Polarization and dichroism Amélie Juhin Sorbonne Université-CNRS (Paris) [email protected] 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) + + Magne6c Linear (MLD) + + Non Reciprocal Linear (NRLD) - - Natural Circular (NCD) + - Magne6c Circular (MCD) - + Magneto-op6cal (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 <M> the local magnetic moment of the absorber 6 ProblemXMCD : The parametrization in LFM Can be measured in ferri / ferromagne6c materials (XMCD = 0 in an6ferromagnets) XAS measured with circularly polarized x-rays on a sample magne6cally polarized by external magne6c 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 rulesMagne6c for circular field B isdichroism set along the orbitalZ axis and spin moment L ⟨�|​​�↓�LZ |�⟩ L< zZ> = Expectaon value of LZ (Z component of orBital momentum operator) 8 XMCD sum rules XAS: Sum rulesMagne6c for circular field B isdichroism set along the orbitalZ axis and spin moment S ⟨�|​​�↓�SZ |�⟩ S< Z> = z ⟨�|​​�↓� |�⟩ (+ <TZ> = TZ ) 9 magne6c dipole operator 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 <M> the local magnetic moment of the absorber XMLD : average value of <M2> 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 coeficient ⎛ 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.