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 ρ Φ= rd)r()r( el ∫ N
Nuclear charge distribution potential
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 6 Evaluation of the interaction integral E ρ Φ= 3rd)r()r( el ∫ N Expansion of the energy (into terms of decreasing magnitude):
0 )( 1)( 2 )( Eel = Eel + Eel + Eel +L monopole term dipole term = 0 quadrupole term point charge
1 ⎛ 2 Φδ ⎞ E )2( = ⎜ ⎟ ρ 3rdxx)r( el ∑⎜ ⎟ ∫ N βα ,βα δδ xx2 ⎝ βα ⎠0
⎛ 2Φ∂ ⎞ Φ = ⎜ ⎟ Tensor of the electric field gradient (EFG) αβ ⎜ ⎟ ⎝ ∂∂ xx βα ⎠
⎛ ΦΦΦ ⎞ ⎛Φ αα'' 00 ⎞ ⎜ αα ßα γα ⎟ Principal axes ⎜ ⎟ transformation ⎜ αβ ββ ΦΦΦ γβ ⎟ ⎜ 0 Φ ββ'' 0 ⎟ ⎜ ⎟ ΦΦΦ ⎜ ⎟ ⎝ αγ βγ γγ ⎠ ⎝ 00 Φ γγ'' ⎠ M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 7 The second order term of the energy
After principal axis-transformation (only diagonal terms):
1 ⎛ 2 Φδ ⎞ 1 nuclear volume E 2)( = ⎜ ⎟ ρ (r 32 rdx) = ρΦ (r 32 rdx) el 2 ∑⎜ δx2 ⎟ ∫ N α 2 ∑ αα ∫ N α α ⎝ α ⎠0 α
Decomposition of the nuclear volume into a spherical and a non-spherical part
1 1 1 E )2( ρΦ= 32 rdr)r( x[)r( −ρΦ+ 322 rd]r el ∑ αα ∫ N ∑ αα ∫ N α 6 α 2 α 3
)a2( )b2( = E + E el ∑ Φ= αα Q αα el α Finite spherical nuclear volume Deviation from sphericity
Î Isotopie-(Isomerie-) shift Î quadrupole splitting
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 8 The electric quadrupole moment Q describes the deviation of the nuclear charge distribution from sphericity
Classical definition:
1 QQ −== 22 ρ )r()3( 3rdrz zz e ∫ N
ρ r)( 3 = Zerd ∫ N
Q > 0 Q = 0 Q < 0
unit : barn (b) = 10-24 cm2
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 9 The effect of the finite spherical charge distribution of the nucleus
1 2 )a( = ρΔΦ 32 rdr)r()(E el 6 r =0 ∫ N
ρel 0)( Ze 2 Possion equation ΔΦ)( r=0 Ψ=−= 0)( 0 εε 0
2 1 2 3 Mean quadratic nuclear radius : r = ρ rd)r(r N Ze ∫ N
Energy-shift caused by the finite nuclear size
2 (2 isomer) Ze 2 2 E = Ψ 0 r)( N el 6 ε 0
nuclear environment nuclear radius parameter
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 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)( = QV Q ∑ αα αα 6 α The tensor of the electric field gradient (EFG) caused by charges outside the nucleus
⎛ VVV ⎞ ⎜ 'z'x'y'x'x'x ⎟ ⎜ VVV 'z'y'y'y'x'y ⎟ ⎜ ⎟ ⎝ VVV 'z'z'y'z'x'z ⎠ Prinipal axes transformation Choice of principal axes V ⎛ xx ⎞ ⎜ ⎟ V ⎜ yy ⎟ xx yy ≤≤ VVV zz ⎜ V ⎟ ⎝ zz ⎠
Since ∑ V αα = 0 the EFG is completely described by 2 parameters: α
(ii) Asymmetry parameter (i) Maximum component Vzz − VV η = xx yy 0, η ≤≤ 1 Vzz M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 12 The electric field gradient (EFG) produced by a point charge x y potential Ze (x,y,z) r (x ,y ,z ) X x‐ component of 0 0 0 the electric field
2 xy‐ component of ∂ V 3 0 −− 0 )yy()xx( Vxy = = Ze 2 2 / 252 the EFG tensor ∂∂ yx (( 0 0 −+−+− 0 ))zz()yy()xx
2 22 diagonal component ∂ V 3 0 −− r)xx( Vxx = 2 = Ze 2 2 / 252 of the EFG tensor ∂x (( 0 0 −+−+− 0 ))zz()yy()xx
3 Vzz = 2 Ze/r − VV η = xx yy =0 Vzz
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 13 The quadrupole splitting of nuclear states z ω
)2( e I Classical description E = QV Larmor precession ∂ E Q ∑ αα αα Q 6 α ∂ z
Transition to quantum mechanics: Spherical Tensor‐Operators , Wigner‐Eckart theorem
eQVzz 2 2 2 Hamiltonian of the QI H = )1I(II3{ η −++− II( )} Q 4I(2I − )1 z x y
Eigenvalues for axial symmetrie of the EFG (η = 0) Quadrupole frequencies - definitions: )m( eQV zz 2 E Q = {3m I(I +− 1) } 4I (2I − )1 eQVzz ωQ = (Mrad/s) 4I(2I − )1 h Axial asymmetry (η≠0): In most cases numerical diagonalisation VQe zz of the QI Matrix
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 − )1I2(I4 EQ hωQ == Vzz = EQ − )12I(4I Qe
-8 Sensivity of HFI techniques: EQ ≥ 10 eV
10 −8 Assumption : Q = 1 b, I = 5/2 Vzz = 10 Ve 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 momentum J zz −⋅α⋅−= )1J2(JJJreV
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 The Magnetic Splitting of Nuclear States
Hamilton Operator: M ⋅−= BµH hf
Energy Eigenvalues: Em I, −= Bµm zz I, m −= γ Bz I, Im z I, −= zN mBµgm
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 E M h L =ω=Δ N ext .3Bµg ⋅= 1015 eV
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 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: == 0 RvmpxRI e r r µ I ,I =γγ=−= gyromagnet 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 ,I =γγ=−= gyromagnet ic ratio, I = ang. momentum 2m0c
Quantum mechanics for free-electron states |I, M=I> :
eh −15 MeV z h μ=γ==μ==μ BIgIIM,IIM,I B ==μ .5 788x10 2mec Gauss
g-factor Bohr magneton
Correspondingly for protons: e MeV h ==μ 3 153x. 10−18 r N 2m c Gauss eh I r N µp g μ== N I 2mpc h µN = nuclear magneton
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 21 Magnetic Dipole Moments r Free particles: p,e = gµ μ N,B I
g-factor Elektron Proton Neutron
Orbital l: gl -1 1 0
Spin s: gs -2.0023 5,5856 -3,8263 anomalous
gs -Dirac theory -2(1+α/2π+ ...) 1 0 α = fine structure constant For magnetic moments in nuclei, the spin-orbit coupling + sl r leads to the g factor
− gg 1 gg ±= LS lfor ± odd Z-even N nuclei l l + 12 2
Experimental results and Schmidt lines
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 22 The magnetic hyperfine fielde Bhf
hf ext += orb + contact + dip + Lorentz + BBBBBBB DM +K
Demagnetisation External Field Lorentz field ≤ 10 – 20 Tesla (100-200 KGauss)
J Dipolar field ⋅−⋅⋅ rJ)rJ(r3 r −= µ2B dip B r 5
max r = 3 Ǻ, J = 10 dip ≈ 1B Tesla
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 23 The magnetic hyperfine field Bhf
hf ext += orb + contact + dip + Lorentz + BBBBBBB DM +K
J Orbital field
−= 2 −3 JJNJrµB e- orb B Angular momentum : LJ += S Spin-orbit coupling JNJ
4f-elements (Gd, Tb, Dy…) : J ≤ 10 3 Borb ≤ 10 Tesla
The Fermi contact field
2 Field produced by s-electrons at the nuclear site: BC= 8π/3 μ0μB|Ψ↑ (0)|
8π 2 2 In ferromagnets: B (Au in Fe) Bcontact = 0 µµ B ∑[ ψns↑ (0) − ψns↓ ](0) C 3 n = 10 2 Tesla 24 spin-polarisation
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 24 Magnetic hyperfine fields in solids
Example: The magnetic 3d- and rare earth (4f-) metals
3d ferromagnets Fe, Co, Ni : Stoner model
4f ferromagnets: RKKY theory of indirect coupling
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 25 Fe, Co, Ni - Stoner model of itinerant d-electron magnetism
3d-electrons Bandstructure Spin up ⇑ of transition metals s-elektrons (schematic) energy ⇑ − dd ⇓ = ⇑ − ss ⇓ = 0nn;0nn Spin down ⇓ Density of states
EF = Fermi-energy
Exchange interaction =− ⋅SSJH ex ex ⇓⇑
2 2 Spin up ⇑ nn SS ⇓⇑ =− Ψ S⇑(0) − Ψ S⇓ ( ) >00
energy Fermi contact field
Spin down ⇓ − > 0nn d-moment E = Fermi-energy ⇑ dd ⇓ Density of states Density of states F μd = ⇑ − dd ⇓)nn( μB
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 26 Hyperfine fields of different probe atoms in ferromagnetic Fe
Au 100 Bhf (T) Hyperfine fields of the diamagnetic 50 Cd probes Au, Cd, Cu Cu in the 3d ferromagnets Fe, Co, Ni
Ni Co Fe µ3d M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 27 Illustration of the state of hyperfine field theory
Hyperfine fields at 4d and 5sp impurities in bcc iron Phys.Rev. B62(2000) 46 H. Haas, Cottenier and
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 28 The magnetic rare earth elements
The electronic configuration of the free atom : (Xe) 4f n 5d2 6s1
Charge state in solids: mostly R3+ -(Xe) 4fn except Ce, Eu, Yb
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 29 Crystal and magnetic structures of the rare earth metals
hcp dhcp basal-plane ferromagnet cone
helix longitudinal-wave structure
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 30 The 4f wave function 4f electrons:
• Inner electrons, highly localized
Wigner-Seitz within the Wigner Seitz cell radius • strongly anisotropic charge distribution • well protected by the outer shells
The 4f charge distribution • pronounced chemical similarity • orbital angular momentum unquenched
• strong crystal field interaction magnetically hard materials • huge orbital fields at the R nuclei • very little 4f-4f overlapp
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 31 Magnetic properties of R3+ ions
Russel-Saunders Coupling
10 L 10 J magnetic moment ( moment magnetic 8 S 8 L (g-1)J J 6 6
4 4
μ (g-1)J 2 2 B
) S spin and angular momentum angular spinand 0 0 Ce Nd Sm Gd Dy Er Yb La Pr Pm Eu Tb Ho Tm Lu
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 32 RKKY theory of indirect 4f-4f coupling
E Exchange interaction between the 4f electrons and the s-conduction electrons E
DOS N(E) E F r r DOS N(E) H sf −= 2Γsf Ss EF
Spin polarisation of the s-conduction electrons nn >↓−↑ 0
2 9 Γπ Sz zfs Kasuya, Yoshida ρ )( nnr −↓=−↑= ∑ 2 F −⋅⋅ RrkF i ))(( 4EF i
B ∝ Γ zfshf ∑ 2 RkFS iF )( 0 i
z −= 1)( JgS spin polarisation polarisation spin spin polarisation
0 distance hf fs −Γ∝ ∑ RkFJgB iF )2()1( long-range, oscillating i
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 33 111 Spin dependence of the Cd hyperfine field in RCo2 and RAl2
hf −Γ∝ ∑ 21 iF )Rk(F)g(B i
60
Gd 111 50 Tb Cd | (MHz) Dy m Sm Ho 1.5 119 ν RCo | Er 2 Sn 40 27Al Nd Tm RAl 30 2 Pr (Gd) 1.0 Γ (g-1)J / 20 Γ 0.5 10 Pr Nd Sm Gd Tb Dy Ho Er 0 -3 -2 -1 0 1 2 3 4 (g-1)J
M. Forker, Nuclear Methods in Solid State Research, CBPF 2012 34