Using Stimulated Echo in Magnetic Resonance for Research of Correlation and Exchange

1 Introduction

Spin relaxation is a key parameter characterizing the dynamics of spins due to their interaction with the environment. A lot of physical processes can contribute to relaxation. For mixtures, each component of the exponential decay of the signal can have an eigenvalue of the relaxation time, and consequently, the total nuclear quadrupole resonance (NQR) signal from the mixture will be a superposition of many exponential functions. The resulting spectrum of relaxation times allows separation of signals from different components and helps to identify different phases and to control changes in the composition and physical properties of the mixture.

In magnetic resonance, it is necessary that the spins of different molecular groups have different precession frequencies. In many cases, the inhomogeneity of the sample is not resolved in a usual Fourier spectrum but, nevertheless, is reflected in the relaxation times of the T₁ and T₂ spins. Both methods can work on one material, allowing the first experiment to be used as a control one.

One of the key shifts in magnetic resonance technology in recent decades has been the widespread introduction of two-dimensional (2D) and multidimensional methods for the study of relaxation and diffusion. Unlike 2D frequency Fourier spectroscopy, in the 2D relaxation experiment, an inversion of the 2D Laplace transform is required, which is much more complicated than the inversion of the 2D Fourier transform. But this 2D method significantly expands the spectral resolution of relaxation and diffusion and opens new possibilities for studying spin dynamics and molecular motion. For example, chemical exchange and complex dynamics of diffusion can be studied with the help of T₂–T₂ and T₁–T₂ experiments. However, such approaches have not yet been applied to the study of motion in solids.

Two-dimensional exchange relaxometry of nuclear magnetic resonance (NMR) [1] is in many respects similar to the 2D exchange frequency NMR [2] and NQR [3]. Nuclear magnetic resonance relaxometry provides information on diffusion and molecular dynamics and requires minimal field uniformity, in contrast to conventional methods of NMR spectroscopy and NMR images. This method is now finding more and more applications, in connection with the development of stable algorithms for the inverse 2D Laplace transform [4]. The results of applying 2D exchange spectrometry and T₂–T₂ exchange NMR relaxometry using the example of a simple two-component system, feasibility, and limitations of both methods were discussed in Lee et al. [5].

Two-dimensional exchange relaxometry can allow a complex study of the distribution of nuclear spins in crystallites, the exchange rates of magnetisations, and the trajectories of the exchange motion between different positions that differ in relaxation times. The concept of ‘‘exchange’’ implies not only molecular rearrangements such as rotational reorientations, where these movements are accompanied by changes in the relaxation times of NQR, but also the spin diffusion of the magnetisation between different positions of the nuclei.

Although one-dimensional distributions of T₁ and T₂ relaxation times can characterize the physical properties of media, like one-dimensional frequency spectroscopy, they provide limited information in the presence of several components and correlations between different properties. Multidimensional relaxation techniques are capable of eliminating these limitations. Methods such as T₂-diffusion (D–T₂) [6], diffusion–diffusion (D–D) [7], and T₁–T₂ [8] were used for detailed analysis of pore geometry, identification of exchange, and liquid identification by NMR method.

The T₂–T₂ relaxation exchange experiment not only includes information on the 1D measurement of T₂ but also provides additional information on the molecular dynamics on the bonds, as well as information on the T₁ spin–lattice relaxation time [9]. Information on the exchange from the T₂–T₂ topogram is more accessible than from other NMR relaxometry methods, as it is contained in the observed cross peaks. A sample with two different components of T₂ relaxation leads to a T₂–T₂ spectrum with two separate peaks on the main diagonal and a pair of off-diagonal peaks arising from the exchange between these two components.

There are attempts to directly analyze the signal in the time domain [10] for the qualitative determination of the presence of an exchange by the curvature of the signal obtained in dependence to the difference of observations.

In most cases, the exchange and spin evolution cannot be considered separately. This situation is very different from what usually occurs in the exchange NMR Fourier spectroscopy. Interpretation of exchange 2D relaxation maps is much less simple than 2D frequency spectra. In contrast to the case of 2D Fourier exchange spectroscopy, 2D exchange relaxation maps can never be interpreted as an absolute exchange case. Exchange here can never be limited to a period of mixing. Relaxation and exchange should be considered simultaneously.

An alternative method of performing the exchange T₂–T₂ NMR experiment was used in D’Eurydice et al. [11], in which the mixing time was changed, not the number of pulses in the first Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence, which significantly shortened the experiment time.

Unlike the methods used in 2D frequency spectroscopy, where correlation maps are obtained by Fourier transforms, the 2D relaxometry relies on a numerical 2D inversion of Fredholm integrals of the first kind.

In Glotova et al. [12], a new technique for recording 2D spectra was used for the first time, based on the use of nonuniform time intervals in the applied pulse sequences. The method made it possible to shorten the measurement time of the 2D signal interferogram by reducing the number of measured points at time intervals, nonuniformly following samples.

The studies carried out in Sinyavsky and Kostrikova [13] open up new possibilities for the exchange T₂–T₂ relaxometry of NQR, study of spin diffusion, polymorphic transformations, and the state of the surface of microscopic crystals, to study filled microporous media that are promising composite materials for science and industry, thanks to their ability to regulate the dimensions and the mutual arrangement of embedded particles and various pore geometries. The method of quantitative evaluation of the exchange constant of magnetisations without registering cross peaks on the T₂–T₂ topogram makes it possible to substantially reduce both the experimental time and the computer processing time of the results.

Stimulated echo (STE) is widely used in volume selective spectroscopy, in diffusion spectroscopy, to process NMR images, in 2D exchange frequency spectroscopy. A detailed description of the formation of the STE can be found in many articles, for example, in Burstein [14]. Stimulated echo signals are generated by three (or more) radiofrequency (RF) pulses. One can verify that the received signal is indeed an STE by turning off one RF pulse. If it is an STE signal, it will disappear, because all three 90-degree pulses participate in its formation.

In order to get rid of additional echo signals, you can only include the STE signal in the data collection window. Alternatively, you can apply a magnetic field gradient that will dephase the unnecessary signals of the spin echo, leaving the STE signal undistorted. To obtain a nondistorted STE with maximum intensity, before feeding the second RF impulse, the spins must be completely out of phase in the transverse plane. The phase of spins after the third RF impulse is shifted by 180 degrees in comparison with the phase after the first impulse. The maximum STE signal is half the maximum amplitude of the spin echo of Hahn. The STE is reduced by T₁ spin–lattice relaxation between the second and third impulses and due to the effects of T₂ spin–spin relaxation between the first and second impulses and after the third impulse.

It is easily seen that the available publications on 2D exchange relaxometry refer only to NMR spectroscopy for liquids in porous media. Further development of ideas about the influence of molecular motion is connected with the study of powders, polymers, and solids in porous matrices and in microcomposite materials by 2D relaxation NQR methods with inversion of the integral transformation.

The task of this article was to elucidate regularities in the formation of correlation and exchange 2D maps while using STE, including the participation of heavy nuclei with electric quadrupole interactions in molecular crystals. Investigated are possibilities of 2D relaxation NQR with the inversion of the integral transformation to obtain unambiguous quantitative information on the intensities and trajectories of exchange in molecular crystals and on the relationship of relaxation times to the corresponding positions of atoms.

2 Theory

The traditional impulse sequence used to produce an STE consists of three 90-degree RF impulses separated by time intervals t₁ and t₂ (Fig. 1).

Figure 1:

Impulse sequence to record the stimulated echo.

The first RF impulse creates a transverse magnetisation, i.e. stimulates coherences that evolve, decaying with the time of transverse relaxation in the course of t₁ time. The second impulse interrupts evolution and fixes coherences in populations of energy levels. During the time interval t₂, along with the longitudinal relaxation, the magnetisations of states differing in relaxation times are mixed. The third impulse again excites coherences and their evolution after the past process of exchange of frequencies and relaxation times for different states.

For simplified calculation of signals during the exchange, we have chosen a kinetic model with two states. Let the initial magnetisation of two states be equal to: M0=(M0a M0b). Spin–spin and spin–lattice relaxation matrices will be written as follows:

where the relaxation rates of the two states are equal to: R1a=1T1a, R1b=1T1b, R2a=1T2a,R2b=1T2b. The exchange matrix for the two states, when the initial fractions of states are not the same, is written in the form:

We will assume approximately that the exchange between the states takes place on the time interval t₂ (mixing time). The spin–lattice relaxation matrix taking into account the exchange will be written as follows:

and after diagonalization becomes

For the sake of simplicity, let us assume that the exchange coefficients are equal to k_a = k_b = k. Then

and the rotation matrix is given by

where P=(R1a−R1b)2+4k2.

We will consider the case of exact resonance and describe the behaviour of the components of the nuclear magnetisation in the intervals between impulses by the Bloch equations:

Before the first RF pulsing, the components of the magnetisation of the states were equal:

After the first impulse,

Before the second impulse, the component of this vector will become equal to

The second 90-degree RF impulse will result in the following vectors:

At the time of the third impulse, M_z components of the magnetisations can be calculated using the following formula:

After finding the exponentials from the diagonalised matrices in (12), returning to the original representation using the rotation matrix Q, we find the M_z components of the magnetisations before the start of the third impulse and then after the third impulse. The amplitude of the STE signal at time 2t₁ + t₂, determined by the sum of the magnetisations of both states, will be

The first and fourth terms in this formula give the cross peaks ab and ba, correspondingly, on T₁–T₂ topogram. The diagonapeaks a and b give the second and the third terms in (13), respectively.

The amplitudes of the peaks on the topogram can be written in the following form:

where R1d=R1a−R1b. By simple transformations, one can obtain an exchange constant formula that can be calculated using the coordinates and amplitudes of the peaks from the experiment:

For simplicity, in order to avoid cumbersome formulas, in this article we restricted ourselves to the case of exact resonance. Off-resonance effects for 2M-exchange frequency spectroscopy of ³⁵Cl NQR were considered in our work [3].

3 Results and Discussion

For the experimental studies, the Tecmag Apollo spectrometer was used, with the TNMR software (United States Tecmag, Inc., Houston, TX, USA). For the inversion of the integral transform, the general algorithms Non-Negative Least Squares (NNLS) and Singular Value Decomposition (SVD) are used [15].

For the standard impulse sequence of the STE (Fig. 1), in the presence of one spin system state and the absence of exchange, the NQR signal will be described by the following expression:

where S(t1,t2) – measured signal as a function of two times.

In the case of a two-phase state where a substantial content of resonant nuclei can be associated with both phases and contribute to the NQR signal, in the presence of exchange between the states, in order to obtain the 2D distribution of the relaxation times f (T₁, T₂), one should use inversion of integral transformations, using (13). The dependence of the amplitudes of the diagonal and cross peaks on the T₁–T₂ relaxation topogram on the exchange constant k, according to (14), is illustrated in Figure 2.

Figure 2:

Dependences of the intensities of diagonal and cross peaks on the T₁–T₂ topogram on the exchange constant k (M_0a = 0.75, M_0b = 0.25, T_1a = 200 ms, T_2a = 2 ms, T_1b = 20 ms, T_2b = 0.2 ms).

Figure 3:

Simulation of T₁–T₂ correlation topogram for the two states in the absence of exchange (a) and in the presence of exchange (b). Initial values: M_0a = M_0b = 0.5, T_1a = 200 ms, T_2a = 2 ms, T_1b = 20 ms, T_2b = 0.2 ms, k = 0 I k = 0.05 ms⁻¹, respectively).

Figure 4:

Correlation between relaxation times T₁ and T₂, obtained from the dependency of the intensity of the ³⁵Cl NQR stimulated echo signal for the two phases of the polymorphous 1,4 dichlorobenzene (a) and ³⁵Cl NQR spectrum at a temperature of T = 298 K for p-C₆H₄Cl₂, which impregnated in its molten condition a tree [16] (b).

Simulation of the 2D relaxogram for two states in the absence and in the presence of exchange using (13) and inversion of the 2D Laplace transform is shown in Figure 3a and b. As can be seen from these figures, if there is an exchange, cross peaks appear on the topogram, and the diagonal peaks move along the horizontal axis. This bias is due to the fact that, for the sake of simplicity, the presence of exchange in the model was assumed only in a long time interval t₂.

Quantitatively, the value of the exchange constant can be calculated by (15), using the coordinates of the peaks and their amplitudes from the experiment. To obtain the coefficient k using (15), it is sufficient to measure the coordinates of the peaks 1/λ_a and 1/λ_b and the amplitudes A_a and A_ab using a 2D correlation map. The k value obtained from the 2D correlation map (Fig. 3b) and (15) is 0.052 ms⁻¹, which well corresponds to an input value of 0.050 ms⁻¹.

In the presence of noise, the accuracy of the method decreases. As is known, the method of relaxometry with the inverse Laplace transformation works correctly by a sufficiently large signal-to-noise ratio (at least 5–10).

The experimental T₁–T₂ relaxogram, obtained by the STE method for the NQR ³⁵Cl case in two-phase paradichlorobenzene, as a result of the inversion of 2D Laplace transform from the signal S(t1,t2) , is shown in Figure 4a. The experiment is essentially three-dimensional. The time of the experiment depends on the relaxation times and the number of accumulations. The experiment shown in Figure 4a was represented by an array of 30 × 30 points. The number of accumulations was 1024. The time of the experiment was about 28 h. Such time is typical of the most multidimensional experiments.

It was shown in [16] that in porous materials, the pores of which were filled with molten paradichlorobenzene, no p-C₆H₄Cl₂ transition was observed in the pores with time from the β phase to the α phase. Thus, in the pores, the β phase is stable. In Figure 4b, the NQR ³⁵Cl line corresponding to the β phase belongs to the substance in the pores, and the α-phase line to the paradichlorobenzene on the surface of the wooden sticks. Retention of the β phase in porous materials in [16] is explained by the effect of limited pore sizes, which prevent the structure from being rearranged into the α phase.

The presence of two phases of 1,4-dichlorobenzene having different relaxation times is presumably accompanied by an exchange of magnetisations at the boundary of the media. In Figure 4a, we present the results of a more successful STE experiment, where exchange cross peaks are observed on a 2D T₁–T₂ map. Calculated by means of (15) using the coordinates and amplitudes of the 2D peaks, the exchange coefficient is k = 0.42 ms⁻¹.

The accuracy of the method depends on both the signal-to-noise ratio of the measured 2D interferogram S(t1,t2)S and the number of points in the 2D inverse Laplace transformation. The method also requires good stabilization of the sample temperature. In order for the method to be acceptable by the duration of the experiment, it should be applied for the case of samples with not very large relaxation times. The distributions of relaxation times for different states should be well resolvable on a 2D map.

4 Conclusions

Thus, the kinetic equations describing evolution of magnetisations due to the relaxation and exchange processes for a two-phase spin system have been analyzed. Expressions have been obtained for the coordinates of the diagonal and cross peaks T₁ and T₂ and their amplitudes for the case of exchange between two positions, which allow us to determine the exchange constants k from the 2D experiment of the STE. Simulations of distributions of relaxation times for different values of the k parameter have been performed. Based on the STE sequence, a 2D relaxation experiment for the ³⁵Cl NQR for a two-phase 1,4-dichlorobenzene has been realized.

The results presented here make it possible to optimize and meaningfully carry out experimental studies of dynamic processes, including participation of heavy nuclei in molecular crystals. The formula obtained in this article will allow us to determine the exchange constants for the case of two states; however, the proposed experimental technique can be extended to the case of exchange between several states. Experiments on 2D exchange relaxometry have a great potential, providing information on molecular dynamics, bonds, and diffusion between different states. The 2D relaxometry method for visualization of the exchange will be useful in many cases, when the usual frequency 2D spectra, by their simplicity, do not reflect complexity of the materials being studied. In NMR, for example, it is the exchange of water molecules through cell membranes that separate structures in which the relaxation times vary significantly.