Reading. What is EPR (ESR)? Spectroscopy: The Big Picture. Electron Paramagnetic Resonance: Hyperfine Interactions. Chem 634 T.

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1 Electron Paramagnetic Resonance: yperfine Interactions hem 63 T. ughbanks Reading Drago s Physical Methods for hemists is still a good text for this section; it s available by download (zipped, password protected): For EPR: hapters 9 and 3. Also good: Inorganic Electronic Structure and Spectroscopy, Vol., hapter by Bencini and Gatteschi; and Vol., hapter by Solomon and anson. Background references: rton, Electron Paramagnetic Resonance; Abragam & Bleaney Electron Paramagnetic Resonance of Transition Ions; arrington & McLachlan, Introduction to Magnetic Resonance; Wertz & Bolton, Electron Spin Resonance. What is EPR (ESR)? Electron Paramagnetic (Spin) Resonance Applies to atoms and molecules with one or more unpaired electrons An applied magnetic field induces Zeeman splittings in spin states, and energy is absorbed from radiation when the frequency meets the resonance condition, hν = E /λ ~ cm, ν = c/λ ~ 0 0 s (microwave, Gz) Spectroscopy: The Big Picture EPR 3 3

2 What information do we get from EPR? hemists are mostly interested in two main pieces of information: g-values and hyperfine couplings, though spin-relaxation information can also be important to more advanced practitioners. g-values: these are, in general, the structure-dependent (i.e., direction-dependent) proportionality constants that relate the electron spin resonance energy to the direction of the applied magnetic field: hν = g x,y,z. The direction-dependence is determined by electronic structure. yperfine coupling: arise from interactions between magnetic nuclei and the electron spin and give information about the delocalization of the unpaired 5 What background do we need? We need an understanding of the magnetic properties of atoms and/or open-shell ions. In particular, we will need to understand the role of spin-orbit coupling, which is essential for understanding g-value anisotropy. Quantum mechanical tools: manipulation of angular momentum operators and understanding of how perturbation theory works. Basic ideas from M theory and ligand field theory must be familiar Experimental Aspects EPR performed at fixed frequency (i.e., field is varied). Some common frequencies: ü X-band: ν ~ 9.5 Gz, ( s ; λ ~ 3 cm); typical required field is ~ 300 G = 0.3 T - most common (hν/ = G for ν = Gz) K-band: ν ~ Gz (. 0 0 s ; λ ~ cm); typical required field is ~ 8600 G = 0.86 T - less common (hν/ = G for ν =.00 Gz) ü Q-band : ν ~ 35 Gz ( s ; λ ~ 0.8 cm); typical required field is ~ 500 G =.5 T (hν/ = 89.5 G for ν = Gz) Solvents: generally avoid water, alcohols, high dielectric constants (why?) 7 Experimental Aspects Form of samples: Gases, liquids, solids, crystals, frozen solutions Glasses are best for solution samples measured below solvent freezing points because they give homogeneity, whereas solutions have cracks and frozen crystallites and scatter radiation ote: glasses can form with pure solvents or mixtures. Unsymmetrical molecules have a greater tendency to form glasses. (e.g., cyclohexane crystallizes, methylcyclohexane tends to form glasses). -bonding promotes crystallization

3 Experimental Aspects ontainers: Pure silica ( quartz ), not Pyrex (borosilicate glass); the latter absorbs microwaves to too great an extent onstant, homogeneous magnetic fields. Microwave generator (klystron) - sets up a standing wave, for which the frequency is fixed by the geometry of the cavity. To improve the S/ ratio (reduce the effect of /f noise), the field is generally modulated (00-00 kz) and detected with a lock-in detector (amplifier). Phasesensitive detection gives a true differential of line shapes. For this reason, EPR spectra are normally displayed as derivatives this is actually convenient since it accentuates features of the absorption in which we are interested anyway. 9 EPR: Fixed microwave frequency, sweep the field In MR, frequencies are varied at fixed fields The power absorbed is measured as a function of field: The Experiment MDULATED PWER SUPPLY MAGET hν S KLYSTR MIRWAVE GEERATR STADIG MIRWAVE Modulated Field (t) = o cos(ωt); where =field t = some time constant and cos(ωt) = modulation frequency The field is modulated to improve the S/ ratio 0 -atom Spin amiltonian st term: electronic Zeeman nd term: nuclear Zeeman 3rd term: Fermi ontact hyperfine (isotropic) magnitude depends on the electron density of the unpaired electron on the nucleus, ψ (0). 9 = ; g = = e! m e c = erg/g J/T Spin = S g µ I + as I a = 8π 3 g µ ψ (0) µ = e! m p c = erg/g * J/T erg ψ (0) =.83 electrons/å 3 a = Mz = J * This is the nuclear magneton - a constant that is not nucleus specific and is not the nuclear moment symbol used by Drago. 0 Electronic Zeeman interaction for -atom If a hydrogen atom is placed in a magnetic field, = z z, ˆ the electron s energy will depend on its m S value. The Zeeman interaction between the applied field and the magnetic moment of the electron is illustrated as: m S = uclear spin neglected! m S = =.003

4 -atom Spin Energies g µ g µ a a -atom Spectrum hv = hv E(α e ) E(β e ) = E(α e ) E(β e ) g µ + a ge g µ a = + g µ a ge + g µ + a (neglects nd -order hyperfine) S g µ I +as I as z I z a(s x I x + S y I y ) = a(s I + S I ) + + turns on mixing! + a = a! a = G ~ 0 Mz Electronic Zeeman g µ g µ uclear Zeeman a a st -order hyperfine nd -order 3 hyperfine 3 Zero-field -atom Exact -atom Spin amiltonian as i I = a(s x I x + S y I y + S z I z ) = a( (S + I + S I + ) + S z I z ) as i I α e = a( )( ) α e = a α α e as i I β e = a( )( ) β e = a β β e ( ) ( ) as i I α e ± β e = a( (S I + S I ) + S I ) α + + z z e ± β e = a(s I + S I ) α + + ( e ± β e ) + a ( ) ( ) α ( e ± β e ) ± a β ( e ± α e ) a α ( e ± β e ) ( ) for + : = a α e + β e ( ) for : = 3 a α e β e 5 5 α e α e β e β e α e α e β e β e ( ) + a E Δ ( + Δ e ) a E a 0 0 a Δ ( + Δ e ) a E Δ ( Δ e ) + a E Δ Δ e where Δ e = and Δ = g µ Solutions: ( ) E = + a ± Δ Δ e E = a ± a + Δ + Δ e 6 from: arrington & McLachlan, Introduction to Magnetic Resonance 6

5 -atom EPR; variable field -atom EPR; variable field 7 from: arrington & McLachlan, Introduction to Magnetic Resonance 7 8 from: arrington & McLachlan, Introduction to Magnetic Resonance 8 most EPR measurements are 9 from: arrington & McLachlan, Introduction to Magnetic Resonance 9 A ote on units of energy Three different units of energy (none of them really are energy!) are used in EPR: cm (/λ), Gz or z or Mz (ν), and gauss (G). The conversions are done as follows: cm c ( cm/s) z or cm Gz z h/ ( G-s) G or Mz G or cm 0697 G The conversion to gauss yields the field necessary to induce an equivalent Zeeman splitting for a free electron. Example: a zero field splitting parameter of 0.0 cm could also be reported as 3.03 Gz or 08 G. 0 0

6 Methyl radical splitting diagram Methyl radical splitting diagram E = + 3 / = + / = / = 3 / The four transitions for the methyl radical. +m I states are lowest for m s = / and the m I state lowest for m s = /, from the I S term. The four transitions for the methyl radical. +m I states are lowest for m s = / and the m I state lowest for m s = /, from the I S term. = 3 / = / = + / = + 3 / From Drago, Physical Methods - Fig. 9-7 corrected. Plotted to reflect the constantfrequency experimental conditions. Methyl radical Spectrum rganic π Radicals The four transitions for the methyl radical. +m I states are lowest for m s = / and the m I state lowest for m s = /, from the I S term. a = Qρ ; Q!.5 G 3 3

7 ESR Spectrum: aphthalide Anion ESR Spectrum: aphthalide Anion aα = 3.79 Mz~.9 G aβ =.9 Mz ~.76 G aphthalide Anion - SM 0.9 (0.5) (-0.63) Bg umbers are coefficients from ESR (ückel) 8 7 8

8 uclei with I > / A nucleus with spin I splits the electron resonance into I + peaks. Again, the electronic Zeeman interaction is much larger than the hyperfine interaction. [V( ) 5 ] + : V +, d, S = / ; 5V(00%), I = 7 / 9 haracteristic vanadyl EPR g-tensor is only modestly anisotropic (g =.9658) itronylnitroxide; Two Delocalized Electrons in lusters (I,I ) For n identical nuclei, hyperfine splitting yields ni + lines. hyperfine = as (I + I ) (, ) (0,0) (,) (,0) (,0) (0,) (0, ) (,) (, ) Ph Ph o Se o o 59 o: 00% abundance I = 7 / (3 7 / )+ = expected lines SM (a ) 3 Single crystal, 77K, z 3 3 3

9 Anisotropic hyperfine coupling Simulated EPR spectrum of a normal copper complex, tetragonal u( ) 6 + (at X-band, n = 9.50 Gz). (A) EPR absorption; (B) first derivative spectrum without and () with copper hyperfine splitting. 63u 69.% I = 3 / A 0 =.95 Gz 65u 30.9% I = 3 / A 0 = 5.30 Gz From Drago (corrected) Excited T g state mixes into 6 A g 55Mn (00%) Mn + doped into MgV 6 g x =.00 ± g y =.009 ± 0.00 g z =.0005 ± D x = 8±5 G; D y = -87±5 G; D z = -306±0 G; from.. g and. alvo, an. J. hem. 50, 369 (97). Assign all the transitions with initial and final values of M S and m I! 3 3 Multiple hyperfine splittings rganic Radicals in Solids A gorgeous case of accidental degeneracy: multiplets; each multiplet has lines with relative intensity ratio: ::3::5:6:5::3:: u II 63 u (I = 3/) 00% enriched (I = ) (I = /) EPR external standard, DPP (diphenylpicrylhydrazide), g =.0037 ± Spin = S g µ I + S T I st term: electronic Zeeman (g-tensor anisotropy neglected) nd term (S T I): nuclear Zeeman 3rd term: yperfine includes the isotropic Fermi contact term and the anisotropic, dipolar, part of the hyperfine (S T I): S T I = S T I + as I

10 Dipolar yperfine Interaction The energy of dipole-dipole interactions falls off as the cube of the distance between dipoles - whether the dipoles are electric or magnetic. ˆr = r r Dipolar yperfine Interaction The electron is distributed over a volume defined by an orbital while the nucleus can be regarded as fixed at a point. Thus, the dipolar interaction is summed (integrated) over the electron s probability distribution, ψ. The classical energy of interaction between two magnetic dipoles, µ e and µ, is E dipolar = µ µ e 3(µ r)(µ r) e ~ r 3 r 5 r 3 where r is the vector separating the dipoles. The corresponding QM amiltonian is S I 3(S r)(i r) dipolar = g µ r 3 r 5 ψ µ µ e 3(µ r)(µ r) e ψ r 3 r 5 ψ yperfine Anisotropy: (S T ) I dominant over nuclear Zeeman, I Dipolar yperfine: rigin of nd -order transitions First-order EPR transitions we've seen involve the flipping of only the electron spin, α e β e and α e β e This is so because = 0 and = = If the axis of quantization changes, both spins can flip α = sin θ and α α = cos θ Therefore, for some orientations, satellite transitions are observed. Detailed study yields signs of the tensor components. If the dipolar hyperfine interaction dominated over the externally applied field (it often doesn t), it would determine the axis of quantization of the nucleus. The hyperfine tensor determines the net angle between electron s axis (along S) and the nucleus s axis, along S T. 39 When dipolar hyperfine interaction is comparable with the Zeeman field (a common case), it would determine the axis of quantization of the nucleus changes depending on whether electron is spin-up or spin-down. The hyperfine tensor determine the net angle between electron s axis (along S) and the nucleus s axis, along S T

11 yperfine Anisotropy Edipolar = 3( µ e r)( µ r) T = T + a ~ 3 r3 r where r is the vector separating the dipoles. The corresponding QM amiltonian is dipolar The classical energy of interaction between two magnetic dipoles, µ e and µ, is dipolar = ge g µ S T I The classical energy of interaction between two magnetic dipoles, µ e and µ, is µe µ yperfine Anisotropy S I 3(S r)(i r) = ge g µ 3 r r 3x 3xy T = 3xz 3xy 3xz r 3y 3yz 3yz r 3z Matrix elements of T should be interpreted as expectation values of the operators shown over the electronic wavefunction. (The origin is at the nucleus at which the spin I resides.) 3cos θ 3( µ e r)( µ r) ~ 3 r3 r where r is the vector separating the dipoles. The corresponding QM amiltonian is Edipolar = µe µ S I 3(S r)(i r) dipolar = ge g µ 3 r Matrix elements of T should be interpreted as expectation values of the operators shown over the electronic wavefunction. (The origin is at the nucleus at which the spin I resides.) θ π θ π = sin 3 = 35.6 θ = cos 3 = yperfine Anisotropy; Example: Malonate Radical -field orientations for single-crystal EPR experiment hν (x-ray)! to molecular plane & bond, to π orbital to molecular plane SM = 6 The malonate radical is obtained by irradiation of malonic acid with x-rays. The trapped radical can be studied with single-crystal EPR - which reveals the anisotropy of the hyperfine tensor, T. (We can initially ignore the Zeeman interaction of the proton, I, i.e., consider only the first-order effects resides.) 3 See arrington & McLachlan, pp B = 9! to molecular plane, to π orbital & bond A = 9 Principal axes of T and observed hyperfine splittings (Mz)

12 V 3 See Wertz & Bolton, p. 30. yperfine Anisotropy; Example: [V() 5 ] 3 The EPR spectrum of the [V() 5 ] 3 ion in a single crystal of KBr is shown above; the magnetic field is aligned along the [00] axis of the KBr host. a) What are the values of A and A? (int: There are two overlapping spectra shown here. Think about the way in which the [V() 5 ] 3 ion is likely to substitute for K + and Br ions in the host; the relative intensities of the two spectra should settle which spectrum corresponds to 5the direction and which corresponds to the direction.) 5 [V() 5 ] 3, cont. (b) Explain the relative magnitudes of the hyperfine constants. (c) Identify all the lines in the spectrum. g =.97; what is the value of g? (d) Draw a d-orbital splitting diagram for this ion (π-effects are important too) and use the g-value information to determine as many of the d-orbital energy splittings as you can from the information you have so far (for V +, ζ = 8 cm ) assuming the orbitals have pure d-character. (e) onsider the effects of covalence in the calculation of the g-values for this ion and discuss how the important ligand bondinffects should influence both g and g. 6 6