1.a Magnetic Resonance in Solids - Uni Stuttgart · 2005. 11. 8. · 2) nuclei: spin magnetic moment for all nuclei with I ≠0 3) muons, positrons, neutrons: elementary magnetic

08/11/2005 GKMR lecture WS2005/06 Denninger magresintro.ppt1

1.a Magnetic Resonance in SolidsThe methods of Magnetic Resonance (MR) measure with high sensitivity the magnetic dipole moments and their couplings to the environment.

Since the coupling between the magnetic dipoles µi and the electromagnetic field is rather small, MR uses very specialised and highly sensitive measurement techniques.

Carriers of magnetic moments:

1) electrons: spin magnetic moment

orbital magnetic moment (bound electrons)

2) nuclei: spin magnetic moment for all nuclei with I ≠ 0

3) muons, positrons, neutrons: elementary magnetic moments

Electron Spin Resonance (ESR,EPR)

Electrons in atoms, molecules and in the solid state have a very strong exchange interaction leading to spin-pairing (Pauli principle). Thus ESR depends on the existence of unpaired electrons.


Important sources of unpaired electrons in solids are:

1. Radicals

C

HH

H H

C

H

H H

CH4, methan *CH3, methyl radical

Unpaired electron

Molecules (or molecular fragments) with broken bonds. The chemical bonds are not completely saturated and some “dangling” bonds carry magnetic moments.

Radiation damage: In solids, such dangling bonds result e.g. from radiation damage associated with UV, X-ray radiation, γ-ray , neutron, electron, proton bombardment.

Si

Si

Si

Si

Si

Si

Si

Si Si

Si

Si

Si

SiSi

Si Si Si

H H

Defects:in semiconductors broken bonds (dangling bonds) are partially saturated by e.g. hydrogen.

The remaining dangling bonds are paramagnetic.

Such dangling bonds are known e.g. in silicon and in SiO2.

H


Donors , acceptors: Donors and acceptors in semiconductors usually have unpaired electrons.

Defects: Defects in semiconductors and in insulators are often paramagnetic.

Intermediate products: Intermediate products of chemical reactions in the solid state are often paramagnetic. Examples: Intermediate products of polymerisation reactions; Oligomers with radical character.

2. Elements with incomplete inner electron shells

(d-transition elements, f-rare earth elements)

CdAgPdRhRuTcMoNbZrYZnCuNiCoFeMnCrVTiSc

HgAuPtIrOsReWTaHf

3d4d

5d

LrNoMdFmEsCfBkCmAmPuNpUPaThLuYbTmErHoDyTbGdEuSmPmNdPrCe4f

5f


These elements can be either in:

Metal-ligand complexes

Ions incorporated in solid materials:

or

e.g. Cr3+ in Al2O3: ruby

ESR can basically determine:

Density of the ions, number of ions in the sample.

Position of insertion, symmetry of the lattice position.

Couplings, line widths dynamics, lattice vibrations

3. Excited electronic statese.g. triplet states in solids: energy transport phenomena, life time of excited states,

optically created probes.


4. Conduction electrons, conduction holes in metals and semiconductors

Especially in unconventional “organic metals”, organic semiconductors and inorganic semiconductors.

3-dimensional

“Normal” metal

2-dimensional

Hetero structures, quantum wells

1-dimensional

Quantum wires

Polymer chains

Questions: susceptibility, phase transitions, dynamics, dimensionality

line width: motional/exchange narrowing

localisation/delocalisation, coupling to nuclei

ESR is a very sensitive spectroscopic technique. In favourable cases, one can detect 108 to 1010 spins. Optically detected MR can detect 107 spins. In very special systems, single spin detection is possible ( time averaging).

10-1410

123

1010 spins 1.6 10 mol6 10 mol −

= ⋅⋅

High field spectrometers can detect 108 spins ≈ 10-16 mol.


1.1 Experimental foundations of ESRAll systems with angular momentum S, J, L, I, posses a corresponding magnetic dipole moment. This magnetic moment µ is fundamentally coupled to the angular momentum.

Electrons: s Bˆ

s g Sµ µ= ⋅ ⋅

Bˆ

ll g Lµ µ= ⋅ ⋅

Spin magnetic moment

Orbital magnetic momentJ B

ˆJ g Jµ µ= ⋅ ⋅

Nuclei: Kˆ

I Ig Iµ µ= ⋅ ⋅

µB : Bohr magneton24

Be

9.27410154 102e J

Tmµ −= = ⋅

µK : nuclear magneton 27K 5.0507866 102 p

e JTMµ −= = ⋅

gS, gl, gJ : g factor, spectroscopic splitting factor, for atoms: Landé factor.

gI : nuclear g factor

In atoms, central potential, l·s coupling, the total momentum is: j = l + s

jls( 1) ( 1) ( 1)1

2 ( 1)j j s s l lg

j j+ + + − +

= +⋅ ⋅ +

This Landé factor is only valid for free atoms with a central potential.


rr

z

1I e T= − ⋅

In classical electrodynamics, one calculates the magnetic moment of a charged particle moving in a circle with radius r:

2z ( )I rµ π= ⋅

areaMagnetic moment current

vv rr

ω ω= × = 12I e eTωπ= − ⋅ = − ⋅ The current I is proportional

to the charge –e and the angular velocity ω

The angular momentum Jz is: 2z vJ m r m r ω= ⋅ ⋅ = ⋅ ⋅

22 2z 2 2 2 z

e e eI r r r Jmπ ω ωµ π π− − −= ⋅ ⋅ = ⋅ = ⋅ = ⋅

~

The angular momentum is measured in units of ħ. zzJ J= ⋅ z 2 ze Jmµ −= ⋅

B 2emµ = Bohr magneton z zB Jµ µ= − ⋅ This applies directly to orbital

angular momenta in quantum mechanics.


2 zs B Sµ µ= − ⋅ ⋅ The spin moment S is connected to a magnetic moment µs which is twice as large as the orbital moment. This factor of 2 is explained by the Dirac-theory.

The general formulation introduces the g-factor. This spectroscopic splitting factor describes the relation between the magnetic moment and the angular momentum. The negative sign of the equation is chosen for particles with negative charge like the electron. The g-factor is than a positive quantity.

Bg Jµ µ= − ⋅ ⋅ g = 1: orbital magnetic moment

g = 2: spin magnetic moment

The g-factor can usually be measured with very high precision in ESR-experiments.

The basic equation for the determination of the g-factor for S=1/2 is:

µw B 0h f g Bµ⋅ = ⋅ ⋅

microwave frequency

magnetic field

Frequencies can be determined and controlled to a precision of at least 10-6. Thus the precision of the g-factor determination is usually dominated by the determination of the magnetic field B0 (at the sample).


Digression: Free electron, quantum-electrodynamics corrections to the g-factor.

The electron is the lightest elementary particle with a magnetic moment. The electron couples to the zero-point fluctuations of the electromagnetic fields in vacuum. This coupling leads to consequences like the Lamb-shift of the bound electron states and to the corrections of the g-factor.

The free electron in vacuum has two eigenstates with respect to its spin: |+1/2> and |-1/2>

As a consequence of the magnetic coupling to the electromagnetic field fluctuations, the electron can temporarily “flip” into the other spin state and back again. The lowest order quantum-electrodynamic process of this kind is the emission and re-absorption of a photon.

|+1/2>

|+1/2>|-1/2>

Photon, S = 1

Feynman-diagram, first order.

There are two coupling vertices with the electromagnetic field.

The coupling strength of this first order process is ∝ α, the Sommerfeld fine structure constant. α ≅ 1/137.

The energy splitting between |+1/2> an |+1/2> is increased by this process. The g-factor in this order is:

2 (1 ) 2.0023232g απ= ⋅ + ≅


Photon

|↓>

|↓>|↑>

In a magnetic field B0 the spin of the electron can be flipped by the absorption/emission of virtual magnetic photons.

|↓> Spin down

|↑> Spin up

These „fluctuations“ of the electron‘s spin state lead to measurable corrections of the spin splitting of free electrons (and bound electrons, too) in a magnetic field .

|↓>

|↑>

≈ 2.002319·µB·B0= 2·µB·B0

corrections of the energetic spin splitting

∆E = gs·µB·B0

E


By incorporating higher order processes and calculating Feynman-diagrams up to 8th order, one can calculate the theoretical value of the g-factor to 10 decimal places.

Comparison between the experimental value and the theoretical predictions by QED:

gexperiment = 2.00231930442 (08) The (08) is the error of the last two digits.

gtheory = 2.00231930492 (40) The (40) is the error of the last two digits.

Source:

Richard Feynman, QED

..00231931.2)328.021(2 2

2=

πα⋅−π

α+⋅≈sg

This value, the g-factor of the free electron, can be measured experimentally with very high precision. Such experiments are called g – 2 experiments.

First measurement: Polykarp Kusch, 1947 Nobelprize 1955

Precision measurement gs – 2:

J. Wesley, A. Rich

Phys. Rev. A4, 1341 (1971)


Size of the spin splitting, experimental realisation of the magnetic fields.

B 0h f g Bµ⋅ = ⋅ ⋅ B

0

f ghBµ⋅=

For free electrons (g = 2) :0

28 GHzT

fB ≅

Which magnetic fields (with sufficient homogeneity) can be reached by which magnet systems? Small distance d for sufficient

homogeneity.a) Permanent magnet systems:

B0 up to 1 T

Problems: temperature dependence

low homogeneity

small sweep range

Permanent magnet systems are employed for small (and relatively cheap) ESR systems. These systems usually are only employed for spin systems near g = 2 and for special purposes, where small and transportable systems have to be used.


b) Electromagnets with iron yoke:

B0 up to 2 T

Problems: very heavy

high electrical power consumption

field regulation necessary

c) Superconducting magnets:

B0 up to 6-8 T. For NMR up to 20 T (narrow bore)

Problems: need liquid helium

sweeping is slow and consumes helium

field regulation is tricky

d) Water cooled high current magnets: 25 T – 35 T

d) Pulsed magnetic fields: up to 150 T in the ms time range.


Bruker electromagnet BE25 for ESR up to 1.5T

Bruker electromagnet BE25 for ESR up to 1.5T

Tapered Polepieces (with polecaps and cryostat)


Supercon 8T, solenoid system, superconsweep coils ± 80 mT, MAGNEX

Supercon 6T, split coil system, room temp. sweep coils ± 30 mT, MAGNEX


Typical frequency bands in ESR spectroscopy and the standard designations:

3.4 T3 mm94 GHzW2.5 T4.3 mm70 GHzV

1.25 T8.5 mm35 GHzQ0.86 T1.25 cm24 GHzK0.35 T3 cm10 GHzX0.1 T10 cm3 GHzC

36 mT30 cm1 GHzL

B0 (g=2)λFrequencyBand

Up to the highest frequencies: Sample extension « wave length λThis is completely different to the case of e.g. optical spectroscopy. ESR is practically always concerned with electromagnetic fields in the near-field limit.

A further major difference: The magnetic field component is decisive.


Coupling of the magnetic moment µ= -gµBS to the field B0 and to the alternating magnetic field B1(t)

In the standard configuration for magnetic resonance, one applies a large and nearly static magnetic field B0 along the z-direction. A much smaller alternating magnetic field

is applied along a perpendicular direction, e.g. the x-direction. 011( ) cos(2 )B t B f tπ= ⋅ ⋅

z

x

y

011( ) cos(2 )B t B f tπ= ⋅ ⋅

B0

Sample tube

In the quantum mechanical description, the magnetic moment µ couples to the magnetic fields B0 and B1 via operators derived from the energy of µ in a magnetic field:

0 1ˆ ˆ ˆ ˆH B B Bµ µ µ= − ⋅ = − ⋅ − ⋅

Leads to transitionsSplits the

energy levels


00 Bˆˆ ˆ ( )H B g S Bµ µ= − ⋅ = + ⋅ ⋅

We first consider only the Zeeman-term -µ·B0

This is one of the simplest spin Hamilton operators encountered in magnetic resonance. We further assume for simplicity, that the g-factor is a scalar, and that all electronic interactions of the electron are just included in the g-factor g.

If this is the only spin-interaction, the direction of quantisation is clearly the direction of the magnetic field B0. By convention, this static field is applied along the z-axis:

0

0

0

0

B

B⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

zB 0ˆH g B Sµ= + ⋅ ⋅ We need the eigenstates and the

eigenvalues of the operator Sz.^

A spin with spin quantum number S has 2·S+1 different eigenstates denoted by the magnetic quantum number mS. We denote states by the Dirac notation as: |S mS>

Examples: S =1/2 : |1/2 +1/2> |1/2 -1/2>

S =5/2 : |5/2 +3/2> |5/2 -1/2> etc.

There are 2·S+1 eigenstates. These states span a Hilbert space of dimension 2·S+1.


For a given S, one very often denotes the states by mS only: |mS>

e.g. S = 5/2: |-5/2> , |-3/2> , |-1/2> , |+1/2> , |+3/2> , |+5/2>

The Hilbert space of spins is a “playground” for quantum mechanical calculations. The number of states is finite (even very small), and the energy spectrum is limited.

There are two operators for a spin S, which have simultaneous eigenvalues and can be

measured independently: and2S ˆzS 2ˆ ( 1)S SS Sm S S Sm= + ⋅

ˆz S SSS Sm m Sm= ⋅

zB 0ˆH g B Sµ= + ⋅ ⋅Thus the operator has the energy eigenvalues E for the stationary

states in a magnetic field B0

0 SBE g B mµ= ⋅ ⋅ ⋅ mS = -S, -S+1, …, S-1, S

B0

|3/2>

|1/2>

|-1/2>

|-3/2>

Energy level diagram for S =3/2 in a magnetic field

E


Ions of the d-transition metals (Hund’s rule)

e.g. Fe3+ 5-d e-:

Mn2+: 5-d e-:

Examples for systems with different spin quantum numbers S:

S =1/2: Single, unpaired electrons e-

Radicals ( *CH3 , *NH2 )

Ions ( C6H6+ , C6H6- , C60

-

Atoms with 1e-: H, Na, K, Cs

Conduction electrons in metals, semiconductors

S =1: Excited states of paired electrons

e.g. triplet states in molecules (metastable)

S =1/2,… 5/2:S = 5/2

S = 5/2

S =1/2,… 7/2: Ions of the rare earth elements

e.g. Eu2+ 7-f e-:

Gd3+: 7-f e-: S = 7/2

S = 7/2


Transition between these states are induced by magnetic dipole radiation.

The selection rules for these magnetic dipole transitions are: ∆mS = ±1

B0

E |+1/2>

|-1/2>

B B0 01 12 2

( ) ( )E g B g Bµ µ−∆ = ⋅ ⋅ ⋅ − = ⋅ ⋅

B0h

E gf Bh

µ∆ ⋅= = ⋅ Transition frequency By irradiating the sample with a alternating

magnetic field B1(t) with this frequency, one induces transitions.

0 01 11( ) cos(2 ) or sin(2 )B t B f t B f tπ π= ⋅ ⋅ ⋅ ⋅ Which direction does B1 has to

have with respect to B0 ?


The transition rates between stationary quantum states can be calculated very generally:

A time dependent perturbation leads to transitions between the states |m> , |n>1

ˆ ( )H t1

ˆ ( )H tm n←⎯⎯→ if:

|m>

|n>

∆E = h·f1) The perturbation has frequency components at the frequency f. f = ∆E/h.

1ˆ ( )H t

2) The matrix element of the perturbation does not vanish.

1ˆm H n

The transition rate Wmn between the states |m> and |n> is:

2 2

mn 12

(2 ) ˆ ( )W m H n p fhπ

= ⋅ ⋅

Transition rate Matrix elementFrequency distribution

of the perturbation

[ ] 2mn 2 2

11 1 Js HzJ sW = = ⋅ ⋅The transition rate has

the dimension 1/time.


The frequency distribution p(f) of the perturbation can have (and generally has) two contributions:

1) The perturbation H1(t) is not monochromatic, and the Fourier-transform of the time dependence leads to a distribution p(f). This is particularly valid for a stochastic perturbation. Processes leading to relaxation are of this kind.

2) The energy difference ∆E = h·f is distributed. The ultimate limit is e.g. the finite lifetime of the excited state |n>. A lifetime τ of the transition leads to a frequency distribution p(f) which is a Lorentzian function with width ∆f ∝ 1/τ.

In general both contributions are operative and have to be considered. Then p(f) is the convolution of the two contributions. Very often, one contribution dominates. This is e.g. the case if the perturbation is monochromatic. For a homogeneous spin system, the lifetime of the state dominates.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Time (s)

-6 -4 -2 0 2 4 60.0

0.1

0.2

0.3

0.4

g(f)

/(1/H

z)

(f-f0) /Hz

Lifetime τ = 0.2 s

∆f = 1/(2πτ) = 0.796 Hz

p(f)/

(1/k

Hz)


Direct calculation of the matrix element for the case S =1/21ˆm H n

|-1/2>

|+1/2>

∆E = h·f

The operators S2 and Sz have the simultaneous eigenstates |+1/2> and |-1/2>.

We define the operators S+ and S- by:

yxˆ ˆ ˆS S iS+ = + yxˆ ˆ ˆ-S S iS− =

1/2 1/2S + − = + 1/2 0S + + =

1/2 0S − − = 1/2 1/2S − + = −

We see directly, that only components with Sx and Sy are responsible for transitions between the states |+1/2> and |-1/2>.

1

1( ) 0 cos(2 )0

BB t f tπ

⎛ ⎞⎜ ⎟= ⋅ ⋅⎜ ⎟⎜ ⎟⎝ ⎠

1 1

0( ) cos(2 )

0B t B f tπ

⎛ ⎞⎜ ⎟= ⋅ ⋅⎜ ⎟⎜ ⎟⎝ ⎠

or

Alternating magnetic fields along the x or y direction induce the transitions, if the main field B0 is along the z-direction.

1 1( ) tB t B e ω−= ⋅

x: real component

y: imaginary componentx

y

Since the magnetization in magnetic resonance rotates around the magnetic field B0, it is appropriate to use rotating fields B1(t).


If a spin system is characterized by a life time τ, the normalized frequency distribution function is:

20

( )( )1

2p ff f

f

τ=

⎛ ⎞−+ ⎜ ⎟∆⎝ ⎠

⋅ 12f π τ∆ = ⋅

with This is a Lorentzian line shape.

-6 -4 -2 0 2 4 60.0

0.1

0.2

0.3

0.4

g(f)

/(1/H

z)

(f-f0) /Hz

12f π τ∆ = ⋅

In resonance, at f=f0, the spectral density p(f0) = 2τ.

The matrix element for magnetic dipole transitions with is:11( ) cos(2 )B t B f tπ= ⋅ ⋅

1B x B112

ˆ1/ 2 1/ 2g B S g Bµ µ+ ⋅ ⋅ − = ⋅

An alternating field amplitude B1 = 10-4 T = 1 Gauß leads to a matrix element of:

9.2741·10-28J for g=2.

A typical lifetime τ for an electron spin system is τ = 56.8 ns. (This is for a line width of 1 Gauß = 0.1 mT).


2 2

mn 12

(2 ) ˆ ( )W m H n p fhπ

= ⋅ ⋅

8.6·10-55 J2113.6·10-9 s(in resonance)

Wmn = 8.785·106 1/s (in resonance)

|-1/2>

|+1/2>

∆E = h·fRelaxation rates

W↑ W↓

Wmn

W↑ ≈ W↓ ≈ 1/(2τ) = 8.80·106 1/s

The transition rate by the magnetic dipole transitions is nearly identical to the relaxation transition rates responsible for the life time. This situation leads to the partial saturation of the transition. Saturation effects are very important in ESR (and in NMR). These effects will be considered in more detail later in later lectures.

Documents

1.a Magnetic Resonance in Solids - Uni Stuttgart · 2005. 11. 8. · 2) nuclei: spin magnetic moment for all nuclei with I ≠0 3) muons, positrons, neutrons: elementary magnetic