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We report on an analysis of a well known three-pulse sequence for generating and detecting spin I=1 quadrupolar order when various pulse errors are taken into account. In the situation of a single quadrupolar frequency, such as the case found in a single crystal, we studied the potential leakage of single and/or double quantum coherence when a pulse flip error, finite pulse width effect, RF transient or a resonance offset is present. Our analysis demonstrates that the four-step phase cycling scheme studied is robust in suppressing unwanted double and single quantum coherence as well as Zeeman order that arise from the experimental artifacts, allowing for an unbiased measurement of the quadrupolar alignment relaxation time, T1Q. This work also reports on distortions in quadrupolar alignment echo spectra in the presence of experimental artifacts in the situation of a powdered sample, by simulation. Using our simulation tool, it is demonstrated that the spectral distortions associated with the pulse artifacts may be minimized, to some extent, by optimally choosing the time between the first two pulses. We highlight experimental results acquired on perdeuterated hexamethylbenzene and polyethelene that demonstrate the efficacy of the phase cycling scheme for suppressing unwanted quantum coherence when measuring T1Q. It is suggested that one employ two separate pulse sequences when measuring T1Q to properly analyze the short time behavior of quadrupolar alignment relaxation data.
Deuteron quadrupolar alignment echo spectroscopy is a powerful tool for probing molecular motions with relaxation times in the range of to T1, corresponding to a correlation time of 10−5s and greater [1, 2]. The method has been successfully applied to study structure and dynamics in a wide range of systems of interest in polymer and chemical physics, biophysics and materials science. Two of the earliest works in the field that implemented the approach involved a study of molecular dynamics in liquid crystals  and motional constraints in polyethylene [1, 2]. The approach has been used to study slow tetrahedral jumps in solid hexamethylenetetramine , the dynamics of gramicidin , to investigate the localized dynamics in DNA fragments , and more recently hydration dependent dynamics in RNA .
The pulse sequence for generating and detecting quadrupolar order consists of three pulses with shifted phases as shown in Fig. 1. Quadrupolar order is excited by the second pulse, evolves during the delay T and is subsequently converted into an observable signal by the last pulse. By varying T, the relaxation time of quadrupolar order, T1Q, can be measured. This pulse sequence intrinsically generates quadrupolar order and double quantum coherence at the same time even under ideal experimental conditions . The unwanted double quantum coherence can be readily suppressed from the detected signal by phase cycling .
One of the prerequisites for performing T1Q experiments on rigid solids is the use of high RF power to excite the nuclear spin ensemble bandwidth, which may exceed 100kHz. The use of high RF power inevitably introduces a variety of pulse artifacts, such as phase transients. Other unavoidable experimental errors include a flip angle error, finite pulse width effect and off-resonance artifact. It is well known from the coherence pathway formalism, that perfect spin rotations as well as the pulse and receiver phase are crucial for the selection of certain orders of quantum coherence . A. Jerschow has reported on a spherical tensor based formalism that quantifies the effects of imperfect rotations when selecting a quantum coherence of interest . In this work, we studied the effects of pulse errors on the alignment echo three-pulse sequence via the fictitious spin- operators developed by Vega and Pines . For the case that only one quadrupolar frequency, ωQ, is present in the spin ensemble (as is the case for a single crystal, for example), we investigated the potential leakage of multiple quantum coherence due to pulse errors. The results show that a simple four-step phase cycling scheme is robust for selecting quadrupolar order if the errors are small, though the signal intensity may be reduced. For the case of a powder distribution of ωQ, we simulated the quadrupolar alignment echo spectra and demonstrated that finite pulse width effects, phase transients and pulse flip errors all distort the powder line shape but that the distortions may be minimized to some degree by a judicious choice of spacing between the first two pulses. Lastly, we highlight experimental data on perdeuterated hexamethylbenzene and polyethelene that demonstrate the effectiveness of a simple four-step phase cycling scheme that was investigated. It is suggested that one perform two experiments with two different phase cycles to verify that the short time behavior of the quadrupolar alignment relaxation data does not include accidentally leaked double quantum coherence.
The fictitious spin- operators were introduced by Vega and Pines to conveniently describe the interaction of a spin I=1 nucleus with a Radio Frequency (RF) field . In this formalism, the basis consists of nine operators given by
From these definitions the various orders of multiple quantum coherence are as follows
In the following, we express the density operator by the above fictitious spin- operators and assume an ensemble of spin I=1 nuclei in a large, static magnetic field. In this situation the spin system is subject to the secular part of the first order quadrupolar interaction as well as a possible resonance offset. In the rotating frame the Hamiltonian is
where Δ is the resonance offset, e2qQ/h is the quadrupolar coupling constant, η is the asymmetry parameter and θ and are the usual Euler angles with respect to the azimuthal axis . We will use the notation ωQ,0 to distinguish with ωQ
which is half the observed splitting of a single crystal doublet when θ=0, or equal to the distance between the peaks in a Pake pattern when η = 0. Neglecting relaxation processes the time evolution of the density matrix is
where H is time independent. In what follows, the initial density matrix ρ(0) is taken as ρ(0)=Iz. The observable signal is determined by computing
where the operator for the receiver, R±y, is defined as R± ±Iy,1 iIx,1.
Before highlighting the effects of various errors on the alignment echo three-pulse sequence, we will show that the phase cycling scheme given in Table 1 (from reference ) eliminates double quantum coherence under ideal experimental conditions. In Table 1, ϕi and ϕr represent the phases of the ith pulse and receiver, respectively. In the discussion that follows, ρn(t) represents the density operator for the nth step of the phase cycle given in Table 1. For ideal experimental conditions we assume the experiment is performed on-resonance, thus Δ = 0 in Eq.(3). The propagator for an RF pulse with flip angle β and phase ϕ according to Eq.(7) is
where β = π/2 for the first pulse and β = π/4 for the second and last pulses. The factor of 2 in the spin rotation propagator comes from the definition of the fictitious spin- operators. For the computations that follow we assume for maximum conversion to quadrupolar order and will expand on this assumption in the proceeding section. By using Eq.(3), (6), (7) and (9) the various states of the spin system at t=τ + T− for each step of the four-step phase cycling are
In the above expression, Q = D = 1. The symbols Q and D have been inserted as coefficients to label the quadrupolar ordered term and double quantum coherence terms respectively. These labels show the presence and magnitude of different quantum coherence created after the second delay T, and will be used to track their evolution into detectable signals in the remaining steps of the pulse sequence. It is important to note that the pulse sequence even in the absence of artifacts creates double quantum coherence. Following the last pulse, the signals at 2τ + T for each step of the phase cycle are
In the above expressions the superscripts denote each step of the phase cycle in Table 1. The signal after phase cycling is thus
The result of this computation demonstrates that the four-step phase cycling scheme given in Table 1 cancels double quantum coherence entirely and that the detected signal results only from the quadrupolar ordered state. In the four subsections below we describe the effect of various pulse errors as well as off-resonance effects. It will be shown that these errors may introduce extra terms, such as single quantum coherence and Zeeman order in the intermittent states. Without proper phase cycling, some of these terms may be transformed into an observable signal. However, with the four-step phase cycling scheme given in Table 1, to a good approximation, all terms other than quadrupolar order are suppressed in the detected signal. For the case of a flip error, finite pulse width effect and pulse transient we assume an on-resonance condition with Δ = 0 in Eq.(3).
For the case of a pulse flip error, where the flip angle of any given pulse deviates from a perfect rotation, we study the possible leakage of various multiple quanta in the detected signal. The propagator for the RF pulse with a flip error is modeled as
In the above expression δβ is the flip error for a pulse with flip angle β. For the π/4 pulses the error δ45 is taken to be half of that of the π/2 pulse, δ90. The phase of each pulse in the cycle, ϕ, is again listed in Table 1. Using this model and the procedure described above, the various states of the spin system at time τ + T− for each step of the phase cycling are
In the above expressions Q denotes quadrupolar order, D denotes double quantum coherence, Z denotes Zeeman order, and S1, S2, S3, S4 all denote single quantum coherence. Following the third pulse the signals detected at t=T + 2τ for each step of the phase cycling are
and the signal after phase cycling is
From the above expressions it is clear that Zeeman order (labeled by Z) which appears at time t=τ + T− in Eq.(14a) and (14b) does not contribute to the phase cycled signal in Eq. (17); the first two steps of the phase cycling are able to cancel single quantum coherence (labeled S2 and S4). Together with the last two steps of the phase cycling scheme, double quantum coherence (labeled D) is also canceled. The only term which is created and detected is the quadrupolar ordered term (labeled Q). Thus, the computation shows that even in the presence of a pulse flip error there is no leakage of quantum coherence other than quadrupolar order after phase cycling. The final detected signal is real and is amplitude modulated by cos3(δ90). Note that when the error δ90 = 0 we find that the results agree with the situation of no error, given in Eq. (12). The signal intensity as a function of the π/2 flip angle error, δ90, in the range of 0 ~ 10° is plotted in Fig. 2. The figure shows that the relative reduction of the signal intensity is less then 5% when the flip error is as large as 10°. The pulse flip error may be experimentally measured by a flip-flip sequence .
In the situation of a rigid solid, the quadrupolar coupling constant for the deuteron can be as large as 200kHz and therefore the evolution under the quadrupolar interaction during the RF pulses cannot be ignored. The propagator for an RF pulse of width tp thus includes the quadrupolar interaction,
In the following we assume that ωQtp is a small quantity. This assumption is justified under the conditions where the π/2 pulse width tp is on the order of 1 to 3 μs, and the quadrupolar frequency ωQ/2π is less than 20 kHz, so that ωQtp ≤ 0.12π < π/2. We have performed a Taylor expansion of the solution to second order of this small quantity to simplify the complex algebraic expressions and give insight into the dynamics of the spin system in the presence of this error. Following the evolution under the sequence of pulses and delays shown in Fig. 1, one obtains the following density matrices before the last pulse for each step of the phase cycle
Again, the symbols Q and D1 indicate the magnitude of quadrupolar order and double quantum coherence respectively, S1, S2, S3 and S4 denote single quantum coherence and Z1 denotes Zeeman order. In addition, the term D2 is a second double quantum coherence term created due to finite pulse width effects.
Upon implementing the four-step phase cycling scheme again given in Table 1, the detected signal at t = 2τ + T is given by
The result shows that when finite pulse widths are taken into account the detected signal contains Zeeman order as well as quadrupolar order. However, the Zeeman order contribution to the signal intensity is cubic in tp and can be ignored to second order. The quadrupolar order contribution to the signal intensity decreases as the pulse width increases, and is plotted in Fig. 3 as a function of the pulse width tp for several values of the quadrupolar frequency ωQ/2π from 2kHz to 20kHz. The figure highlights that the relative reduction of the signal intensity is less than 5% for the largest value of ωQtp we considered ωQtp/2π=0.06.
In this section we model the effects of RF transients. In a previous work , we adopted a model after A. J. Vega , where any RF pulse of our pulse sequence includes a transient effect. The model consists of an orthogonal RF field that is applied to the spin system before and after the main pulse. The propagator for an antisymmetric pulse transient is given by
where the first and last propagators represent a pulse transient that rotate the ensemble by an angle αt and have phases orthogonal to that of the main pulse. For example, if the main pulse is about the x-axis then the transient is about the + y-axis before and - y-axis after the pulse. We performed a computation that accounts for pulse transients on all pulses in the spin alignment sequence, assuming that the transient flip angle αt is the same for all pulses. Because αt will typically be small relative to the main pulse duration we have taken a Taylor expansion to second order in our density matrix calculation. The states of the spin system before the last pulse for each step of the phase cycling are
The notation again for the various symbols Q, D1, S1, S2, S3, S4 and Z1, D2 are the same as those in the previous section on finite pulse errors. The computation shows the creation of single quantum coherence, double quantum coherence and Zeeman order in addition to the quadrupolar order terms resulting from RF transients. After implementing the four-step phase cycling one obtains the following expression for the signal at t=2τ + T
The result shows that the detected signal in the presence of RF transients arises only from the quadrupolar ordered state. However, the detected signal contains both real and imaginary components whose individual amplitudes depend on the transient flip angle αt. The real part and the magnitude of the complex signal as functions of αt in the range of 0 to 30° are plotted in Fig. 4. When the flip angle of the RF transient is as large as 30° the signal intensity on the real channel reduces to approximately 70%, and the magnitude of the complex signal reduces to approximately 75% of the signal intensity under ideal conditions. Experimentally RF transients may be characterized by a flip-flop sequence by measuring the modulation frequency induced by the transient , . From average Hamiltonian theory, the first order term of Magnus expansion for our transient model in the flip-flop sequence is
where τc is cycle time of the multiple pulse sequence and the modulation induced by the RF transient in a stroboscopic measurement is given by 
The relation between J1 and the transient flip angle αt in our model is
Lastly we consider the case when the internal Hamiltonian has a nonzero resonance offset, i.e. Δ ≠ 0 in Eq.(3). We assume the propagators for the RF pulses are the same as that for the ideal condition given in Eq.(9). The density matrices at t = τ + T− for the various steps in the phase cycling are
With the four-step phase cycling the final detected signal is
The above result shows that in the presence of a resonance offset that the detected signal still only arises from quadrupolar order with no additional quantum coherence leaked into the detected signal. There are two observed effects that the resonance offset error has on the signal; there is a phase and amplitude modulation of the signal that is proportional to Δ/ωQ. Because the resonance offset is typically much smaller than the quadrupolar frequency, the effect on the detected signal is negligible. Fig. 5 highlights the variation in the amplitude of the signal as a function of the resonance offset for four different values of quadrupolar frequencies, ωQ/2π, from 2kHz to 20kHz. The results show the reduction in signal in the alignment echo amplitude in the presence of an off-resonance effect as large as 40 Hz is negligible for quadrupolar frequencies ωQ/2π=20 kHz.
In the previous section the analysis was performed for a single value of ωQ and the delay τ was set such that τ = π/2ωQ. Under these conditions, it is straightforward to show that one achieves optimal conversion to quadrupolar order under ideal experimental conditions. For situations where there is more than one quadrupolar frequency, such as in the case of polycrystal or powdered samples, the above condition cannot be achieved and it is challenging to obtain analytical expressions for a distribution of ωQ. In this section, we consider simulation results that illustrate the effects of a given pulse error on the alignment echo spectra acquired by the three-pulse sequence. Broadband excitation of quadrupolar spectra may be achieved by composite pulses and alleviate the stringent requirement that τ = π/2 ωQ to some extent, but are not the subject of this work [16, 17, 18]. The parameters we used in the simulation are ωQ,0/2π = 64 [kHz], η = 0 with τ = 25μs or τ = 30μs. For each value of τ, the spectra for three types of errors were individually investigated. The simulations are performed using a home written algorithm in Matlab and the error models described in the previous section. In all of our simulations we considered 360 powder averages, T=100μs, a 350 μs acquisition window and a dwell time of 0.5μs. The spectra shown in Fig. 6 and Fig. 7 are obtained with a Gaussian broadening of 2kHz and zero padding the data once.
Fig. 6 and Fig. 7 show the simulated FID and spectra for τ = 25μs and τ = 30μs respectively under various pulse error conditions. It should be noted that the vertical scale of the spectra shown in Fig. 6 is half that in Fig. 7. Referring to Fig. 6b one observes that the effect of a flip error is to introduce an asymmetry into the spectra. A finite pulse width effect, shown in Fig. 6c through Fig. 6e, in the range of 1μs to 3μs reduces the signal intensity and introduces an asymmetry in the spectra as well. Fig. 6f and Fig. 6g show results for two different RF transients. The main effect of an RF transient in addition to causing an asymmetry distortion is that it introduces some additional artifacts in the center of the spectra, and as the transient is made larger this feature becomes more pronounced. With the introduction of any of the three errors we modeled, an imaginary component is introduced in the FID.
Fig. 7, highlights results for τ = 30μs; the data demonstrate that the effects of the various errors appear to be larger than when τ=25μs. The effect of a flip error shown in Fig.7b also introduces an asymmetry in the spectra, but to a greater extent than that of Fig. 6b where τ = 25μs. The effect of a finite pulse width, shown in Fig. 7c through Fig. 7e also distorts the spectra, but to a greater extent than that shown in the results of Fig. 6c through Fig. 6e. RF transients still result in a spectral distortion in the center of the spectra, however, the asymmetry appears to be much greater than the condition when τ = 25μs shown in Fig. 6f and Fig. 6g. Together, the simulations show that the spectral distortions due to pulse errors may be minimized, to some extent, by judiciously choosing the time between the first two pulses.
An intuitive method of choosing the optimal value of τ is to consider the maxima of the function
which is the powder average of the prefactor of the signal intensity, under ideal conditions. To be clear, the amplitude modulation arises from the generation and subsequent readout of quadrupolar order. A graph of this function is shown in Fig. 8 for the case ωQ,0/2π = 64 [kHz]. Referring to the figure, it is evident that the optimal choice for τ is approximately 9μs. Other than a reduction in signal intensity, the spectra simulated with this choice of τ essentially reproduced that shown for the next maximum, τ=25μs. While the shorter τ choice may maximize the conversion to quadrupolar order, this choice may result in additional distortions in the acquired spectra due to instrumental dead times and probe ring down effects in experiments. Further, Fig. 8 accounts for the observation that the signal intensity at τ = 30μs is smaller than that at τ = 25μs, as the shorter choice in τ allows for more efficient conversion of single quantum coherence to quadrupolar alignment with the π/4 pulse. In practice one may measure the value of ωQ and η with a solid echo and generate a similar figure for determining an optimal choice in τ.
We performed a measurement of the quadrupolar alignment relaxation rate on a perdeuterated hexamethylbenzene(HMB) powdered sample, and a sample of deuterated polyethelene(PE). We measured ωQ,0/2π=16.1 kHz in the HMB sample and ωQ,0/2π=120.5 kHz in the PE sample. The 2H NMR signals were obtained at 27.55MHz, using a Tecmag Apollo solid state NMR system and a home-built NMR probe. The pulse length was 2.8μs for the π/2 pulse with τ set to 35μs in all our experiments. In signal averaging, 80 scans were accumulated for the HMB sample and 1000 scans were accumulated for the PE sample with a recycle delay of 2s. All the experiments were performed at room temperature. In order to probe the effectiveness of the suppression of double quantum coherence we implemented the four-step phase cycling scheme in Table 1 as well as a two step phase cycling scheme consisting of the first two steps in Table 1; the later scheme suppresses single quantum coherence but not double quantum coherence. The integrated spectral intensity as a function of the delay T are shown in Fig. 9 for HMB and Fig. 10 for PE, on a logarithmic scale to reveal the details of the short time behavior. The purpose of the experimental data presented is to demonstrate the efficacy of the four-step phase cycling scheme under real experimental conditions and verify some of the results of our analysis.
Data for the HBM sample using the two-step phase cycling scheme was fitted by two exponentials, shown as the solid line through the triangle points, with time constants 57.5ms and 0.5ms. The four-step phase cycled data, however, is well fit by a single exponential and the fit is shown as the solid line through the circle points, with a time constant of 57.5ms. It is clear that the 0.5ms fast decay component from the two-step phase cycling scheme is double quantum coherence. The absence of other components needed to fit both data sets suggest that there is no single quantum coherence or Zeeman order leakage under our experimental conditions. Furthermore, the four-step phase cycled data indicate that there is no double quantum coherence or other terms that are leaked into the detected signal other than the quadrupolar ordered state by using this scheme. Experimental data acquired on PE are highlighted in Fig. 10 and demonstrate similar results as that found on HMB. Data acquired with two step phase cycling scheme are shown as triangle points; the fit shown as a solid line includes three time constants: 147.7ms, 3.8ms and 0.09ms. Using the four step phase cycling scheme we observe only two time constants: 147.7ms and 3.8ms. The double quantum signal that has a decay time constant of 0.09ms observed in the two step phase cycling experiment is suppressed in the four step phase cycling experiment. It is evident that the pulse phase cycling scheme is also robust for this sample that has a quadrupolar frequency approximately 7.4 times larger than the HMB sample. The bi-exponential decay in the spin alignment echo experiment of PE has been previously studied by H. Speiss and coworkers, and is attributed to ‘rigid’ and ‘mobile’ deuterons in crystalline and amorphous regions of the polymer . In our experimental setup, we estimated the pulse flip errors and transients with flip-flip and flip-flop pulse sequences . The flip error for the 90° pulse in our system was determined to be approximately 3.5° and the pulse transient flip angle was determined to be approximately 11°. While an 11° degree phase transient is within the limits of our Taylor expansion computed in the previous section, we found that the phase cycling scheme was robust in suppressing unwanted double quantum coherence. While the pulse flip error was 3.5°, the analysis showed that the four-step phase cycling scheme is also robust against this error and suppresses unwanted double quantum coherence when ωQτ = π/2. Finally, in our experimental setup ωQtp π/2, so no double quantum or single quantum or Zeeman order leakage is expected to be realized, when ωQτ = π/2. We note that the experimental artifacts commonly encountered in other systems may be greater than that found on our system, and it behooves the experimenter to consider running two separate experiments with two separate phase cycles to verify that the short time decay in T1Q experiments is indeed quadrupolar order.
In this work we report on the effects of a variety of experimental imperfections on spin I=1 quadrupolar alignment echo spectroscopy. We studied the potential leakage of unwanted quantum coherence as well as the reduction in signal intensity for four different types of commonly encountered experimental artifacts. For the case of a single quadrupolar frequency (in the case of a single crystal), the findings indicate that by implementing a well known four-step phase cycling scheme that pulse flip errors, finite pulse width effects, RF transients and off-resonance effects will not introduce unwanted quantum coherences. Off-resonance effects have the smallest effect and are negligible in many rigid or semi-rigid solids. Pulse flip errors and RF transients reduce the signal intensity for typical experimental errors to a similar extent. Finite pulse width effects may have substantial effects if the quadrupolar interaction is large, resulting in a reduction in signal intensity comparable in magnitude to the effects of the pulse flip errors and transients. We studied the various artifacts that may arise in the situation of a powder distribution, via simulation. The artifacts associated with the pulse errors may be minimized to some extent by a judicious choice in τ, where single quantum coherence is maximally converted to quadrupolar order. Lastly, it is demonstrated experimentally that the potential leakage of double quantum coherence in the short time decay of quadrupolar alignment relaxation studies may be probed by performing two separate experiments; one cycle that intentionally allows for detecting the double quantum signal and a second cycle that only allows for detecting quadrupolar order.
G. S. Boutis acknowledges support from the Professional Staff Congress of the City University of New York and NIH grant number 7SC1GM086268-03. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences or the National Institutes of Health.
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