Hyperfine interactions

Karlheinz SchwarzInstitute of Materials Chemistry

TU Wien

Some slides were provided by Stefaan Cottenier (Gent)

nuclear point chargesinteracting with

electron charge distribution

hyperfine interaction

all aspects of the nucleus-electron interaction

which go beyondan electric point charge for a nucleus.

=

Definition :

electricpoint

charge

electricpoint

charge

• volume• shape

N

S

electricpoint

charge

• volume• shape• magnetic moment

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• …

• Definitions

• magnetic hyperfine interaction

• electric quadrupole interaction

• isomer shift

• summary

Content

N

S

N

S

N

S

N

S

N

S

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

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

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

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

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

-1.5

-1

-0.5

0

0.5

1

1.5

0 30 60 90 120 150 180

Ener

gy (u

nits

: µB)

E

-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

-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 :

nuclear property electron property

N

S

(vector) (vector)

interaction energy (dot product) :

Btot = Bdip + Borb + Bfermi + Blat

Source of magnetic fields at a nuclear site in an atom/solid

Btot = Bdip + Borb + Bfermi + Blat

Bdip = electron as bar magnet

Source of magnetic fields at a nuclear site in an atom/solid

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

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

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

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 ?

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

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

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+

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

YBaFe2O5 HFF, IS and EFG with GGA+U, LDA/GGA

CO

VM

HFF(Fe2+)

HFF(Fe3+)

HFF(Fe2.5+)

• Definitions

• magnetic hyperfine interaction

• electric quadrupole interaction

• isomer shift

• summary

Content

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

• Force on a point charge:

• Force on a point charge:

• Force on a general charge:

-Q

-Q

-Q

-Q

-Q

-Q

-Q

2

-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

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

-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

nuclear property electron property(tensor – rank 2 )

interaction energy (dot product) :

(tensor – rank 2 )

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

≥

First-principles calculation of EFG

, 1192

Previous: point charge model andSternheimer factor to experimental value

Electronic structure of Li3N

• The charge anisotropy around N differs strongly between • muffin-tin and • full-potential

• affecting the EFG.

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)

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

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)

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

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)

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)

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

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

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

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

NMR shielding, chemical shift:

• Definitions

• magnetic hyperfine interaction

• electric quadrupole interaction

• isomer shift

• summary

Content

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

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

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.