Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC3148855

Formats

Article sections

- Abstract
- 1. Introduction
- 2. Case I: Theoretical treatment for the case of a single quadrupolar frequency
- 3. Case II: Simulations of alignment echo powder line shapes in the presence of pulse errors
- 4. Experimental
- 5. Conclusion
- References

Authors

Related links

J Magn Reson. Author manuscript; available in PMC 2012 August 1.

Published in final edited form as:

Published online 2011 May 18. doi: 10.1016/j.jmr.2011.05.003

PMCID: PMC3148855

NIHMSID: NIHMS304444

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 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, *T*_{1Q}. 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 *T*_{1Q}. It is suggested that one employ two separate pulse sequences when measuring *T*_{1Q} 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
${T}_{2}^{}$ to *T*_{1}, corresponding to a correlation time of 10^{−5}s 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 [3] and motional constraints in polyethylene [1, 2]. The approach has been used to study slow tetrahedral jumps in solid hexamethylenetetramine [4], the dynamics of gramicidin [5], to investigate the localized dynamics in DNA fragments [6], and more recently hydration dependent dynamics in RNA [7].

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, *T*_{1Q}, can be measured. This pulse sequence intrinsically generates quadrupolar order and double quantum coherence at the same time even under ideal experimental conditions [3]. The unwanted double quantum coherence can be readily suppressed from the detected signal by phase cycling [3].

One of the prerequisites for performing *T*_{1Q} 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 [8]. A. Jerschow has reported on a spherical tensor based formalism that quantifies the effects of imperfect rotations when selecting a quantum coherence of interest [9]. In this work, we studied the effects of pulse errors on the alignment echo three-pulse sequence via the fictitious spin-
$\frac{1}{2}$ operators developed by Vega and Pines [10]. For the case that only one quadrupolar frequency, *ω _{Q}*, 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

The fictitious spin-
$\frac{1}{2}$ operators were introduced by Vega and Pines to conveniently describe the interaction of a spin *I*=1 nucleus with a Radio Frequency (RF) field [10]. In this formalism, the basis consists of nine operators given by

$$\begin{array}{l}{I}_{x,1}=\frac{1}{2}{I}_{x},\phantom{\rule{1em}{0ex}}{I}_{x,2}=\frac{1}{2}({I}_{y}{I}_{z}+{I}_{z}{I}_{y}),\phantom{\rule{1em}{0ex}}{I}_{x,3}=\frac{1}{2}({I}_{z}^{2}-{I}_{y}^{2})\\ {I}_{y,1}=\frac{1}{2}{I}_{y},\phantom{\rule{1em}{0ex}}{I}_{y,2}=\frac{1}{2}({I}_{z}{I}_{x}+{I}_{x}{I}_{z}),\phantom{\rule{1em}{0ex}}{I}_{y,3}=\frac{1}{2}({I}_{x}^{2}-{I}_{z}^{2})\\ {I}_{z,1}=\frac{1}{2}{I}_{z},\phantom{\rule{1em}{0ex}}{I}_{z,2}=\frac{1}{2}({I}_{x}{I}_{y}+{I}_{y}{I}_{x}),\phantom{\rule{1em}{0ex}}{I}_{z,3}=\frac{1}{2}({I}_{y}^{2}-{I}_{x}^{2})\end{array}$$

(1)

From these definitions the various orders of multiple quantum coherence are as follows

$$\begin{array}{ll}{I}_{x,1},{I}_{y,1},{I}_{x,2},{I}_{y,2}:& \text{single quantum}\\ {I}_{x,3},{I}_{y,3}:& \text{quadrupolar order and double quantum}\\ {I}_{x,3}+{I}_{y,3}=-{I}_{z,3},{I}_{z,2}:& \text{double quantum}\hfill \\ {I}_{z,1}:& \text{Zeeman order}\\ {I}_{x,3}-{I}_{y,3}:& \text{quadrupolar order}.\end{array}$$

In the following, we express the density operator by the above fictitious spin-
$\frac{1}{2}$ 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

$$H={H}_{RF}+{H}_{\mathit{\text{int}}}$$

(2)

with

$$\begin{array}{l}{H}_{\mathit{\text{int}}}=-\mathrm{\Delta}{I}_{z}+\frac{1}{3}{\omega}_{Q}\phantom{\rule{0.2em}{0ex}}[3{I}_{z}{I}_{z}-I(I+1)]\\ \phantom{\rule{1.8em}{0ex}}=-2\mathrm{\Delta}{I}_{z,1}+\frac{2}{3}{\omega}_{Q}\phantom{\rule{0.2em}{0ex}}({I}_{x,3}-{I}_{y,3})\end{array}$$

(3)

$${\omega}_{Q}=\frac{3{e}^{2}qQ}{4I(2I-1)}$$

(4)

where Δ is the resonance offset, *e*^{2}*qQ/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 [11]. We will use the notation *ω _{Q}*

$${\omega}_{Q,0}=\frac{3{e}^{2}qQ}{4I\left(2I-1\right)}$$

(5)

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

$$\rho (t)=U\rho (0){U}^{-1}$$

(6)

with

$$U=exp(-\mathit{\text{iHt}})$$

(7)

where *H* is time independent. In what follows, the initial density matrix *ρ*(0) is taken as *ρ*(0)=*I _{z}*. The observable signal is determined by computing

$$\mathit{\text{Sig}}=\text{Tr}(\rho {R}_{\pm y})$$

(8)

where the operator for the receiver, *R*_{±y}, is defined as *R*_{±} ±*I*_{y,1} *iI*_{x,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 [3]) eliminates double quantum coherence under ideal experimental conditions. In Table 1, *ϕ _{i}* and

$${U}_{RF}(\beta ,\varphi )=exp(-i\beta 2{I}_{\varphi})$$

(9)

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-
$\frac{1}{2}$ operators. For the computations that follow we assume
$\tau =\frac{\pi}{2{\omega}_{Q}}$ 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

$${\rho}^{1,2}(\tau +{T}_{-})=Q({I}_{x,3}-{I}_{y,3})/2+D({I}_{y,3}+{I}_{x,3})/2$$

(10a)

$${\rho}^{3,4}(\tau +{T}_{-})=Q({I}_{x,3}-{I}_{y,3})/2-D({I}_{x,3}+{I}_{y,3})/2$$

(10b)

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 2*τ* + *T* for each step of the phase cycle are

$$Si{g}^{1,2}\left(2\tau +T\right)=1/8(-D-3Q)$$

(11a)

$$Si{g}^{3,4}(2\tau +T)=1/8(D-3Q)$$

(11b)

In the above expressions the superscripts denote each step of the phase cycle in Table 1. The signal after phase cycling is thus

$$\mathit{\text{Sig}}(2\tau +T)=\sum _{i=1}^{4}Si{g}^{i}(2\tau +T)=(-3/2)Q$$

(12)

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

$${U}_{RF}(\beta ,\varphi ;{\delta}_{\beta})=exp[-i(\beta +{\delta}_{\beta})2{I}_{\varphi}]$$

(13)

In the above expression *δ _{β}* is the flip error for a pulse with flip angle

$$\begin{array}{l}{\rho}^{1,2}(\tau +{T}_{-})=\frac{{I}_{y,3}-{I}_{x,3}}{2}Q+\frac{{I}_{y,3}+{I}_{x,3}}{2}D\\ \phantom{\rule{4.6em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{z,1}Z-{I}_{y,2}{S}_{4}-{I}_{y,1}{S}_{3}\\ \phantom{\rule{4.6em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{x,1}{S}_{1}-{I}_{x,2}{S}_{2}\end{array}$$

(14a)

$$\begin{array}{l}{\rho}^{3,4}(\tau +{T}_{-})=-\frac{{I}_{x,3}-{I}_{y,3}}{2}Q-\frac{{I}_{x,3}+{I}_{y,3}}{2}D\\ \phantom{\rule{4.6em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{z,1}Z+{I}_{x,2}{S}_{4}-{I}_{x,1}{S}_{3}\\ \phantom{\rule{4.6em}{0ex}}+\phantom{\rule{0.2em}{0ex}}{I}_{y,1}{S}_{1}-{I}_{y,2}{S}_{2}\end{array}$$

(14b)

where

$$\begin{array}{l}Q=D={cos}^{2}({\delta}_{90})\\ Z=cos\frac{1}{4}(\pi +2{\delta}_{90})\\ {S}_{1}=cos(T{\omega}_{Q})sin{\delta}_{90}sin\frac{1}{4}(\pi +2{\delta}_{90})\\ {S}_{2}=sin(T{\omega}_{Q})sin{\delta}_{90}sin\frac{1}{4}(\pi +2{\delta}_{90})\\ {S}_{3}=\frac{1}{2}sin(T{\omega}_{Q})sin2{\delta}_{90}\\ {S}_{4}=\frac{1}{2}cos\left(T{\omega}_{Q}\right)sin2{\delta}_{90}\end{array}$$

(15)

In the above expressions *Q* denotes quadrupolar order, *D* denotes double quantum coherence, *Z* denotes Zeeman order, and *S*_{1}, *S*_{2}, *S*_{3}, *S*_{4} all denote single quantum coherence. Following the third pulse the signals detected at *t*=*T* + 2*τ* for each step of the phase cycling are

$$\begin{array}{l}Si{g}^{1}(2\tau +T)=\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\left[-(D+3Q)cos({\delta}_{90})-4i{S}_{2}cos\frac{1}{4}(\pi +2{\delta}_{90})+4{S}_{4}sin({\delta}_{90})\right]\\ Si{g}^{2}(2\tau +T)=\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\left[-(D+3Q)cos({\delta}_{90})+4i{S}_{2}cos\frac{1}{4}(\pi +2{\delta}_{90})-4{S}_{4}sin({\delta}_{90})\right]\\ Si{g}^{3}(2\tau +T)=\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\left[(D-3Q)cos({\delta}_{90})+4i{S}_{4}cos\frac{1}{4}(\pi +2{\delta}_{90})+4{S}_{2}sin({\delta}_{90})\right]\\ Si{g}^{4}(2\tau +T)=\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\left[(D-3Q)cos({\delta}_{90})-4i{S}_{4}cos\frac{1}{4}(\pi +2{\delta}_{90})-4{S}_{2}sin({\delta}_{90})\right]\end{array}$$

(16)

and the signal after phase cycling is

$$\begin{array}{l}\mathit{\text{Sig}}(2\tau +T)=\sum _{i=1}^{4}Si{g}^{i}(2\tau +T)\\ \phantom{\rule{5em}{0ex}}=-\frac{3}{2}Qcos{\delta}_{90}\end{array}$$

(17)

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 *S*_{2} and *S*_{4}). 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 cos^{3}(*δ*_{90}). Note that when the error *δ*_{90} = 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, *δ*_{90}, 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 [12].

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 *t _{p}* thus includes the quadrupolar interaction,

$${U}_{RF}(\beta ,\varphi ;{t}_{p})=exp\phantom{\rule{0.2em}{0ex}}[-i(\beta 2{I}_{\varphi}+{H}_{\mathit{\text{int}}}{t}_{p})]$$

(18)

In the following we assume that *ω _{Q}t_{p}* is a small quantity. This assumption is justified under the conditions where the

$$\begin{array}{l}{\rho}^{1,2}(\tau +{T}_{-})=-{I}_{x,1}{S}_{1}+{I}_{x,2}{S}_{2}+{I}_{y,1}{S}_{3}-{I}_{y,2}{S}_{4}+{I}_{z,1}{Z}_{1}\\ \phantom{\rule{18em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}-\frac{{I}_{y,3}-{I}_{x,3}}{2}Q-\frac{{I}_{y,3}+{I}_{x,3}}{2}{D}_{1}\end{array}$$

(19a)

$$\begin{array}{l}{\rho}^{3,4}(\tau +{T}_{-})=-{I}_{y,1}{S}_{1}+{I}_{y,2}{S}_{2}+{I}_{x,1}{S}_{3}+{I}_{x,2}{S}_{4}+{I}_{z,1}{Z}_{1}\\ \phantom{\rule{18em}{0ex}}+\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}+\frac{{I}_{x,3}-{I}_{y,3}}{2}Q+\frac{{I}_{x,3}+{I}_{y,3}}{2}{D}_{1}\end{array}$$

(19b)

where

$$\begin{array}{l}Q={D}_{1}=\frac{-8{\pi}^{2}+(8+\pi (4+\pi )){\omega}_{Q}^{2}{t}_{p}^{2}}{8{\pi}^{2}}\\ {D}_{2}=\frac{{\omega}_{Q}{t}_{p}}{\sqrt{2}\pi}\\ {Z}_{1}=\frac{{\omega}_{Q}^{2}{t}_{p}^{2}}{\sqrt{2}{\pi}^{2}}\\ {S}_{1}=\frac{{\omega}_{Q}{t}_{p}sin(T{\omega}_{Q})}{\sqrt{2}\pi}\\ {S}_{2}=\frac{{\omega}_{Q}{t}_{p}cos(T{\omega}_{Q})}{\sqrt{2}\pi}\\ {S}_{3}=\frac{4\pi (2+\pi ){\omega}_{Q}{t}_{p}cos(T{\omega}_{Q})-6\pi {\omega}_{Q}^{2}{t}_{p}^{2}sin(T{\omega}_{Q})}{8{\pi}^{2}}\\ {S}_{4}=\frac{6\pi {\omega}_{Q}^{2}{t}_{p}^{2}cos(T{\omega}_{Q})+4\pi (2+\pi ){\omega}_{Q}{t}_{p}sin(T{\omega}_{Q})}{8{\pi}^{2}}\end{array}$$

(20)

Again, the symbols *Q* and *D*_{1} indicate the magnitude of quadrupolar order and double quantum coherence respectively, *S*_{1}, *S*_{2}, *S*_{3} and *S*_{4} denote single quantum coherence and *Z*_{1} denotes Zeeman order. In addition, the term *D*_{2} 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* = 2*τ* + *T* is given by

$$\begin{array}{l}\mathit{\text{Sig}}(2\tau +T)=\text{Tr}\phantom{\rule{0.2em}{0ex}}(\sum _{i=1}^{4}{\rho}^{i}(T+2\tau ){R}_{i})& \phantom{\rule{5em}{0ex}}\approx \frac{1}{4}\phantom{\rule{0.2em}{0ex}}\left[i\sqrt{2}{\omega}_{Q}{Z}_{1}{t}_{p}+Q\phantom{\rule{0.2em}{0ex}}\left(6-\frac{3{\omega}_{Q}^{2}{t}_{p}^{2}}{{\pi}^{2}}\right)\right]\\ \phantom{\rule{5em}{0ex}}\approx \frac{3}{16}\phantom{\rule{0.2em}{0ex}}\left[-8+\frac{(12+4\pi +{\pi}^{2})\phantom{\rule{0.2em}{0ex}}({\omega}_{Q}{t}_{p}^{2})}{{\pi}^{2}}\right]\end{array}$$

(21)

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 *t _{p}* 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

In this section we model the effects of RF transients. In a previous work [13], we adopted a model after A. J. Vega [14], 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

$$\begin{array}{l}{U}_{RF}(\beta ,\varphi ;{\alpha}_{t})=exp(-i{\alpha}_{t}2{I}_{\varphi +\pi /2}^{t})\\ \phantom{\rule{16em}{0ex}}\times exp(-i\beta 2{I}_{\varphi})\times exp(i{\alpha}_{t}2{I}_{\varphi +\pi /2}^{t})\end{array}$$

(22)

where the first and last propagators represent a pulse transient that rotate the ensemble by an angle *α _{t}* and have phases orthogonal to that of the main pulse. For example, if the main pulse is about the

$$\begin{array}{l}{\rho}^{1,2}(\tau +{T}_{-})={I}_{x,1}{S}_{1}+{I}_{x,2}{S}_{2}+{I}_{y,1}{S}_{3}+{I}_{y,2}{S}_{4}+{I}_{z,1}{Z}_{1}\\ \phantom{\rule{17em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}-\frac{{I}_{y,3}-{I}_{x,3}}{2}Q+\frac{{I}_{y,3}+{I}_{x,3}}{2}{D}_{1}\end{array}$$

(23a)

$$\begin{array}{l}{\rho}^{3,4}(\tau +{T}_{-})=-{I}_{y,1}{S}_{1}+{I}_{y,2}{S}_{2}+{I}_{x,1}{S}_{3}-{I}_{x,2}{S}_{4}+{I}_{z,1}{Z}_{1}\\ \phantom{\rule{17em}{0ex}}+\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}+\frac{{I}_{x,3}-{I}_{y,3}}{2}Q-\frac{{I}_{x,3}+{I}_{y,3}}{2}{D}_{1}\end{array}$$

(23b)

where

$$\begin{array}{l}Q={\alpha}_{t}^{2}-1\\ {D}_{1}=1-3{\alpha}_{t}^{2}\\ {D}_{2}=\sqrt{2}{\alpha}_{t}\\ {Z}_{1}=\frac{{\alpha}_{t}^{2}}{\sqrt{2}}\\ {S}_{1}=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\left(\sqrt{2}{\alpha}_{t}^{2}cos(T{\omega}_{Q})+2{\alpha}_{t}sin(T{\omega}_{Q})\right)\\ {S}_{2}=\frac{1}{2}\left(-2{\alpha}_{t}cos(T{\omega}_{Q})+\sqrt{2}{\alpha}_{t}^{2}sin(T{\omega}_{Q})\right)\\ {S}_{3}=\left(-1+\sqrt{2}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}^{2}sin(T{\omega}_{Q})\\ {S}_{4}=\left(-1+\sqrt{2}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}^{2}cos(T{\omega}_{Q})\end{array}$$

(24)

The notation again for the various symbols *Q*, *D*_{1}, *S*_{1}, *S*_{2}, *S*_{3}, *S*_{4} and *Z*_{1}, *D*_{2} 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*=2*τ* + *T*

$$\begin{array}{l}\mathit{\text{Sig}}(2\tau +T)=\text{Tr}\phantom{\rule{0.2em}{0ex}}(\sum _{i=1}^{4}{\rho}^{i}\left(T+2\tau \right){R}_{i})& \phantom{\rule{4.8em}{0ex}}\approx \frac{3}{4}Q\phantom{\rule{0.2em}{0ex}}\left[2-2i\phantom{\rule{0.2em}{0ex}}\left(-1+\sqrt{2}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}+\phantom{\rule{0.2em}{0ex}}\left(-3+2\sqrt{2}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}^{2}\right]\\ \phantom{\rule{4.8em}{0ex}}\approx -\frac{3}{2}+\frac{3}{2}i\phantom{\rule{0.2em}{0ex}}\left(-1+\sqrt{2}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}+\left(\frac{15}{4}-\frac{3}{\sqrt{2}}\right)\phantom{\rule{0.2em}{0ex}}{\alpha}_{t}^{2}\end{array}$$

(25)

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 *α _{t}*. The real part and the magnitude of the complex signal as functions of

Absolute value of the alignment echo intensity at *t* = 2*τ* + *T* with errors (*S*) divided by the signal intensity without errors (*S*_{0}) as a function of the transient flip angle *α*_{t} in degrees. The dashed line represents the real component of **...**

$${H}_{T}^{0}={J}_{1}/{\tau}_{c}({I}_{z,1}-{I}_{y,1})$$

(26)

where *τ _{c}* is cycle time of the multiple pulse sequence and the modulation induced by the RF transient in a stroboscopic measurement is given by [13]

$$\omega =-\sqrt{2}{J}_{1}/{\tau}_{c}.$$

(27)

The relation between *J*_{1} and the transient flip angle *α _{t}* in our model is

$${J}_{1}=-{\alpha}_{t}$$

(28)

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

$$\begin{array}{l}{\rho}^{1,2}(\tau +{T}_{-})=-{I}_{x,2}{S}_{4}+{I}_{x,1}{S}_{3}-{I}_{y,2}{S}_{2}-{I}_{y,1}{S}_{1}\\ \phantom{\rule{17.5em}{0ex}}-\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}-\frac{{I}_{x,3}-{I}_{y,3}}{2}Q+\frac{{I}_{x,3}+{I}_{y,3}}{2}D\end{array}$$

(29a)

$$\begin{array}{l}{\rho}^{3,4}(\tau +{T}_{-})=-{I}_{y,2}{S}_{4}-{I}_{y,1}{S}_{3}+{I}_{x,2}{S}_{2}-{I}_{x,1}{S}_{1}\\ \phantom{\rule{17.5em}{0ex}}+\phantom{\rule{0.2em}{0ex}}{I}_{z,2}{D}_{2}-\frac{{I}_{x,3}-{I}_{y,3}}{2}Q-\frac{{I}_{x,3}+{I}_{y,3}}{2}D\end{array}$$

(29b)

where

$$\begin{array}{l}Q=cos\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)\\ {D}_{1}=cos\phantom{\rule{0.2em}{0ex}}\left(2T\mathrm{\Delta}\right)cos\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)-\sqrt{2}sin\left(2T\mathrm{\Delta}\right)sin\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)\\ {D}_{2}=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\left[cos\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)sin\left(2T\mathrm{\Delta}\right)+\sqrt{2}cos\left(2T\mathrm{\Delta}\right)sin\phantom{\rule{0.2em}{0ex}}\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)\right]\end{array}$$

(30)

$$\begin{array}{l}{S}_{1}=\frac{sin(T{\omega}_{Q})sin(T\mathrm{\Delta})sin\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)}{\sqrt{2}}\hfill \\ {S}_{2}=\frac{cos(T{\omega}_{Q})sin(T\mathrm{\Delta})sin\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)}{\sqrt{2}}\hfill \\ {S}_{3}=\frac{cos(T\mathrm{\Delta})sin(T{\omega}_{Q})sin\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)}{\sqrt{2}}\hfill \\ {S}_{4}=\frac{cos(T{\omega}_{Q})cos(T\mathrm{\Delta})sin\left(\frac{\pi \mathrm{\Delta}}{2{\omega}_{Q}}\right)}{\sqrt{2}}\hfill \end{array}$$

(31)

With the four-step phase cycling the final detected signal is

$$\begin{array}{l}\mathit{\text{Sig}}(2\tau +T)=\text{Tr}\phantom{\rule{0.2em}{0ex}}(\sum _{i=1}^{4}{\rho}^{i}\left(T+2\tau \right){R}_{i})& \phantom{\rule{5em}{0ex}}=-\frac{3}{2}{e}^{-\frac{i\pi \mathrm{\Delta}}{2{\omega}_{Q}}}Q\end{array}$$

(32)

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/*ω _{Q}*. 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,

In the previous section the analysis was performed for a single value of *ω _{Q}* and the delay

Simulated phase cycled alignment echo FID and the corresponding real component of the spectra for the case of *τ* = 25*μ*s (other parameters are provided in the text). a) Ideal experimental conditions b) A flip error with *δ*_{β} **...**

Simulated phase cycled alignment echo FID and the corresponding real component of the spectra for the case of *τ* = 30*μ*s (other parameters are provided in the text). a) Ideal experimental conditions b) A flip error with *δ*_{β} **...**

Fig. 6 and Fig. 7 show the simulated FID and spectra for *τ* = 25*μ*s and *τ* = 30*μ*s 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 1*μ*s to 3*μ*s 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* = 30*μ*s; the data demonstrate that the effects of the various errors appear to be larger than when *τ*=25*μs*. 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* = 25*μ*s. 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* = 25*μ*s 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

$${\int}_{0}^{\pi}{\mathit{\text{sin}}}^{2}\phantom{\rule{0.2em}{0ex}}\left({\omega}_{Q,0}\frac{3{cos}^{2}\theta -1}{2}\tau \right)\phantom{\rule{0.2em}{0ex}}\mathit{\text{sin}}\left(\theta \right)d\theta $$

(33)

which is the powder average of the prefactor of the signal intensity, under ideal conditions. To be clear, the
${\mathit{\text{sin}}}^{2}\phantom{\rule{0.2em}{0ex}}\left({\omega}_{Q,0}\frac{3{cos}^{2}\theta -1}{2}\tau \right)$ 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 *ω _{Q}*

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 *ω*_{Q,0}/2*π*=16.1 kHz in the HMB sample and *ω*_{Q,0}/2*π*=120.5 kHz in the PE sample. The ^{2}H 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 2.8*μ*s for the *π*/2 pulse with *τ* set to 35*μ*s 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.

Results from relaxation measurements shown on a logarithmic scale (in time) for two different phase cycling schemes in powdered perdeuterated hexamethylbenzene. As discussed in the text, the two step phase cycling scheme allows for detection of double **...**

Results from relaxation measurements shown on a logarithmic scale (in time) for two different phase cycling schemes in deuterated polyethelene. As discussed in the text, the two step phase cycling scheme allows for detection of double quantum coherence **...**

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 [19]. In our experimental setup, we estimated the pulse flip errors and transients with flip-flip and flip-flop pulse sequences [12]. 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 *ω _{Q}τ* =

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.

**Publisher's Disclaimer: **This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1. Spiess H. Molecular dynamics of solid polymers as revealed by deuteron NMR. Colloid Polym Sci. 1983;261(3):193–209.

2. Spiess H. Deuteron spin alignment: A probe for studying ultraslow motions in solids and solid polymers. J Chem Phys. 1980;72:6755.

3. Vold R, Dickerson W, Vold R. Application of the Jeener-Broekaert pulse sequence to molecular dynamics studies in liquid crystals. J Mag Res. 1981;43:213–223.

4. Lausch M, Spiess H. Ultraslow tetrahedral jumps in solid hexamethylenetetramine studied by deuteron spin alignment. Chem Phys Let. 1980;71:182–186.

5. Prosser R, Davis J. Dynamics of an integral membrane peptide: a deuterium NMR relaxation study of gramicidin. Biophys J. 1994;66:1429–1440. [PubMed]

6. Hatcher M, Mattiello D, Meints G, Orban J, Drobny G. A solid-state deuterium NMR study of the localized dynamics at the C9pG10 step in the DNA Dodecamer [d (CGCGAATTCGCG)] 2. J Am Chem Soc. 1998;120:9850–9862.

7. Olsen G, Bardaro M, Echodu D, Drobny G, Varani G. Hydration dependent dynamics in RNA. J Bio Mol NMR. 2009;45:133–142. [PubMed]

8. Bodenhausen G, Kogler H, Ernst R. Selection of coherence-transfer pathways in NMR pulse experiments. J Mag Res. 1984;58:370–388. 1969.

9. Jerschow A. Nonideal rotations in nuclear magnetic resonance: Estimation of coherence transfer leakage. J Chem Phys. 2000;113:979–986.

10. Vega S, Pines A. Operator formalism for double quantum NMR. J Chem Phys. 1977;66:5624–5644.

11. Abragam A. The Principles of Nuclear Magnetism. Oxford University Press; USA: 1983.

12. Haeberlen U. Adv Magn Reson. Academic; New York:

13. Sun C, Boutis G. Simulation studies of instrumental artifacts on spin I= 1 double quantum filtered NMR spectroscopy. J Mag Res. 2010;205:102–108. [PMC free article] [PubMed]

14. Vega A. Controlling the effects of pulse transients and RF inhomogeneity in phase-modulated multiple-pulse sequences for homonuclear decoupling in solid-state proton NMR. J Mag Res. 2004;170:22–41. [PubMed]

15. Rhim W, Elleman D, Schreiber L, Vaughan R. Analysis of multiple pulse NMR in solids. II. J Chem Phys. 1974;60:4595–4604.

16. Wimperis S, Bodenhausen G. Broadband excitation of quadrupolar order by modified Jeener-Broekaert sequences. Chem Phys Lett. 1986;132:194–199.

17. Wimperis S. Broadband and narrowband composite excitation sequences. J Mag Res. 1990;86:46–59. 1969.

18. Hoatson G. Broadband composite excitation sequences for creating quadrupolar order in 2H NMR. J Mag Res. 1991;94:152–159.

19. Hentschel D, Sillescu H, Spiess H. Deuteron nmr study of chain motion in solid polyethylene. Polymer. 1984;25(8):1078–1086.

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |