Hyperfine interaction

Transcription

1 Hyperfine interaction The 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. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

2 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 B n (r), which is called the nuclear field. The electronic magnetic moment interacts with this nuclear field B n (r). B 0 The interaction energy is: A = -µ e B n (r) B n There are two contributions: µ n µ e. The wave function ψ(r) of the electron does not vanish at the nuclear position ( s-states in atomic terminology). ψ(0) 0. A µ = 8π ( g µ ) ( g µ ) ψ(0) 4π 3 0 s n K e B This is the Fermi-contact interaction.. The wave function of the electron has an angular dependence and vanishes at the nuclear position: ψ(0) 0 (e.g. a p-function). A µ 4π 5 g g r 0 = ( p n µ ) ( ) 3cos K e µ B θ 3 Averaging over the electronic wave function GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

3 If both contributions are present: A = A + A θ s p (3cos ) B 0 θ θ Is the angle between the magnetic field B 0 and the p-function lobe. p-function In gases and liquids, only the isotropic part A s 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 A s is retained. In the general case: the interaction is a nd rank tensor, the hyperfine tensor. A = A sin θ cos Φ + A sin θ sin Φ + A cos θ XX YY ZZ θ and Φ are the angles between the coordinate system axes and the tensor principle axes X,Y,Z. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 3

4 In general situations, an electronic spin S and a nuclear spin I are coupled by an interaction ˆ H = Iˆ A Sˆ The second rank tensor A is called the hyperfine tensor and it hfi can be described by its three principal axes values A XX, A YY, A ZZ 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: H ˆ = ai ˆ Sˆ = hai ˆ Sˆ 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 33 Cs with the nuclear magnetic moment of the Cs I=7/ nucleus. hfi B 0 Electron spin nuclear spin The Hamiltonian of an electronic spin ^ S and a nuclear spin ^I in a magnetic field B 0 along the z-direction: Hˆ = g µ BS ˆ g µ BI ˆ + hais ˆ ˆ e B 0 z n K 0 z GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 4

5 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,m S > and the nuclear spin states I, m I > ψ = Sm, Im, m S = -S, -S+,,S-,S m I = -I, -I+,,I-,I S I There are (S+) (I+) base states. In order to avoid a too clumsy notation, one usually omits the S, I in the base states and writes: ψ = mm = m m S I S I For the case S =/ I =/, we have 4 base states: = + + = + 3 = + 4 = The operators S z and I z are diagonal in the base states. Sˆ z ˆ = Sz + + = + + = z ˆ S = + ˆ S z 3 = 3 S ˆ z = 4 4 ˆ I z = + I ˆ z = ˆ I z 3 = + 3 I ˆ z = 4 4 GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 5

6 H ˆ hfi = hai ˆ S ˆ = ha ( Iˆ ˆ ˆ ˆ ˆ ˆ x Sx + Iy Sy + Iz Sz) H hfi = ha Iˆ ˆ ˆ ˆ ˆ ˆ z Sz + ( I+ S- + I- S+ ) ˆ ( ) S ˆ 3 = S ˆ 4 = S ˆ = 3 S ˆ - = 4 I ˆ = I ˆ 4 = 3 I ˆ = I ˆ - 3 = 4 S operates on the electronic spin only leaving the nuclear spin state unaffected. Similarly I only operates on the nuclear spin. IS ˆ ˆ + - = I ˆ 4 + = 3 IS ˆ ˆ = Iˆ- = ˆ ( ) H = ha Iˆ Sˆ + ( Iˆ Sˆ ˆ ˆ - + I- S ) hfi z z + + Is diagonal with matrix elements ±ha/4 Couples the states > and 3> with matrix elements ha/ Note: For quantum mechanical calculations with spins, it is generally vary practical to express the operators S x, S y, I x, I y in terms of the shift operators S +, S -, I +, I -. The matrix elements of these operators are easy to evaluate: S + shifts a state m S > to m S +>, unless m S > is the highest state. S - shifts m S > to m S ->, unless m S > is the lowest state. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 6

7 Matrix elements of the Hamilton operator H = g µ BS ˆ g µ BI ˆ + hais ˆ ˆ e B 0 z n K 0 z 3 4 = + + = + = + = = + + = + 3 = + 4 = geµ B B ha gnµ K B geµ B B 0 ha 0 0 ha + gnµ K B 0 4 geµ B B0 ha 0 0 ha gnµ K B 0 4 geµ B B ha + gnµ K B GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 7

8 E/h (GHz) 0 For high field B 0, the base states are practically the eigenstates of the Hamiltonian. Thus at high field, the states can be classified by >, > etc. > > B 0 /T > 3> GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 8

9 E/h (GHz) Selection rules for ESR-transitions: m S = ± m I = 0 > 3> and > 4> are the ESR transitions in high field B 0 > > B 0 /T -5-0 ESR spectrum B ha g µ e B 0 4> 3> GKMR ESR lecture WS005/06 Denninger B hyperfine_interaction.ppt 9

10 In low field B 0, 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. f=ghz Transitions which apparently do not obey the selection rules m S = ±, m I = 0, are called A/h =.4 GHz f=ghz f=ghz B 0 /T 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. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 0

11 What a the states in the limit of B 0 and B 0 = 0? The limit of very large B 0 is: the base states >, >, 3>, 4> are the eigenstates. States > and 3> have the S and I spin anti-parallel, thus they are lower in energy through the hyperfine interaction. Visualisation of the state coefficients 3 4 > > 4> 3> GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

12 For B 0 = 0, the solutions are: a singlet with total spin 0 ( + + ) a triplet with total spin +,, + ( + ) + + These are very general properties of two spins coupled by an interaction of the form hai S. Triplet E=h A Singlet GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

13 Visualisation of the coefficients of the states: 3 4 Negative coefficients are drawn downwards GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 3

14 The following operators are frequently found in calculations of spin systems: ˆ, ˆ z, ˆ x, ˆ ˆ ˆ y, +, - J J J J J J The quantum states are written as: JmJ The operator symbol J stands for any angular momentum operator like L, S, I, J = L+S, F = J+I etc. J is the spin number, m J the projection of the spin onto the quantisation axis, usually taken as the z-axis. Fundamental relations for the operators and the states are: ˆ J = ( + ) J Jm J J Jm J Jm = m Jm ˆz J J J J Diagonal with eigenvalue J(J+) Diagonal with eigenvalue m J J ˆ Jm J ( J ) m ( m ) Jm + + J = + + J J J Shifts one state up. Jˆ Jm J ( J ) m ( m ) Jm- J = + J J J ˆ ˆ ˆ + = Jx + ij J ˆ J ˆ ij ˆ = x y y J ˆ ( ˆ ˆ x J J ) = + + y + J J ˆ = i ( J ˆ J ˆ ) Shifts one state down. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 4

15 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. J J, J + -/> +/> <-/ <-5/ <-3/ <-/ <+/ <+3/ <+5/ <+/, J J, J + -> 0> +> <- < 0 < /> -3/> -/> +/> +3/> +5/> J, J + <- <- < 0 <+ <+ J, J + -3/> -/> +/> +3/> <-3/ <-/ <+/ <+3/ > -> 0> +> +> GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 5

16 J, J + <-3 <- <- < 0 <+ <+ <+3-3> -> -> 0> +> +> +3> J=7/ J, J + <-7/ <-5/ <-3/ <-/ <+/ <+3/ <+5/ <+7/ J=3-7/> -5/> -3/> -/> +/> +3/> +5/> +7/> GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 6

17 Hyperfine interaction in high field B 0 : if the electron Zeeman energy g e µ B B 0 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 B 0. The electronic field B e acting on the nuclei is either in the same direction as or opposed to B 0, 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 >, >, 3>, 4>. E = + + = + ha geµ B B 0 gnµ K B ha geµ B B + 0 gnµ K B 0 4 ha gµ B + e B 0 gµ B ha e B 0 4 = 3 = + ha + geµ B B + 0 gnµ K B0 4 ha geµ B B 0 gnµ K B0 4 GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 7

18 The ESR signal consists of two lines, the splitting is ha measured in field units. ESR signal st derivative B 0 B = ha g µ e B 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. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 8

19 Coupling with one nuclear spin I : I+ equidistant lines of practically equal intensity ESR signal ( st der.) I=3/ Magnetic field offset (mt) GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 9

20 Hyperfine interaction with a number of equivalent nuclei Without hyperfine coupling x I=/ ESR signal ( st der.) ESR signal ( st der.) Magnetic field offset (mt) Magnetic Field offset (mt) x I=/ x I=/ ESR signal ( st der.) ESR signal ( st der.) Magnetic Field offset (mt) Magnetic Field offset (mt) GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 0

21 The hyperfine structure is only resolved in the ESR spectra if A > line width. atom I e B H 3 C 9 F 9 Si 35 Cl ha /( g µ ) s As / 50.8 mt 4 MHz /.0 mt 3.08 GHz / 70.0 mt 48.6 GHz /.8 mt 3.4 GHz 3/ 66.5 mt 4.66 GHz 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 (978) GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

22 S=/, I= 6 states Spectrum in low field: 3 lines Not quite equidistant GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt

23 Spectrum in high field: 3 equidistant lines. GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 3

24 High resolution hyperfine spectra of radicals in solution DPPH: α, α -diphenyl-β-picryl hydrazyl Main coupling: equivalent N-atoms with I=. 5 lines. Small couplings: to the protons. α-pentachlorophenyl, α-phenylβ-picryl hydrazyl GKMR ESR lecture WS005/06 Denninger hyperfine_interaction.ppt 4

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