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Hyperfine interactions
Karlheinz SchwarzInstitute of Materials Chemistry
TU Wien
Some slides were provided by Stefaan Cottenier (Gent)
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nuclear point chargesinteracting with
electron charge distribution
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hyperfine interaction
all aspects of the nucleus-electron interaction
which go beyondan electric point charge for a nucleus.
=
Definition :
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electricpoint
charge
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electricpoint
charge
• volume• shape
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N
S
electricpoint
charge
• volume• shape• magnetic moment
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How to measure hyperfine interactions ?
This talk:• Hyperfine physics• How to calculate HFF with WIEN2k
• NMR• NQR• Mössbauer spectroscopy• TDPAC• Laser spectroscopy• LTNO• NMR/ON• PAD• …
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• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
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N
S
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N
S
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N
S
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N
S
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N
S
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N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
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-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
Classical Quantum(=quantization)
e.g. I =1
m =+1
m =0
m =-1
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-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
Classical Quantum(=quantization)
e.g. I =1
m =+1
m =0
m =-1
Hamiltonian :
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nuclear property electron property
N
S
(vector) (vector)
interaction energy (dot product) :
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Btot = Bdip + Borb + Bfermi + Blat
Source of magnetic fields at a nuclear site in an atom/solid
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Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
Source of magnetic fields at a nuclear site in an atom/solid
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Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
Source of magnetic fields at a nuclear site in an atom/solid
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Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
BFermi = electron at nucleus
2s
2p
Source of magnetic fields at the nuclear site in an atom/solid
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Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
BFermi = electron at nucleus
2s
2p
Source of magnetic fields at a nuclear site in an atom/solid
Blat = neighbours as bar magnets
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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 contributionin case.outputdipan
more info:UG 7.8 (lapwdm)UG 8.3 (dipan)
How to do it in WIEN2k ?
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Mössbauer spectroscopy:
Isomer shift: δ = α (ρ0Sample – ρ0
Reference); α=-.291 au3mm s-1
proportional to the electron density ρ at the nucleus
Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip
Bcontact = 8π/3 µB [ρup(0) – ρdn(0)] … spin-density at the nucleus
… orbital-moment
… spin-moment
S(r) is reciprocal of the relativistic mass enhancement
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Verwey transition in YBaFe2O5
CO structure: Pmma VM structure: Pmmma:b:c=2.09:1:1.96 (20K) a:b:c=1.003:1:1.93 (340K)
Fe2+ and Fe3+ chains along b Fe2.5+
a b
c
Charged orderedFe2+ Fe3+
Valence mixedFe2.5+
Ba
Y
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DOS: GGA+U vs. GGAGGA+U GGA
insulator, t2g band splits metallic single lower Hubbard-band in VM splits in CO with Fe3+ states lower than Fe2+
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Difference densities ∆ρ=ρcryst-ρatsup
CO phase VM phaseFe2+: d-xzFe3+: d-x2
O1 and O3: polarized toward Fe3+
Fe: d-z2 Fe-Fe interactionO: symmetric
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YBaFe2O5 HFF, IS and EFG with GGA+U, LDA/GGA
CO
VM
HFF(Fe2+)
HFF(Fe3+)
HFF(Fe2.5+)
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• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
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Electric Hyperfine-Interaction
between nuclear charge distribution (σ) and external potential
Taylor-expansion at the nuclear position∫= dxxVxE n )()(σ
+
∂∂∂
+
∂∂
+
=
∑ ∫ ∫
∫∑
ijji
ji
ii i
dxxxxxx
V
dxxxx
V
ZVE
)()0(21
)()0(
2
0
σ
σ
direction independent constant
electric field xnuclear dipol moment (=0)
electric fieldgradient xnuclear quadrupol moment Q
higher terms neglected
nucleus with charge Z, but not a sphere
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• Force on a point charge:
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• Force on a point charge:
• Force on a general charge:
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-Q
-Q
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-Q
-Q
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-Q
-Q
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-Q
2
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-Q
-Q
-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
2
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θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
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-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
m=±1
m=0
e.g. I = 1
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nuclear property electron property(tensor – rank 2 )
interaction energy (dot product) :
(tensor – rank 2 )
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Electric field gradients (EFG)
Nuclei with a nuclear quantum number I 1 have an electrical quadrupole moment Q
Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site
Experiments NMR NQR Mössbauer PAC
heQ /Φ≈νEFG traceless tensorΦ
VV ijijij2
31
∇−=Φ δ
jiij xx
VV∂∂
∂=
)0(2
0=++ zzyyxx VVVwith traceless
⇒
cc
bc
ac
cb
bb
ab
ca
ba
aa
VVV
VVV
VVV
zz
yy
xx
VV
V00
0
0
00 |Vzz| |Vyy| |Vxx|≥ ≥
//////
zz
yyxx
VVV −
=η
EFG Vzzasymmetry parameterprincipal component
Nuclear electronic
≥
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First-principles calculation of EFG
, 1192
Previous: point charge model andSternheimer factor to experimental value
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Electronic structure of Li3N
• The charge anisotropy around N differs strongly between • muffin-tin and • full-potential
• affecting the EFG.
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Nuclear quadrupole moment of 57Fe, 3545
From the slope between the theoreical EFG and experimental quadrupole
splitting ΔQ (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 inliterature (Q=0.082 b)
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theoretical EFG calculations:
∫ ∑=′′−
′=
∂∂∂
=LM
LMLMcji
ij rYrVrdrr
rrVxx
VV )ˆ()()()()0(2 ρ
2220
2220
20
21
21
)0(
VVV
VVV
rVV
xx
yy
zz
+−∝
−−∝
=∝
EFG is a tensor of second derivatives of VC at the nucleus:
[ ]
[ ]222 )(211
)(211
)(
3
3
320
zyzxzyxxyd
dzz
zyxp
pzz
dzz
pzzzz
dddddr
V
pppr
V
VVdrr
YrV
−+−+∝
−+∝
+=∝
−
∫ρ
Cartesian LM-repr.
EFG is proportional to differences in orbital occupations
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High temperature superconductor
YBa2Cu3O7
Electronic structure Charge density, EFG
EFG (electric field gradient)
K.Schwarz, C.Ambrosch-Draxl, P.Blaha,Phys.Rev. B 42, 2051 (1990)
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EFG at O sites in YBa2Cu3O7
Interpretation of the EFG (measured by NQR) at the oxygen sites
px py pz Vaa Vbb Vcc
O(1) 1.18 0.91 1.25 -6.1 18.3 -12.2
O(2) 1.01 1.21 1.18 11.8 -7.0 -4.8
O(3) 1.21 1.00 1.18 -7.0 11.9 -4.9
O(4) 1.18 1.19 0.99 -4.7 -7.0 11.7
ppp
zz rnV ><∆∝ 3
1)(
21
yxzp pppn +−=∆
Asymmetry count EFG (p-contribution)
EFG is proportional to asymmetric charge distributionaround given nucleus
Cu1-d
O1-py
EF
partly occupied
y
x
z
non-bonding O-p,C
O1
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EFG (1021 V/m2) in YBa2Cu3O7
Site Vxx Vyy Vzz η Y theory -0.9 2.9 -2.0 0.4 exp. - - - - Ba theory -8.7 -1.0 9.7 0.8 exp. 8.4 0.3 8.7 0.9 Cu(1) theory -5.2 6.6 -1.5 0.6 exp. 7.4 7.5 0.1 1.0 Cu(2) theory 2.6 2.4 -5.0 0.0 standard LDA calculations give exp. 6.2 6.2 12.3 0.0 good EFGs for all sites except Cu(2) O(1) theory -5.7 17.9 -12.2 0.4 exp. 6.1 17.3 12.1 0.3 O(2) theory 12.3 -7.5 -4.8 0.2 exp. 10.5 6.3 4.1 0.2 O(3) theory -7.5 12.5 -5.0 0.2 exp. 6.3 10.2 3.9 0.2 O(4) theory -4.7 -7.1 11.8 0.2 exp. 4.0 7.6 11.6 0.3
K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) D.J.Singh, K.Schwarz, P.Blaha, Phys.Rev. B46, 5849 (1992)
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Cu partial charges in YBa2Cu3O7
a transfer of 0.07 e into the dz2 would increase the EFG from -5.0 by
bringing it to -11.6 inclose to the Experimental value (-12.3 1021 V/m2)
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How to do it in WIEN2k ?
Electric-field gradient
In regular scf file::EFGxxx:ETAxxxMain directions of the EFG
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)
} 5 degrees of freedom
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Ba3Al2F12
Rings formed by four octahedra sharing
corners
α-BaCaAlF7
Isolated octahedra
Ca2AlF7, Ba3AlF9-Ib,
α-BaAlF5
Isolated chains of octahedra linked by
corners
α-CaAlF5, β-CaAlF5,
β BaAlF γ
EFGs in fluoroaluminates10 different phases of known structures from CaF2-AlF3,
BaF2-AlF3 binary systems and CaF2-BaF2-AlF3 ternary system
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νQ and ηQ calculations using XRD data
Attributions performed with respect to the proportionality between |Vzz|and νQ for the multi-site compounds
Calculated ηQ
0,0 0,2 0,4 0,6 0,8 1,0
Exp
erim
enta
l ηQ
0,0
0,2
0,4
0,6
0,8
1,0
AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9
α-BaCaAlF7
RegressionηQ,mes. =ηQ, cal.
Calculated Vzz (V.m-2)
0,0 1,0e+21 2,0e+21 3,0e+21
Exp
erim
enta
l νQ (H
z)
0,0
2,0e+5
4,0e+5
6,0e+5
8,0e+5
1,0e+6
1,2e+6
1,4e+6
1,6e+6
1,8e+6AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ibβ-Ba3AlF9 α-BaCaAlF7 Régression
ηQ,exp = 0,803ηQ,cal R2 =
0,38 νQ = 4,712.10-16 |Vzz| with R2 =
0,77
Important discrepancies when structures are used which were determined from X-ray powder diffraction data
![Page 62: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/62.jpg)
M.Body, et al., J.Phys.Chem. A 2007, 111, 11873 (Univ. LeMans)
Very fine agreement between experimental and calculated values
Calculated Vzz (V.m-2)
0,0 5,0e+20 1,0e+21 1,5e+21 2,0e+21 2,5e+21 3,0e+21
Exp
erim
enta
l νQ (H
z)
0,0
2,0e+5
4,0e+5
6,0e+5
8,0e+5
1,0e+6
1,2e+6
1,4e+6
1,6e+6
1,8e+6
AlF3 α-CaAlF5 β-CaAlF5
Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9 α-BaCaAlF7 Regression
νQ = 5,85.10-16 Vzz R2 = 0,993
Calculated ηQ
0,0 0,2 0,4 0,6 0,8 1,0
Exp
erim
enta
l ηQ
0,0
0,2
0,4
0,6
0,8
1,0
AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9
α-BaCaAlF7 RegressionηQ,exp. =ηQ, cal.
ηQ, exp = 0,972 ηQ,cal
R2 = 0,983
νQ and ηQ after structure optimization
![Page 63: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/63.jpg)
NMR shielding, chemical shift:
![Page 64: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/64.jpg)
• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
![Page 65: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/65.jpg)
![Page 66: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/66.jpg)
Mössbauer spectroscopyIsomer shift: charge transfer too small in LDA/GGAHyperfine fields: Fe2+ has large Borb and BdipYBaFe2O5 HFF, IS and EFG with GGA+U, LDA/GGA
CO
VM
![Page 67: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/67.jpg)
Cu(2) and O(4) EFG as function of r
EFG is determined by the non-spherical charge density inside sphere
Cu(2)
O(4)
∫∫∑ =∝= drrrdrr
YrVYrr zzLM
LMLM )()()()( 20320 ρρρρ
final EFG
r
r r
rr
![Page 68: Lecture of K.Schwarz on hyperfine interactions](https://reader031.vdocuments.net/reader031/viewer/2022013113/5890551f1a28ab3e038c27b1/html5/thumbnails/68.jpg)
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 nodeor within 1 bohr are important.