1.a Magnetic Resonance in Solids

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1 .a Magnetic Resonance in Solids The methods of Magnetic Resonance (MR) measure with high sensitivity the magnetic dipole moments and their couplings to the environment. nce the coupling between the magnetic dipoles µ i and the electromagnetic field is rather small, MR uses very specialised and highly sensitive measurement techniques. Carriers of magnetic moments: ) electrons: spin magnetic moment orbital magnetic moment (bound electrons) ) 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.

2 Important sources of unpaired electrons in solids are:. Radicals Molecules (or molecular fragments) with broken bonds. The chemical bonds are not completely saturated and some dangling bonds carry magnetic moments. H H C CH 4, methan H H H C H H Unpaired electron *CH 3, methyl radical 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. 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 O. H H H

3 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.. Elements with incomplete inner electron shells (d-transition elements, f-rare earth elements) 3d Sc Ti V Cr Mn Fe Co Ni Cu Zn 4d Y Zr Nb Mo Tc Ru Rh Pd Ag Cd 5d Hf Ta W Re Os Ir Pt Au Hg 4f Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 5f Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 3

4 These elements can be either in: Metal-ligand complexes or Ions incorporated in solid materials: e.g. Cr 3+ in Al O 3 : 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 states e.g. triplet states in solids: energy transport phenomena, life time of excited states, optically created probes. 4

5 4. Conduction electrons, conduction holes in metals and semiconductors Especially in unconventional organic metals, organic semiconductors and inorganic semiconductors. 3-dimensional Normal metal -dimensional Hetero structures, quantum wells -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 0 8 to 0 0 spins. Optically detected MR can detect 0 7 spins. In very special systems, single spin detection is possible ( time averaging). 0 = 60 mol spins.6 0 mol 3 High field spectrometers can detect 0 8 spins 0-6 mol. 5

6 . Experimental foundations of ESR All 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: µ = ˆ s µ B s g S µ µ ˆ = g l l B L Nuclei: µ = g µ ˆ K I µ B : Bohr magneton µ K : nuclear magneton g S, g l, g J : g I : I I µ µ Spin magnetic moment Orbital magnetic moment = e B me = = e K M = p µ = g µ J J 4 7 J T J T J B ˆ g factor, spectroscopic splitting factor, for atoms: Landé factor. nuclear g factor In atoms, central potential, l s coupling, the total momentum is: j = l + s g jls = + j( j+ ) + s( s+ ) l( l+ ) j ( j+ ) This Landé factor is only valid for free atoms with a central potential. 6

7 z In classical electrodynamics, one calculates the magnetic moment of a charged particle moving in a circle with radius r: µ π z = I ( r ) r I r = e T Magnetic moment current area v = ω r ω = v r ω I = e = e T π ~ The angular momentum J z is: The current I is proportional to the charge e and the angular velocity ω Jz = mr v = mrω µ z = π Ir = πω e r = eω r = e J π m z The angular momentum is measured in units of ħ. J z = Jz µ e z = J m z µ = e B Bohr magneton m µ z µ J B z = This applies directly to orbital angular momenta in quantum mechanics. 7

8 µ = µ The spin moment S is connected to a magnetic moment µ s which is s S B z twice as large as the orbital moment. This factor of 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. µ = g µ J B g = : orbital magnetic moment g = : 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=/ is: h f = g µ B µw B 0 microwave frequency magnetic field Frequencies can be determined and controlled to a precision of at least 0-6. Thus the precision of the g- factor determination is usually dominated by the determination of the magnetic field B 0 (at the sample). 8

9 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: +/> and -/> 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. -/> +/> Photon, S = 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 /37. +/> The energy splitting between +/> an +/> is increased by this process. The g-factor in this order is: g = ( + α ).0033 π 9

10 > > Photon > In a magnetic field B 0 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. E > corrections of the energetic spin splitting E = g s µ B B 0 > = µ B B µ B B 0 0

11 By incorporating higher order processes and calculating Feynman-diagrams up to 8 th order, one can calculate the theoretical value of the g-factor to 0 decimal places. g s ( + α 0.38 α ) π π = Comparison between the experimental value and the theoretical predictions by QED: g experiment = (08) The (08) is the error of the last two digits. g theory = (40) The (40) is the error of the last two digits. Source: Richard Feynman, QED This value, the g-factor of the free electron, can be measured experimentally with very high precision. Such experiments are called g experiments. First measurement: Polykarp Kusch, 947 Nobelprize 955 Precision measurement g s : J. Wesley, A. Rich Phys. Rev. A4, 34 (97)

12 ze of the spin splitting, experimental realisation of the magnetic fields. f h f = g µ B B g µ B 0 = B h 0 For free electrons (g = ) : Which magnetic fields (with sufficient homogeneity) can be reached by which magnet systems? a) Permanent magnet systems: B 0 up to T f B 0 8 GHz T Small distance d for sufficient homogeneity. 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 = and for special purposes, where small and transportable systems have to be used.

13 b) Electromagnets with iron yoke: B 0 up to T Problems: very heavy high electrical power consumption field regulation necessary c) Superconducting magnets: B 0 up to 6-8 T. For NMR up to 0 T (narrow bore) Problems: need liquid helium sweeping is slow and consumes helium field regulation is tricky d) Water cooled high current magnets: 5 T 35 T d) Pulsed magnetic fields: up to 50 T in the ms time range. 3

14 Bruker electromagnet BE5 for ESR up to.5t Bruker electromagnet BE5 for ESR up to.5t Tapered Polepieces (with polecaps and cryostat) 4

15 Supercon 8T, solenoid system, supercon sweep coils ± 80 mt, MAGNEX Supercon 6T, split coil system, room temp. sweep coils ± 30 mt, MAGNEX 5

16 Typical frequency bands in ESR spectroscopy and the standard designations: Band Frequency λ B 0 (g=) L GHz 30 cm 36 mt C 3 GHz 0 cm 0. T X 0 GHz 3 cm 0.35 T K 4 GHz.5 cm 0.86 T Q 35 GHz 8.5 mm.5 T V 70 GHz 4.3 mm.5 T W 94 GHz 3 mm 3.4 T Up to the highest frequencies: Sample extension «wave length λ This 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. 6

17 Coupling of the magnetic moment µ= -gµ B S to the field B 0 and to the alternating magnetic field B (t) In the standard configuration for magnetic resonance, one applies a large and nearly static magnetic field B 0 along the z-direction. A much smaller alternating magnetic field B 0 () t = B cos( π f t ) is applied along a perpendicular direction, e.g. the x-direction. y Sample tube x B () t = B cos( π f t ) 0 z In the quantum mechanical description, the magnetic moment µ couples to the magnetic fields B 0 and B via operators derived from the energy of µ in a magnetic field: B 0 H ˆ = ˆ µ B = ˆ µ B ˆ µ B Splits the energy levels 0 Leads to transitions 7

18 We first consider only the Zeeman-term -µ B 0 H ˆ = ˆ µ B =+ gµ ( S B ) 0 B ˆ 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. 0 If this is the only spin-interaction, the direction of quantisation is clearly the direction of the magnetic field B 0. By convention, this static field is applied along the z-axis: B 0 0 = 0 B 0 Ĥ = + gµ B Sˆ We need the eigenstates and the B 0 z eigenvalues of the operator S^ z. A spin with spin quantum number S has S+ different eigenstates denoted by the magnetic quantum number m S. We denote states by the Dirac notation as: S m S > Examples: S =/ : / +/> / -/> S =5/ : 5/ +3/> 5/ -/> etc. There are S+ eigenstates. These states span a Hilbert space of dimension S+. 8

19 For a given S, one very often denotes the states by m S only: m S > e.g. S = 5/: -5/>, -3/>, -/>, +/>, +3/>, +5/> 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: Ŝ and S ˆz Sˆ Sm = S( S+ ) Sm S S Thus the operator E Ĥ = + gµ B Sˆ B 0 B z 0 S S Sm = m Sm ˆz S S S has the energy eigenvalues E for the stationary states in a magnetic field B 0 E = g µ B m m S = -S, -S+,, S-, S 3/> Energy level diagram for S =3/ in a magnetic field /> -/> -3/> B 0 9

20 Examples for systems with different spin quantum numbers S: S =/: ngle, unpaired electrons e - Radicals ( *CH 3, *NH ) Ions ( C 6 H 6 +, C 6 H 6-, C - 60 Atoms with e - : H, Na, K, Cs Conduction electrons in metals, semiconductors S =: Excited states of paired electrons e.g. triplet states in molecules (metastable) S =/, 5/: S =/, 7/: Ions of the d-transition metals (Hund s rule) e.g. Fe 3+ 5-d e-: S = 5/ Mn + : 5-d e-: S = 5/ Ions of the rare earth elements e.g. Eu + 7-f e-: S = 7/ Gd 3+ : 7-f e-: S = 7/ 0

21 Transition between these states are induced by magnetic dipole radiation. The selection rules for these magnetic dipole transitions are: m S = ± E +/> B E = ( g µ B ) ( ) = g µ B 0 B 0 B 0 f E g µ B -/> = h = h B 0 Transition frequency By irradiating the sample with a alternating magnetic field B (t) with this frequency, one induces transitions. B ( t ) = B cos( π f t ) or B sin( π f t ) Which direction does B 0 0 has to have with respect to B 0?

22 The transition rates between stationary quantum states can be calculated very generally: H ˆ () t A time dependent perturbation leads to transitions between the states m>, n> m ˆ () H t n if: n> ) The perturbation H ˆ has frequency () t components at the frequency f. f = E/h. E = h f m> mhˆ n ) The matrix element of the perturbation does not vanish. The transition rate W mn between the states m> and n> is: ( π ) mn = ˆ ( ) W m H n p f h Transition rate The transition rate has the dimension /time. [ ] Matrix element s Js W = = mn J Hz Frequency distribution of the perturbation

23 The frequency distribution p(f) of the perturbation can have (and generally has) two contributions: ) The perturbation H (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. ) 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 /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 Lifetime τ = 0. s p(f)/(/khz) g(f) /(/Hz) f = /(πτ) = Hz Time (s) (f-f 0 ) /Hz 3

24 mhˆ n Direct calculation of the matrix element for the case S =/ The operators S and S z have the simultaneous eigenstates +/> and -/>. We define the operators S + and S - by: Sˆ Sˆ isˆ ˆ ˆ ˆ + = x + y S = Sx - isy E = h f +/> -/> Ŝ + / = + / Ŝ + + / = 0 Ŝ / = 0 Ŝ + / = / We see directly, that only components with S x and S y are responsible for transitions between the states +/> and -/>. B B( t) = 0 cos( π f t ) 0 nce the magnetization in magnetic resonance rotates around the magnetic field B 0, it is appropriate to use rotating fields B (t). or 0 B() t = B cos( π f t ) 0 y x Alternating magnetic fields along the x or y direction induce the transitions, if the main field B 0 is along the z-direction. t B () t = B e ω x: real component y: imaginary component 4

25 If a spin system is characterized by a life time τ, the normalized frequency distribution function is: p( f) = τ with f = ( f f ) π τ f 0 + This is a Lorentzian line shape. In resonance, at f=f 0, the spectral density p(f 0 ) = τ The matrix element for magnetic dipole transitions with B () t = B cos( π f t ) is: g(f) /(/Hz) 0. f = π τ + / gµ B Sˆ / = gµ B B x B 0. An alternating field amplitude B = 0-4 T = Gauß leads to a matrix element of: J for g= (f-f 0 ) /Hz A typical lifetime τ for an electron spin system is τ = 56.8 ns. (This is for a line width of Gauß = 0. mt). 5

26 ( π ) mn = ˆ ( ) W m H n p f h W mn = /s (in resonance) J s (in resonance) Relaxation rates W W W W /(τ) = /s W mn +/> E = h f -/> 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. 6

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