Hyperfine interactions Mössbauer, PAC and NMR Spectroscopy: Quadrupole splittings, Isomer shifts, Hyperfine fields (NMR shifts)

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1 Hyperfine interactions Mössbauer, PAC and NMR Spectroscopy: Quadrupole splittings, Isomer shifts, Hyperfine fields (NMR shifts) Peter Blaha Institute of Materials Chemistry TU Wien

2 Definition of Hyperfine Interactions hyperfine interaction = all aspects of the nucleus-electron interaction which go beyond an electric point charge for a nucleus and is measured at the nucleus (affects the nucleus)

3 description of the nucleus: electric point charge (Z/r) nucleus with volume, shape and magnetic moment

4 How to measure hyperfine interactions? NMR NQR Mössbauer spectroscopy TDPAC

5 Electric Hyperfine-Interaction between nuclear charge distribution () and external potential E n ( x) V ( x) dx Taylor-expansion at the nuclear position E V Z 0 direction independent constant (monopole interaction) i V (0) x i ( x) x i dx electric field x nuclear dipol moment (=0) 1 2 ij 2 V (0) x x i j ( x) x i x j dx electric fieldgradient x nuclear quadrupol moment Q higher terms neglected nucleus with charge Z, but not a sphere

6 Electric monopole interaction Mössbauer Isomer Shift : integral over nuclear radius of electron density x nuclear charge nuclear radii are different for ground and excited state E 2 2 (0) (0) R R A A A E Q Q (0) (0) Q A Q bcc Fe ( 57 Fe)=-0.24 mm/s a 3 0 large (0) neg. IS Fe 2+, Fe 3+, :RTOxxx = (0)

7 V V V E 2 aa ba ca cb cc ij V ij Q Quadrupole interaction 1 Q( 57 Fe)=0.16 barn similarity transformation Vab Vac Vxx 0 0 Vbb Vbc 0 Vyy 0 V V 0 0 V zz V ij : traceless 3x3 tensor of electric field gradient EFG (2 nd derivative of V(0)) with V xx V yy V zz 0 V zz V yy V xx V ij 2 V (0) x x i j EFG characterized by principal component V zz and asymmetry parameter / V xx / / V / V zz / yy / QS 1 2 eqvzz 1 2 3

8 First-principles calculation of EFG

9 1 E 2 ij V ij Q

10 spatial decomposition : zz zz ( r) Y 3 r 20 theoretical EFG calculations We write the charge density and the potential inside the atomic spheres in a lattice-harmonics expansion V V ( r) LM d LM 3 r ( r) Y LM LM sphere sphere ( rˆ); V ij LM 20( r) dr 3 r ( r) Y r 2 V (0) ; x x 3 i LM Y j 20 d 3 V r c ( r) interstitial int erstital ( r) r r ( r) Y 3 r dr 20 d 3 r orbital decomposition : ( r) 20 nk lm k, n. l, l, m, m * nk lm Y 20 drˆ p p; d d;( s d) contr. V zz V pp zz V dd zz... interstitial

11 theoretical EFG calculations V V zz pp zz V pp zz 1 r V 3 p dd zz ( p x interstitial p y ) p z V dd zz 1 r 3 d d xy d x ( d d ) y xz yz d z 2 EFG is proportial to differences of orbital occupations, e.g. between p x,p y and p z. if these occupancies are the same by symmetry (cubic): V zz =0 with axial (hexagonal, tetragonal) symmetry (p x =p y ): =0 In the following various examples will be presented.

12 EFG (10 21 V/m 2 ) in YBa 2 Cu 3 O 7 Site Vxx Vyy Vzz Y theory exp Ba theory exp Cu(1) theory exp Cu(2) theory standard LDA calculations give exp good EFGs for all sites except Cu(2) O(1) theory exp O(2) theory exp O(3) theory exp O(4) theory exp K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) D.J.Singh, K.Schwarz, K.Schwarz, Phys.Rev. B46, 5849 (1992)

13 EFG in YBa 2 Cu 3 O 7 Interpretation of the EFG at the oxygen sites p x p y p z V aa V bb V cc z O(1) O(2) y O(3) x O(4) E F Asymmetry count EFG (p-contribution) Cu 1 -d n p p z 1 ( p 2 x p y ) V p zz n 1 r p 3 p O 1 -p y EFG is proportional to asymmetric charge distribution around given nucleus partly occupied

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15 Cu(2) and O(4) EFG as function of r EFG is determined by the non-spherical charge density inside sphere ( r) Y20 ( r) LM ( r) YLM Vzz dr 20( r) r dr 3 r LM Cu(2) r r final EFG O(4) r r r

16 EFG contributions: Depending on the atom, the main EFG-contributions come from anisotropies (in occupation or wave function) semicore p-states (eg. Ti 3p much more important than Cu 3p) valence p-states (eg. O 2p or Cu 4p) valence d-states (eg. TM 3d,4d,5d states; in metals small ) valence f-states (only for localized 4f,5f systems) usually only contributions within the first node or within 1 bohr are important.

17 LDA problems in Cuprates Undoped Cuprates (La 2 CuO 4, YBa 2 Cu 3 O 6 ) are nonmagnetic metals instead of antiferromagnetic insulators Both, doped and undoped cuprates have a planar Cu EFG which is by a factor of 2-3 too small We need a method which giver a better description of correlated 3d electrons: can LDA+U fix these problems??

18 Magn. moments and EFG in La 2 CuO 4 AMF FLL LDA+U gives AF insulator with reasonable moment U of 5-6 ev gives exp. EFG GGAs mimic a U of 1-2 ev (EV-GGA more effective than PBE, but very bad E-tot!!)

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22 Magnetic hyperfine interaction Zeman - interaction between magnetic moment I of the nucleus and the external magnetic field B (at the nucleus, produced by the spin-polarized e - in a FM) B ( 0) (0) B proportional to the spindensity at the nucleus B often proportional to the magnetic moment of an atom in a solid.

23 magnetic fields at nucleus:

24 Magnetic fields at the nucleus:

25 How to do it in WIEN2k:

26 Verwey transition in YBaFe 2 O 5 charge ordered (CO) phase: valence mixed (VM) phase: Pmma a:b:c=2.09:1:1.96 (20K) Pmmm a:b:c=1.003:1:1.93 (340K) c Fe 2+ and Fe 3+ form chains along b a b contradicts Anderson charge-ordering conditions with minimal electrostatic repulsion (checkerboard like pattern) has to be compensated by orbital ordering and e - -lattice coupling

27 DOS: GGA+U vs. GGA GGA+U GGA single insulator, lower Hubbard-band t 2g band splits in VM splits in CO with Fe 3+ states metallic lower than Fe 2+ GGA+U insulator metal

28 Difference densities = cryst - at sup CO phase VM phase Fe 2+ : d-xz Fe 3+ : d-x 2 O1 and O3: polarized toward Fe 3+ Fe: d-z 2 Fe-Fe interaction O: symmetric

29 Mössbauer spectroscopy CO VM

30 Isomer shift: charge transfer too small in LDA/GGA CO VM

31 Hyperfine fields: Fe 2+ has large B orb and B dip CO VM

32 EFG: Fe 2+ has too small anisotropy in LDA/GGA CO VM

33 conversion of exp. data to EFGs Mössbauer: {e Q V zz (1+ 2 /3) 1/2 } / 2 ; (e Q c V zz ) / 2 E (E ) / c Q( 57 Fe)=0.16 b; E =14410 ev Vzz [10 21 V/m 2 ] = 6 * [mm/s] NMR: v Q = (3 e Q V zz ) / {2 h I (2 I 1)} I.. nuclear spin quantum n. V zz [10 21 V/m 2 ] = v Q [MHz] / Q [b] Q( 49 Ti)=0.247 b