hyperfine interaction - uni stuttgart...gkmr esr lecture ws2005/06 denninger...

GKMR ESR lecture WS2005/06 Denninger hyperfine_interaction.ppt 1

Hyperfine interactionThe notion hyperfine interaction (hfi) comes from atomic physics, where it is used for the interaction of the electronic magnetic moment with the nuclear magnetic moment.

In magnetic resonance this interaction is encountered frequently and is responsible for a number of important aspects of ESR and NMR spectra:

In high resolution ESR spectroscopy of e.g. radicals in solution the hyperfine interaction leads to resolved hyperfine splitting of the ESR.

If the hyperfine interaction is larger than other unresolved contributions to the ESR line width, the hyperfine splitting is resolved in the ESR of localised centres in the solid state.

Unresolved hyperfine interactions with many coupled nuclei are often a major contribution to the ESR line widths in solid state ESR.

If the electronic spin is exchanged rapidly due to the exchange interaction or the motion of conduction electrons, the NMR is shifted by the averaged hyperfine interaction of the electrons. This is the Knight shift of the NMR.


The hyperfine interaction can be visualized as the motion of the electron in the magnetic dipole field of the nucleus. The nuclear magnetic moment leads to a magnetic field Bn(r), which is called the nuclear field. The electronic magnetic moment interacts with this nuclear field Bn(r).

µe

µn

Bn

B0 The interaction energy is: A = -µe·Bn(r)

There are two contributions:

1. The wave function ψ(r) of the electron does not vanish at the nuclear position ( s-states in atomic terminology). ψ(0) ≠ 0.

20es n K B

8 ( ) ( ) (0)4 3 g gA µ π µ µ ψπ= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

This is the Fermi-contact interaction.

2. The wave function of the electron has an angular dependence and vanishes at the nuclear position: ψ(0) → 0 (e.g. a p-function).

20en K Bp 3

12 ( ) ( ) 3cos 14 5 g gr

A µ µ µ θπ= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ −Averaging over the electronic wave function


B0θ

p-function

If both contributions are present:

2ps (3cos 1)A A A θ= + ⋅ −

θ Is the angle between the magnetic field B0 and the p-function lobe.

In gases and liquids, only the isotropic part As is observable, since the rapid reorientation of the atoms or molecules averages the anisotropic contributions to zero.

In the solid state, the site symmetry of the nuclear position is important:

For cubic site symmetry, only As is retained.

In the general case: the interaction is a 2nd rank tensor, the hyperfine tensor.

2 2 2 22 2 2 2 2XX YY ZZsin cos sin sin cosA A A Aθ θ θ= ⋅ ⋅ Φ + ⋅ ⋅ Φ + ⋅

θ and Φ are the angles between the coordinate system axes and the tensor principle axes X,Y,Z.


In general situations, an electronic spin S and a nuclear spin I are coupled by an interaction

hfiˆˆˆ I A SH = ⋅ ⋅ The second rank tensor A is called the hyperfine tensor and it

can be described by its three principal axes values AXX, AYY, AZZ along the principal axis system X,Y,Z.^ ^ ^

In many situations, the hyperfine interaction is a scalar, independent of the spin direction. This case is e.g. the Fermi contact interaction between s-like electrons at the nuclear site.

In this situation, the hfi leads to a scalar interaction term: hfiˆ ˆˆ ˆˆ a I S h A I SH = ⋅ ⋅ = ⋅ ⋅ ⋅

The hfi coupling constant a = h·A is often given in frequency units: e.g. A = 65 MHz.

It is interesting to note at that point, that our fundamental definition and the experimental realisation of the time unit second is realised by the hyperfine interaction of the s-electron in 133Cs with the nuclear magnetic moment of the Cs I=7/2 nucleus.

B0 Electron spin

nuclear spin

The Hamiltonian of an electronic spin S and a nuclear spin I in a magnetic field B0 along the z-direction:

^^

ne z zB K0 0ˆ ˆˆ ˆ ˆH g B S g B I h A I Sµ µ= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅


This Hamiltonian is solved by diagonalisation, leading to the energy eigenvalues and the eigenstates. As the base states for the quantum mechanical treatment, we take product states between the electron spin states |S,mS> and the nuclear spin states |I, mI>

S I, ,S m I mψ = ⋅ mS = -S, -S+1,…,S-1,S mI = -I, -I+1,…,I-1,I

There are (2S+1)·(2I+1) base states.

In order to avoid a too clumsy notation, one usually omits the S, I in the base states and

writes: IS S Im m m mψ = = ⋅

1 12 21 = + + 1 1

2 22 = + − 1 12 23 = − + 1 1

2 24 = − −

For the case S =1/2 I =1/2, we have 4 base states:

zz1 11 11 1ˆ ˆ

2 2 2 22 21 1S S= + + = ⋅ + + = ⋅ z

1ˆ2

2 2S = + ⋅ z1ˆ2

3 3S = − ⋅ z1ˆ2

4 4S = − ⋅

z1ˆ2

1 1I = + ⋅ z1ˆ2

2 2I = − ⋅ z1ˆ2

3 3I = + ⋅ z1ˆ2

4 4I = − ⋅

The operators Sz and Iz are diagonal in the base states.


hfi x y z zx yˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( )ˆ h A I S h A I S I S I SH = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ + ⋅ + ⋅

hfi z z + +1ˆ ˆ ˆˆ ˆ ˆ( )- -2

ˆ ( )h A I S I S I SH = ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅

+ 3 1S = + 4 2S = ˆ 1 3-S = ˆ 2 4-S =

+ 2 1I = + 4 3I = ˆ 1 2-I = ˆ 3 4-I =

+ +2ˆˆ - 3ˆ 4I S I= = + 3ˆˆ 1- 2ˆ-I S I= =

hfi z z + +1ˆ ˆ ˆˆ ˆ ˆ( )- -2

ˆ ( )I S I S I SH h A= ⋅ ⋅ + ⋅ ⋅ + ⋅⋅

Is diagonal with matrix elements ±hA/4

Couples the states |2> and |3> with matrix elements hA/2

Note: For quantum mechanical calculations with spins, it is generally vary practical to express the operators Sx, Sy, Ix, Iy in terms of the shift operators S+, S-, I+, I-. The matrix elements of these operators are easy to evaluate: S+ shifts a state |mS> to |mS+1>, unless |mS> is the highest state. S- shifts |mS> to |mS-1>, unless |mS> is the lowest state.

S operates on the electronic spin only leaving the nuclear spin state unaffected. Similarly I only operates on the nuclear spin.


1 12 21 = + +

1 112 2

= + +

1 12 22 = + − 1 1

2 23 = − + 1 12 24 = − −

1 122 2

= + −

1 132 2

= − +

1 142 2

= − −

ne z zB K0 0ˆ ˆˆ ˆH g B S g B I h A I Sµ µ= ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅Matrix elements of the

Hamilton operator

e B 0

n K 0

e B 0

n K 0

e B 0

n K 0

e B 0

n K 0

12

12

12

12

12

12

12

12

0 0 0

4

0 02

4

0 02

4

0 0 0

4

g B

hAg B

g B hAhAg B

g BhAhAg B

g B

hAg B

µ

µ

µ

µ

µ

µ

µ

µ

⎛ ⎞⋅⎜ ⎟⎜ ⎟⎜ ⎟− ⋅ +⎜ ⎟⎜ ⎟⎜ ⎟+ ⋅⎜ ⎟⎜ ⎟

−⎜ + ⋅ ⎟⎜ ⎟⎜ ⎟

− ⋅⎜ ⎟⎜ ⎟⎜ ⎟− ⋅ −⎜ ⎟⎜ ⎟⎜ ⎟− ⋅⎜ ⎟⎜ ⎟⎜ ⎟+ ⋅ +⎜ ⎟⎝ ⎠


|1>|2>

|3>|4>

B0/T1.00.5

E/h (GHz)

10

-10

5

-5

For high field B0, the base states are practically the eigenstates of the Hamiltonian. Thus at high field, the states can be classified by |1>, |2> etc.


|1>|2>

|3>|4>

B0/T1.00.5

E/h (GHz)

10

-10

5

-5

Selection rules for ESR-transitions: ∆mS = ±1 ∆mI = 0

|1> ↔ |3> and |2> ↔ |4> are the ESR transitions in high field B0

B0ESR spectrum

e B

hAB g µ∆ ≅


In low field B0, the eigenstates clearly are no longer the base states.

As the states are now a mixture of the 4 base states, there can be in principle 6 magnetic resonance transitions. However, it depends on the frequency ∆E = h·f, which are observed.

A/h = 1.4 GHzB0/T

f=1GHz

f=1GHz

f=1GHz

Transitions which apparently do not obey the selection rules ∆mS= ±1, ∆mI = 0, are called

‘forbidden’ transitions.

However, there is nothing forbidden with these transitions. The states coupled by these transitions are mixtures of the base states, and the magnetic dipole transitions can occur with the correct selection rules.


What a the states in the limit of B0→∞ and B0 = 0 ?

The limit of very large B0 is: the base states |1> , |2> , |3> , |4> are the eigenstates.

|1>|2>

|3>|4>

States |2> and |3> have the S and I spin anti-parallel, thus they are lower in energy through the hyperfine interaction.

1 2 3 4

Visualisation of the state coefficients


For B0 = 0, the solutions are: a singlet with total spin 0

a triplet with total spin 1

These are very general properties of two spins coupled by an interaction of the form hAI·S.

11 1 1 11 1 1 1, ,2 2 2 2 2 2 2 22( )+ + ⋅ + − + − + − −

1 1 11 12 2 2 22

( )⋅ + − − − +

Triplet

Singlet

∆E=h·A


Visualisation of the coefficients of the states:

1 2 3 4

Negative coefficients are drawn downwards


The following operators are frequently found in calculations of spin systems:

2y -z x +

ˆ ˆ ˆ ˆ ˆ ˆ, , , , ,J J J J J JThe operator symbol J stands for any angular momentum operator likeL , S , I , J = L+S , F = J+I etc.

The quantum states are written as:

JJm J is the spin number, mJ the projection of the spin onto the quantisation axis, usually taken as the z-axis.

Fundamental relations for the operators and the states are:

2J J

ˆ ( 1)J Jm J J Jm= ⋅ +

J JJzJ Jm m Jm= ⋅

J JJ Jˆ ( 1) ( 1) +1J Jm J J m m Jm+ = ⋅ + − ⋅ + ⋅

J JJ Jˆ -( 1) ( 1) 1J Jm J J m m Jm− = ⋅ + − ⋅ − ⋅

yxˆ ˆ ˆJ J iJ+ = +

yxˆ ˆ ˆJ J iJ− = −

+12

ˆ ˆ ˆ( )xJ J J−= + +2ˆ ˆ ˆ( )y

iJ J J−−= −

Diagonal with eigenvalue J(J+1)

Diagonal with eigenvalue mJ

“Shifts” one state up.

“Shifts” one state down.


Whereas the diagonal operators can easily be calculated, matrix element tables for the operators J+ and J- are very convenient for practical quantum mechanical calculations.

0 11 0

,J J+ − |-1/2> |+1/2>

<-1/2|

<+1/2|

0 0

02

2

2

0 2 0

,J J+ − |-1> | 0> |+1>

<-1|

<+1|

< 0|

0 0 0

0 0

0

3

4

34 0

3

0 0 3 0

,J J+ − |-1/2> |+1/2> |+3/2>|-3/2>

<-1/2|

<+1/2|

<+3/2|

<-3/2|

0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0

5

8

9

8

5

8

9

8

50 0

0 0 0 0 5 0

|-5/2> |-3/2> |-1/2> |+1/2> |+3/2> |+5/2>

<-5/2|

<-3/2|

<-1/2|

<+1/2|

<+3/2|

<+5/2|

,J J+ −

0 0 0 0

0 0 0

0 0

4

6

6

0

0 0 0

0

4

6

6

4

0 0 4 0

|-2> |-1> | 0> |+1> |+2>

<-2|

<-1|

< 0|

<+1|

<+2|

,J J+ −


0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0

7

12

15

16

15

10 0

0 0 0

7

12

15

16

15

10 0 0

0 0 0 0

2

2 7

0 0 7 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

6

0

10

12

0 0 0

12

10 0

6

10

12

12

1

0 0 0

0

6

0 0 6 0

|-3> |-2> |-1> | 0> |+1> |+2> |+3>,J J+ −

<-3|

<-2|<-1|

< 0|

<+1|

<+2|

<+3|

,J J+ − |-7/2> |-5/2> |-3/2> |-1/2> |+1/2> |+3/2> |+5/2> |+7/2>

<-7/2|

<-5/2|

<-3/2|

<-1/2|

<+1/2|

<+3/2|

<+5/2|

<+7/2|

J=3

J=7/2


Hyperfine interaction in high field B0: if the electron Zeeman energy ge·µB·B0 is large compared to the hyperfine interaction, than the analysis is relatively simple:

The electronic spin is quantized along the axis z of the magnetic field B0. The electronic field Be acting on the nuclei is either in the same direction as or opposed to B0, and the nuclear spin is quantized along z as well. The hyperfine interaction is thus ±hA/4, depending on parallel or anti-parallel spins.

In mathematical terms this means, that only the diagonal matrix elements of the operators have to be taken into account. The eigenstates are the base states |1> , |2> ,|3> , |4>.

1 12 21 = + +

1 12 22 = + −

1 12 23 = − +

1 12 24 = − −

ne B K0 011

2 2 4hAg B g Bµ µ⋅ − ⋅ +

ne B K0 011

2 2 4hAg B g Bµ µ+⋅ ⋅ −

ne B K0 011

2 2 4hAg B g Bµ µ+− ⋅ ⋅ +

ne B K0 011

2 2 4hAg B g Bµ µ− ⋅ − ⋅ −

E

e B 0 2hAg Bµ ⋅ −

e B 0 2hAg Bµ ⋅ +


B0

ES

R s

igna

l 1st

der

ivat

ive

The ESR signal consists of two lines, the splitting is hA measured in field units.

e B

hAB g µ∆ =

Note: the nuclear Zeeman energy plays no role in high fields, since the nuclear spin is not “flipped” in an ESR transition.

If the hyperfine interaction is resolved in the ESR-spectrum, it can be directly read of the spectrum.

Unfortunately, the hyperfine interaction is often small and not resolved, and very often there is hyperfine coupling to many nuclei, making the ESR spectrum fairly complicated.


-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

ES

R s

igna

l (1st

der

.)

Magnetic field offset (mT)

Coupling with one nuclear spin I : 2·I+1 equidistant lines of practically equal intensity

I=3/2


Hyperfine interaction with a number of equivalent nuclei.

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

ES

R s

igna

l (1st

der

.)

Magnetic field offset (mT)

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

ES

R s

igna

l (1st

der

.)

Magnetic Field offset (mT)

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

ES

R s

igna

l (1st

der

.)


-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

ES

R s

igna

l (1st

der

.)


Without hyperfine coupling

1x I=1/2

2x I=1/2

3x I=1/2


The hyperfine structure is only resolved in the ESR spectra if A > line width.

4.66 GHz166.5 mT3/235Cl

3.41 GHz121.8 mT1/229Si

48.16 GHz1720.0 mT1/219F

3.108 GHz111.0 mT1/213C

1422 MHz50.8 mT1/21H

atom I e Bs /( )h A g µ⋅ ⋅ sA

Selected atomic values for the hyperfine interaction

Calculated values for the complete periodic table:

J.R. Morton , K.F. Preston , Journal of Magnetic Resonance 30, 577 (1978)


S=1/2, I=1 6 states

Spectrum in low field: 3 lines

Not quite equidistant


Spectrum in high field: 3 equidistant lines.


High resolution hyperfine spectra of radicals in solution

DPPH:

α, α’-diphenyl-β-picryl hydrazyl

α-pentachlorophenyl, α-phenyl-β-picryl hydrazyl

Main coupling: 2 equivalent N-atoms with I=1. 5 lines.

Small couplings: to the protons.

hyperfine interaction - uni stuttgart...gkmr esr lecture ws2005/06 denninger...

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