Three pulse recoupling and phase jump matching

Abstract

The paper describes a family of novel recoupling pulse sequences, called three pulse recoupling. These pulse sequences can be employed for both homonuclear and heteronuclear recoupling experiments and are robust to dispersion in chemical shifts and rf-inhomogeneity. These recoupling pulse sequences can be used in design of two-dimensional solid state NMR experiments that use powdered dephased antiphase coherence (γ preparation) to encode chemical shifts in the indirect dimension. Both components of this chemical shift encoded gamma-prepared states can be refocused into inphase coherence by a recoupling element. This helps to achieve sensitivity enhancement in 2D NMR experiments by quadrature detection.

1. Introduction

Nuclear magnetic resonance (NMR) spectroscopy opens up the possibility of studying insoluble protein structures such as membrane proteins, fibrils, and extracellular matrix proteins which are difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [2–6]. Recoupling pulse sequences that enable transfer of magnetization between coupled spins is the workhorse of all these experiments, either as a means to obtain structural information (e.g., internuclear distances) or as a means to improve resolution as building blocks in multiple-dimensional correlation experiments. The present paper describes some new methodology development for design of recoupling pulse sequences and demonstration of their use in so called gamma-preparation experiments [7,14] that enhance the sensitivity of 2D solid-state NMR experiments by a factor of $\sqrt{2}$.

The paper is organized as follows. In Section 2, we describe a novel approach to homonuclear recoupling that recouple dipolar coupled spins under magic angle spinning (MAS) experiments. We call them three pulse recoupling. These experiments are broadband and robust to rf-inhomogeneity. This work extends recently developed techniques for broadband homonuclear recoupling as reported in the [14,1,16–18]. In Section 3, we describe these methods in the context of heteronuclear experiments. In the context of heteronuclear spins, the recoupling is achieved by matching the syncronized phase jumps on the two rf-channels (analogous to Hartmann–Hahn matching of the rf-power commonly seen in heteronuclear recoupling experiments [22]). In Section 4, we show how these recoupling blocks can be used to prepare a powder dephased, antiphase coherence and refocus both components of the chemical shift encoded γ prepared states (following t₁ evolution) into inphase coherence. Section 5, describes experimental verification of the proposed techniques.

2. Three pulse recoupling in homonuclear spins

Consider, two homonuclear spins I and S under magic angle spinning condition [20]. In a rotating frame, rotating with both the spins at their common Larmor frequency, the Hamiltonian of the spin system takes the form

$$H(t)={\omega }_I(t)I_z+{\omega }_S(t)S_z+{\omega }_{IS}(t)(3I_z{S}_z-I·S)+2\pi A(t)\times (cos\varphi (t)F_x+sin\varphi (t)F_y),$$ (1)

where the operator F_x = I_x + S_x, and ω_I(t) and ω_S(t) represent the chemical shift for the spins I and S respectively and ω_IS(t) represents the time varying couplings between the spins under magicangle spinning. These interactions may be expressed as a Fourier series

$${\omega }_{\lambda }(t)=\sum _{m=-2}^2{\omega }_{\lambda }^{m}exp(im{\omega }_{r}t),$$ (2)

where ω_r is the spinning frequency (in angular units), while the coefficients ω_λ(λ = I, S, IS) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole–dipole coupling constant and through this the internuclear distance [23].

The term I · S in (1), commutes with the rf-field Hamiltonian, and in the absence of the chemical shifts, it averages to zero under MAS.

Consider the rf irradiation on homonuclear spin pair whose amplitude is chosen as $A(t)=\frac{C}{2\pi }$, where C = nω_r and phase ϕ(t), such that

$$\stackrel{.}{\varphi }(t)=\frac{C\pi }{2n}\left(\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right),$$ (3)

where δ(t) is a delta function ( ${\int }_{-\epsilon }^{t}f(\tau )\delta (\tau )d\tau =f(0)$ for all ε ≥ 0, t > 0) and corresponds to instantaneous phase increment of 1 radian. See Fig. 1.

Fig. 1.

Top left panel shows ramp of phase in Eq. (5). The Top right panel shows time derivative of phase in Eq. (5). Area under this curve gives the total phase change. The bottom panel shows the accumulation of phase area at the crossed points when the magnetization is rotated in the interaction frame of CF_x as in Eq. (5).

In the modulation frame, of the phase ϕ(t), the rf-field Hamiltonian takes the form

$$H^{rf}(t)=C{F}_x-\stackrel{.}{\varphi }{F}_z,$$ (4)

where C is in the angular frequency units and we choose C ≫ ω_I(t),ω_S(t),ω_IS(t),ω_r. In the interaction frame of the irradiation along x axis, with the strength C, the chemical shifts of the spins are averaged out. The rf-field Hamiltonian of the spin system transforms to

$$H_I^{rf}(t)=-\underset{\stackrel{.}{\varphi }(t)}{\underbrace{\frac{C\pi }{2n}\left(\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right)}}(F_{z}cos(Ct)+F_{y}sin(Ct)),$$ (5)

which accumulates an integral only when delta functions peak at $Ct=\frac{\pi }{2}$ and $Ct=\frac{3\pi }{2}$. The net rf rotation generated by $H_I^{rf}(t)$ is therefore $-\frac{\pi }{n}{F}_y$, which corresponds to a phase advance of $\frac{\pi }{2n}$ at $Ct=\frac{\pi }{2}$ and phase decrement of $\frac{\pi }{2n}$ at $Ct=\frac{3\pi }{2}$ (a negative phase decrement is multiplied by $sin(\frac{3\pi }{2})$, which makes it a positive accumulation). See Fig. 1. A net rotation of $\frac{\pi }{n}$ in time ${\tau }_c=\frac{{\tau }_r}{n}$ corresponds to a net effective field of $\frac{{\omega }_r}{2}$. We have an effective field along −y direction. The modulation frame of ϕ(t) returns to origin (see Figs. 2 and 3).

Fig. 2.

Top panel in the figure shows the build up of transfer of magnetization in a basic ¹³C–¹³C correlation experiment on a 400 mHz (proton frequency) static field, using the TPR pulse unit with 18° as phase increment and no compensating phase decrement. Simulation uses internuclear distance of 1.52 Å and powder avaraging. a, b, c corresponds to inhomogeneity value ε of 0, .01 and .05 respectively. Middle Panel in the figure shows the build up basic ¹³C–¹³C correlation experiment using the TPR pulse unit with 18° as phase increment combined with an 18° decrement unit. a, b, c, d corresponds to inhomogeneity value ε of 0, .01, .05, and .1 respectively. In practice, the rf-inhomogeneity results in a weighted sum of these different ε. Offset of spin pair is assumed to be on resonance. Bottom panel in the figure shows the build up of basic ¹³C–¹³C correlation experiment using the TPR pulse unit with 18° as phase increment combined with an 18° decrement unit, with different chemical shift of the target spin. a, b, c, d, corresponds to chemical shift of 0, 5, 10 and 15 kHz on the target spin. In the bottom panel, we include a CSA value of 8.9 and 13.9 ppm for the two spins respectively with anisotropy parameter .13 and .98, representing the C_α–C_β spin pair in alanine respectively.

Fig. 3.

Top panel in the figure shows the build up basic ¹³C–¹⁵N correlation experiment on a 400 mHz (proton frequency) static field, using the TPR pulse unit with 27° and 9° as phase increment and no compensating phase decrement on ¹³C and ¹⁵N respectively. a, b, c corresponds to inhomogeneity value ε of 0, .01 and .05 on Carbon channel. Middle Panel in the figure shows the build up using the TPR pulse unit with compensating phase decrement unit. Bottom panel in the figure shows the build up of basic ¹³C–¹⁵N correlation experiment using the TPR pulse unit, with different chemical shift of the ¹³C spin. a, b, c, d, corresponds to chemical shift of 0, 5, 7.5 and 10 kHz of the ¹³C spin.

The above may be implemented with a three pulse sequence (for n = 5)

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{18°}{\left(\frac{\pi }{2}\right)}_{0°}.$$

where $\frac{\pi }{2}$ is the flip angle which takes $\frac{{\tau }_r}{4n}$ units of time with pulse amplitude of C = nω_r, so that in a rotor period τ_r, we have n of these units.

When pulse amplitude has inhomogeneity, then, in the modulation frame, of the phase ϕ(t), the rf-field Hamiltonian takes the form

$$H^{rf}(t)=C(1+\epsilon )F_x-\stackrel{.}{\varphi }{F}_z,$$ (6)

where C is in the angular frequency units and we choose C ≫ ω_I(t),ω_S(t),ω_IS(t),ω_r. In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to

$$H_{I_a}^{rf}(t)=\epsilon C{F}_x-\underset{\stackrel{.}{\varphi }(t)}{\underbrace{\frac{C\pi }{2n}\left(\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right)}}(F_{z}cos(Ct)+F_{y}sin(Ct)),$$ (7)

which has an additional factor of εCF_x, which accumulates to first order an evolution $\epsilon 2\pi F_x-\frac{\pi }{n}{F}_y$, which for an inhomogeneity of ε = .05 and n = 5, corresponds to an evolution $\frac{\pi }{10}{F}_x-\frac{\pi }{5}{F}_y$ which limits the transfer efficiency. The inhomogeneity factor can be cancelled by following this pulse with a pulse of amplitude −C, where the phase is first decremented followed by increment, to accumulate same rf-Hamiltonian in the interaction frame of the x phase pulse.

$$\stackrel{.}{\varphi }(t)=\frac{C\pi }{2n}\left(-\delta \left(Ct-\frac{\pi }{2}\right)+\delta \left(Ct-\frac{3\pi }{2}\right)\right)$$ (8)

In the interaction frame of the irradiation along x axis, with the strength C, the rf-field Hamiltonian of the spin system transforms to

$$H_{I_b}^{rf}(t)=-\epsilon C{F}_x+\underset{\stackrel{.}{\varphi }(t)}{\underbrace{\frac{C\pi }{2n}\left(+\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right)}}(F_{z}cos(Ct)-F_{y}sin(Ct)),$$ (9)

Which accumulates an integral only when delta functions peak at $Ct=\frac{\pi }{2}$ and $Ct=\frac{3\pi }{2}$. The net rf rotation is therefore $-\frac{\pi }{n}{F}_y$ and −CF_x cancel to first order.

The above may be implemented with a three pulse sequence (for n = 5)

$${\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{180°-18°}{\left(\frac{\pi }{2}\right)}_{180°}.$$

The pulse sequence takes the form

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{18°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{162°}{\left(\frac{\pi }{2}\right)}_{180°}.$$

We have calculated ${\int }_0^{{\tau }_c}{H}_{I_a}^{rf}(t)dt$ and ${\int }_0^{{\tau }_c}{H}_{I_b}^{rf}(t)dt$, which are first order contributions to generated rotation by time varying Hamiltonian $H_{I_a}^{rf}(t)$ and $H_{I_b}^{rf}(t)$. We can calculate the second order terms to write the total evolution as

$$exp\left({\int }_0^{{\tau }_c}{H}_{I_a}^{rf}(t)dt+\frac{1}{2}{\int }_0^{{\tau }_c}[H_{I_a}^{rf}(t),{\int }_0^t{H}_{I_a}^{rf}(\tau )d\tau ]dt\right).$$

Let u(t) be a unit step function that is zero for t ≤ 0 and 1 for t > 0.

$${\int }_0^t{H}_{I_a}^{rf}(\tau )d\tau =\epsilon C{F}_{x}t-\frac{\pi }{2n}(u(t-\pi /2C)+u(t-3\pi /2C))F_y$$ (10)

$$\begin{array}{l}\frac{1}{2}{\int }_0^{{\tau }_c}[H_{I_a}^{rf}(t),{\int }_0^t{H}_{I_a}^{rf}(\tau )d\tau ]=+i{F}_z\left(\frac{\pi \epsilon C{\tau }_c}{2n}-\frac{\pi \epsilon }{2n}\frac{\pi }{2}-\frac{\pi \epsilon }{2n}\frac{3\pi }{2}\right)\\ =0\end{array}$$ (11)

$${\int }_0^t{H}_{I_b}^{rf}(\tau )d\tau =-\epsilon C{F}_{x}t-\frac{\pi }{2n}(u(t-\pi /2C)+u(t-3\pi /2C))F_y$$ (12)

$$\begin{array}{l}{\int }_0^{{\tau }_c}[H_{I_b}^{rf}(t),{\int }_0^t{H}_{I_b}^{rf}(\tau )d\tau ]=+-i{F}_z\left(\frac{\pi \epsilon C{\tau }_c}{2n}-\frac{\pi \epsilon }{2n}\frac{\pi }{2}-\frac{\pi \epsilon }{2n}\frac{3\pi }{2}\right)\\ =0\end{array}$$ (13)

Therefore we prepare effective Hamiltonians $\epsilon C{F}_x-\frac{\pi }{n{\tau }_c}{F}_y=\epsilon C{F}_x-\frac{\pi }{{\tau }_r}{F}_y=\epsilon C{F}_x-\frac{{\omega }_r}{2}{F}_y$, followed by the Hamiltonian $-\epsilon C{F}_x-\frac{{\omega }_r}{2}{F}_y$. Toggling between them eliminates εCF_x. The second order Hamiltonian is of order $\epsilon \frac{\pi }{2}{\omega }_r{F}_z$. We have a recoupling field along $-\frac{{\omega }_r}{2}{F}_y+\epsilon \frac{\pi }{2}{\omega }_r{F}_z$.

To evaluate the effect of recoupling field on the coupling Hamiltonian, we consider the pulse sequence

$${\left[{\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{18°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{162°}{\left(\frac{\pi }{2}\right)}_{180°}\right]}_n.$$

The rf-field alternates between C(cos ϕ(τ)F_x + sin ϕ(τ)F_y) and −C(cos−ϕ(τ)F_x + sin−ϕ(τ)F_y) every τ_c units of time, where ϕ is as in Fig. 1 and for t ∈ [kτ_c, (k + 1)τ_c ], τ = t − kτ_c, τ ∈ [0, τ_c]. In the modulation frame of the rf-phase which alternates between ϕ(τ) and −ϕ(τ), rf-Hamiltonian alternates between CF_x − ϕ̇ (τ)F_z and −CF_x + ϕ̇ (τ)F_z, every τ_c units of time where ϕ̇ (τ) is as in Fig. 1. By transforming into the interaction frame of irradiation along x axis which alternates between CF_x and −CF_x, we prepare $H_{I_a}^{rf}(\tau )$ and $H_{I_b}^{rf}(\tau )$ every τ_c units of time. The coupling Hamiltonian alternates between

$$H_{I+}^{DD}(t)=\frac{3}{2}{\omega }_{IS}(t)(I_z{S}_z+I_y{S}_y)+\frac{3}{2}{\omega }_{IS}(t)((I_z{S}_z-I_y{S}_y)cos(2C\tau )+(I_z{S}_y+I_y{S}_z)sin(2C\tau )),$$ (14)

$$H_{I-}^{DD}(t)=\frac{3}{2}{\omega }_{IS}(t)(I_z{S}_z+I_y{S}_y)+\frac{3}{2}{\omega }_{IS}(t)((I_z{S}_z-I_y{S}_y)\times cos(2C(-\tau ))+(I_z{S}_y+I_y{S}_z)sin(2C(-\tau ))),$$ (15)

Now, transforming the coupling Hamiltonian in Eqs. (14) and (15), into the interaction frame of the rf-field Hamiltonian $H_I^{rf}$, we act with an effective field $-\frac{{\omega }_r}{2}{F}_y$, and only retain terms that give static contribution to the effective Hamiltonian, i.e., terms oscillating with frequency C are dropped and the residual Hamiltonian takes the form (neglecting the terms oscillating at frequency 2ω_r as they are not recoupled)

$$H_{II}(t)={\kappa }_h\{cos({\omega }_{r}t)cos({\omega }_{r}t+\gamma )(I_n{S}_n-I_x{S}_x)+sin({\omega }_{r}t)\times cos({\omega }_{r}t+\gamma )(I_n{S}_x+I_x{S}_n)\},$$ (16)

which averages to

$${\overline{H}}_{II}=\frac{{\kappa }_h}{2}\{cos(\gamma )(I_n{S}_n-I_x{S}_x)+sin(\gamma )(I_n{S}_x+I_x{S}_n)\},$$ (17)

where ${\kappa }_h=\frac{3}{4\sqrt{2}}{b}_{IS}sin(2\beta )$ sin(2β), where b_IS is the dipole coupling constant and n̂ is the effective direction

$$\widehat{n}=cos\theta \widehat{z}+sin\theta \widehat{y},$$

where θ is of order πε and m̂ = cos θ ŷ − sin θẑ, and m̂ × n̂ = x̂. For small ε, n̂ = ẑ.

The rf-interaction frame, $H_I^{rf}$ prepares an effective field $-\frac{{\omega }_r}{2}{F}_y$, every τ_c units of time. Transformation of the coupling Hamiltonian can be evaluated every τ_c units of time [14], by realizing that ${\int }_0^{T}U(t)JU(t)dt={\sum }_k{\int }_{k-1{\tau }_c}^{k{\tau }_c}{U}_k(t)U((k-1){\tau }_c)JU{((k-1){\tau }_c)}^{\prime }{U}_k{(t)}^{\prime }dt$, where U((k − 1)τ_c) is the interaction frame propagator at time (k − 1)τ_c and is same as evolution under effective field $-\frac{{\omega }_r}{2}{F}_y$ at this time, and U_k(t) evolves in time [(k − 1)τ_c, kτ_c] with $U_k(t)=I+o(\frac{{\omega }_r}{2}{\tau }_c)$. For ω_r ≪ C, we can neglect the second factor and assume we evolve the coupling Hamiltonian under an effective field, $-\frac{{\omega }_r}{2}{F}_y$.

This pulse sequence is broadband as a large value of C averages out the chemical shift [16]. The high power pulse sequence is made robust to rf-inhomogeneity by combining phase increment and decrement units. The resulting pulse sequence is called TPR, Three Pulse Recoupling, as in Fig. 1.

In Fig. 1, we chose C = 50 KHz and ω_r = 10 kHz which gives n = 5. We can use TPR with larger spinning frequencies. C = 100 kHz and ω_r = 20 kHz which gives n = 5. Similarly C = 120 kHz and ω_r = 40 kHz which gives n = 3. In the latter two cases, large power on the carbon can help in decoupling proton-carbon interactions so a seprate decoupling field on proton may not be needed.

We can evaluate the effect of TPR pulse sequence on the chemical shift anisotropy in the interaction frame of irradiation along x axis which alternates between CF_x and −CF_x. The chemical shift anisotropy of spin I can be expressed as $({\omega }_I^{\pm 1}exp(\pm i{\omega }_r)+{\omega }_I^{\pm 2}exp(\pm i2{\omega }_r))I_z$, we can decompose the operator I_z = I⁺ + I⁻, where I^± = I_z ± iI_y, where exp(−iCI_xt)I^± exp(+iCI_xt) = exp(±iCt)(I^±). Using this, For n = 5, the CSA averages every two rotor periods, where TPR toggles between ±C every τ_c. Proceeding to interaction frame, $H_I^{rf}$, by acting with an effective field $-\frac{{\omega }_r}{2}{F}_y$, we find that for n = 6, the CSA averages to zero in first order.

The effective evolution resulting from the pulse sequence,

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{\alpha }{\left(\frac{\pi }{2}\right)}_{0°},$$

can be directly evaluated using pauli matrices σ_α.

$$\begin{array}{l}cos\frac{\pi }{4}I-i2sin\frac{\pi }{4}{\sigma }_x(-i2(cos\alpha {\sigma }_x+sin\alpha {\sigma }_y))cos\frac{\pi }{4}I-i2sin\frac{\pi }{4}{\sigma }_x=-(cos\alpha I+i2sin\alpha {\sigma }_y)\\ {\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{180-\alpha }{\left(\frac{\pi }{2}\right)}_{180°},\end{array}$$

is evaluated as

$$cos\frac{\pi }{4}I+i2sin\frac{\pi }{4}{\sigma }_x(i2(cos\alpha {\sigma }_x-sin\alpha {\sigma }_y))cos\frac{\pi }{4}I+i2sin\frac{\pi }{4}{\sigma }_x=-(cos\alpha I+i2sin\alpha {\sigma }_y)$$

The resulting propagator on two spins is exp(i2αF_y), an effective rotation on two spins along −y direction, with 2α flip angle, where α is radian equivalent of degrees as understood. For the presented sequence in above section α = 18°, i.e. $\frac{\pi }{2n}$, for n = 5, as in Fig. 2.

Let, $exp(-i{\oint }_0^{t}H(\tau )d\tau )$ denote Unitary propogator generated by time varying Hamiltonian H. We can evaluate the evolution

$$V(t)=exp\left(-i\underset{0}{\overset{t}{\oint }}{H}_{I_a}^{rf}d\tau \right)=exp({\mathit{iCF}}_{x}t)exp(i\varphi (t)F_z)U(t).$$

where U(t) is evolution of the pulse sequence ${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{\alpha }{\left(\frac{\pi }{2}\right)}_{0°}$. For $0\le t\le \frac{{\tau }_c}{4}$, we have, ϕ(t) = 0 and U(t) = exp(−iCF_xt), resulting in V(t) = I. For $\frac{{\tau }_c}{4}<t\le \frac{3{\tau }_c}{4}$, we have ϕ(t) = α, $\theta =Ct-\frac{C{\tau }_c}{4}$,

$$V(t)=exp\left(i{F}_x\left(\frac{\pi }{2}+\theta \right)\right)exp(i\alpha F_z)exp(-i(F_{x}cos\alpha +F_{y}sin\alpha )\theta )exp\left(-i{F}_x\frac{\pi }{2}\right)$$ (18)

$$=exp\left(i{F}_x\frac{\pi }{2}\right)exp(i\alpha F_z)exp\left(-i{F}_x\frac{\pi }{2}\right)$$ (19)

$$=exp\left(i{F}_x\frac{\pi }{2}\right)exp(i\alpha F_z)exp\left(-i{F}_x\frac{\pi }{2}\right)=exp(i\alpha F_y)$$ (20)

For $\frac{3{\tau }_c}{4}<t\le {\tau }_c$, we have ϕ(t) = 0,

$$\begin{array}{l}V(t)=exp\left(i{F}_x\left(\frac{3\pi }{2}\right)\right)exp(-i(F_{x}cos\alpha +F_{y}sin\alpha )\pi )exp\left(-i{F}_x\frac{\pi }{2}\right)\\ =exp(i2\alpha F_y)\end{array}$$ (21)

We can calculate the same for the phase decrement sequence

$$W(t)=exp\left(-i\underset{0}{\overset{t}{\oint }}{H}_{I_a}^{rf}dt\right)=exp(-{\mathit{iCF}}_{x}t)exp(i\varphi (t)F_z)U(t).$$

where U(t) is evolution of the pulse sequence ${\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{180-\alpha }{\left(\frac{\pi }{2}\right)}_{180°}$. For $0\le t\le \frac{{\tau }_c}{4}$, we have, ϕ(t) = 0 and U(t) = exp(iCF_xt), resulting in V(t) = I. For $\frac{{\tau }_c}{4}<t\le \frac{3{\tau }_c}{4}$, we have ϕ(t) = −α, $\theta =t-\frac{{\tau }_c}{4}$,

$$W(t)=exp\left(-i{F}_x\left(\frac{\pi }{2}+\theta \right)\right)exp(-i\alpha F_z)exp(i(F_{x}cos\alpha -F_{y}sin\alpha )\theta )exp\left(i{F}_x\frac{\pi }{2}\right)$$ (22)

$$=exp\left(-i{F}_x\frac{\pi }{2}\right)exp(-i\alpha F_z)exp\left(i{F}_x\frac{\pi }{2}\right)=exp(i\alpha F_y)$$ (23)

For $\frac{3{\tau }_c}{4}<t\le {\tau }_c$, we have ϕ(t) = 0,

$$\begin{array}{l}W(t)=exp\left(-i{F}_x\left(\frac{3\pi }{2}\right)\right)exp(i(F_{x}cos\alpha -F_{y}sin\alpha )\theta )exp\left(i{F}_x\frac{\pi }{2}\right)\\ =exp(i2\alpha F_y)\end{array}$$ (24)

3. Phase matching and heteronuclear recoupling

Consider two coupled heteronuclear spins I and S under magic angle spinning condition [22]. The spins are irradiated with rf fields at their Larmor frequencies along say the x direction. In a double-rotating Zeeman frame, rotating with both the spins at their Larmor frequency, the Hamiltonian of the system takes the form

$$H(t)={\omega }_I(t)I_z+{\omega }_S(t)S_z+{\omega }_{IS}(t)2I_z{S}_z+H^{rf}(t),$$ (25)

where ω_I(t),ω_S(t), and ω_IS(t) represent time-varying chemical shifts for the two spins I and S and the coupling between them, respectively. These interactions can be expressed as a Fourier series ${\omega }_{\lambda }(t)={\sum }_{m=-2}^2{\omega }_{\lambda }^{m}exp(im{\omega }_{r}t)$ exp(imω_rt), where ω_r is the spinning frequency (in angular units), while the coefficients ω_λ, (λ = I, S) reflect the dependence on the physical parameters like the isotropic chemical shift, anisotropic chemical shift, the dipole–dipole coupling constant and through this the internuclear distance [23,24].

Consider the rf irradiation on heteronuclear spin pair, where amplitude on spin I and S is chosen as $A_{\lambda }(t)=\frac{C}{2\pi }(\lambda =I,S)$ such that C = nω_r, and phase modulation, given by

$${\stackrel{.}{\varphi }}_I(t)=\frac{3C\pi }{4n}\left(\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right)$$ (26)

$${\stackrel{.}{\varphi }}_S(t)=\frac{C\pi }{4n}\left(\delta \left(Ct-\frac{\pi }{2}\right)-\delta \left(Ct-\frac{3\pi }{2}\right)\right)$$ (27)

where phase jump is three halves and half of the modulation considered in previous section. As a result, the effective fields prepared on two channels will be $\frac{3{\omega }_r}{4}$ and $\frac{{\omega }_r}{4}$, with average $\frac{{\omega }_r}{2}$, which will recouple a double quantum Hamiltonian. In implementation, this would correspond to a phase increment which is three halves and half of the phase increment in previous section, i.e., 27° and 9°. To mitigate the effect of inhomogeneity of rf-field, the basic pulse unit is accompanied with a phase decrement unit. In nutshell the pulse sequence on channels I and S (say C¹³ and N¹⁵) is following respectively.

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{27°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{153°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (28)

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{9°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{171°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (29)

In the frame of the rf-field along x axis, that toggles between ±C(I_x + S_x), the coupling Hamiltonian is averaged to

$$H_{I\pm }^{DD}(t)={\omega }_{IS}(t)(I_z{S}_z+I_y{S}_y)+{\omega }_{IS}(t)((I_z{S}_z-I_y{S}_y)cos(2C\tau )\pm (I_z{S}_y+I_y{S}_z)sin(2C\tau )),$$ (30)

With effective rf-field $-\frac{3{\omega }_r}{4}{I}_y-\frac{{\omega }_r}{4}{S}_y$ written as $-{\omega }_r\frac{(I_y+S_y)}{2}-\frac{{\omega }_r}{2}\frac{(I_y-S_y)}{2}$, the coupling Hamiltonian is further averaged to

$${\overline{H}}_{II}={\kappa }_d\underset{}{\underbrace{\{(I_z{S}_z-I_x{S}_x)cos(\gamma )+(I_x{S}_z+I_z{S}_x)sin(\gamma )\}}}.$$ (31)

We call the above experiments heteronuclear TPR.

4. Gamma-preparation

We now show how TPR recoupling pulse sequences can be used in design of two-dimensional solid state NMR experiments that use powdered dephased antiphase coherence (γ preparation) to encode chemical shifts in the indirect dimension. Both components of this chemical shift encoded gamma-prepared states can be refocused into inphase coherence by a recoupling element. This helps to achieve sensitivity enhancement in 2D NMR experiments by quadrature detection.

Notation

Let S_α(β, γ) denote the rotation of operator S_α around the axis β by angle γ, where α, β ∈ {x, y, z}, i.e., S_α(β, γ) = exp(−iγS_β)S_α exp(iγS_β). For example S_x(z, γ) = S_x cos γ+ S_y sin γ. It is straightforward to verify relations of the following kind

$$[-i{S}_x(\beta ,\gamma ),-i{S}_y(\beta ,\gamma )]=-i{S}_z(\beta ,\gamma ).$$

It also follows from definition that for any unitary transformation U,

$$U{S}_{\alpha }(\beta ,\gamma )U^{\prime }=S_{U\alpha U^{\prime }}(U\beta U^{\prime },\gamma ),$$

where S_UαU′ (UβU′, γ), denotes rotation of operator US_αU′ around US_βU′.

Let the unit vectors l,m, n constitute a right handed coordinate system (l × m = n). Then

$${\mathcal{H}}_{ZQ}^l(\gamma )=I_m{S}_m(l,\gamma )+I_n{S}_n(l,\gamma )=I_m(l,-\gamma )S_m+I_n(l,-\gamma )S_n.$$ (32)

and

$${\mathcal{H}}_{DQ}^l(\gamma )=I_m{S}_m(l,\gamma )-I_n{S}_n(l,\gamma )=I_m(l,\gamma )S_m-I_n(l,\gamma )S_n.$$ (33)

are used to denote the zero quantum and multiple (double) quantum operators.

Consider the recoupling effective field $-\frac{{\omega }_r}{2}{F}_y$ as in Section 2, obtained using TPR pulse sequence. This prepares a recoupling Hamiltonian,

$${\mathcal{H}}_{DQ}^y(\gamma )=I_z{S}_z(y,\gamma )-I_x{S}_x(y,\gamma ).$$ (34)

and with bracketing $\frac{\pi }{2}$ pulses along −x and x, we prepare an Hamiltonian (z̄ = −z),

$${\mathcal{H}}_{DQ}^{\overline{z}}(\gamma )=I_y{S}_y(\overline{z},\gamma )-I_x{S}_x(\overline{z},\gamma ).$$ (35)

Now, consider the following experiment,

$$I_x\underset{I}{\overset{{({\mathcal{H}}_{DQ}^{\overline{z}}(\gamma ))}_{\pi }}{\to }}-2I_z{S}_y(\overline{z},\gamma )\underset{II}{\overset{{\omega }_s{S}_z}{\to }}-2I_z{S}_y(\overline{z},\gamma -{\omega }_s{t}_1)\underset{\mathit{III}}{\overset{{(H_{DQ}^{\overline{z}}({\gamma }^{\prime }))}_{\pi }}{\to }}-I_x(z,\gamma -{\gamma }^{\prime }-{\omega }_s{t}_1)$$ (36)

If the preparation Hamiltonian ${\mathcal{H}}_{DQ}^{\overline{z}}(\gamma )$ is applied for an integral number of rotor periods, then

$${\gamma }^{\prime }=\gamma +{\omega }_r{t}_1.$$

The final state following the refocusing is

$$-I_x(z,-({\omega }_r+{\omega }_s)t_1)=-(I_{x}cos(-({\omega }_r+{\omega }_s))t_1+I_{y}sin(-({\omega }_r+{\omega }_s))t_1.$$

Remark 1

The magnetization in t₁ evolution precesses with chemical shift, −(ω_r + ω_s), where ω_s is measured with respect to carrier frequency. The Gamma-Preparation peaks in the 2D spectra are reflected around the carrier frequency and further downshifted by rotor frequency. The spectra is obtained as described in the following and shown in Fig. 4B for the ¹³C_α–¹³C_β correlation experiment.

Fig. 4.

(A) shows the pulse sequence for a ¹³C_α–¹³C_β correlation experiment with TPR as the recoupling element. (B) Shows the pulse sequence for a ¹³C_α–¹³C_β correlation experiment using gamma preparation, with TPR as the recoupling element. (C) Shows the pulse sequence for a ¹³C_α–¹⁵N correlation experiment using gamma preparation, with TPR as the recoupling element. In (C) the last $\frac{\pi }{2}$ pulse on the carbon is toggled from −x to x during acquisition of second scan which is added and subtracted with first scan as described in (B).

Fig. 4B shows how two Gamma-Preparation experiments are processed. The Gamma-Preparation magnetization after the t₁ evolution is given by −I_x cos(−(ω_r + ω_s)t₁) − I_y sin(−(ω_r + ω_s)t₁). Following this with a π pulse in a second experiment gives −I_x cos(−(ω_r + ω_s)t₁) + I_y sin(−(ω_r + ω_s)t₁). In Fig. 4B, π pulse is implemented with two consecutive $\frac{\pi }{2}$ pulses whose phase cancels to give an echo signal, and adds to give an anti-echo signal. Adding the two results and subtracting them, gives magnetization −2I_x cos(−(ω_r + ω_s)t₁) and −2I_y sin(−(ω_r + ω_s)t₁), prior to the t₂ evolution. Following t₂ evolution, recorded signal has the form, s_A(t₁, t₂) = 2 cos(−(ω_S + ω_r)t₁) exp(−jω_It₂) and s_B(t₁, t₂) = 2 sin(−(ω_S + ω_r)t₁) exp(−jω_It₂), for the two experiments respectively. TPPI processing of both, gives two absorptive 2D-spectra. Fig. 7A shows one of these spectra. The two spectra can be phased and added to get sensitivity enhancement as shown in Fig. 7B and C.

Fig. 7.

Top panel shows the 2D gamma-preparation spectrum for the pulse sequence shown in figure for ¹³C_α–¹³C_β correlation as described in the text. The frequencies in the indirect dimension are reflected around the carrier and down-shifted by rotor frequency of 10 kHz (100 ppm at 400 kHz) as described in the text. The carrier is at around 40 ppm. Middle panel shows the comparison of peak intensity of a slice through a cross-peak of a straight ¹³C_α–¹³C_β, correlation obtained with TPR, with magnetization evolving on C_β during indirect evolution as in bottom panel of Fig. 5, vs a slice through a larger gamma-preparation cross peak, where magnetization begins from C_β and precesses on C_α during t₁. Bottom panel shows the comparison of peak intensity of a slice through a cross-peak of a straight ¹³C_α–¹³C_β, correlation obtained with TPR, as in bottom panel of Fig. 5, with magnetization evolving on C_α during indirect evolution, vs a slice through a larger gamma-preparation cross peak, where magnetization begins from C_α and precesses on C_β during t₁.

We can apply Gamma preparation to ¹³C(I)–¹⁵N(S) Heteronuclear correlation experiment, where we prepare antiphase coherence of the S spin N¹⁵ that evolves during t₁. The preparation can be done with an effective field −ω_rS_y, which excited both double and zero quantum coherence frames and creates an effective Hamiltonian 2I_zS_z(y, γ) which can be tilted to 2I_yS_y(z, −γ), with bracketing $\frac{\pi }{2}$ pulses. The effective field can be obtained with a TPR pulse sequence.

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{36°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{144°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (37)

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{0°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (38)

on the ¹⁵N and ¹³C channels respectively. We call this TPR1 in Fig. 4C. After t₁ evolution, the refocusing is done with the pulse sequence described in the main text which is

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{27°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{153°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (39)

$${\left(\frac{\pi }{2}\right)}_{0°}{(\pi )}_{9°}{\left(\frac{\pi }{2}\right)}_{0°}{\left(\frac{\pi }{2}\right)}_{180°}{(\pi )}_{171°}{\left(\frac{\pi }{2}\right)}_{180°}.$$ (40)

on ¹³C and ¹⁵N channels respectively. We call this TPR2 in Fig. 4C. Now, consider the following experiment, starting from magnetization on I spin ¹³C,

$$I_x\underset{I}{\overset{{(I_y{S}_y(\overline{z},\gamma ))}_{\pi }}{\to }}-2I_z{S}_y(\overline{z},\gamma )\underset{II}{\overset{{\omega }_s{S}_z}{\to }}-2I_z{S}_y(\overline{z},\gamma -{\omega }_s{t}_1)\underset{\mathit{III}}{\overset{{({\mathcal{H}}_{DQ}^{\overline{z}}({\gamma }^{\prime }))}_{\pi }}{\to }}-I_x(z,\gamma -{\gamma }^{\prime }-{\omega }_s{t}_1).$$ (41)

The processing steps are same as described above for homonuclear experiment. The pulse sequence is as shown in Fig. 4C.

5. Experimental results

All experiments [14] were performed on a 400 MHz spectrometer (¹H Larmor frequency of 400 MHz) equipped with a triple resonance 4 mm probe. Uniformly ¹³C, ¹⁵N-labeled sample of alanine and uniformly ¹³C, ¹⁵N-labeled sample of glycine was used in the full volume of standard 4 mm rotor at ambient temperature using $\frac{{\omega }_r}{2\pi }=10\text{kHz}$ sample spinning, for homonuclear and heteronuclear experiments respectively. The experiments used 3s recycling delay. All 2D experiments were processed using TPPI processing.

Top panel of Fig. 5A shows the build up curve for transfer of magnetization, starting from one of the carbons, for ¹³C_α–¹³C_β correlation experiment with TPR as the recoupling element. The experiment uses an initial ramped CP for ¹H to ¹³C cross polarization. The TPR recoupling block is designed for a nominal power of 50 kHz as described in the text. In experiment, this power is optimized to give maximum transfer efficiency. The experiment used CW decoupling on protons of $\frac{{\omega }_{rf}^H}{2\pi }=100\text{kHz}$ during the transfer. Bottom panel of Fig. 5A shows a ¹³C_α–¹³C_β 2D correlation spectrum obtained using the TPR as the recoupling element. The experiment used 1024 points in both direct and indirect dimension with dwell time of 25 μs.

Fig. 5.

Top panel shows the build up curve for transfer of magnetization, shown every 2 rotor periods, at 10 kHz spinning, starting from one of the carbons, for ¹³C_α–¹³C_β correlation experiment, in uniformly labelled sample of Alanine, with TPR as the recoupling element as described in the text. Bottom panel shows a ¹³C_α–¹³C_β 2D correlation spectrum obtained using the TPR as the recoupling element.

Top panel of Fig. 6 shows the build up of transfer of magnetization, shown every 8 rotor periods for the ¹³C_α → ¹⁵N experiment with TPR as the recoupling element. The experiment uses an initial ramped CP for ¹H to ¹³C cross polarization. The TPR recoupling block is designed for a nominal power of 50 kHz as described in the text. In experiment, this power is optimized to give maximum transfer efficiency. The experiment used CW decoupling on protons of $\frac{{\omega }_{rf}^H}{2\pi }=100\text{kHz}$ during the transfer. Middle panel of Fig. 6 shows the build up of transfer of magnetization, shown every 8 rotor periods for the ¹⁵N → ¹³C experiment with TPR as the recoupling element. The experiment uses an initial ramped CP for ¹H to ¹⁵N cross polarization. The TPR block used for this transfer is same as above. Bottom panel of Fig. 6 shows the 2D spectrum for ¹³C_α–¹⁵N experiment with TPR as the recoupling element. The magntization precesses on ¹⁵N during indirect evolution. The experiment used 1024 points in both direct and indirect dimension with dwell time of 25 μs.

Fig. 6.

Top panel shows the build up of transfer of magnetization, shown every 8 rotor periods, at 10 kHz spinning, for the ¹³C_α → ¹⁵N experiment, in uniformly labelled sample of glycine, with TPR as the recoupling element as described in text. Middle panel shows the build up of transfer of magnetization, shown every 8 rotor periods for the ¹⁵N→¹³C experiment with same TPR as the recoupling element. Bottom panel shows the 2D spectrum for ¹³C_α–¹⁵N experiment with TPR as the recoupling element. The magntization precesses on ¹⁵N during indirect evolution.

Top panel of Fig. 7 shows the 2D gamma-preparation spectrum for the pulse sequence shown in Fig. for ¹³C_α–¹³C_β correlation as described in the text. The frequencies in the indirect dimension are reflected around the carrier and rotor down-shifted as described in the text. The carrier is at around 40 ppm. The experiment uses an initial ramped CP for ¹H to ¹³C cross polarization, followed by gamma-preparation experiment with TPR as recoupling block, designed with a nominal power of 50 kHz as described in the text. The experiment used CW decoupling on protons of $\frac{{\omega }_{rf}^H}{2\pi }=100\text{kHz}$ during the transfer. Middle panel of Fig. 7 shows the comparison of peak intensity of a slice through a cross-peak of a straight ¹³C_α–¹³C_β, correlation obtained with TPR, with magnetization evolving on C_β during indirect evolution as in bottom panel of Fig. 5, vs a slice through a gamma-preparation cross peak, where magetization begins from C_β and precesses on C_α during t₁. The gamma-preparation peak was processed as shown in 4B. Bottom panel of Fig. 7 shows the comparison of peak intensity of a slice through a cross-peak of a straight ¹³C_α–¹³C_β, correlation obatained with TPR, as in bottom panel of Fig. 5, with magnetization evolving on C_α during indirect evolution, vs a slice through a gamma-preparation cross peak, where magetization begins from C_α and precesses on C_β during t₁. The gamma-preparation peak was processed as shown in Fig. 4B. In both cases, we see noticeable enhancement in sensitivity. Gamma-preparation experiment requires collecting two scans as shown in Fig. 4. For sensitivity comparison, we doubled the number of scans in a straight ¹³C_α–¹³C_β, correlation obtained with TPR, as in top panel of Fig. 5. The experiment used 1024 points in both direct and indirect dimension with dwell time of 25 μs.

Fig. 8 shows the gamma preparation peak for the 2D hetrocuclear experiment which prepares an antiphase coherence on nitrogen to encode its chemical shift with preparation and refocusing done with TPR pulse sequences with power of 50 kHz, with rotor frequency of 10 kHz and n = 5. The carrier on nitrogen is on-resonance. The pulse sequence is as shown in Fig. 4C with preparation and refocusing pulse sequences labelled as TPR 1 and TPR 2. The experiment used 1024 points in both direct and indirect dimension with dwell time of 25 μs. Choice of a different preparation pulse is made to reduce preparation time by half.

Fig. 8.

Figure shows the gamma preparation peak for the 2D hetrocuclear experiment which prepares an antiphase coherence on nitrogen to encode its chemical shift with preparation and refocussing done with TPR pulse sequence.

6. Conclusion

In this paper we introduced a class of recoupling pulse sequences, which rest on the principle of second oscillating field. A strong field is used to eliminate chemical shifts and make the sequence broadband. Furthermore this strong rf-field is used to demodulate a second oscillating field which performs recoupling. In our design, the second oscillating field comes about by principled phase changes which are described in the paper. The recoupling sequences presented in the paper for homonuclear and heteronuclear spin systems are broadband and robust to rf-inhomogeneity. Furthermore, they are easy to program and implement on the spectrometer. The sequence have been used as building block for Gamma-preparation experiments that perform simultaneous transfer of both the components of magnetization in a transfer experiment and hence enhance sensitivity by Quadrature detection.