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J Magn Reson. Author manuscript; available in PMC 2010 April 1.

Published in final edited form as:

Published online 2008 December 16. doi: 10.1016/j.jmr.2008.12.011

PMCID: PMC2715147

NIHMSID: NIHMS106561

The publisher's final edited version of this article is available at J Magn Reson

See other articles in PMC that cite the published article.

We describe some new developments in the methodology of making heteronuclear and homonuclear recoupling experiments in solid state NMR insensitive to rf-inhomogeneity by phase alternating the irradiation on the spin system every rotor period. By incorporating delays of half rotor periods in the pulse sequences, these phase alternating experiments can be made *γ* encoded. The proposed methodology is conceptually different from the standard methods of making recoupling experiments robust by the use of ramps and adiabatic pulses in the recoupling periods. We show how the concept of phase alternation can be incorporated in the design of homonuclear recoupling experiments that are both insensitive to chemical-shift dispersion and rf-inhomogeneity.

An important application of solid state nuclear magnetic resonance (NMR) spectroscopy is in structural analysis of “insoluble” protein structures such as membrane proteins, fibrils, and extracellular matrix proteins which are exceedingly difficult to analyze using conventional atomic-resolution structure determination methods, including liquid-state NMR and X-ray crystallography [1, 2, 3, 4, 5]. The goal of studying increasingly complex molecular systems is a strong motivation for the development of improved solid-state NMR methods. The paper describes a set of new principles for design of heteronuclear and homonuclear dipolar recoupling experiments that are both broadband and insensitive of rf-field inhomogeneities. The merits of the proposed techniques with respect to state of the art methods is discussed.

For solids, the internal Hamiltonian not only contains isotropic interactions, such as isotropic chemical shifts and scalar *J* couplings, but also anisotropic (i.e., orientation dependent) chemical shifts and dipole-dipole coupling interactions in the case of coupled spin-1/2 nuclei. This implies that each molecule/crystallite in a “powder” sample may exhibit different nuclear spin interactions leading to severe line broadening and thereby reduced spectral resolution and sensitivity. This problem may be alleviated using magic-angle spinning (MAS), which averages these interactions and hereby results in high-resolution conditions for solid samples. However, this also results in loss of useful parts of the anisotropic interactions like dipolar couplings, which carry information about distances between nuclei and can help in obtaining structural information. This has triggered the development of dipolar recoupling techniques [6, 11, 12, 13, 14, 15, 16], which selectively reintroduce these couplings to enable measurement of internuclear distances, torsion angles, and transfer of magnetization from spin to spin in the molecule. Such recoupling experiments are the building blocks of essentially all biological solid-state NMR experiments using “powder” samples. However, the recoupling experiments are sensitive to the amplitude of the radio-frequency fields and the orientation dependence of the dipolar coupling interaction.

The latter challenge motivates the present paper, where we present methods for making dipolar recoupling experiments insensitive to rf-inhomogeneity and Hartmann-Hahn mismatch by phase alternating the irradiation on the spins every rotor period. In [7], it was shown that this simple technique makes recoupling experiments less sensitive to the dispersion in the strength of the radio-frequency field. These experiments however do not achieve uniform transfer efficiency for all *γ*, where *γ* represents the rotation of the crystallite around the rotor axis. By introducing, half rotor period delays in the middle of the proposed phase-alternating recoupling blocks, the recoupling is made insensitive to the angle *γ*. One standard technique for making recoupling experiments robust to Hartmann Hahn mismatch is by using ramps or shaped adiabatic pulses on the rf-power during the recoupling period. The described methodology is conceptually very different. The main contribution of this paper is the development of broadband homonuclear recoupling experiments that are robust to rf-inhomogeneity by using this concept of phase alternation. The main ideas described here are further developments of the recent work [38] on broadband homonuclear recoupling. We show that by using phase alternation, we can make experiments in [38], less sensitive to rf-inhomogeneity.

Phase alternating pulse sequences have appeared in solid state NMR experiments in numerous context before [7, 8, 9, 10]. The main contribution of this paper is further development of this methodology as a means to design robust recoupling experiments in various contexts.

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

$$H(t)={\omega}_{I}(t){I}_{z}+{\omega}_{S}(t){S}_{z}+{\omega}_{IS}(t)2{I}_{z}{S}_{z}+{\omega}_{rf}^{I}(t){I}_{x}+{\omega}_{rf}^{S}(t){S}_{x},$$

(1)

where *ω _{I}* (

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

(2)

where *ω _{r}* is the spinning frequency (in angular units) and
${\tau}_{r}={\scriptstyle \frac{2\pi}{{\omega}_{r}}}$ is the rotor period. The coefficients

$$\begin{array}{l}\overline{H}=\frac{1}{4}\{({\omega}_{IS}^{-(p+q)}+{\omega}_{IS}^{(p+q)})(2{I}_{z}{S}_{z}-2{I}_{y}{S}_{y})\\ +({\omega}_{IS}^{-(p-q)}+{\omega}_{IS}^{(p-q)})(2{I}_{z}{S}_{z}+2{I}_{y}{S}_{y})\\ -i({\omega}_{IS}^{-(p+q)}-{\omega}_{IS}^{(p+q)})(2{I}_{z}{S}_{y}+2{I}_{y}{S}_{z})\\ i({\omega}_{IS}^{-(p-q)}-{\omega}_{IS}^{(p-q)})(2{I}_{z}{S}_{y}-2{I}_{y}{S}_{z})\}\end{array}$$

(3)

Choosing *p* − *q* = −1 and |*p* + *q*| > 2, we prepare the effective Hamiltonian,

$${H}_{\mathit{dcp}}(\gamma )=\kappa \{cos(\gamma )({I}_{z}{S}_{z}+{I}_{y}{S}_{y})-sin(\gamma )({I}_{z}{S}_{y}-{I}_{y}{S}_{z})\},$$

(4)

where *λ* as before is the Euler angle discriminating the crystallites by rotation around the rotor axis. This is the standard DCP experiment [6]. The scaling factor

$$\kappa =\frac{1}{2\sqrt{2}}{b}_{IS}sin(2\beta ),$$

(5)

depends on the dipole-dipole coupling constant *b _{IS}* and the angle

The effective Hamiltonian *H _{dcp}* mediates the coherence transfer

In the interaction frame of the rf-irradiation, with nominal rf-strength *pω _{r}* and

$$\stackrel{\sim}{H}(\gamma )={H}_{\mathit{dcp}}(\gamma )+{\epsilon}^{-}{\omega}_{r}\frac{{I}_{x}-{S}_{x}}{2}+{\epsilon}^{+}{\omega}_{r}\frac{{I}_{x}+{S}_{x}}{2},$$

(6)

where
${\epsilon}^{-}={\scriptstyle \frac{{\epsilon}_{I}-{\epsilon}_{S}}{2}}$ and
${\epsilon}^{+}={\scriptstyle \frac{{\epsilon}_{I}+{\epsilon}_{S}}{2}}$. Let operators *X*^{−}*, Y*^{−} and *Z*^{−} represent the zero quantum operators
${\scriptstyle \frac{{I}_{x}-{S}_{x}}{2}}$, *I _{z} S_{z}* +

The figure depicts the zero quantum frame, where the transfer *I*_{x} → *S*_{x} is simply viewed as the inversion of the initial operator *X*^{−} → − *X*^{−}. In the presence of rf-inhomogeneity, the inversion is incomplete, as the **...**

Fig. 2A shows the efficiency *η* of ^{15}N → ^{13}C, coherence transfer for the DCP experiment as a function of the mixing time *τ _{d}*. The build up curve is shown for different (increasing) values of

The above three panels show efficiency *η* of ^{15}N → ^{13}C, coherence transfer for the DCP, MOIST and PATCHED experiment as a function of the mixing time *τ*_{d}, with experimental conditions corresponding to MAS experiments with 10 kHz **...**

We now describe the concept of phase-alternating pulse sequences as introduced in [7] for compensating rf-inhomogeneity in the system. Consider the pulse sequence in Fig. 3A, where the phase on both *I* and *S* channel is changed between +*x* and −*x* every rotor period. In [7], this experiment is called MOIST. To analyze the effect of this pulse sequence, in Eq. 6, we can omit the operator *X*^{+}, as it commutes with all operators in the zero quantum frame. In the interaction frame of the rf-field, the effective Hamiltonian in the first rotor period is

In Fig. A is shown the basic idea of the MOIST experiment, where the phase of the rf-field is switched every *τ*_{r} units of time. In Fig. B is shown the PATCHED experiment, where introducing delays of
${\scriptstyle \frac{{\tau}_{r}}{2}}$, preceding and past every fourth **...**

$${H}_{1}={H}_{\mathit{dcp}}(\gamma )+{\epsilon}^{-}{\omega}_{r}{X}^{-},$$

(7)

and the effective Hamiltonian in the second rotor period is

$${H}_{2}={H}_{\mathit{dcp}}(-\gamma )-{\epsilon}^{-}{\omega}_{r}{X}^{-},$$

(8)

The effective Hamiltonian from the phase alternation is

$$\frac{{H}_{1}+{H}_{2}}{2}\sim \frac{\kappa}{2}cos(\gamma )({I}_{z}{S}_{z}+{I}_{y}{S}_{y}),$$

where we have used the small flip angle approximation (*κ τ _{r}* 1) and assumed that

$$exp(-i{H}_{1}{\tau}_{1})exp(-i{H}_{2}{\tau}_{r})\sim exp(-i({H}_{1}+{H}_{2}){\tau}_{r}).$$

This Hamiltonian is not *γ* encoded. The resulting efficiency of transfer is lower as compared to the *γ* encoded experiments. Fig. 2B shows the transfer efficiency as a function of the mixing time for the phase alternating pulse sequence for different degrees of rf-inhomogeneity. Comparison to Fig. 2A shows that this sequence is more robust to rf-inhomogeneity.

We now present a method to make this phase alternating pulse sequence *γ* encoded. We call this experiment PATCHED (**P**hase **A**lterna**t**ing experiment **C**ompensated by **H**alf rotor p**E**riod **D**elays). Fig. 3B shows the pulse sequence, consisting of a five rotor period building block, which is subsequently repeated many times, depending on the desired mixing period.

In the interaction frame of the nominal rf-irradiation, the first and third rotor period prepares the same effective Hamiltonian *H*_{1} as in Eq. (7). The second rotor period prepares the same effective Hamiltonian *H*_{2} as in Eq. (8). Following the third rotor period, a half rotor period delay moves the crystallites with angle *γ* to *γ* + *π*. In the final recoupling period, one prepares the effective Hamiltonian,

$${H}_{4}={H}_{\mathit{dcp}}(-(\gamma +\pi ))-{\epsilon}^{-}{\omega}_{r}{X}^{-}.$$

(9)

Again, using the small angle approximation, we sum up the four Hamiltonians to get the effective Hamiltonian , such that

$$\begin{array}{l}4\overline{H}=({H}_{\mathit{dcp}}(\gamma )+{\epsilon}^{-}{\omega}_{r}{X}^{-})+({H}_{\mathit{dcp}}(-\gamma )-{\epsilon}^{-}{\omega}_{r}{X}^{-})+\\ ({H}_{\mathit{dcp}}(\gamma )+{\epsilon}^{-}{\omega}_{r}{X}^{-})+({H}_{\mathit{dcp}}(-\gamma -\pi )+{\epsilon}^{-}{\omega}_{r}{X}^{-}).\end{array}$$

The resulting effective Hamiltonian to leading order is then

$$\overline{H}=\frac{2\kappa}{5}\{cos(\gamma )({I}_{z}{S}_{z}+{I}_{y}{S}_{y})-sin(\gamma )({I}_{z}{S}_{y}-{I}_{y}{S}_{z})\}.$$

This is a *γ* encoded Hamiltonian, simply scaled down by a factor of
${\scriptstyle \frac{2}{5}}$. To leading order, the effect of inhomogeneous rf-field gets canceled. Fig. 2B and Fig. 2C shows the transfer efficiency as a function of the mixing time for the presented phase-alternating and gamma-encoded phase alternating pulse sequence for different rf-inhomogeneity. The sequence achieves the same transfer efficiency as the *γ* encoded pulse sequences and are robust to rf-inhomogeneity.

Fig. 4 shows the experimental results comparing transfer efficiency as a function of rf-mismatch of the three experiments DCP, MOIST and PATCHED experiment.

The above figure shows the experimental spectra for DCP, Phase-alt (MOIST) and PATCHED experiments for ^{15}*N* →^{13}*C*_{α}, transfer for a powder ^{13}*C*_{α}, ^{15}*N* labeled glycine using MAS spinning at
${\scriptstyle \frac{{\omega}_{r}}{2\pi}}=8\phantom{\rule{0.16667em}{0ex}}\text{Khz}$ and CW decoupling **...**

The methods presented in the previous section, generalize in a straightforward way to homonuclear spins. To fix ideas, we begin by considering two homonuclear spins *I* and *S* under magic angle spinning condition. A classical technique for recoupling the spins is the HORROR experiment [13]. The spins are irradiated with an rf-field strength of
${\scriptstyle \frac{{\omega}_{r}}{2}}$, where *ω _{r}* is the rotor frequency. In a rotating frame, rotating with both the spins at their Larmor frequency, the Hamiltonian of the spin system takes the form

$$H(t)={\omega}_{I}(t){I}_{z}+{\omega}_{S}(t){S}_{z}+{\omega}_{IS}(t)(3{I}_{z}{S}_{z}-I\xb7S)+\frac{{\omega}_{r}}{2}{F}_{x},$$

(10)

where the operator *F _{x}* =

The term *I* · *S* in Eq. (10), commutes with the rf-Hamiltonian,
${\scriptstyle \frac{{\omega}_{r}}{2}}{F}_{x}$, as it averages under MAS. We will, therefore, drop this term in the subsequent treatment. In the interaction frame of the rf irradiation, the internal Hamiltonian from Eq.(1), takes the form

$${H}_{I}(t)=\frac{3}{2}{\omega}_{IS}(t){Z}^{-}+\frac{3}{2}{\omega}_{IS}(t)({Z}^{+}cos({\omega}_{r}t)+{Y}^{+}sin({\omega}_{r}t)),$$

(11)

where, *X*^{−}*, Y*^{−}*, Z*^{−} are zero quantum operators and *X*^{+}*, Y*^{+}*, Z*^{+} are multiple quantum operators as defined before. To begin, we assume that *ω _{I}* (

$${H}_{\mathit{hom}}(\gamma )={\kappa}_{h}[cos(\gamma )({I}_{z}{S}_{z}-{I}_{y}{S}_{y})-sin(\gamma )({I}_{z}{S}_{y}+{I}_{y}{S}_{z})],$$

(12)

where the scaling factor

$${\kappa}_{h}=\frac{3}{4\sqrt{2}}{b}_{IS}sin(2\beta )$$

(13)

depends on the dipole-dipole coupling constant *b _{IS}*. In the multiple quantum frame spanned by the operators

In practice, *ω _{I}* (

The inhomogeneity is modeled by a parameter *ε*, such that the rf- Hamiltonian of the system takes the form

$${H}^{rf}=\frac{{\omega}_{r}}{2}(1+\epsilon ){F}_{x}.$$

(14)

In the interaction frame of rf-irradiation along *x* axis, with strength
${\scriptstyle \frac{{\omega}_{r}}{2}}$, the Hamiltonian of the system in (10) averages to

$${H}_{I}={H}_{\mathit{hom}}(\gamma )+\epsilon \frac{{\omega}_{r}}{2}{F}_{x}.$$

(15)

The Hamiltonian gives a poor transfer of the the initial operator *X*^{+} → −*X*^{+}, (and therefore also for the transfer of state *I _{x}* →

In Fig. A is shown the phase alternating HORROR experiment, where the phase of the rf-field is switched every *τ*_{r} units of time. In Fig. B is shown a modified version of the sequence in Fig. A, where introducing delays of
${\scriptstyle \frac{{\tau}_{r}}{2}}$, preceding **...**

Analogous to experiments in Fig. 3A and Fig. 3C, are the homonuclear recoupling experiments shown in Fig. 5A and Fig. 5B. The experiment in Fig. 5B is phase alternating, *γ* encoded version of the experiment in 5A. The analysis of these phase alternating homonuclear experiments is very similar to their heteronuclear counterparts. For sake of completion, we present the analysis here.

In Fig. 5B, in the interaction frame of the x-phase rf-irradiation with strength ${\scriptstyle \frac{{\omega}_{r}}{2}}$, in the first and third rotor period, we prepare the effective Hamiltonian,

$${H}_{1}={H}_{3}={H}_{\mathit{hom}}(\gamma )+\frac{\epsilon {\omega}_{r}}{2}{F}_{x}.$$

(16)

In the second rotor period, an effective Hamiltonian

$${H}_{2}={H}_{\mathit{hom}}(-\gamma )-\frac{\epsilon {\omega}_{r}}{2}{F}_{x},$$

(17)

is prepared.

In the fourth recoupling period (following the half rotor period delay), the effective Hamiltonian,

$${H}_{4}={H}_{\mathit{hom}}(-\gamma -\pi )-\frac{\epsilon {\omega}_{r}}{2}{F}_{x},$$

(18)

is produced. The net Hamiltonian, to leading order, is then

$$\overline{H}=\frac{{H}_{1}+{H}_{2}+{H}_{3}+{H}_{4}}{4}=\frac{2\kappa}{5}\{cos(\gamma )({I}_{z}{S}_{z}-{I}_{y}{S}_{y})+sin(\gamma )({I}_{z}{S}_{y}+{I}_{y}{S}_{z})\}.$$

(19)

Observe, to leading order, the inhomogeneity in the field gets canceled by phase alternation. The recoupling Hamiltonian is scaled down by a factor of
${\scriptstyle \frac{2}{5}}$, compared to standard HORROR experiment. Fig. 6A, shows the transfer efficiency vs mixing time curves for the HORROR experiment shown in the Fig. 5A with different level of rf-inhomogeneity. Fig. 6B shows the transfer efficiency vs mixing time curves for the phase alternating HORROR experiment shown in Fig. 5B. Note, both experiments in Fig. 5A and Fig. 5B are narrowband and work in the regime *ω _{r}*

In practice, *ω _{I}* (

Consider the rf-irradiation on homonuclear spin pair whose amplitude is chosen as
$A(t)={\scriptstyle \frac{C}{2\pi}}$ and the offset.Δ*ω*(*t*) = *ω _{r}* sin(

$$\phi (t)=\frac{2{\omega}_{r}}{C}{sin}^{2}(Ct)=\frac{{\omega}_{r}}{C}(1-cos(Ct)).$$

so that the rf-Hamiltonian, takes the form

$${H}^{rf}(t)=C{F}_{x}-{\omega}_{r}sin(Ct){F}_{z},$$

(20)

where *C* is in angular frequency units and we choose *C ω _{I}* (

$${H}_{I}^{DD}(t)=\frac{3}{2}{\omega}_{IS}(t)({I}_{z}{S}_{z}+{I}_{y}{S}_{y})+\frac{3}{2}{\omega}_{IS}(t)(({I}_{z}{S}_{z}-{I}_{y}{S}_{y})cos\mathit{Ct}+({I}_{z}{S}_{y}+{I}_{y}{S}_{z})sin\mathit{Ct}),$$

(21)

and the rf-Hamiltonian of the spin system transforms to

$${H}_{I}^{rf}(t)=\frac{{\omega}_{r}}{2}(-cos2Ct\phantom{\rule{0.16667em}{0ex}}{F}_{z}+2{sin}^{2}Ct\phantom{\rule{0.16667em}{0ex}}{F}_{y}),$$

(22)

which averages over a period *τ _{c}* = 2&

$${H}_{I}^{rf}=\frac{{\omega}_{r}}{2}{F}_{y}.$$

(23)

Now, transforming into the interaction frame of the rf Hamiltonian
${H}_{I}^{rf}$, we only retain terms that give secular contribution to the effective Hamiltonian, i.e., terms oscillating with frequency *C* are dropped and the residual Hamiltonian takes the form

$${H}_{II}(t)={\kappa}_{h}\{cos({\omega}_{r}t)cos({\omega}_{r}t+\gamma )({I}_{z}{S}_{z}-{I}_{x}{S}_{x})+sin({\omega}_{r}t)cos({\omega}_{r}t+\gamma )({I}_{z}{S}_{x}+{I}_{x}{S}_{z})\},$$

(24)

which over a rotor period averages to

$${\overline{H}}_{II}=\frac{{\kappa}_{h}}{2}\{cos(\gamma )({I}_{z}{S}_{z}-{I}_{x}{S}_{x})-sin(\gamma )({I}_{z}{S}_{x}+{I}_{x}{S}_{z})\}.$$

(25)

where *κ _{h}* is as in Eq. (12). The Hamiltonian

This cosine modulated pulse sequence is broadband as a large value of *C* averages out the chemical shift [38]. The main drawback of the high power cosine modulated pulse sequence is its sensitivity to rf-inhomogeneity. To analyze the effect of inhomogeneity, let *ε* denote the inhomogeneity parameter. Then, rewriting *H _{rf}* in Eq. (20) gives

$${H}^{rf}(t)=C(1+\epsilon ){F}_{x}-{\omega}_{r}sin(Ct){F}_{z}.$$

(26)

In the interaction frame of irradiation along *x* axis with strength *C*, the rf-Hamiltonian averages to
$\epsilon C{F}_{x}+{\scriptstyle \frac{{\omega}_{r}}{2}}{F}_{y}$. Now, proceeding to the interaction frame of the rf Hamiltonian
${\scriptstyle \frac{{\omega}_{r}}{2}}{F}_{y}$, the total Hamiltonian of the spin system averages to

$${H}_{\mathit{eff}}={\overline{H}}_{II}+\epsilon C{F}_{x},$$

(27)

where * _{II}* is as in the equation (25). In this interaction frame, the Hamiltonian

Here, we present a method to eliminate this effect of the rf-inhomogeneity by simply phase alternating every
${\tau}_{c}={\scriptstyle \frac{2\pi}{C}}$ units of time, by making (*t* + *τ _{c}* = π − (

$${H}_{1}^{rf}(t)=(1+\epsilon )C{F}_{x}-{\omega}_{r}sin(Ct){F}_{z};\phantom{\rule{0.16667em}{0ex}}\phantom{\rule{0.16667em}{0ex}}\phantom{\rule{0.16667em}{0ex}}{H}_{2}^{rf}(t)=-(1+\epsilon )C{F}_{x}+{\omega}_{r}sin(Ct){F}_{z}$$

(28)

The top panels in Fig. 7 show the phase modulation of the rf-field over a rotor period for the CMRR and PAMORE pulse sequences respectively.

The top two panels show the phase of the rf irradiation as a function of time (in the units of *τ*_{r}) for the CMRR (I) and PAMORE (II) pulse sequences when *C* = 8 *ω*_{r}. The bottom two panel show numerical simulations of the transfer efficiency **...**

To analyze the effect of the pulse sequence, whereby the rf Hamiltonian is switched between
${H}_{1}^{rf}(t)$ and
${H}_{2}^{rf}(t)$, let us first transform into the interaction frame of the Hamiltonian that switches between *CF _{x}* and −

Therefore, in the first *τ _{c}*, one obtains an effective rf-Hamiltonian,

$${H}_{1}^{rf}=\epsilon C{F}_{x}+\frac{{\omega}_{r}}{2}{F}_{y},$$

(29)

followed by the Hamiltonian

$${H}_{2}^{rf}=-\epsilon C{F}_{x}+\frac{{\omega}_{r}}{2}{F}_{y}.$$

(30)

The effective rf-Hamiltonian one prepares is

$${\overline{H}}^{rf}=\frac{{H}_{rf}^{1}+{H}_{rf}^{2}}{2}=\frac{{\omega}_{r}}{2}{F}_{y}.$$

(31)

If *C* is chosen large enough, the coupling Hamiltonian
${H}_{I}^{DD}(t)$, in the interaction frame takes the form,
${\scriptstyle \frac{3{\omega}_{IS}(t)}{2}}({I}_{z}{S}_{z}+{I}_{y}{S}_{y})$, where fast oscillating component at frequency *C* have been neglected, as they average out. Now, proceeding to the interaction frame of the rf Hamiltonian in Eq. (31), the total Hamiltonian of the spin system averages to * _{II}* as in the equation (25). To leading order, the effect of inhomogeneity is canceled out.

Fig. 7 shows simulations comparing the homonuclear recoupling transfer efficiency obtained using CMRR and PAMORE pulse sequence for three different values of *ε* (0,.02,.05).

The PAMORE pulse sequence can be further compensated against dispersion in the angle *β* by adiabatically changing the amplitude of the sinusoidally varying offset, i.e,

$$\phi (t)=m(t)\pi +{(-1)}^{m(t)}\frac{a(t)}{C}(1+cos\mathit{Ct}),$$

, where *m*(*t*) switches between 0 and 1 every *t _{c}* period. This modulation of the phase is depicted in Fig. 7II. This modulation then corresponds to switching between Hamiltonians

$${H}^{rf}(\pm )=\pm \{(1+\epsilon )C{F}_{x}-a(t)sin(Ct){F}_{z}\}.$$

(32)

where *a*(*t*) is a shaped pulse as described below.

We transform into the interaction frame of the Hamiltonian that switches between *CF _{x}* and −

$${\overline{\mathcal{H}}}^{rf}=\frac{a(t)}{2}{F}_{y}.$$

(33)

and the effective Hamiltonian is

$${\overline{\mathcal{H}}}^{\mathit{eff}}=\frac{3}{2}{\omega}_{IS}(t)({I}_{z}{S}_{z}+{I}_{y}{S}_{y})+\frac{a(t)}{2}(1+\epsilon ){F}_{y}.$$

(34)

Now, as *a*(*t*) is swept adiabatically (or ramped) from *ω _{r}* − Δ to

All experiments were performed on a 360 MHz spectrometer (^{1}H Larmor frequency of 360 MHz) equipped with a triple resonance 4 mm probe. The data from uniformly ^{13}C, ^{15}N -labeled samples of glycine (purchased from Cambridge Isotope Laboratories, Andover, MA) was obtained using the full volume of standard 4 mm rotor at ambient temperature using
${\scriptstyle \frac{{\omega}_{r}}{2\pi}}=8\phantom{\rule{0.16667em}{0ex}}\text{kHz}$ sample spinning. The experiments used 3s recycling delay, 8 scans, ^{1}H to ^{15}N cross-polarization with rf-field strength of 54 (^{1}H) and 58–68 (^{15}N ramped) kHz, respectively and duration 1.5 ms.

Fig. 4 shows the experimental ^{13}*C* spectra for DCP, Phase-alt (MOIST) and PATCHED experiments for the ^{15}*N* → ^{13}*C _{α}*, following the

As expected from theory and simulations, the MOIST experiment is much more robust to rf-inhomogeneity compared to the standard DCP experiment. However, it is not a *γ* encoded experiment and therefore has a lower transfer efficiency. The PATCHED experiment is both insensitive to rf-inhomogeneity and *γ* encoded as seen from Fig. 4.

Fig. 8, shows the experimental spectra for CMRR and PAMORE pulse sequences, for the ^{13}*C _{α}* →

The top panel shows the experimental spectra for CMRR and PAMORE pulse sequences, ^{13}*C*_{α} → ^{13b} *CO*, transfers for a powder ^{13}*C*, ^{15}*N* labeled glycine using MAS spinning at
${\scriptstyle \frac{{\omega}_{r}}{2\pi}}=8\phantom{\rule{0.16667em}{0ex}}\text{Khz}$ and CW decoupling on protons of
${\scriptstyle \frac{{\omega}_{}^{}}{}}$ **...**

It is evident from the simulation results in Fig. 7 and the experimental data in Fig. 8 that the PAMORE experiment is significantly less sensitive to rf-inhomogeneity. Here we have used modest rf-power of 48 kHz on the ^{13}*C* channel to make the Homonuclear recoupling broadband. This however requires separate Proton decoupling. By simply increasing the power on the carbon channel, we can eliminate the need for Proton decoupling as the same rf-field on carbon can help to decouple protons. This has been suggested as a merit of the CMRR like pulse sequences as it can significantly reduce sample heating [38]. However, the issue of sensitivity to rf-inhomogeneity becomes even more critical as we use more rf-power on the carbon as argued in Eq. (27). Therefore, the use of PAMORE pulse elements becomes even more critical at high ^{13}C powers.

In this article, we reported some new developments based on the concept of phase alternating pulse sequences for the design of heteronuclear and homonuclear recoupling experiments that are insensitive to rf-inhomogeneity. The principle was finally applied to the development of improved homonuclear recoupling experiments.

The techniques presented here can be incorporated in more elaborate experiment design that can also compensate for the dispersion in the angle *β* in Eq. (5) and Eq. (13). In our recent work [20], we showed that the problem of dipolar recoupling in the presence of anisotropies in *β* and the strength of rf-field to a problem of control of single spin in the presence of rf-inhomogeneity and Larmor dispersion respectively. Using this analogy, we demonstrated how ideas of composite pulse sequences [21, 22, 23] from liquid-state NMR can be incorporated in dipolar recoupling experiments to make them insensitive to angle *β* and rf-inhomogeneity. It is now possible to construct a family of dipolar recoupling experiments of increasing length and degree of compensation that ultimately achieve 100% transfer efficiency for all orientations of the dipolar coupling tensor. However, these composite pulse sequences tend to be long, especially in the presence of rf-inhomogeneity which translates into large Larmor dispersion in the single spin picture. Therefore, elaborate designs are needed to compensate for this rf-inhomogeneity. Based on the methods presented in this paper, significantly shorter composite re-coupling pulse sequences can be engineered based on phase alternating re-coupling blocks. This essentially reduces the problem to compensating only for dispersion in angle *β*, which takes significantly shorter composite dipolar recoupling experiments.

Furthermore, using gradient ascent algorithms as described in [20, 26], it is possible to engineer the shape of *a*(*t*) in Eq. (34), which optimizes the compensation against *β* and rf-inhomogeneity. We will address this problem in future work.

The research was supported in part by ONR 38A-1077404, AFOSR FA9550-05-1-0443 and NSF 0724057 and in part by the grants from the National Institutes of Biomedical Imaging and Bioengineering (EB003151 and EB002026).

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