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Hyperfine Interactions

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Hyperfine Interactions Hyperfine Interactions Interaction between the electromagnetic moments of a nucleus and electromagnetic fields acting on the nucleus I = nuclear spin µ = magn. dipole moment Q = electric quadrupole moment e- μ, Q, I Q μ, Electric field Magnetic field electric HFI magnetic HFI Hyperfine Interactions Electric and magnetic fields at nuclear sites may be produced by: (i) the electrons of the atom under consideration Î Hyperfine structure of optical transitions (ii) External sources (magnetic fields ) Î nuclear structure studies (iii) The electrons of nearby atoms Î Information on chemical and solid state properties M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 2 Hyperfine structure of electronic states of atoms eV 10-4 eV 10-6 eV Bohr Fine structure Hyperfine structure Energy levels Spin-orbit coupling nuclear effect OrderGrößenordnung of magnitude of der energies Energien eV 10 ‐4 eV 10‐6 eV M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 3 Hyperfine splitting of the D2‐line of Na Detection in the resonance radiation of a Na‐atomic beam excited by a frequency‐variable dye laser splitting Na‐atomic beam Theorie Experiment dye laser beam M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 4 Hyperfine Interactions in Condensed Matter Electric HF interactions Magnetic HF interactions Static Static non-cubic solids (metals, Ferromagnets semiconductors,isolators Paramagnets at low temperatures Defects in cubic solids Knight Shift Dynamic Dynamic Paramagnets at finite temperatures Atomic motion in solids, liquids and Paramagnetische Lösungen gases e.g. metal-hydrogen systems Spinfluctuations in ferromagnets M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 5 Electric Hyperfine Interaction Interaction between the charge distribution of a nucleus ρN(r)and the potential Φ(r) created by the charges surrounding the nucleus ρ (r) N Φ(r) 3 Interaction energy: E=( rρ ) ( Φ r ) d r el ∫ N Nuclear charge distribution potential M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 6 Evaluation of the interaction integral E=( rρ ) ( Φ r )3 d r el ∫ N of the energy Expansion (into terms of decreasing magnitude): ()0 ()1 ()2 Eel = Eel + Eel + Eel +L monopole term dipole term = 0 quadrupole term tpoin egchar 1 ⎛ δ2 Φ ⎞ E( 2 )= ⎜ ⎟( rρ ) x x3 d r el ∑⎜ ⎟ ∫ N α β 2α, βδ x δ x ⎝ α β⎠0 ⎛∂2 Φ ⎞ Φ = ⎜ ⎟ Tensor of the electric field tadiengr (EFG) αβ ⎜ ⎟ ⎝∂xα ∂ x β⎠ ΦΦΦ⎛ ⎞ ⎛Φ 0 0 ⎞ ⎜ αα ßα γα ⎟ Principal axes ⎜ α'' α ⎟ transformation ΦΦΦ⎜ αβ ββ γβ ⎟ ⎜ 0 Φβ'' β 0 ⎟ ΦΦΦ⎜ ⎟ ⎜ ⎟ ⎝ αγ βγ γγ ⎠ ⎝ 0 0 Φγ'' γ ⎠ CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 7 The second of the energyorder term After principal axis-transformation (only diagonal terms): 1 δ⎛ 2 Φ⎞ 1 nuclear volume E()2 = ⎜ ⎟ ρ)(r x2 d 3 = r Φ) ρ(r x2 d 3 r el 2 ∑⎜ δx 2 ⎟ ∫ N α 2 ∑ αα ∫ N α α ⎝ α ⎠0 α part and a non-spherical of the nuclear volume into a spherical Decomposition 1 1 1 E(= 2 ) Φ( r )ρ 2 r+ 3 d r Φ( r ρ )2 [ xr ] 2 − d 3 r el ∑ αα ∫ N ∑ αα ∫ N α 6 α 2 α 3 ( 2 a ) (E 2 b ) Q = E + el =∑ Φαα αα el α Finite spherical nuclear volume Deviation from sphericity Î Isotopie-(Isomerie-) shift Î quadrupole splitting CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 8 quadrupole moment The electric Q describes the deviation of the nuclear charge distribution from sphericity Classical definition: 1 QQ= =( 3z2 r − ) 2 ρ ( r d3 ) r zz e ∫ N ρ ()dr 3 r= Ze ∫ N Q > 0 Q = 0 < 0Q unit : barn (b) = 10-24 cm2 CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 9 sphericalThe effect of the finite charge distribution of the nucleus 1 E(2 a ( )= ΔΦ ) ( rρ 2 ) r 3 d r el 6 r =0 ∫ N ρel()0 Ze 2 Possion equation ()ΔΦ=r=0 − =()0 Ψ ε0 ε0 2 1 2 3 Mean quadratic nuclear radius : r =r (ρ r ) d r N Ze ∫ N by the finite nuclear size caused Energy-shift 2 (2 isomer) Ze 2 2 Eel = Ψ(0 )N r 6ε0 nuclear environment nuclear radius parameter CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 10 Isotope- and Isomer shift 2 (2a) Ze 2 2 Eel = Ψ 0 r)( N 6ε0 Spherical nucleus with Point charge nucleus radii RA, RB in states A, B M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 11 The quadrupole interaction e E()2 = VQ Q ∑ αα αα 6 α (EFG)The tensor of the electric field gradient caused by charges outside the nucleus VVV⎛ ⎞ x'x'⎜ x'y' x'z'⎟ y'x'VVV⎜ y'y' y'z'⎟ ⎜ ⎟ z'x'VVV⎝ z'y' z'z'⎠ Prinipal axes transformation Choice of principal axes ⎛V ⎞ ⎜ xx ⎟ ⎜ Vyy ⎟ VVVxx≤yy ≤ zz ⎜ ⎟ ⎝ Vzz ⎠ ∑ V αα Since= 0 by 2 parameters:the EFG is completely described α (ii) Asymmetry parameter (i) Maximum component Vzz VV− η = xx yy,≤ 0 η ≤1 Vzz CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 12 gradient (EFG) produced by a point chargefield The electric x y ialtenpot Ze ,z)(x,y r (x ,y ,z ) X x‐ componetn of 0 0 0 the electric field 2 xy‐ componetn of ∂ V ( x3 − x0 ) ( y − 0 y ) Vxy = = Ze 2 2 2 5/ 2 the EFG tensor ∂xx ∂ y x)−(( 0 (y + − y)0 + (z0 − z)) 2 2 2 onaldiag componetn ∂ V (3 x− x0 ) − r Vxx = 2 = Ze 2 2 2 5/ 2 of the EFG tensor ∂x x)−(( 0 (y + − y)0 + (z0 − z)) 3 Vzz = 2 Ze/r VV− η = xx yy =0 Vzz CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 13 The quadrupole splitting of nuclear states z ω e I Classical description E )2( = VQ ∂E Q ∑ αα αα Larmor precession 6 α Q ∂z Transition to quantum mechanics: Spherical Tensor‐Operators , Wigner‐Eckart theorem eQVzz 2 2 2 Hamiltonian of the QI H = { 3 I− I + ( I 1+η(II ) )} − Q 4I(2I − )1 z x y Eigenvalues for axial symmetrie of the EFG (η = 0) Quadrupole frequencies - definitions: ( m ) eQVzz 2 EQ = {3m −I(I +1)} 4I(2I − )1 eQVzz ωQ = (Mrad/s) 4I(2I − )1 h Axial asymmetry (η≠0): e Q V In most cases numerical diagonalisation ν = zz (MHz) of the QI Matrix <I,m|H |I,m‘> required Q Q h M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 14 Electric quadrupole interaction Larmor precession Quadrupole splitting z ω EQ (ħ ωQ) I 10 m = ± 5/2 ħ ω I = 5/2 2 ħ ω3 ∂E -2 m = ± 3/2 Q ∂z ħ ω1 -8 m = ± 1/2 0 1 asymmetry η M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 15 Order of magnitude of electric field gradients eQVzz 4 I ( 2− I 1 ) EQ=hωQ = Vzz = EQ 4I ( 2I− 1 ) e Q -8 Sensivity of HFI techniques: EQ ≥ 10 eV 10 −8 Assumption : Q = 1 b, I = 5/2 Vzz = 10 e V e ⋅10−24 cm2 17 2 Vzz ≥ 10 V/cm Sources of electric field gradients of sufficient strength • EFG produced by external charges is too weak • Charge distribution in solids solid Vzz • Deformation of closed elctronic shells (1- γ ) solid ∞ Vzz Sternheimer-Korrektur γ∞ = 10 …. 70 −3 • Unclosed electronic shells with angular momentumV J = erzz − J ⋅ JJ(2J1) α ⋅ − 3+ 9 18 2 • Example: Rare earth ions (4f-elements); Dy : 4f , J = 15/2, Vzz = 6·10 V/cm M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 16 Phase Identification and Structure Information by Measurement of Electric Quadrupole Interaction - Example: ZrO2 ZrO2-structures Frequency and asymmetry νq ≠ 0 monoclinic η≠0 ν ≠ 0 tetragonal q η= 0 cubic νq = 0 2 -ZrO t 1.0 fraction of of fraction The m Æ t phase transition of ZrO2 0.5 0.0 1000 1100 1200 1300 1400 1500 T (K) M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 17 Magnetic Hyperfine interaction: The interaction between nuclear magnetic dipole moments µ and magnetic fields B acting on the nucleus ħωm ~ μ·B External field Bext Hyperfine Field Bhf B ≤ 103 T Bext ≤ 20 T hf nuclear physics condensed matter physics determination of • unpaired spin density magnetic moments (magnetically ordered systems) • Orbital and dipolar fields (ferro- and paramagnetic systems) M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 18 Nuclear Statesof The Magnetic Splitting Hamilton Operator: HM= − µ ⋅hf B Energy Eigenvalues: Em=mI, − µz z I, Bm= −γ Bz I,mmz II, = g − µN Bz m Zeeman Splitting Larmor-Precession equidistant -I ≤ m≤ I Bz m = I I I m = I-1 ωL 0 ΔE m = - I +1 m = - I ΔE = g µNBz Larmor frequency = ħ ωL ωL = ΔE /ħ= - g µNBz/ ħ -12 Order of magnitude: Bext = 100 kG, g = 1, µN = 3.15 10 eV/ G −7 ΔEM =hg ωL µ =N Btex =15 3 ⋅ . 10eV CBPF 2012h, rcaResein Methods Solid State r NucleaM. Forker, 19 The nuclear parameter: the magnetic dipole moment µ Moving charges = currents → magnetic moment Classical example : Charge -e on an orbit with angular momentum I: µ Definition of the magnetic moment: e, m µ = (1/c) circular current x area R v Angular momentum: I R= x p = 0 m v R e r r =µ −I = γI , gyromagnet γ = ic ratio, I= ang. momentum 2m0c M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 20 Magnetic Dipole Moments e r r Classical:= µ −I = γI , gyromagnet γ = ic ratio, I= ang.
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