Heteronuclear Spin Decoupling in Magic-Angle-Spinning Solid-State NMR. Matthias Ernst ETH Zürich

Transcription

1 Heteronuclear Spin Decoupling in Magic-Angle-Spinning Solid-State NMR Matthias Ernst ETH Zürich

2 Heteronuclear Decoupling in Liquid-State NMR Matthias Ernst ETH Zürich Heteronuclear J-coupled two-spin system RF irradiation CW I I S S RF irradiation of the I spins leads to a collapse of the J-split multiplet on the S spins. Decoupling strategies: - continuous-wave (cw) irradiation - noise decoupling - multiple-pulse sequences (MLEV, WALTZ, GARP, DIPSI, FLOPSY,...) - adiabatic inversions (WURST,...)

3 Heteronuclear Decoupling in Liquid-State NMR Matthias Ernst ETH Zürich CW irradiation: off-resonance effects RF irradiation CW I I J J eff S S Residual splitting is reduced to J IS ν I ν for offresonance cw irradiation. I z Scaling corresponds to a projection of the heteronuclear J coupling onto the effective rffield direction. Typical magnitudes of such terms: = 5 Hz, ν I < 5 khz, ν khz. J IS ν I +J ± J eff -J ν ν eff I x

4 Heteronuclear Decoupling in Liquid-State NMR Composite pulses and multiple-pulse sequences RF irradiation I I S S Better inversion over a larger range of chemical-shift offsets. But now compensation of rf-field inhomogeneities or errors in the pulse length are necessary. A.J. Shaka, J. Keeler, and R. Freeman, Evaluation of a New Broadband Decoupling Sequence: WALTZ- J. Magn. Reson. 5, - (98). Matthias Ernst ETH Zürich

5 Heteronuclear Decoupling in Liquid-State NMR Adiabatic inversion pulses RF irradiation I I S S Adiabatic inversion pulses give very good inversion over a large range of chemical shifts. amplitude π phase RF-field amplitude is not a critical -T p / T p / T p T p / -T p / T p / T p T p / parameter in adiabatic pulses. E. Kupce and R. Freeman, Adiabatic Pulses for Wideband Inversion and Broadband Decoupling, J. Magn. Reson. A5, 7-7 (995). Matthias Ernst ETH Zürich 5

6 Matthias Ernst ETH Zürich Detour: Representation of Hamiltonians Cartesian notation of Hamiltonians: a xx a xy a xz ( ) Î k A ( kn, ) Î Î nx = n = Î kx Î ky Î kz a yx a yy a yz Î ny = A kn, a zx a zy a zz Î nz ˆ kn, ( ) ( k, n) Î Scalar-product formulation of the Hamiltonian in a Cartesian basis. Basis transformation leads to spherical-tensor representation: ˆ ( k, n) (, n) = A k = In spherical-tensor notation, rotations of the Hamiltonian are simpler due to the block structure of the rotation matrix. ˆ ( kn, ) =. rotation matrix =. a xx a xy a xz a yx a yy a yz a zx a zy a zz (a xx +a yy +a zz )/ (a xy -a yx )/ (a xz -a zx )/ (a yz -a zy )/ (a xx -a yy -a zz )/ (-a xx +a yy -a zz )/ (a xy +a yx )/ (a xz +a zx )/ (a yz +a zy )/

7 Matthias Ernst ETH Zürich 7 Detour: Origin Of Time-Dependent Hamiltonians System Hamiltonian in the laboratory frame is static if the molecule is static. ˆ = i = q = ( ) q A q () i () i, ˆ, q S z Ut () S z ωt θ S y S y sample rotation interaction-frame transformation S x rotating frame S x interaction frame ˆ = i = q = () i, ( ) q A q () ˆ t () i, q spatial part of Hamiltonian is modulated Examples: - MAS, DAS, DOR ˆ = i = q = ( ) q A q () i () i ˆ, q () t spin part of Hamiltonian is modulated Examples: - interaction-frame transformations We need methods to deal with time-dependent Hamiltonians.

8 Matthias Ernst ETH Zürich 8 Detour: Average Hamiltonian Theory Time-dependent Hamiltonians () t are generated by - interaction-frame transformations - sample rotation, e.g. magic-angle spinning (MAS). S z ωt θ S y For a single time dependence with a cycle time we can write the Hamiltonian as a Fourier series: () t = n n ( ) e inω mt Average Hamiltonian theory (AHT) τ m S x interaction frame = ( ) -- n ( n) ( n) [, ] nω m -- [[ ( n), ( ) ], ( n) ] ( nω m ) n ( ) ( ) ( ) -- k, n ( n) ( k) ( n k) [, [, ]] knω m + Different orders of the AHT approximate the effective Hamiltonian with increasing accuracy. Symmetric sequences ( () t = ( τ c t) ) eliminate the odd orders of the AHT expansion: ( ) is the leading term for the residual line width in all liquid-state decoupling sequences.

9 Theoretical Description of Decoupling in Liquids Matthias Ernst ETH Zürich 9 Spin-system Hamiltonian in the rotating frame is static. Interaction-frame transformation with the rf-field leads to a time-dependent Hamiltonian. = I + S + II + SS + IS interaction-frame transformation: U() t = Tˆ exp i rf ( t ) dt () t = I() t + S + II + SS + IS() t = k t k ( ) e ikω m t S z S y S x rotating frame Ut () S z ωt θ S y S x interaction frame Only the chemical shifts of the I spins and the heteronuclear J couplings become time dependent in the rf-interaction frame. Single time dependence allows the use of average Hamiltonian theory (AHT). Only cross terms between the chemical shift and the heteronuclear coupling can appear in ( ) ( ) while can contain cross terms between all parts of the Hamiltonian.

10 Heteronuclear Decoupling in Liquid-State NMR Matthias Ernst ETH Zürich Decoupling in liquid-state NMR is mostly a question of perfectly inverting the spins over a large range of chemical shifts with minimum rf power. Decoupling sidebands can be observed at multiples of the inverse cycle time. Symmetric sequences ( () t = ( τ c t) ) eliminate the first-order of the AHT. Residual splitting is determined by the second-order (double commutator) contributions. Homonuclear scalar couplings become important only in second-order AHT. DIPSI: Decoupling in the presence of scalar couplings.

11 Heteronuclear Decoupling in Static Solids Matthias Ernst ETH Zürich I Liquids Static Solids J II Hz I I J II D II Hz 5 Hz I J IS 5 Hz J IS D IS 5 Hz Hz S J SS 5 Hz S S D SS J SS 5 Hz 5 Hz S Spin-spin couplings in solids (dipolar couplings) are roughly by a factor of - larger than in liquids (J couplings) for rigid bio-organic substances. - Higher rf-field amplitudes and better broadband inversion schemes are required. - Typical rf-field amplitudes: = 5- khz ν Dipolar couplings are anisotropic and, therefore, orientation dependent.

12 Heteronuclear Decoupling in Static Solids Matthias Ernst ETH Zürich I Liquids Static Solids J II Hz I I J II D II Hz 5 Hz I J IS 5 Hz J IS D IS 5 Hz Hz S J SS 5 Hz S S D SS J SS 5 Hz 5 Hz S Spin part of the homonuclear dipolar coupling ( I z I z I I ) is a second-rank tensor and not a scalar ( I I ) like in the J coupling. - Homonuclear dipolar couplings become time dependent in the rf-interaction frame. - There can now be first-order cross terms in the AHT expansion between the homonuclear and the heteronuclear dipolar coupling that lead to residual line broadening. - Special decoupling sequences optimized for homonuclear dipolar-coupled systems: COMARO (composite magic-angle rotation)

13 Heteronuclear Decoupling in Static Solids Matthias Ernst ETH Zürich I Liquids Static Solids J II Hz I I J II D II Hz 5 Hz I J IS 5 Hz J IS D IS 5 Hz Hz S J SS 5 Hz S S D SS J SS 5 Hz 5 Hz S Dense dipolar-coupling network on the I spins leads to spin diffusion among the I spins. Self decoupling can result in line narrowing due to an exchange-type process between the powder line shapes of the multiplet lines. increasing spin diffusion

14 Theoretical Description of Decoupling in Static Solids Spin-system Hamiltonian in the rotating frame is static. Interaction-frame transformation with the rf-field leads to a time-dependent Hamiltonian. = I + S + II + SS + IS t S z θ S interaction-frame transformation: U() t Tˆ i rf ( t ) dt z = exp ωt Ut () () t = I() t + S + II() t + SS + IS() t = k ( ) e ikω m t Only the chemical shifts of the I spins, the heteronuclear J couplings, and the homonuclear dipolar couplings become time dependent in the rf-interaction frame. Single time dependence allows the use of average Hamiltonian theory (AHT). Cross terms between the heteronuclear coupling and the chemical shift or the homonuclear dipolar coupling can appear in ( ) while ( ) can contain cross terms between all parts of the Hamiltonian. Symmetric sequences ( () t = ( τ c t) ) eliminate the first-order of the AHT. Residual splitting is determined by the second-order contributions. Matthias Ernst ETH Zürich k S x S y rotating frame S x S y interaction frame

15 Matthias Ernst ETH Zürich 5 Development of Decoupling Techniques liquid-state NMR 95 cw decoupling 9 noise decoupling 98 multiple-pulse decoupling 98 MLEV 98 WALTZ 985 GARP 988 DIPSI 995 adiabatic decoupling 995 WURST 997 SWIRL cw decoupling multiple pulse solid-state NMR under MAS 95 cw decoupling 995 multiple-pulse decoupling 995 TPPM SPINAL XiX low-power decoupling Why was the development of decoupling techniques much slower in solid-state NMR under MAS conditions than in liquid-state NMR?

16 Heteronuclear Decoupling in Rotating Solids Matthias Ernst ETH Zürich Static Solids Rotating Solids I J II D II Hz 5 Hz I I J II D II Hz 5 Hz I J IS D IS 5 Hz Hz J IS D IS 5 Hz Hz () t S D SS J SS 5 Hz 5 Hz S S D SS J SS 5 Hz 5 Hz S Spin-system Hamiltonian in the rotating frame is time dependent due to magic-angle spinning (MAS). () t = I () t + S () t + II () t + SS () t + IS () t = n n = ( ) e inω r t This additional time dependence is the source of the difficulties in decoupling under sample rotation.

17 Why Do We Need Decoupling Under Fast MAS? Matthias Ernst ETH Zürich 7 MAS will average in zeroth-order approximation all anisotropic interactions. F V Fγ Fδ Fε,ζ ν r Vα Fα Fβ Vβ Vγ Vγ 8 khz with XiX decoupling 8 khz khz khz Δν / [Hz] 5 CH khz 5 CH 8 khz 8 C chemical shift [ppm] 5 ν r [khz] Higher-orders will lead to a residual line width due to cross terms between heteronuclear and homonuclear dipolar couplings: Δν / ω r. Faster MAS (~ 5 khz) will lead to a liquid-like NMR spectrum. Isotropic J couplings are not averaged. I-spin spin diffusion will lead to a line broadening of the J multiplet lines.

18 Theoretical Description of Decoupling in Rotating Solids Matthias Ernst ETH Zürich 8 Spin-system Hamiltonian in the rotating frame is time dependent due to magic-angle spinning (MAS). () t = I () t + S () t + II () t + SS () t + IS () t = n Interaction-frame transformation with the rf-field introduces a second time-dependence in the Hamiltonian. There are now two frequencies: and. Average Hamiltonian theory requires that n = ( ) e inω r t () t I() t + S() t + II() t + SS() t + IS() t ( n, k) e ikω m t e inω r = = t - the two frequencies are commensurate, i.e., nω m = kω r (simultaneous averaging) or - a separation of time scales, i.e., ω m» ω r or ω m «ω r (sequential averaging). interaction-frame transformation: U() t = Tˆ exp i rf ( t ) dt t ω r n = ω m k

19 Detour: Origin Of Time-Dependent Hamiltonians System Hamiltonian in the laboratory frame is static if the molecule is static. ˆ = i = q = ( ) q A q () i () i, ˆ, q S z θ S z sample rotation + interaction-frame transformation S x S y Ut () ωt S x S y ˆ = i = q = () i, ( ) q A q () ˆ t () i, q () t rotating frame interaction frame Spatial and Spin part of the Hamiltonian are modulated with different frequencies. We need methods to deal with Hamiltonians with multiple time dependencies. If multiples of the two frequencies are matched, we obtain again time-independent parts in the Hamiltonian. Matthias Ernst ETH Zürich 9

20 Matthias Ernst ETH Zürich Multiple Time Dependencies: Floquet Theory Construction of Floquet Hamiltonian ˆ ˆ () t F - Block diagonalization using van-vleck method ˆ Projection back into Hilbert space Λˆ F -

21 Matthias Ernst ETH Zürich Multiple Time Dependencies: Floquet Theory Construction of Floquet Hamiltonian ˆ ˆ () t F Calculation is independent of the detailed structure of the Hilbert-space blocks. - - Block diagonalization using van-vleck method ˆ Projection back into Hilbert space Λˆ F -

22 Matthias Ernst ETH Zürich Effective Hamiltonians from Floquet Theory Single frequency ω m () t = k Construction ( ) of e ikω m t Floquet Hamiltonian k Projection n back into [[ Hilbert ( n), ( ) ], space ( n) ] ˆ F Calculation is independent of the detailed structure of the Hilbert-space blocks (k). Block diagonalization using van-vleck method If we know (k) we can calculate. = ( ) -- n ( n), ( n) [ ] nω m ( nω m ) k, n Λˆ F - - ( n), ( k), ( n k) [ [ ]] knω m - - -

23 Matthias Ernst ETH Zürich Effective Hamiltonians from Floquet Theory Two frequencies ω r and ω m () t = n k Construction (, ) of e inω r t e ikω m t Floquet Hamiltonian n, k ˆ F Calculation is independent of the detailed structure of Block diagonalization the Hilbert-space blocks (n,k) using. van-vleck method If we know (n,k) we can calculate n k = n, k n ω r + k ω m = νω r + κω m Projection (, ) back -- into Hilbert space n, k ν, κ ( n ν, k κ) Λˆ F ( ν, κ) [, ] νω r + κω m + -

24 Matthias Ernst ETH Zürich Effective Hamiltonians from Floquet Theory We can calculate effective Hamiltonians for Hamiltonians with multiple time dependencies in a way very similar to AHT. What is different when going from a single modulation frequency to two (or more) modulation frequencies? one modulation frequency two modulation frequencies ( n) ( n) ( ) -- [, ] = n, k = nω m n n, k ( ) n, k -- ν, κ ( n ν, k κ) ( ν, κ) [, ] νω r + κω m resonance conditions: n ω r + k ω m = With multiple modulation frequencies, resonance conditions at n ω r + k ω m = appear. They lead to time-independent terms in first-order or second-order perturbation theory. At these resonance conditions we do not average out certain components n k of the Hamiltonian. (, ) ( n, k ) ( ) There are first-order n k and second-order resonance conditions. (, )

25 Matthias Ernst ETH Zürich 5 Theoretical Description of Decoupling in Rotating Solids Time-dependent interaction-frame Hamiltonian has (at least) two modulation frequencies () t ( n, k) e ikω m t e inω r = t n = k S z ωt θ S y S x n k = n, k (, ) -- n, k ν, κ ( ν, k κ) n ( ν, κ), νω r + κω m + interaction frame Factors determining the observable line width in solid-state NMR decoupling: (, ) - Residual coupling ( ( ) ) is given by the commutator term because the interactionframe Hamiltonian is not symmetric due to the MAS rotation. (, ) ( n, k ) - Resonance conditions ( n k and ( ) ) between the two modulation frequencies can lead to large terms which can be beneficial or detrimental to the decoupling process. - I-spin spin diffusion leads to an additional averaging of the residual couplings.

26 Continuous-Wave Decoupling in Rotating Solids line intensity HORROR condition.7 (ω = ω r /)..5.. fractional rotary resonance (ω = ω r /) rotary resonance (ω = nω r ) ω /(π) [khz] higher-order rotary resonance (ω = nω r ) decoupling Glycine: H N-CH -COOH ν r = 8.5 khz ω /ω r ω /(π) Matthias Ernst ETH Zürich

27 Theory Of CW Decoupling Under MAS Matthias Ernst ETH Zürich 7 Interaction-frame transformation with RF: Ut () = exp iω I mz N m = We find a single frequency ω in the interaction frame: interaction-frame representation I mx I mx cos( ω t) + I my sin( ω t) I my I my cos( ω t) I mx sin( ω t) Fourier coefficients.5 FT Only the first side-diagonal in the rf-modulation space ( ω t ( π) k = ω ω k = ± ) is occupied.

28 Continuous-Wave Decoupling in Rotating Solids Matthias Ernst ETH Zürich 8 Residual coupling is given by a cross term between the I-spin CSA and the heteronuclear dipolar couplings. (, ) ( ) = -- κ ω ( ν) ( ν) ( ν) ( ν) I ωsi + ω ωi SI Sz I νω r + κω z ν, κ Resonance conditions: first-order resonance conditions with ( n, k ) ( n, k ) + and second-order resonance conditions ( n, k ) ( n, k ) with ( ) + ( ). HORROR condition at ω = ω r : - Recouples I-spin homonuclear dipolar couplings. - Line intensity is increased due to I- spin spin diffusion (self decoupling). line intensity HORROR condition.7 (ω = ω r /) -- ω ( ) + + ( ) - - ( m I Im + ω mi Im ). < m ω /(π) [khz] ω /ω r decoupling

29 Continuous-Wave Decoupling in Rotating Solids Matthias Ernst ETH Zürich 9 Residual coupling is given by a cross term between the I-spin CSA and the heteronuclear dipolar couplings. (, ) ( ) = -- κ ω ( ν) ( ν) ( ν) ( ν) I ωsi + ω ωi SI Sz I νω r + κω z ν, κ Resonance conditions: first-order resonance conditions with ( n, k ) ( n, k ) + and second-order resonance conditions ( n, k ) ( n, k ) with ( ) + ( ). First-order rotary-resonance condition at ω = ω r and ω = ω r. - Recouples heteronuclear dipolar couplings. - Lines are broadened and intensity is reduced. line intensity ω I rotary resonance (ω = nω r ) ω /(π) [khz] ( ) I + ( ) - ( ) + + ω I I + ω SI Sz I ω /ω r ω SI ( ) - Sz I decoupling

30 Continuous-Wave Decoupling in Rotating Solids Matthias Ernst ETH Zürich Residual coupling is given by a cross term between the I-spin CSA and the heteronuclear dipolar couplings. (, ) ( ) = -- κ ω ( ν) ( ν) ( ν) ( ν) I ωsi + ω ωi SI Sz I νω r + κω z ν, κ Resonance conditions: first-order resonance conditions with ( n, k ) ( n, k ) + and second-order resonance conditions ( n, k ) ( n, k ) with ( ) + ( ). Second-order rotary-resonance condition at ω = ω r and ω = ω r. - Recouples heteronuclear dipolar couplings. - Lines are broadened and intensity is reduced. line intensity ω /(π) [khz] - * [ ω ω S m S z I z I m + ω S m S z I z I m ] r, m ω /ω r higher-order rotary resonance (ω = nω r ) decoupling

31 Continuous-Wave Decoupling in Rotating Solids Residual coupling is given by a cross term between the I-spin CSA and the heteronuclear dipolar couplings. (, ) ( ) = -- κ ω ( ν) ( ν) ( ν) ( ν) I ωsi + ω ωi SI Sz I νω r + κω z ν, κ Resonance conditions: first-order resonance conditions with ( n, k ) ( n, k ) + and second-order resonance conditions ( n, k ) ( n, k ) with ( ) + ( ). - HORROR condition at ω = ω r. - first-order rotary-resonance condition at ω = ω r and ω = ω r. - second-order rotary-resonance condition at ω = ω r and ω = ω r. I-spin spin diffusion is active everywhere and scales with ω r. line intensity HORROR condition.7 (ω = ω r /) fractional rotary resonance (ω = ω r /) rotary resonance (ω = nω r ) ω /(π) [khz] Matthias Ernst ETH Zürich ω /ω r higher-order rotary resonance (ω = nω r ) decoupling

32 Continuous-Wave Decoupling in Rotating Solids Continuous-wave decoupling is a terrible decoupling sequence with a large residual coupling. Residual coupling increases with increasing B field strength (CSA!). D C D C D C + 5 N _ H Cl - Rotary-resonance conditions have to be avoided: cw khz B = T - High-power decoupling: ω > ω r Low-power decoupling: ω ω r for high MAS frequencies. I-spin spin diffusion averages the residual coupling: cw khz B =7 T - Observable line width increases with increasing spinning frequency: spin diffusion is slowed down. - Low-power decoupling at the HORROR condition leads to a narrower line width. High-power decoupling: Δν / decreases with ν. no decoupling B =7 T Low-power decoupling: decreases with. Δν / Matthias Ernst ETH Zürich ν r ω/(π) [Hz]

33 Rotor-Synchronized Sequences Matthias Ernst ETH Zürich rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Rotor-synchronized R and C sequences allow us to select certain components of spin interactions. Analysis is based on a symmetry-driven version of average Hamiltonian theory. Very powerful tool for tailoring the effective Hamiltonian under MAS. Malcolm H. Levitt Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR, Encyclopedia of Nuclear Magnetic Resonance, Volume 9, 5-9 ().

34 Rotor-Synchronized Sequences Matthias Ernst ETH Zürich rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Rotor-synchronized R and C sequences allow us to select certain components of spin interactions. Analysis is based on a symmetry-driven version of average Hamiltonian theory. Very powerful tool for tailoring the effective Hamiltonian under MAS. Malcolm H. Levitt Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR, Encyclopedia of Nuclear Magnetic Resonance, Volume 9, 5-9 ().

35 Rotor-Synchronized Sequences Matthias Ernst ETH Zürich 5 rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Rotor-synchronized R and C sequences allow us to select certain components of spin interactions. Analysis is based on a symmetry-driven version of average Hamiltonian theory. Very powerful tool for tailoring the effective Hamiltonian under MAS. Malcolm H. Levitt Symmetry-Based Pulse Sequences in Magic-Angle Spinning Solid-State NMR, Encyclopedia of Nuclear Magnetic Resonance, Volume 9, 5-9 ().

36 Decoupling Using Rotor-Synchronized Sequences Matthias Ernst ETH Zürich rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Ideal case of eliminating all interactions (time suspension) does not exist.

37 Decoupling Using Rotor-Synchronized Sequences Matthias Ernst ETH Zürich 7 rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Ideal case of eliminating all interactions (time suspension) does not exist. Isotropic homonuclear J II coupling cannot be eliminated unless selective pulses are used.

38 Decoupling Using Rotor-Synchronized Sequences Matthias Ernst ETH Zürich 8 rotor position C7 R8 C C C C9 n C R R 9 R8 5 9 R 7 radio frequency C π/7 C π/7 C π/7 C π/7 C 8π/7 C π/7 C π/7 select σ I J II D IS CSA I D II J IS Ideal case of eliminating all interactions (time suspension) does not exist. Isotropic homonuclear J II coupling cannot be eliminated unless selective pulses are used. Recoupling the homonuclear dipolar coupling can have advantages for heteronuclear decoupling.

39 Rotor-Synchronized Decoupling: C Matthias Ernst ETH Zürich 9 C performs quite well and gives comparable line widths to other decoupling sequences. RF-field requirement ω = ω r dictates -field strength for given MAS frequency. B Synchronization is not a strict requirement: in the range of -5 khz one obtains 9% of the maximum intensity. Effective flip angle of π is very critical. ω /(π) [khz] τ p [μs] Experimental line height of CH group in sodium propionate. ( π) = khz, ω ( π) = khz, ω r τ p B = 7.57 μs. C = ( π) φ, Δφ = -, = 9. T.

40 Non Rotor-Synchronized Decoupling Sequences Matthias Ernst ETH Zürich Rotor synchronization of the pulse sequence is not always desirable. CW TPPM XiX +φ -φ π ω /(π) ω /(π) ω /(π) τ p π/ωm n τ p π/ω m n ω r, ω ω r, ω m, ω α ω r, ω m There are many modifications of the TPPM sequence: - frequency-modulated and phase-modulated (FMPM) - small phase angle rapid cycling (SPARC); small phase incremental alternation (SPINAL) - CPM m-n; amplitude-modulated TPPM (AM-TPPM); GT-n - continuous modulation (CM) TPPM - swept-frequency TPPM (SW f -TPPM)

41 XiX Decoupling Under MAS Matthias Ernst ETH Zürich Two pulses with 8 phase shift. 8 Pulse duration is important not flip angle. Insensitive to rf-field inhomogeneities. τ p n ν D C D C + 5 N _ H Cl - Optimum performance around τ p.85τ r and τ p.85τ r. Performance minima at τ p = nτ r ( C n recoupling sequence). line intensity.8... τ p /τ r 5 Sample: [d9]-trimethyl- 5 N- ammonium chloride, ω r ( π) = khz, ω ( π) = khz (black), ω ( π) = 5 khz (blue). D C

42 Theory of XiX Decoupling Under MAS Analytical interaction-frame transformation with RF: Ut () = exp iβ() t I mz N m = ω I mx I mx cos( β() t ) + I my sin( β() t ) I my I my cos( β() t ) I mx sin( β() t ) Flip-angle is time dependent: β() t π ω m -- = -- π ( k + ) cos( ( k + )ω m t) k = We find integer multiples of the modulation frequency ω m in the interaction frame: interaction-frame representation Fourier coefficients β() t cos( β() t ) sin( β() t ) 5.5 FT ω m t ( π) k = ω ω m We will find resonances between ω m = π τ p and ω r. Matthias Ernst ETH Zürich

43 XiX Decoupling Under MAS Matthias Ernst ETH Zürich Residual coupling is given by a cross term between the homonuclear and the heteronuclear dipolar couplings. ( ν) ( ν) ωi I m ν ω SI ( ) ( ν) ωi I m (, ) ( ) -- i ω SI ( a νω r + κω κ, x b κ, x + a κ, y b κ, y )S z I z I my m ν, κ m Resonance conditions: first-order resonance conditions at n = ±, ±. τ p τ r = k (half rotor cycle) and τ p τ r = k (quarter rotor cycle) τ p /τ r = -k /( n ) First-order resonance conditions are very strong and rf-field and spinning frequency independent. Recouples heteronuclear dipolar couplings. Strength of recoupling depends on the Fourier coefficients. a k line intensity.8... τ p /τ r 5 (, ) ( n, k ) n k = + = Re ω ( n ) SI ak x (, S z I x + a k, ys z I ) y

44 XiX Decoupling Under MAS Matthias Ernst ETH Zürich Residual coupling is given by a cross term between the homonuclear and the heteronuclear dipolar couplings. ( ν) ( ν) ωi I m ν ω SI (, ) ( ) -- i ω SI ( a νω r + κω κ, x b κ, x + a κ, y b κ, y )S z I z I my m ν, κ m Resonance conditions: second-order resonance conditions at n = ±, ±, ±, ±. τ p τ r = k (one sixth rotor cycle) and τ p τ r = k 8 (one eights rotor cycle) Second-order resonance conditions decrease with increasing MAS frequency and increasing rf-field amplitude. ( ) ( ν) ωi I m Cross terms between I-spin CSA tensors and heteronuclear dipolar couplings leads to a second-order coupling term. line intensity.8... τ p /τ r = -k /( n ) τ p /τ r 5 ( n, k ) ( n, k ) ( ) + ( ) Im ω SI ( +) ( +) ωi κ ( a k κ, xa κ, y a κ, x a k κ, y) S ω r + κω z I z m

45 XiX Decoupling Under MAS Matthias Ernst ETH Zürich 5 Residual coupling is given by a cross term between the homonuclear and the heteronuclear dipolar couplings. ( ν) ( ν) ωi I m (, ) ( ) -- i ω SI ( a νω r + κω κ, x b κ, x + a κ, y b κ, y )S z I z I my m ν, κ m Resonance conditions: first-order (n = ±,±) and second-order resonances at n = ±, ±. First-order resonance conditions are very strong and rf-field and spinning frequency independent. Second-order resonance conditions decrease with increasing MAS frequency and increasing rf-field amplitude. I-spin spin diffusion is present everywhere as a second-order contribution and scales with ω r. ν ω SI ( ) ( ν) ωi I m line intensity.8... τ p/τ r = -k /( n ) τ p/τ r (, ) ( ) ν, κ m < p ( ν) ( ν) ω I I ωi m I p ( ν) ( ν) ω I I ωi m I p ( b νω r + κω κ, x + b κ, y )I z ( I m Ip I mip ) m

46 XiX Decoupling Under MAS Matthias Ernst ETH Zürich XiX decoupling is a sequence that gives a very small residual couplings for fast MAS and high rf-field amplitudes: CH group τ p τ r =.85, ω r ( π) = khz, ( π) = 5. ω Resonance conditions at τ p τ r = k z with z =,,, and 8 have to be avoided. The proximity to resonance conditions and the strength of these resonance conditions limits the achievable line width in XiX decoupling. count residual splitting [Hz] I-spin spin diffusion is present but does not play a major role due to the small magnitude of the residual couplings. 9.5 XiX decoupling works best for high MAS frequencies ( ω r ( π) > 5 khz) and large ratios of ω ω r. A good starting point for the local optimization is =.85. τ p τ r ω I /ω r τ p /τ r = -k /

47 TPPM Decoupling Under MAS ν r = khz ν = khz ν r = 5 khz ν = 5 khz +φ φ.5.5 ν n τ p τ p [μs] τ p [μs] φ [ ] φ [ ] ν r = 5 khz ν = 5 khz ν r = 8 khz ν = 9 khz τ p [μs] τ p [μs] φ [ ] φ [ ] Matthias Ernst ETH Zürich 7

48 TPPM Decoupling Under MAS Matthias Ernst ETH Zürich 8 TPPM consists of two pulses with a phase shift of φ. TPPM decoupling works well over a large range of spinning frequencies and rf-field amplitudes. +φ φ ν khz ν r khz ( max) τ p μs φ ( max) ( max) τ p Optimum phase angle changes significantly with experimental parameters. Optimum pulse length is always close to a 8 pulse. Improvement over cw decoupling increases with increasing spinning frequency. τ π φ ( max) ( max) I( τ p, ) Icw ( ) τ p n xx xy xz ν

49 Theory of TPPM Decoupling Under MAS There is no analytical interaction-frame transformation with the RF. t Ut () = Tˆ exp i rf ( t )dt I mx I mx f xx () t + I my f xy () t + I mz f xz () t α One finds two frequencies ω m = π τ p and ω α = -- ω with. π m α = acos( cosβcos φ + sin φ ) Fourier coefficients can only be calculated numerically. Interaction-frame Hamiltonian has now three time dependencies () t ( n, k, ) e ikω m t e inω r t e i ω α = t n = k, S z ωt θ S y S x n k = n, k (,, ) -- n, k ν, κ ( ν, k κ, λ) n interaction frame ( ν, κ, λ), νω r + κω m + λω α + Triple-mode Floquet description is required. Matthias Ernst ETH Zürich 9

50 TPPM Decoupling Under MAS Residual coupling is given by a cross term between the I-spin CSA tensor and the heteronuclear dipolar couplings. (, ) ( ) i p ν = ( ν) ( ν) ω Ip S ωip Residual coupling has four parts which cannot be zeroed all ( n) simultaneously. The q xy component is always zero. The magnitude of the residual coupling depends on the relative orientation of the two tensors. The smallest residual couplings are found near a pulse length of a π pulse and for a phase angle between 5 and. ( ν) ( ν) ω Ip S ωip ( ν) ( ν) ν ( + )( q xx Ipx S z + q xy Ipy S z + q xz ) ( ) q xx τ p [μs] ( ) q xx τ p [μs] ( ) Ipz S z φ [ ] φ [ ] Matthias Ernst ETH Zürich 5 5 ( ) q xz τ p [μs] φ [ ] φ [ ] ( ) q xz τ p [μs] ν r = 5 khz,ν = khz 5

51 TPPM Decoupling Under MAS Residual coupling is given by a cross term between the I-spin CSA tensor and the heteronuclear dipolar couplings. (, ) ( ) Resonance conditions: i p ν = ( ν) ( ν) ω Ip S ωip - n ω r = ω m (straight lines in a) - n ω r = ω α (curved lines in a) The heteronuclear dipolar coupling is recoupled in first order ( n =,) or second order ( =,). n These resonance conditions lead to a broadening of the lines and are detrimental to the decoupling. ( ν) ( ν) ω Ip S ωip ( ν) ( ν) ν ( + )( q xx Ipx S z + q xy Ipy S z + q xz ) a) τ p [μs] c) τ p [μs] ( ) Ipz S z φ [ ] φ [ ] φ [ ] φ [ ] Matthias Ernst ETH Zürich τ p [μs] τ p [μs] b) d) ν r = 5 khz,ν = khz

52 TPPM Decoupling Under MAS Residual coupling is given by a cross term between the I-spin CSA tensor and the heteronuclear dipolar couplings. (, ) ( ) i p ν = ( ν) ( ν) ω Ip S ωip ( ν) ( ν) ω Ip S ωip ( ν) ( ν) ν ( + )( q xx Ipx S z + q xy Ipy S z + q xz ) ( ) Ipz S z Resonance conditions: - n ω r = ± ω m + ω α (b). a) 9 8 b) 9 8 ν r = 5 khz,ν = khz This is a purely homonuclear recoupling condition because the ( ±, + ) a xμ Fourier coefficients are always zero. The magnitude of the τ p [μs] φ [ ] φ [ ] homonuclear terms determines the observed line width in connection with the residual coupling φ [ ] φ [ ] φ [ ] Matthias Ernst ETH Zürich 5 τ p [μs] c) τ p [μs] 75 τ p [μs] τ p [μs] 7 d) 5 5

53 TPPM Decoupling Under MAS Residual coupling is given by a cross term between the I-spin CSA tensor and the heteronuclear dipolar couplings. (, ) ( ) Resonance conditions: - n ω r = ± ω m ± ω α (c) - n ω r = ± ω m + ω α (d) i p ν = ( ν) ( ν) ω Ip S ωip The heteronuclear dipolar coupling is recoupled in second order ( =,,,). n These recoupling conditions are weaker than the first-order ones. ( ν) ( ν) ω Ip S ωip ( ν) ( ν) ν ( + )( q xx Ipx S z + q xy Ipy S z + q xz ) a) τ p [μs] c) τ p [μs] ( ) Ipz S z φ [ ] φ [ ] φ [ ] φ [ ] Matthias Ernst ETH Zürich τ p [μs] τ p [μs] b) d) ν r = 5 khz,ν = khz

54 TPPM Decoupling Under MAS TPPM decoupling is a sequence that has small residual couplings that come from cross term between I-spin CSA and dipolar-coupling tensors. Cross terms between the heteronuclear and the homonuclear dipolar couplings are only important for φ = 9. Some resonance conditions reintroduce the heteronuclear dipolar coupling ( n ω r = ω α ) and have to be avoided. Others reintroduce the homonuclear dipolar couplings of the I spins ( n ω r = ± ω m + ω α ) and are beneficial for the decoupling process. I-spin spin diffusion is present everywhere but is emphasized on the homonuclear resonance conditions. ν r = 5 khz,ν = khz simulations for a CH system τ p [μs] τ p [μs] all interactions φ [ ] no I-spin CSA tensors φ [ ] no homonuclear dipolar couplings Matthias Ernst ETH Zürich 5 τ p [μs] 7 5 φ [ ]...

55 Conclusions Matthias Ernst ETH Zürich 55 Resonance conditions between sample spinning and spin rotations makes decoupling in solid-state NMR under sample rotation more complicated. Observable line width in rotating solids is determined by (, ) -the residual coupling terms ( ). - the influence of nearby resonance conditions n k and -the I-spin spin diffusion induced self decoupling. (, ) ( n, k ) ( ) Leading term for the residual coupling in solids is the commutator term while in static samples the double commutator determines the line width. The residual coupling in TPPM and cw decoupling is dominated by the I-spin CSA cross term with the heteronuclear dipolar coupling. In XiX decoupling, the cross term between homonuclear and a heteronuclear dipolar couplings is the most important term. Resonance conditions can be bad, e.g., heteronuclear couplings leading to additional line broadening or good, e.g., homonuclear couplings leading to self decoupling. Self decoupling due to I-spin spin diffusion leads to additional narrowing of the residual couplings.

56 Practical Considerations Matthias Ernst ETH Zürich 5 CW decoupling should not be used in rotating solids. TPPM decoupling and XiX decoupling give both better performance (smaller residual couplings). TPPM decoupling can be used over a large range of spinning frequencies and rf-field amplitudes. Optimization of TPPM is critical: two-parameter optimization of pulse length and phase angle φ! τ p Modified TPPM sequences like SPINAL or SW f -TPPM are more stable under certain experimental conditions. There are no experimental studies that compare the performance of these sequences over a large range of experimental parameters ( B, ν r, and ν ) XiX decoupling works best for high MAS frequencies ( > khz) and high rf-field amplitudes ( ν > 5ν r ). Optimization of XiX decoupling is a local one parameter optimization around =.85. τ p τ r ν r

57 Other Contributions To Experimental Line Width Technical problems: VT air - Temperature gradients turbine Heating by MAS -B -field homogeneity (shim) rf coil B θ m = Setting of magic angle air bearings optical fibres drive air bearing air Sample preparation - Sample heterogeneity - Susceptibility effects Spin-dynamics - Decoupling efficiency - S-spin homonuclear couplings (rotational-resonance effects) - Relaxation effects lyophilized recrystallized. fast solvent evaporation recrystallized, slow evaporation in controlled humidity 5 C 7 5 ppm 5 5 N ppm ω r /(π) = khz ω r /(π) = khz ω r /(π) = 5 khz ω r /(π) = khz.5 ω r /(π) = khz ω [khz] 5 ν [khz] Matthias Ernst ETH Zürich 57

58 Acknowledgements Matthias Ernst ETH Zürich 58 University of California, Berkeley, USA Andrew Kolbert, Seth Bush Alexander Pines MPI für Med. Technik, Heidelberg Herbert Zimmermann National Institute of Chemical Physics and Biophysics, Tallinn, Estonia Ago Samoson, Jaan Past University of Nottingham, UK Helen Geen University of Durham, UK Paul Hodgkinson ETH Zürich, Switzerland Andreas Detken, Ingo Scholz Beat H. Meier

59 References Reviews about decoupling in solids: [] M. Ernst, Heteronuclear spin decoupling in solid-state NMR under magic-angle sample spinning, J. Magn. Reson. (). [] P. Hodgkinson, Heteronuclear decoupling in the NMR of solids, Prog. Nucl. Magn. Reson. Spectrosc. (5) 97. AHT [] M. Mehring, Principles of High Resolution NMR in Solids, nd edition, Springer, Berlin, 98. [] U. Haeberlen, High Resolution NMR in Solids: Selective Averaging, Academic Press New York, 97. Multimode Floquet theory [5] M. Baldus, T. O. Levante, B. H. Meier, Numerical simulation of magnetic resonance experiments: concepts and applications to static, rotating and double rotating experiments, Z. Naturforsch. 9a (99) [] E. Vinogradov, P. K. Madhu, S. Vega, A bimodal Floquet analysis of phase-modulated lee-goldburg high-resolution proton magic-angle spinning NMR experiments, Chem. Phys. Lett. 9 () 7. [7] E. Vinogradov, P. K. Madhu, S. Vega, Phase-modulated Lee-Goldburg magic-angle spinning proton nuclear magnetic resonance experiments in the solid state: A bimodal Floquet theoretical treatment, J. Chem. Phys. 5 () [8] R. Ramesh, M. S. Krishnan, Effective Hamiltonians in Floquet theory of magic-angle spinning using van Vleck transformation., J. Chem. Phys. () [9] M. Ernst, A. Samoson, B. H. Meier, Decoupling and recoupling using continuous-wave irradiation in magic-angle-spinning solid-state NMR: A unified descriptionusing bimodal Floquet theory, J. Phys. Chem. (5). [] J. R. Sachleben, J. Gaba, L. Emsley, Floquet-van vleck analysis of heteronuclear spin decoupling in solids: The effect of spinning and decoupling sidebands on the spectrum., Solid State NMR 9 () 5. [] R. Ramachandran, V. S. Bajaj, R. G. Griffin, Theory of heteronuclear decoupling in solid-state nuclear magnetic resonance using multi pole-multimode Floquet theory, J. Chem. Phys. (5) 5. [] M. Ernst, H. Geen, B. H. Meier, Amplitude-modulated decoupling in rotating solids: A bimodal Floquet approach, Sol. State Nucl. Magn. Reson. 9 (). [] M. Leskes, R. S. Thakur, P. K. Madhu, N. D. Kurur, S. Vega, A bimodal Floquet description of heteronuclear dipolar decoupling in solid-state nuclear magnetic resonance, J. Chem. Phys. 7 (7) 5. [] I. Scholz, B. H. Meier, M. Ernst, Operator-based triple-mode Floquet theory in solid-state NMR., J. Chem. Phys. 7 (7) 5. Matthias Ernst ETH Zürich 59

60 References CW decoupling [5] H. J. Reich, M. Jautelat, M. T. Messe, F. J. Weigert, J. D. Roberts, Off resonance decoupling, J. Am. Chem. Soc. 9 (99) 75. [] B. Birdsall, N. J. M. Birdsall, J. Feeney, Off resonance decoupling, J. Chem. Soc. Chem. Commun. 97 (97). [7] I. J. Shannon, K. D. M. Harris, S. Arumugan, High-resolution solid state C NMR studies of ferrocene as a function of magic angle sample spinning frequency, Chem.Phys.Lett. 9 (99) [8] M. Ernst, H. Zimmermann, B. H. Meier, A simple model for heteronuclear spin decoupling in solid-state NMR., Chem. Phys. Lett. 7 () [9] G. Sinning, M. Mehring, A. Pines, Dynamics of spin decoupling in carbon--proton NMR, Chem. Phys. Lett. (97) 8 8. [] M. Mehring, G. Sinning, Dynamics of heteronuclear spin coupling and decoupling in solids., Phys. Rev. B 5 (977) [] M. Ernst, S. Bush, A. C. Kolbert, A. Pines, Second-order recoupling of chemical-shielding and dipolar-coupling tensors under spin decoupling in solidstate NMR, J. Chem. Phys. 5 (99) [] M. Ernst, A. Samoson, B. H. Meier, Decoupling and recoupling using continuous-wave irradiation in magic-angle-spinning solid-state NMR: A unified descriptionusing bimodal Floquet theory, J. Phys. Chem. (5). TPPM decoupling and variants [] A. E. Bennett, C. M. Rienstra, M. Auger, K. V. Lakshmi, R. G. Griffin, Heteronuclear decoupling in rotating solids, J. Chem. Phys. (995) [] Z. H. Gan, R. R. Ernst, Frequency- and phase-modulated heteronuclear decoupling in rotating solids, Solid State NMR 8 (997) [5] Y. L. Yu, B. M. Fung, An efficient broadband decoupling sequence for liquid crystals, J. Magn. Reson. (998) 7. [] B. M. Fung, A. K. Khitrin, K. Ermolaev, An improved broadband decoupling sequence for liquid crystals and solids, J. Magn. Reson. () 97. [7] A. Khitrin, B. M. Fung, Design of heteronuclear decoupling sequences for solids, J. Chem. Phys. () [8] K. Takegoshi, J. Mizokami, T. Terao, H decoupling with third averaging in solid NMR, Chem. Phys. Lett. () 5 5. [9] G. Gerbaud, F. Ziarelli, S. Caldarelli, Increasing the robustness of heteronuclear decoupling in magic-angle sample spinning solid-state NMR, Chem. Phys. Lett. 77 () 5. [] G. DePaepe, A. Lesage, L. Emsley, The performance of phase modulated heteronuclear dipolar decoupling schemes in fast magic-angle-spinning nuclear magnetic resonance experiments, J. Chem. Phys. 9 () 8 8. [] R. S. Thakur, N. D. Kurur, P. K. Madhu, Swept-frequency two-pulse phase modulation for heteronuclear dipolar decoupling in solid-state NMR, Chem. Phys. Lett. () 59. [] A. Khitrin, T. Fujiwara, H. Akutsu, Phase-modulated heteronuclear decoupling in NMR of solids., J. Magn. Reson. () 5. Matthias Ernst ETH Zürich

61 References Other decoupling sequences [] G. De Paepe, P. Hodgkinson, L. Emsley, Improved heteronuclear decoupling schemes for solid-state magic angle spinning NMR by direct spectral optimization, Chem. Phys. Lett. 7 () [] M. Eden, M. H. Levitt, Pulse sequence symmetries in the nuclear magnetic resonance of spinning solids: Application to heteronuclear decoupling, J. Chem. Phys. (999) [5] J. Leppert, O. Ohlenschläger, M. Görlach, R. Ramachandran, Adiabatic heteronuclear decoupling in rotating solids, J. Biomol. NMR 9 () 9. [] G. De Paepe, D. Sakellariou, P. Hodgkinson, S. Hediger, L. Emsley, Heteronuclear decoupling in NMR of liquid crystals using continuous phase modulation, Chem. Phys. Lett. 8 () 5 5. XiX decoupling [7] P. Tekely, P. Palmas, D. Canet, Effect of proton spin exchange on the residual C MAS NMR linewidths. Phase-modulated irradiation for efficient heteronuclear decoupling in rapidly rotating solids, J. Magn. Reson. Ser. A 7 (99) 9. [8] A. Detken, E. H. Hardy, M. Ernst, B. H. Meier, Simple and efficient decoupling in magic-angle spinning solid-state NMR: the XiX scheme, Chem. Phys. Lett. 5 () 98. [9] M. Ernst, H. Geen, B. H. Meier, Amplitude-modulated decoupling in rotating solids: A bimodal Floquet approach, Sol. State Nucl. Magn. Reson. 9 (). Low-power decoupling under fast MAS [] M. Ernst, A. Samoson, B. H. Meier, Low-power decoupling in fast magic-angle spinning NMR, Chem. Phys. Lett. 8 () 9. [] M. Ernst, A. Samoson, B. H. Meier, Low-power XiX decoupling in MAS NMR experiments, J. Magn. Reson. () 9. [] M. Ernst, M. A. Meier, T. Tuherm, A. Samoson, B. H. Meier, Low-power high-resolution solid-state NMR of peptides and proteins, J. Am. Chem. Soc. () [] M. Kotecha, N. P. Wickramasinghe, Y. Ishii, Efficient low-power heteronuclear decoupling in C high-resolution solid-state NMR under fast magic angle spinning., Magn. Reson. Chem. 5 (7) S S. [] X. Filip, C. Tripon, C. Filip, Heteronuclear decoupling under fast MAS by a rotor-synchronized hahn-echo pulse train, J. Magn. Reson. 7 (5) 9. Matthias Ernst ETH Zürich