Hyperfine interactions

Transcription

1 Hyperfine interactions Karlheinz Schwarz Institute of Materials Chemistry TU Wien Some slides were provided by Stefaan Cottenier (Gent)

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3 nuclear point charges interacting with electron charge distribution

4 Definition : hyperfine interaction = all aspects of the nucleus-electron interaction which go beyond an electric point charge for a nucleus.

5 electric point charge

6 electric point charge volume shape

7 electric point charge volume shape magnetic moment S N

8 How to measure hyperfine interactions? NMR NQR Mössbauer spectroscopy TDPAC Laser spectroscopy LTNO NMR/ON PAD This talk: Hyperfine physics How to calculate HFF with WIEN2k

9 Content Definitions magnetic hyperfine interaction electric quadrupole interaction isomer shift summary

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12 N S

13 S N

14 S N

15 S N

16 S N

17 E S 1.5 Energy (units: µ B ) N

18 E S 1.5 Energy (units: µ B ) N

19 E S 1.5 Energy (units: µ B ) N

20 E S 1.5 Energy (units: µ B ) N

21 E S 1.5 Energy (units: µ B ) N

22 E 1.5 Energy (units: µ B )

23 Classical Quantum (=quantization) E m =-1 Energy (units: µ B ) m =0 m = e.g. I =1

24 Classical Quantum (=quantization) E m =-1 Energy (units: µ B ) m =0 m = e.g. I =1 Hamiltonian :

25 nuclear property (vector) electron property (vector) S N interaction energy (dot product) :

26 Source of magnetic fields at a nuclear site in an atom/solid B tot = B dip + B orb + B fermi + B lat

27 Source of magnetic fields at a nuclear site in an atom/solid B tot = B dip + B orb + B fermi + B lat B dip = electron as bar magnet

28 Source of magnetic fields at a nuclear site in an atom/solid B tot = B dip + B orb + B fermi + B lat B dip = electron as bar magnet B orb = electron as current loop L I r -e v M orb B orb

29 Source of magnetic fields at the nuclear site in an atom/solid B tot = B dip + B orb + B fermi + B lat B dip = electron as bar magnet B orb = electron as current loop L I r -e v B Fermi = electron at nucleus M orb B orb 2s 2p

30 Source of magnetic fields at a nuclear site in an atom/solid B tot = B dip + B orb + B fermi + B lat B dip = electron as bar magnet B orb = electron as current loop L I r -e v B Fermi = electron at nucleus M orb B orb 2s B lat = neighbours as bar magnets 2p

31 How to do it in WIEN2k? Magnetic hyperfine field In regular scf file: :HFFxxx (Fermi contact contribution) After post-processing with LAPWDM : orbital hyperfine field ( 3 3 in case.indmc) dipolar hyperfine field ( 3 5 in case.indmc) in case.scfdmup After post-processing with DIPAN : lattice contribution in case.outputdipan more info: UG 7.8 (lapwdm) UG 8.3 (dipan)

32 Mössbauer spectroscopy: Isomer shift: δ = α (ρ 0 Sample ρ 0 Reference ); α=-.291 au 3 mm s -1 proportional to the electron density ρ at the nucleus Magnetic Hyperfine fields: B tot =B contact + B orb + B dip B contact = 8π/3 µ B [ρ up (0) ρ dn (0)] spin-density at the nucleus orbital-moment spin-moment S(r) is reciprocal of the relativistic mass enhancement

33 Verwey transition in YBaFe 2 O 5 Charged ordered Valence mixed Fe 2+ Fe 3+ Fe 2.5+ CO structure: Pmma VM structure: Pmmm a:b:c=2.09:1:1.96 (20K) a:b:c=1.003:1:1.93 (340K) Ba c Y a b Fe 2+ and Fe 3+ chains along b Fe 2.5+

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

35 Difference densities ρ=ρ cryst -ρ 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

36 YBaFe 2 O 5 HFF, IS and EFG with GGA+U, LDA/GGA CO HFF(Fe 2+ ) HFF(Fe 3+ ) VM HFF(Fe 2.5+ )

37 Content Definitions magnetic hyperfine interaction electric quadrupole interaction isomer shift summary

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

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40 Force on a point charge:

41 Force on a point charge: Force on a general charge:

42 -Q -Q

43 -Q -Q

44 -Q -Q

45 -Q 2

46 Energy (units e 2 /4πε 0 ) -3,97-3,98-3,99-4,00-4,01-4,02-4,03-4,04-4,05 θ (deg) E-tm0 -Q 2 -Q

47 θ (deg) Energy (units e 2 /4πε 0 ) -3,97-3,98-3,99-4,00-4,01-4,02-4,03-4,04-4, E-tm0

48 θ (deg) Energy (units e 2 /4πε 0 ) -3,97-3,98-3,99-4,00-4,01-4,02-4, E-tm0 m=0 e.g. I = 1-4,04-4,05 m=±1

49 nuclear property (tensor rank 2 ) electron property (tensor rank 2 ) interaction energy (dot product) :

50 Electric field gradients (EFG) Nuclei with a nuclear quantum number I moment Q 1 have an electrical quadrupole Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site Experiments NMR NQR Mössbauer PAC ν Nuclear eq Φ / h electronic Φ Φ ij EFG traceless tensor = V ij 1 δ ij 3 2 V V ij 2 V (0) = x x i j with V xx + Vyy + Vzz = 0 traceless V V V aa ba ca V V V ab bb cb V V V ac bc cc V xx 0 0 V 0 yy 0 V 0 0 zz V zz V yy V xx EFG V zz principal component η = / V xx / / V / V zz / yy asymmetry parameter /

51 First-principles calculation of EFG, 1192 Previous: point charge model and Sternheimer factor to experimental value

52 Electronic structure of Li 3 N The charge anisotropy around N differs strongly between muffin-tin and full-potential affecting the EFG.

53 Nuclear quadrupole moment of 57 Fe, 3545 From the slope between the theoreical EFG and experimental quadrupole splitting Δ Q (mm/s) the nuclear quadrupole moment Q of the most important Mössbauer nucleus is found to be about twice as large (Q=0.16 b) as so far in literature (Q=0.082 b)

54 theoretical EFG calculations: EFG is a tensor of second derivatives of V C at the nucleus: V V V ij zz p zz 2 V (0) = x x i ( r) Y ρ 3 r 1 r 3 j p 20 V c ( r) = ρ( r ) dr = r r [ 1 ] ( p + p ) p 2 dr = x V y p zz + V d zz z LM V LM ( r) Y LM ( rˆ) Cartesian LM-repr. V V V zz yy xx V 20 ( r = V V ) V + V V d zz 1 r 3 d [ ] d + d 1 ( d + d d xy x 2 xz yz 2 2 ) y z 2 EFG is proportional to differences in orbital occupations

55 High temperature superconductor YBa 2 Cu 3 O 7 Electronic structure Charge density, EFG EFG (electric field gradient) K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B 42, 2051 (1990)

56 EFG at O sites in YBa 2 Cu 3 O 7 Interpretation of the EFG (measured by NQR) at the oxygen sites p x p y p z V aa V bb V cc z O(1) O(2) O(3) O(4) Asymmetry count 1 n p = pz ( px + p 2 y ) EFG (p-contribution) p 1 V zz n p < > 3 p r E F O 1 -p y x O 1 Cu 1 -d y non-bonding O-p,C EFG is proportional to asymmetric charge distribution around given nucleus partly occupied

57 EFG (10 21 V/m 2 ) in YBa 2 Cu 3 O 7 Site Vxx Vyy Vzz η 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, P.Blaha, Phys.Rev. B46, 5849 (1992)

58 Cu partial charges in YBa 2 Cu 3 O 7 a transfer of 0.07 e into the d z2 would increase the EFG from -5.0 by bringing it to inclose to the Experimental value ( V/m 2 )

59 How to do it in WIEN2k? Electric-field gradient In regular scf file: :EFGxxx :ETAxxx Main directions of the EFG }5 degrees of freedom Full analysis printed in case.output2 if EFG keyword in case.in2 is put (UG 7.6) (split into many different contributions) more info: Blaha, Schwarz, Dederichs, PRB 37 (1988) 2792 EFG document in wien2k FAQ (Katrin Koch, SC)

60 EFGs in fluoroaluminates 10 different phases of known structures from CaF 2 -AlF 3, BaF 2 -AlF 3 binary systems and CaF 2 -BaF 2 -AlF 3 ternary system Isolated chains of octahedra linked by corners Isolated octahedra Rings formed by four octahedra sharing corners α-baalf 5 α-caalf 5, β- CaAlF 5, α-bacaalf 7 Ca 2 AlF 7, Ba 3 AlF 9 - Ib, Ba 3 Al 2 F 12

61 ν Q and η Q calculations using XRD data ν Q = 4, V zz with R 2 = 1,8e+6 AlF 3 0,77 α-caalf Attributions performed with respect to the proportionality between 5 V zz β-caalf 5 1,0 Ca and ν Q for the multi-site compounds 2 AlF 7 0,38 AlF 3 α-baalf 5 α-caalf 5 β-baalf 5 1,6e+6 β-caalf 5 γ-baalf 5 η Q,exp = 0,803 η Q,cal R 2 = 1,4e+6 Ca 2 AlF 7 α-baalf 5 0,8 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib β-baalf 5 β-ba 3 AlF 9 Experimental ν Q (Hz) 1,2e+6 1,0e+6 8,0e+5 6,0e+5 γ-baalf 5 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib β-ba 3 AlF 9 α-bacaalf 7 Régression Experimental η Q 0,6 0,4 α-bacaalf 7 Regression η Q,mes. =η Q, cal. 4,0e+5 0,2 2,0e+5 0,0 0,0 1,0e+21 2,0e+21 3,0e+21 Calculated V zz (V.m -2 ) 0,0 0,0 0,2 0,4 0,6 0,8 1,0 Calculated η Q Important discrepancies when structures are used which were determined from X-ray powder diffraction data

62 M.Body, et al., J.Phys.Chem. A 2007, 111, (Univ. LeMans) ν Q and η Q after structure optimization 1,8e+6 1,0 1,6e+6 1,4e+6 ν Q = 5, V zz R 2 = 0,993 0,8 η Q, exp = 0,972 η Q,cal R 2 = 0,983 1,2e+6 Experimental ν Q (Hz) 1,0e+6 8,0e+5 6,0e+5 AlF 3 α-caalf 5 β-caalf 5 Ca 2 AlF 7 α-baalf 5 Experimental η Q 0,6 0,4 AlF 3 α-caalf 5 β-caalf 5 Ca 2 AlF 7 α-baalf 5 β-baalf 5 β-baalf 5 γ-baalf 5 4,0e+5 2,0e+5 0,0 0,0 5,0e+20 1,0e+21 1,5e+21 2,0e+21 2,5e+21 3,0e+21 Calculated V zz (V.m -2 ) γ-baalf 5 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib β-ba 3 AlF 9 α-bacaalf 7 Regression 0,2 α-bacaalf 7 Regression η Q,exp. =η Q, cal. 0,0 0,0 0,2 0,4 0,6 0,8 1,0 Calculated η Q Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib β-ba 3 AlF 9 Very fine agreement between experimental and calculated values

63 NMR shielding, chemical shift:

64 Content Definitions magnetic hyperfine interaction electric quadrupole interaction isomer shift summary

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66 Mössbauer spectroscopy Isomer Hyperfine YBaFe shift: fields: charge Fe 2+ 2 O 5 HFF, IS and transfer has EFG large with too BGGA+U, small orb and in B LDA/GGA dip CO VM

67 Cu(2) and O(4) EFG as function of r EFG is determined by the non-spherical charge density inside sphere ρ( r) Y20 ρ( r) = ρlm ( r) YLM Vzz dr = ρ20( r) r dr 3 r LM Cu(2) r r final EFG O(4)

68 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.