,3 -0.4 L 3 4K, +/-6.8T → XMCD (
) 1.0 � -0.6 XAS right 710 715 720 725 730 735
┬ units Photon Energy (eV) . Energy (eV)
0.5 arb XAS (arb. unit) (arb. XAS XAS (
0.0 710 715 720 725 730 735 EnergyEnergy (eV) (eV) 7 XMCD sum rules
XAS: Sum rulesMagne c for circular field B isdichroism set along the orbitalZ axis and spin moment
L ⟨�|�↓�LZ |�⟩ L< zZ> = Expecta on value of LZ (Z component of orbital momentum operator) 8 XMCD sum rules
XAS: Sum rulesMagne c for circular field B isdichroism set along the orbitalZ axis and spin moment
S ⟨�|�↓�SZ |�⟩ S< Z> = z ⟨�|�↓� |�⟩ (+
Widely applied in XMCD, less applied in other types of dichroisms
XMCD : average value of
XMLD : average value of
XNCD : mixture between states with different parity (orbital pseudodeviator)
XNLD : anisotropy of charge distribution (quadrupole / hexadecapole moments)
Dichroism is not straightforward to predict / calculate…
Let’s start with X-ray Natural Linear Dichroism (XNLD)
10 The XAS cross-section
2 2 σ (ω) = 4π α!ω∑ f Ô i δ (!ω − E f + Ei ) f ,i
Photon polarization Photon wave vector
i i
11 Electric operators Magnetic operators Dipole spin position operator
Negligible for X-rays Sizeable for K edge XMCD N. Bouldi, PRB 96 (2017)
quadrupole octupole 2 ! !⎛ i ! ! 1 ! ! 2 ⎞ ! ! k ! !! ! k ! ! ! ! 2 f ε.r ⎜1+ k.r − k.r ⎟ i = f ε.r i +i f ε.ru.r i − f ε.r (u.r) i ⎝ 2 6 ( ) ⎠ 2 6
! ! k Photon energy in u 1 = k = 2 α!ω Rydberg k 1 Fine structure constant ≅ 127 Selection rules ! ! ! ! The expansion of and in real spherical harmonics gives : ε.r u.r
! ! m 4π m ε.r = (−1) 3 rY1 (Ω) Ω = (θ,ϕ) For example, polarization along z, wave vector along x :
! ! 4π 0 0 ε .r = z = r cosθ = 3 rY1 = c10rY1 ℓo = 1 mo= 0
! !! ! 2 2 −1 1 ε .ru.r = zx = r sinθ cosθ cosϕ = c21r (Y2 −Y2 ) ℓo = 2 mo= + 1 and -1
The transition matrix is :
Radial integral Gaunt coef icient ⎛ R ⎞⎛ ⎞ f o i = c a f (E) b r, E g (r)r2+ℓo dr ⎜ Y m Ω Y mo Ω Y mi Ω dΩ⎟ ℓomo ∑ ℓm ⎜ ∫ ℓ ( ) ℓi ⎟⎜ ∫∫ ℓ ( ) ℓo ( ) ℓi ( ) ⎟ ℓ,m ⎝ 0 ⎠⎝Sphere ⎠
Non zero, only for some ℓ and m à gives the selection rules
13 Case of K-edge (1s initial state): ℓi = 0 = mi
dipole component and polarization along z : ℓo = 1 mo= 0 the only non-zero matrix element is for ℓ =1 m= 0
à one probes the pz inal states projected onto the absorbing atom
! ! 1 1 dipole component and polarization along y : ε.r = y = c11r(Y1 +Y−1)
à one probes the py inal states projected onto the absorbing atom
If pz and py electron density are different : one can measure XNLD
àXNLD is due to anisotropy in charge distribution Case of K-edge (1s initial state): ℓi = 0 = mi
quadrupole component, polarization along z, wave vector along x :
one probes the dxz inal states projected onto the absorbing atom
quadrupole component, polarization along (x+y)/sqrt(2), wave vector along (x-y)/sqrt(2)
one probes the dx2-y2 inal states projected onto the absorbing atom
If the dxz and dx2-y2 electron densities are different, one can measure XNLD
z
x
e
B k → → E ⊥ Z
Example : s-p transitions in square planar 3d complex z
k CN 2+ Ni → NC CN E NC 2.0 σ⊥
1.5
σ// 1.0 → → E // Z 0.5
0.0 Absorbance / u.a.
-0.5 - σ⊥ σ// NiNi -K- K edge
-1.0 8300 8320 8340 8360 8380 8400 8420 8440 8460 8480 N N
E / eV C C
k
2+ → Ni E z C C N N
16 Dipole versus quadrupole transitions
k [001] 3+ Octahedral Cr ions in MgAl2O4 ε
Cr K-edge Cr K pre-edge
electric dipole 1sàp transitions electric quadrupole 1sà3d transitions
No XNLD XNLD
Same crystal but different angular dependence 17 One absorbing site versus whole crystal
2 4 2 f o i E E σ (ω) = π α!ω∑ δ (!ω − f + i ) f ,i
What we measure :
XAS signal from the crystal (sum over atoms) : symmetry of the material (space group)
What we directly calculate with an atomic code :
XAS signal from one atom : symmetry of the atomic site (point group symmetry) lower than crystal symmetry
How to make connection between crystal properties and site properties ?
18 One absorbing site versus whole crystal
Our example for today : spinel MgAlMgAl2O2O4 4
• Fd-3m space group (#227): cubic system m-3m point group
• Cubic unit cell contains 32 octahedral sites : 16 are occupied, with Wyckoff position 16c
Do we need to perform 16 calculations ? Can we simplify the problem ?
19 3 translations
3 x 4 = 12
4 equivalent sites by rotation One absorbing site versus whole crystal
Our example for today : spinel MgAl2O4
• Fd-3m space group (n°227): cubic system m-3m point group
• Cubic unit cell contains 32 octahedral sites :
16 are occupied, with Wyckoff position 16c : D3d or -3m symmetry
• Only 4 are crystallographic equivalent (translations do not matter for XAS)
Symmetry elements in Oh point group
Symmetry elements in D3d point group
The C3 axis in D3d is one of the C3 axis in Oh One absorbing site versus whole crystal
Our example for today : spinel MgAl2O4
• Fd-3m space group (n°227): cubic system
• Cubic unit cell
• There are four D3d sites crystallographic equivalent, with their respective C3 axis along [111] [1 11] [11 1] [11 1 ] directions A, B, C and D sites are not « equivalent» for XAS :
A priori they€ yield €different € cross-sections ! €
σ cube = σ A + σ B + σ C + σ D
23
€ Issues related to XNLD calculations
1. Can we predict the angular dependence based on the crystal structure ?
2. Is there an analytical expression of the XAS cross-section :
for electric dipole transitions ? for electric quadrupole transitions ?
3. How does one calculate the spectrum for the whole crystal using a single atom model ?
24 Expression of the angular dependence of XAS, RIXS etc…
Many physical properties can be described by a tensor : for example : electric dipole transition amplitude in XAS : tensor of rank 1 (= a vector) : 31 = 3 components electric dipole transition intensity in XAS : tensor of rank 2 (= a matrix) 32 = 9 components
A tensor of rank N is the generalized form : 3N components
Two formalisms can be used to describe the same physical property :
- Cartesian Tensors - Spherical Tensors
25 Expression of the angular dependence of XAS, RIXS etc…
Take a Cartesian tensor of rank 2 and apply a transformation :
its components will transform linearly into themselves
Now we limit ourselves to rotations
It is possible to make « special » linear combinations the 9 components and from « groups » where they transform into themselves
One is invariant : a scalar (a 0th rank tensor)
A group of 3 transform into themselves : a vector (a 1st rank tensor)
A group of 5 transform into themselves : a 2nd rank tensor
It is not possible to make smaller groups : we call them irreducible tensors 26 Expression of the angular dependence of XAS, RIXS etc…
m Ex : Spherical harmonics Y ℓ are irreducible tensors
m = −ℓ,...,ℓ m By analogy, irreducible tensors are labeled : Tℓ
rank
It is easy to rotate or multiply them.
27 XAS cross-section in Cartesian coordinates : (1) electric dipole transitions
In Quanty (which uses Green function formalism) the electric dipole XAS cross- section is calculated as the Imaginary part of the « conductivity tensor » σ : sxx sxy sxz
Absorp on = -Im[ε.σ.ε] ε : polarization
syx syy syz Real Imaginary = XAS
s s s zx zyω0 zz à The conductivity tensor is calculated once à The absorption cross-section can then very easily be calculated for any ε
28 XAS cross-section in Cartesian coordinates : (1) electric dipole transitions
• For linearly polarized x-rays, this conductivity tensor writes :
with
i = x,y,z ; j= x,y,z
à The « conductivity tensor » σij = a Cartesian tensor of rank 2 (= a matrix):
3 x 3 = 9 components sxx sxy sxz
syx syy syz
szx szy szz
29 XAS cross-section in Cartesian coordinates : (1) electric dipole transitions
Spectrum measured with x-polarized light : ε =(100) (i=j=x) sxx sxy sxz
syx syy syz
szx szy szz XAS cross-section in Cartesian coordinates : (1) electric dipole transitions
Spectrum measured with y-polarized light : ε=(010) (i=j=y) sxx sxy sxz
syx syy syz
szx szy szz sxx sxy sxz
Spectrum measured with z-polarized light : ε = (001) (i=j=z) syx syy syz
szx szy szz But the conduc vity tensor (σ) is a TENSOR
Spectrum measured with ε =1/sqrt(2) (011) i=y,z ; j=y,z sxx sxy sxz
syx syy syz ½ ½ szx szy szz ½ ½ XAS cross-section in Cartesian coordinates : (1) electric dipole transitions
• Simpli ication is possible by considering the symmetry group G of the sample
For any operation S in G, �(�)=�[�(�)]
σ is symmetrized (averaged over all S symmetry operations)
à Spherical average : the case of a powder :
The isotropic spectrum is the trace of the conductivity tensor in any (x,y,z) frame
34 à The case of cubic symmetry
In the cubic (xyz) frame the conductivity tensor writes :
σxx = σyy= σzz
isotropic
Other components are zero à The case of tetragonal symmetry z axis
In the (xyz) tetragonal frame where z is the fourfold axis the conductivity tensor writes :
σxx = σyy≠ σzz
dichroism
Other components are zero Other (equivalent) expressions from the literature for a dichroic crystal :
A crystal with a high symmetry axis z : trigonal or tetragonal system
The conductivity tensor can be written in the principal axis
// means parallel to z (=σzz) ┴ means perpendicular to z (=σxx, σyy)
2 fundamental spectra only are needed in order to know everything
Analytical expression of dipole cross-section for a trigonal or tetragonal crystal :
At magic angle (54.7 ° between ε and z) : one measures the isotropic spectrum 37 à The case of cubic symmetry + magnetic ield Magnetic ield along z
sxx sxy sxz
w0
syx syy syz
H= 4 I 0 szx szy szz -I 4 0 ( 0 0 4 ) à The case of orthorhombic symmetry
In the (xyz) orthrhombic frame the conductivity tensor writes :
σxx ≠ σyy≠ σzz
Other components are zero à The case of monoclinic symmetry
In the (xyz) monoclinic frame the conductivity tensor writes :
σxx ≠ σyy≠ σzz
σxy and σyx are non-zero à The case of triclinic symmetry
In the (xyz) triclinic frame the conductivity tensor writes :
9 components sxx sxy sxz are different w0
syx syy syz
H= 4 2 2 szx szy szz 2 0 1 ( 2 1 8 ) XAS cross-section in Cartesian coordinates : (2) electric quadrupole transitions
For linearly polarized x-rays, the electric quadrupole XAS cross-section writes :
with
à Conductivity tensor : a Cartesian tensor of rank 4 : more complicated than a matrix ! 3 x 3 x3 x 3 = 81 components Note that Quanty calculates only a 5x5 matrix (25 well-chosen wrt symmetry) components
à For a powder :
= the isotropic quadrupole spectrum 42 Expressing the angular dependence with Cartesian tensors quickly becomes heavy :
for electric quadrupole transitions when symmetry is low
What about with spherical tensors ?
Expressions for all symmetry groups (dipole and quadrupole operators) are given in :
C. Brouder, « Angular dependence of x-ray absorption spectra », J. Phys. Condens. Matter 2 701 (1990)
43 XAS cross-section in spherical coordinates : (1) electric dipole transitions
For linearly polarized x-rays, the electric dipole XAS cross-section writes :
with
isotropic anisotropic tensor components = fundamental spectra angular coef icient = energy-dependent functions
à The symmetry of the crystal restricts the possible values of σ(2,m).
44 XAS cross-section in spherical coordinates : (1) electric dipole transitions
Examples z axis de ined as high symmetry axis of the crystal
Cubic isotropic : 1 spectrum to measure / calculate
= σxx= σyy=σzz
Trigonal / Tetragonal dichroism : 2 spectra to measure / calculate
similar to :
Triclinic
trichroism 6 spectra to measure / calculate
45 XAS cross-section in spherical coordinates : (2) electric quadrupole transitions
For linearly polarized x-rays, the electric quadrupole XAS cross-section writes :
1 isotropic
with
5 +9 = 14 anisotropic tensor components
15 independent fundamental spectra (energy-dependent functions) in order to determine (= measure or calculate) σ for any (ε,k)
46 XAS cross-section in spherical coordinates : (2) electric quadrupole transitions
Examples spherical coordinates
(xyz) frame : z axis de ined as high symmetry axis of the crystal x and y are de ined according Tables of Crystallography
Cubic No XNLD
XNLD is not zero for a cubic crystal ! 2 spectra to measure / calculate Trigonal dichroism
4 spectra to measure / calculate Triclinic
15 spectra to measure / calculate 47 Coming back to our example : how to calculate XNLD in practice ?
2 I 4 2 f .r .rk.r i E E σ (ω) = π α!ω∑ ε + ε δ (!ω − f + i ) f ,i 2
2 2 4 2 f .r i E E 2 f .rk.r i E E σ (ω) = π α!ω∑ ε δ (!ω − f + i )+π α!ω∑ ε δ (!ω − f + i ) f ,i f ,i dipole quadrupole Cubic crystal
One spectrum to calculate : Two spectra to calculate : any orientation of ε is possible two independent sets of (ε,k)
48 quadrupole Two spectra to calculate : two independent sets of (ε,k)
[001] k ε
k [001] ε
49 Coming back to our example : how to calculate XNLD in practice ?
2 I 4 2 f .r .rk.r i E E σ (ω) = π α!ω∑ ε + ε δ (!ω − f + i ) f ,i 2
dipole quadrupole Cubic crystal
isotropic Dichroism : 2 fundamental spectra
One D3d site
Dichroism : 2 fundamental spectra
Trichroism : 4 fundamental spectra 50 Coming back to our example : how to calculate XNLD in practice ?
2 I 4 2 f .r .rk.r i E E σ (ω) = π α!ω∑ ε + ε δ (!ω − f + i ) f ,i 2
dipole quadrupole Cubic crystal isotropic dichroism : 2 fundamental spectra Average over A, B, C and D sites Average over A, B, C and D sites
One D3d site
dichroism : 2 fundamental spectra
trichroism : 4 fundamental spectra 51 Be careful : the fundamental spectra (tensor components)of the crystal are not necessarly the same as for a single site.
Here it can be shown that :
cube D3d σ D (0,0) = σ D (0,0)
cube D3d crystal σ Q (0,0) = σ Q (0,0) site 1 σ cube (4,0) = − (7σ D3d (4,0) + 2 70σ D3d (4,3)) Q 18 Q Q
52 rotB=Rz(π/2) site A à site B σ (rot -1(ε)) σD(ε) D B σ (rot -1(ε), rot -1(k)) σQ(ε,k) Q B B
rotC=Rz(3π/2) cube A B C D site A à site C σ D = σ D +σ D +σ D +σ D -1 σD(ε) σ (rot (ε)) D C cube A B C D -1 -1 σQ(ε,k) σQ(rotC (ε), rotC (k)) σ Q = σ Q +σ Q +σ Q +σ Q
rotD=Rz(π) site A à site D ( ) -1 σD ε σD(rotD (ε)) -1 -1 σQ(ε,k) σQ(rotD (ε), rotD (k))
53 Site A Site B Site D Site C
1 calculation
Site A Site D Site B Site C
2 calculations
For quadrupole XNLD, the number of calculations to do has been reduced from :
16 (sites) x 15 (components) to 3 by symmetry considerations
Conclusion : Always look irst for symmetries ! 54 DFT calculations
55 J. Phys. Condens. Matter 20, 455205 (2008) Phys. Rev. B 78, 195103 (2008)
56 Conclusion
• Symmetry considerations and tensor expressions are very helpful :
- to reduce the number of calculations / experiments needed - to know what angular dependence to expect
• XNLD in XAS is well understood
• XNLD in RIXS, XMLD, XNCD… are much more dif icult
57