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J Magn Reson. Author manuscript; available in PMC 2010 April 19.

Published in final edited form as:

PMCID: PMC2856658

NIHMSID: NIHMS177950

Sharon Peled,^{*,}^{1} Ching-Hua Tseng,^{*}^{†} Aaron A. Sodickson,^{*,}^{2} Ross W. Mair,^{†} Ronald L. Walsworth,^{†} and David G. Cory^{*}

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.

A single-shot pulsed gradient stimulated echo sequence is introduced to address the challenges of diffusion measurements of laser polarized ^{3}He and ^{129}Xe gas. Laser polarization enhances the NMR sensitivity of these noble gases by >10^{3}, but creates an unstable, nonthermal polarization that is not readily renewable. A new method is presented which permits parallel acquisition of the several measurements required to determine a diffusive attenuation curve. The NMR characterization of a sample's diffusion behavior can be accomplished in a single measurement, using only a single polarization step. As a demonstration, the diffusion coefficient of a sample of laser-polarized ^{129}Xe gas is measured via this method.

Optical pumping techniques can be used to transfer angular momentum from laser photons to the nuclear spins of the noble gases ^{3}He and ^{129}Xe, aligning their nuclear spin ensembles far more than in a thermal distribution (1, 2). This process enhances the NMR sensitivity of these gases by as much as five orders of magnitude, enabling diverse applications, including fundamental symmetry tests (3, 4), polarized enhancement of other species (5, 6), biomedical imaging (7-9), and gas phase diffusion measurements (10-14). As with imaging, diffusion measurements using laser-polarized nuclei require special consideration because the magnetization is not readily renewable. In an NMR experiment, the diffusion coefficient, *D*, is generally determined from a series of echo attenuation measurements following periods of diffusion in magnetization gratings (15). In the case of laser-polarized gases, standard NMR diffusion techniques are inefficient since the magnetization is exhausted in a single measurement before further echoes with different diffusive attenuation can be acquired. The technique described here avoids this constraint by detecting a series of echoes with different diffusive attenuation factors in a single-shot pulse sequence based on stimulated echoes. A similar sequence, based on spin-echoes, has been proposed and shown to reproduce literature values for the diffusion coefficients of a number of liquids (16).

The first step in measuring a diffusion coefficient is to create a magnetization grating through the application of a field gradient that winds up the phase of the transverse magnetization. In a pulsed gradient stimulated echo (PGSTE) sequence, this grating is partially destroyed by the random motion of the spins due to diffusion. Therefore, only a fraction of the initial transverse magnetization is refocused by the later application of a second field gradient, and an attenuated echo is detected. For a free gas or liquid undergoing Brownian motion, the attenuation of a grating as a function of time is an exponential decay characterized by the diffusion coefficient *D*. The echo amplitude is related to the rate of diffusion, the wave number of the grating *k*(*t*), and the diffusion time *t* as

$$A=\text{exp}[-D{\int}_{0}^{t}{k}^{2}\left({t}^{\prime}\right)d{t}^{\prime}],$$

[1]

with

$$k\left(t\right)-k\left(0\right)=\gamma {\int}_{0}^{t}g\left({t}^{\prime}\right)d{t}^{\prime},$$

[2]

where ** γ** is the gyromagnetic ratio, and

As will be demonstrated below, the high signal-to-noise ratio of a laser-polarized sample enables the use of multiple RF excitations to generate multiple simultaneous *k*-space trajectories that produce a series of echoes from which the time-dependent diffusion behavior may be extracted regardless of the underlying physical mechanisms. The greater efficiency of this method is obtained at a cost of more complex spin dynamics. The full character of these dynamics may be straightforwardly followed via the *k*-space formalism of Ref. (17) as generalized in Ref. (18).

While a complete description of the generalized *k*-space formalism is provided in Ref, (18), a brief introduction to the main points will be presented here. Under the influence of RF pulses and magnetic field gradients, the magnetization evolves as a vector field, forming spatial helices with a pitch—or spatial modulation frequency—described by a wave number *k*. For these experiments, the complete spin dynamics are most conveniently explored in a reciprocal space described by a Fourier decomposition basis set of right- and left-handed helices of transverse magnetization and sinusoidal amplitude modulations of longitudinal magnetization (18). The wave numbers {+*k*_{trans}, –*k*_{trans}, *k*_{long}} describing these three basis functions may be thought of as components of a three-dimensional wave vector **k** describing the instantaneous spatial behavior of the sample. This approach leads to a set of *k*-space trajectories that fully describe the spin dynamics. That is, the spatial behavior of the spin density ** ρ**(

The single-shot pulse sequence for diffusion measurements, based on stimulated echoes, is depicted in Fig. 1. A train of small flip angle pulses—a DANTE sequence (19-21)—initiates *k*-trajectories that evolve under the influence of a field gradient. The spatial modulation thus formed is then projected onto the longitudinal axis by the 90° pulse in order to avoid *T*_{2} dephasing during the long diffusion interval ** τ**. The transverse magnetization remaining after the 90° pulse is destroyed by a crusher gradient along a different axis. After a time

Pulse sequence and *k*-space trajectories. The principle pathways in bold are the only ones to survive in the small flip angle limit and form echoes following the readout pulse. Using this sequence the full diffusive behavior of a sample can be read out **...**

To describe the full spin dynamics of the proposed method, the initial DANTE sequence (19) must be explored in more detail. According to Ref. (18), each RF pulse ** ε** transforms each wave vector of the spatial grating as:

$$\left[\begin{array}{ccc}{\text{cos}}^{2}\left({\scriptstyle \frac{\u220a}{2}}\right)\hfill & {\text{sin}}^{2}\phantom{\rule{thickmathspace}{0ex}}\left({\scriptstyle \frac{\u220a}{2}}\right)\hfill & {\scriptstyle \frac{1}{2}}\phantom{\rule{thickmathspace}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\hfill \\ {\text{sin}}^{2}\phantom{\rule{thickmathspace}{0ex}}\left({\scriptstyle \frac{\u220a}{2}}\right)\hfill & {\text{cos}}^{2}\phantom{\rule{thickmathspace}{0ex}}\left({\scriptstyle \frac{\u220a}{2}}\right)\hfill & {\scriptstyle \frac{1}{2}}\phantom{\rule{thickmathspace}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\hfill \\ \text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\hfill & \text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\hfill & \text{cos}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\hfill \end{array}\right]{\left[\begin{array}{c}\hfill +{k}_{\text{trans}}\hfill \\ \hfill -{k}_{\text{trans}}\hfill \\ \hfill {k}_{\text{long}}\hfill \end{array}\right]}_{{\u220a}^{-}}={\left[\begin{array}{c}\hfill +{k}_{\text{trans}}\hfill \\ \hfill -{k}_{\text{trans}}\hfill \\ \hfill {k}_{\text{long}}\hfill \end{array}\right]}_{{\u220a}^{+.}}$$

[3]

After the first ** ε** RK pulse, a transverse magnetization of amplitude sin(

- leaves the bulk of the magnetization along , with amplitude cos
^{2}()*ε* - introduces a new transverse magnetization component at
*k*= 0, with amplitude cos() sin(*ε*)*ε* - rotates part of the transverse grating at +
*k*_{0}back into a grating with the same wave number*k*_{0}, with amplitude sin^{2}()*ε* - rotates part of the transverse grating at +
*k*_{0}into a transverse grating at –*k*_{0}without altering the wave number*k*_{0}, with amplitude sin() sin*ε*^{2}(/2)*ε* - leaves most of the transverse grating at +
*k*_{0}alone, with amplitude sin() cos*ε*^{2}(/2).*ε*

Therefore, immediately after the second pulse the full spatial modulation of the spin magnetization is described by the Fourier components:

$$\begin{array}{cc}\hfill \text{longitudinal magnetization}\phantom{\rule{1em}{0ex}}& @k=0\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\text{cos}}^{2}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{1em}{0ex}}\to 1\hfill \\ \hfill & @k={k}_{0}\phantom{\rule{1em}{0ex}}{\text{sin}}^{2}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{1em}{0ex}}\to 0\hfill \end{array}$$

$$\begin{array}{cc}\hfill \text{transverse magnetization}\phantom{\rule{1em}{0ex}}& @k=+{k}_{0}\phantom{\rule{1em}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{thickmathspace}{0ex}}{\text{cos}}^{2}\phantom{\rule{thickmathspace}{0ex}}(\u220a\u22152)\phantom{\rule{1em}{0ex}}\to \u220a\hfill \\ \hfill & @k=-{k}_{0}\phantom{\rule{1em}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{thickmathspace}{0ex}}{\text{sin}}^{2}\phantom{\rule{thickmathspace}{0ex}}(\u220a\u22152)\phantom{\rule{1em}{0ex}}\to 0\hfill \\ \hfill & @k=0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{thickmathspace}{0ex}}\text{cos}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{1em}{0ex}}\to \u220a\hfill \end{array}$$

which reduce to the amplitudes in the rightmost column in the limit of a small flip angle pulse ** ε**. This process may be continued for later

$${A}_{n}=\text{exp}\left[-D{\left(\gamma gn\delta \right)}^{2}\left(\tau +\frac{2n\delta}{3}\right)\right].$$

[4]

The maximum *k*-value attained for the *n*th echo is * γgnδ*. The storage time along contributes the term in

Continuing the analysis of the echo amplitudes as above yields a full echo amplitude for the *n*th echo of

$$\frac{{A}_{n}}{2}\phantom{\rule{thinmathspace}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left(\u220a\right)\phantom{\rule{thickmathspace}{0ex}}\text{sin}\phantom{\rule{thickmathspace}{0ex}}\left({\u220a}^{\prime}\right)\phantom{\rule{thickmathspace}{0ex}}{\left[\text{cos}\left(\u220a\right)\right]}^{N-n}{\left[{\text{cos}}^{2}(\u220a\u22152)\right]}^{n-1},$$

[5]

where *N* is the total number of echoes produced by the train of *N* RF pulses. Each ** ε** pulse applied before the

The indirect contributions to the echo amplitudes in Eq. [5] from other superimposed pathways are smaller by at least two orders in ** ε**. For example, inverting a grating to a negative

Figure 2 shows an example of the application of the single-shot technique to the measurement of unrestricted diffusion of 3 atm of isotopically enriched laser-polarized ^{129}Xe in a 25-cc cylindrical glass cell. Laser-polarization was induced using the standard spin-exchange optical pumping technique (1) employing infrared light (795 nm) from a fiber-coupled 15-W Optopower diode laser array projected along the ^{129}Xe-filled glass cell heated to ~90°C. The measured echo amplitudes were first corrected for the effects of nonzero flip angle RK pulses by dividing out the sine and cosine terms of Eq. [5]. Plotting the natural log of the corrected echo amplitudes against the diffusion-encoding parameter *b* = (** γ**gn

Results of an eight-echo single-shot diffusion experiment with a laser-polarized ^{129}Xe sample at 3 atm gas pressure and room temperature. Inset: Plot and linear fit of echo amplitudes against the diffusion-encoding factor *b* = (**δ**gn**δ**)^{2}( **...**

While it is possible to work outside of the small pulse limit by calculating the amplitude of all confounding *k*-space trajectories, the additional complications are significant and the added SNR modest. One could alternatively avoid overlapping echoes from different trajectories by spacing the excitation pulses irregularly. However, for the flip angle ** ε** = 6.8° used here, the amplitudes of these overlapping trajectories are less than 1% of the desired echo pathways and do not significantly alter the measured diffusive attenuation curve.

A simple single-shot method is described and demonstrated for characterizing diffusion behavior. This technique is of particular value for magnetization sources that are not readily renewable, such as laser-polarized ^{3}He and ^{129}Xe. Diffusion studies of laser-polarized samples without such single-shot techniques would be quite time-consuming and for a conventional diffusion sequence would require a calibration procedure to normalize the different polarizations associated with each excitation pulse. The large magnetization available in these systems allows the acquisition of multiple measurements in a single step, thereby circumventing the time-consuming repolarization process. The technique employs multiple RF excitations to generate numerous simultaneous *k*-space trajectories that produce a series of echoes from which the diffusive behavior is determined. The complex spin dynamics that result are easily treated by the general *k*-space formalism of Sodickson and Cory (18).

Support for this work was received from NIH Grant RO1-GM52026, NSF Grant BES-9612237, NASA Grant NAG-4920, Whitaker Foundation Grant RG 95-0228, Smithsonian Institution Scholarly Studies Program, and DOE.

*Note added in proof*. A related measurement of time-independent solution state ^{129}Xe diffusion may be found in (22).

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