Introduction to crystal field multiplet calculations Introduction - PowerPoint PPT Presentation

Introduction to crystal field multiplet calculations Introduction Introduction Multiplets (atomic physics) Multiplets (atomic physics) Multiplets (crystal field) Multiplets (crystal field) 1

2p XAS NiO 2p - XAS Experiment is sharper than LDA Experiment is sharper than LDA L edge 3 empty density of states empty density of states Branching ratio ( L 2 , L 3 intensity Branching ratio ( L 2 , L 3 intensity ratio is not 3:2) ratio is not 3:2) Intensity (arb. units) Experiment shows more Experiment shows more structure (multiplets) than DOS structure (multiplets) than DOS L 2 continuum L 3 edge jump continuum edge jump L edge 2 2p 840 845 850 855 860 865 870 875 880 885 Photon energy (eV) 2

2p XAS (core level spectroscopy) • Excitons and multiplets dominate the spectral line- shape. • It is not sufficient to assume independent electrons interacting with an average potential • One has to consider the interaction between each pair of electrons explicitly. 3

Charge Transfer Multiplet program Quanty, CTM4XAS, Tanaka code (XTLS), etc Used for the analysis of XAS, EELS, PES, IPES, Auger, RIXS, NIXS, etc ATOMIC PHYSICS ⇓ CRYSTAL FIELD THEORY ⇓ CHARGE TRANSFER 4

Quanty - XAS L 23 edges transition metals L 23 edges transition metals M 45 edges rare-earth M 45 edges rare-earth PRB 93 , 165107 (2016) PRB 85 , 165113 (2012) 5

Quanty – RIXS Orbital and dispersing magnetic Orbital and dispersing magnetic transitions in TiOCl transitions in TiOCl Linear Dichroism in Sr 2 IrO 4 Linear Dichroism in Sr 2 IrO 4 6 PRL 107 , 107402 (2011).

Quanty – non-resonant IXS Angular dependence of d-d excitations in NiO Angular dependence of d-d excitations in NiO 7 EPL 96 , 37007 (2011)

Charge Transfer Multiplet program input • Initial State configuration (3d 8 for Ni 2+ ) • Final state configuration (2p 5 3d 9 for Ni 2+ 2p XAS) • Transition operator (dipole for XAS) • Experimental geometry (polarization of the light, sample orientation, magnetic field, temperature, etc.) • H i Hamiltonian of the initial state • H f = H i + H cd + H SOC Hamiltonian of the final state core 8

Charge Transfer Multiplet program What information we can get ? • Ground state symmetry, important quantum numbers, magnetic moment, etc. • Orbital occupation and orbital level splitting • Many body energy level diagram • Physical properties (magnetic anisotropy, susceptibility, magnetization, g factors, etc.) 9

Hamiltonian � � � � � = + ξ � � � ∙ � � + � �� + � ��� + � � ∙ � � + 2� � � �� ��� � 10

Hamiltonian (atomic physics) � � � � � = + ξ � � � ∙ � � + � �� + � ��� + � � ∙ � � + 2� � � �� ��� � 11

Coulomb interaction Expand the operator on Spherical Harmonics and split the operator and Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. the wave functions into an angular and a radial part. with with 12

Coulomb interaction Expand the operator on Spherical Harmonics and split the operator and Expand the operator on Spherical Harmonics and split the operator and the wave functions into an angular and a radial part. the wave functions into an angular and a radial part. with with 13

Coulomb interaction – Slater Integrals Integral to calculate Integral to calculate Expansion on renormalized Spherical Harmonics Expansion on renormalized Spherical Harmonics 14

Coulomb interaction – Slater Integrals Radial part: Slater integrals Radial part: Slater integrals Angular part: Analytical solution Angular part: Analytical solution 15

Coulomb interaction – Slater Integrals d electrons d electrons f - electrons f - electrons 16

Coulomb interaction – Slater Integrals Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – exchange term Core (p) valence (d) interaction – exchange term 17

Coulomb interaction – Slater Integrals Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – direct term Core (p) valence (d) interaction – exchange term Core (p) valence (d) interaction – exchange term 18

Initial state Hamiltonian (atomic multiplet theory) Electron-electron interaction of Electron-electron interaction of Valence Spin-orbit coupling Valence Spin-orbit coupling valence states valence states ξ � � � ∙ � � ∑ ∑ S + S + = k + k L 2 L f F g G 2 1 e 2 1 | | J J k k r 12 k k � 19

Final state Hamiltonian (atomic multiplet theory) Electron-electron interaction of Electron-electron interaction of Valence Spin-orbit coupling Valence Spin-orbit coupling valence states valence states ξ � � � ∙ � � ∑ ∑ S + S + = k + k L 2 L f F g G 2 1 e 2 1 | | J J k k r 12 k k � Core-valence electron interaction Core-valence electron interaction Core Spin-orbit coupling Core Spin-orbit coupling ∑ ∑ + + ξ � � � ∙ � � S S = k + k L L f F g G e 2 2 1 2 1 | | J J k k r 12 k k � 20

Final state Hamiltonian (atomic multiplet theory) Electron-electron interaction of Electron-electron interaction of Valence Spin-orbit coupling Valence Spin-orbit coupling valence states valence states ξ � � � ∙ � � ∑ ∑ S + S + = k + k L 2 L f F g G 2 1 e 2 1 | | J J k k r 12 k k � Core-valence electron interaction Core-valence electron interaction Core Spin-orbit coupling Core Spin-orbit coupling ∑ ∑ + + ξ � � � ∙ � � S S = k + k L L f F g G e 2 2 1 2 1 | | J J k k r 12 k k � Spin-orbit coupling couples L and S quantum numbers Only the total moment J is a good quantum number 21

Atomic multiplet theory Ca 2p XAS Ca 2p XAS 22

Term symbols 2S+1 L 23

Term symbols – single electron Orbital occupation Orbital occupation Angular momentum Angular momentum Term symbol Term symbol → s 1 2 S S=1/2, L=0 → p 1 2 P S=1/2, L=1 → d 1 2 D S=1/2, L=2 24

Term symbols – two electrons Orbital occupation Orbital occupation Angular momentum Angular momentum Term symbol Term symbol S 1s =1/2, S 2s =1/2 → S tot =0,1 1s 1 2s 1 1 S, 3 S → L tot =0 L 1s =0, L 2s =0 25

Term symbols – two electrons Orbital occupation Orbital occupation Angular momentum Angular momentum Term symbol Term symbol 2S+1 L S 2p =1/2, S 3d =1/2 → S tot =0,1 3 P, 3 D, 3 F 2p 1 3d 1 (2p 5 3d 1 ) 1 P, 1 D, 1 F L 2p =1, L 3d =2 → L tot =1, 2 or 3 26

Term symbols – two electrons Orbital occupation Orbital occupation Angular momentum Angular momentum Term symbol Term symbol 2S+1 L J 2p 1 3d 1 S 2p =1/2, L 2p =1 → J 2p =1/2,3/2 1 P 1 , 3 P 0 , 3 P 1 , 3 P 2 1 D 2 , 3 D 1 , 3 D 2 , 3 D 3 (2p 5 3d 1 ) S 3d =1/2, L 3d =2 → J 3d =3/2,5/2 1 F 3 , 3 F 2 , 3 F 3 , 3 F 4 For each term symbol J MAX = S 1 + L 1 J MIN = |S 1 – L 1 | Steps of 1 between J MIN and J MAX 27

Atomic multiplet theory Ca 2p XAS Ca 2p XAS 3 peaks in the spectrum. Why? 28

Atomic multiplet theory Dipole transition Ca 2p XAS Ca 2p XAS 3d 0 → 2p 5 3d 1 Without spin-orbit coupling Initial state symmetry : 3 peaks in the spectrum. Why? 1 S Final state symmetry : 1 P, 1 D, 1 F Selection rules for dipole transition: ΔL=+1 or -1 ΔS=0 Allowed transition < 1 S| ∆ S=0; ∆ L=+1 | 1 P> ≠0 1 peak in the spectrum 29

Atomic multiplet theory Dipole transition Ca 2p XAS Ca 2p XAS 3d 0 → 2p 5 3d 1 With spin-orbit coupling Initial state symmetry : 3 peaks in the spectrum. Why? 1 S 0 Final state symmetry : 1 P 1 , 3 P 0 , 3 P 1 , 3 P 2 , 1 D 2 , 3 D 1 , 3 D 2 , 3 D 3 1 F 3 , 3 F 2 , 3 F 3 , 3 F 4 Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions < 1 S 0 | ∆ J=+1| 1 P 1 , 3 P 1 , 3 D 1 > ≠0 3 peaks in the spectrum 30

Atomic multiplet theory Dipole transition Ca 2p XAS Ca 2p XAS 3d 0 → 2p 5 3d 1 With spin-orbit coupling Initial state symmetry : 1 S 0 Final state symmetry : 1 P 1 , 3 P 0 , 3 P 1 , 3 P 2 , 1 D 2 , 3 D 1 , 3 D 2 , 3 D 3 1 F 3 , 3 F 2 , 3 F 3 , 3 F 4 Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Allowed transitions < 1 S 0 | ∆ J=+1| 1 P 1 , 3 P 1 , 3 D 1 > ≠0 3 peaks in the spectrum 31

Atomic multiplet theory Dipole transition La 3d XAS La 3d XAS 4f 0 → 3d 9 4f 1 With spin-orbit coupling Initial state symmetry : 1 S 0 Final state symmetry : 1 P 1 , 3 P 0 , 3 P 1 , 3 P 2 , 1 D 2 , 3 D 1 , 3 D 2 , 3 D 3 , 1 F 3 , 3 F 2 , Calculated 3 F 3 , 3 F 4 , 1 G 4 , 3 G 3 , 3 G 4 , 3 G 5, 1 H 5 , 3 H 4 , 3 H 5 , 3 H 6 Selection rules for dipole transition: ΔJ=+1 or -1 or 0 J=J’≠0 Experimental Allowed transitions < 1 S 0 | ∆ J=+1| 1 P 1 , 3 P 1 , 3 D 1 > ≠0 3 peaks in the spectrum 32