Konstantin L Ivanov

Thomas Kress

Mathieu Baudin

David Guarin

Daniel Abergel

Geoffrey Bodenhausen

Dennis Kurzbach

Relaxation of long-lived modes in NMR of deuterated methyl groups

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

NOMENCLATURE

DNP

= dynamic nuclear polarization D-DNP = dissolution dynamic nuclear polarization NMR = nuclear magnetic resonance LLS = long-lived state SSI = spin state imbalance

I. INTRODUCTION

In nuclear magnetic resonance (NMR), the combination of long-lived spin states (LLSs) 9,10,[START_REF] Levitt | Singlet nuclear magnetic resonance[END_REF]20,21 with hyperpolarization methods based on dissolution dynamic nuclear polarization (D-DNP) 3 has the potential to preserve nuclear hyperpolarization over periods of time much longer than spin-lattice relaxation times, which opens new avenues for the study of slow chemical exchange 1,8,24 and slow transport phenomena. 2,22,25 An illustrative example of an LLS is the singlet state of a pair of spins with I = 1 / 2 , such as a pair of protons in H 2 molecules or -CH 2 -fragments. Since the intra-pair dipolar coupling is symmetric with respect to spin exchange, it cannot induce a) Authors to whom correspondence should be addressed: Ivanov@tomo.nsc.ru and Kurzbach@ens.fr transitions between the anti-symmetric singlet state and the symmetric triplet states. Such forbidden transitions can only be induced by extra-pair dipole-dipole interactions, paramagnetic effects, anisotropic chemical shifts, or spin rotation effects. However, as dipolar relaxation by intra-pair dipolar couplings is often dominant, the lifetime of imbalances between the populations of singlet and triplet states can be much longer than spin-lattice relaxation times, provided the intramolecular dipolar coupling is efficiently suppressed. For quantitative calculations of the lifetimes of population imbalances between manifolds of different symmetry, cross correlations of the fluctuations of distinct dipole-dipole interactions must be considered, but auto-correlations of dipolar interactions suffice for a qualitative understanding of the flow of populations between manifolds of different symmetry. In 13 C 1 H 3 groups, cross-relaxation between spin states that belong to different irreducible representations A, B, E1, and E2 of the C 3v point group can only be explained by correlated fluctuations of the 1 H-1 H and 1 H- 13 C dipole-dipole interactions. 12 LLSs have been studied 12,13,[START_REF] Meier | [END_REF] in many systems comprising two or more protons or heteronuclei such as 13 C, 19 F, etc. Molecules containing at least two magnetically equivalent deuterium nuclei that have spin S = 1 (in this paper, we shall use the symbol "D" rather than 2 H for simplicity) imbalances between populations of spin states of different symmetry can give rise to LLS, in the sense that their lifetime T LLS (D) can be longer than their Zeeman spin-lattice relaxation times T 1 (D), which are usually short because of rapid quadrupolar relaxation of deuterium nuclei. [START_REF] Jhajharia | [END_REF]16 In the extreme narrowing regime, deuterium spin-lattice relaxation rates R 1 (D) = 1/T 1 (D) are proportional to ω 2 Q τ C where ω Q is the quadrupolar coupling and τ C is the correlation time of molecular tumbling.

To create LLSs in deuterated 13 CD 2 and 13 CD 3 groups, we employed dissolution dynamic nuclear polarization (D-DNP), [START_REF] Jhajharia | [END_REF] which allows one to generate imbalances between the averaged populations of the A-states and E-states, thus giving rise to long-lived states.

A description of LLS in 13 CD 2 groups has been outlined elsewhere. [START_REF] Jhajharia | [END_REF] The nine energy states of the D 2 subsystem can be separated into six sextet states that are symmetric with respect to the permutation of the two deuterium nuclei, and three triplet states are anti-symmetric upon permutation. The LLS has been identified as a "sextet-triplet imbalance" (STI). To evaluate the lifetime T STI , a 18 × 18 relaxation matrix can be constructed containing relaxation rates that have been determined using a model assuming that (i) relaxation was driven by quadrupolar relaxation, the two quadrupolar tensors being assumed to be axially symmetric and aligned along the 13 C-D bonds, and (ii) molecular motion was assumed to be described by anisotropic rotational diffusion so that the two quadrupolar tensors were in effect averaged about the two-fold axis. [START_REF] Jhajharia | [END_REF] The analysis of LLSs in 12 CD 3 groups is more complex since the system comprises 27 states. (Extensions to 13 CD 3 groups that comprise 54 states will not be considered here. If we consider a deuterated methyl group containing a 13 C spin, the stepwise reduction can be conducted in a similar manner, provided one neglects multispin order involving the carbon nucleus.)

The relaxation matrix can be constructed [START_REF] Jhajharia | [END_REF] in the basis of symmetry-adapted spin states as originally devised by Bernatowicz et al. 6 The symmetry-adapted states are discussed in the Appendix. The origin of LLSs can be ascribed to generalized spin-state imbalances (SSIs), which can be identified qualitatively as population imbalances that persist due to the inefficient flow of populations between spin states belonging to different irreducible representations. However, the actual long-lived population modes were not identified, and their lifetimes (i.e., the inverse of their relaxation times) have not yet been determined and remain an open question. One should note that the case of 13 CD 3 groups is significantly more complex compared to the 13 CD 2 case, not only because of the much larger relaxation matrix, but also because the symmetry-adapted states do not merely belong to two, but to four distinct symmetry manifolds (A, B, E1, and E2). Several population imbalances that can potentially lead to LLSs need therefore to be considered, since there are 6 pairs of manifolds.

In this work, we present an approach that allows one to reduce the dimensionality of the relaxation matrix of a D 3 group from 27 × 27 to 3 × 3. This enables the determination of long-lived modes of "lumped populations" as well as their relaxation rates and lifetimes. These modes can be monitored indirectly by inspection of the scalar-coupled multiplet in the 13 C NMR spectrum of a 13 CD 3 group. This indirect observation enables the quantitative interpretation of experimental results and provides an understanding of the key factors governing the relaxation behavior of LLSs in deuterated methyl groups.

An extension to protonated methyl groups is straightforward, although one has to take into account 3 homoand 3 heteronuclear dipole-dipole couplings and 4 chemical shift anisotropy (CSA) tensors and a variety of cross correlations between their fluctuations, rather than merely 3 dominant quadrupolar interactions. Yet, due to the much smaller dimension of the Zeeman space (8 × 8 instead of 27 × 27), the theoretical treatment can proceed without the reduction of the dimension of Liouville space demonstrated here.

II. THEORY

The approach developed here has been described in a previous publication [START_REF] Jhajharia | [END_REF] where the 27 × 27 dimensional relaxation matrix R had been introduced. Only relaxation mechanisms due to quadrupolar couplings have been considered, all other mechanisms being neglected. Two molecular motions are considered in this model: (i) isotropic tumbling of the molecule with a correlation time τ C and (ii) rotation of the methyl group around the C-C bond with a correlation time τ R . The model has been implemented in Mathematica using the SpinDynamica package, adapted from the work by Levitt and co-workers for protons. 4 The R-matrix has been calculated in the Redfield approximation in the extreme narrowing limit. Previous work had revealed two types of long-lived states in 13 CD 2 and 13 CD 3 groups, (i) one in-phase with the attached carbon and (ii) another in antiphase with respect to the carbon nucleus. They give rise to (i) deviations of the relative intensities of the carbon-13 septet from the ideal hightemperature 1:3:6:7:6:3:1 distribution, in such a manner that the central line has varying intensity with respect to the outer lines and (ii) an asymmetry of the septet with respect to its center. The latter effect is associated with 13 C-D multi-spin order terms and is not considered here, since its description requires an extension of the dimension of R to 54 × 54 to include the carbon-13 nucleus. We focus on the LLS associated with population imbalances of the spin states of the 3 deuterons.

Generally, relaxation processes of populations involving N states of the symmetry-adapted basis can be described in Liouville space, where the N × N-dimensional density matrix ρ can be developed to form an N 2 -dimensional columnvector, while the relaxation rates given by the relaxation superoperator R can be expressed by an N 2 × N 2 -dimensional matrix R. The auto-relaxation rate R ii of the population of the ith state corresponds to the diagonal element of the matrix R, whereas the cross-relaxation rate R ji that describes the flow between populations p i and p j corresponds to the off-diagonal element R jj,ii between two elements ρ ii and ρ jj . The secular approximation allows us to neglect all elements that connect populations and spin coherences, apart from zero-quantum coherences (ZQCs) between degenerate states. Hence, we can omit all coherences of order p ≥ 1 and need not consider the entire N 2 × N 2 -dimensional matrix R by taking only the elements that are responsible for the interconversion of the populations and ZQCs. We shall describe the column-vector of state populations P, comprising the elements {P} i = p i , and the relaxation matrix R comprising the following elements:

R ij = -R ii,jj = -R ji when i j; R ii = j i R ij . ( 1
)
Here we consider the R-matrix as symmetric. The time evolution of the P-vector obeys the following differential equation:

dP dt = -R(P -P th ), (2)
where P th stands for the thermal populations of the nuclear spin states.

In CD 3 groups, the elements of the 27 × 27-dimensional matrix R are the rates of auto-and cross-relaxation between the populations of the 27 different states of the 3 deuterium spins, using the symmetry-adapted basis defined by Bernatowicz et al. 6 A graphical representation of this relaxation matrix is depicted in Fig. 1 for the case where τ C τ R together with the set of symmetry-adapted manifolds of spin states. For Fig. 1(a), the states have been ordered according to the irreducible representations, leading to a block-diagonal form, showing the lack of cross-relaxation between different symmetry manifolds; this feature allows LLSs to occur. In Fig. 1(a), the colors reflect the cross-relaxation rates (offdiagonal elements shown in white are very small although they are not necessarily zero). One should note that such an appearance of the R-matrix, corresponding to fast relaxation within manifolds of similar symmetry and negligible relaxation between different manifolds, is expected only when there is a hierarchy of motional correlation times, τ C τ R .

First step in the reduction of the dimension. From Fig. 1, it is obvious that some of the off-diagonal elements of the relaxation matrix are large (represented by dark colors), meaning that the cross-relaxation rates are fast, while some others are small so that cross-relaxation can be neglected. For the individual states within each manifold, we can assume that they rapidly acquire identical populations since all populations within each spin manifold are equilibrated rapidly by crossrelaxation. For example, we can average the populations of the 7 + 3 = 10 states of symmetry A. Likewise, we can average the populations of the 5 + 3 = 8 states of symmetry E1 and of the 5 + 3 = 8 states of symmetry E2. Thus, we can group the state populations by adding populations belonging to the same irreducible representations,

P A = 10 i=1 p i , P B = p 11 , P E1 = 19 i=12 p i , P E2 = 27 i=20 p i . (3)
This amounts to a reduction of the dimension of Liouville space from 27 × 27 to 4 × 4. This assumption allows us to sum the corresponding equations in system (2) and to obtain the equations for the populations P A , P E1 , and P E2 . For example, for the A-states we write

dP A dt = 10 i=1 dp i dt = - 10 i=1 27 j=1 R ij ∆p j = -R 11 ∆P A -R 12 ∆P E1 -R 13 ∆P E2 -R 14 ∆P B . (4)
Here we introduce the elements of the relaxation matrix R in the reduced 4 × 4 space. The elements can be calculated as follows: All other elements of R can be evaluated in a similar way. In contrast to R, the R matrix is no longer symmetric, resulting in unequal populations of the lumped states at thermal equilibrium, as discussed below. By lumping together the populations within the same manifold, we can introduce a new 4-dimensional column-vector of populations P with the elements P A , P E1 , P E2 , and P B and construct a new differential equation with a 4 × 4 relaxation matrix,

R
dP dt = -R (P -P th ). ( 6
)
The elements of P th are the thermal populations of the A, E1, E2, and B states. The new relaxation matrix R can be diagonalized analytically. Its eigenvalues λ k give the effective auto-relaxation rates of the averaged populations. One of the modes, λ 1 , is always zero corresponding to the fact that the sum of all four lumped populations must always remain equal to 1. The corresponding eigenvector V 1 is simply a weighted sum of the four averaged populations,

V 1 = 1 √ 229 10 1 8 8 . ( 7
)
The weights of this vector are equal to the number of states in the corresponding batches of averaged states. The remaining three normalized eigenvectors are found to be,

V 2 = 1 2 1 1 -1 -1 , V 3 = 1 √ 42 5 -1 -4 0 , V 4 = 1 √ 2 0 0 1 -1 . ( 8
)
At thermal equilibrium, only the V 1 mode is populated, since the other three relax to naught due to multiplication by exponents e -λ k t , which tend to zero for large t since the relaxation rates are positive λ k > 0.

Long-lived modes. The non-vanishing λ k rates have been determined analytically by using Mathematica supplemented with a SpinDynamica 4 routine to calculate the R-matrix, to evaluate the R -matrix, and to determine the eigenvalues. Such a treatment yields the following values of λ k :

λ 2 = 3 2 λ 3 , λ 3 = λ 4 = 81 64 τ C τ R ω 2 Q sin 2 θ 17τ C + 5τ R + 3(5τ C + τ R ) cos 2θ (τ C + τ R )(4τ C + τ R ) , (9)
where θ = 70.5 • is the angle between the three-fold symmetry axis of the methyl group and the CD-bonds so that sin θ = 2 √ 2 3

and cos θ = 1 3 , yielding

λ 2 = 9τ C τ R (2τ C + τ R ) 2(τ C + τ R )(4τ C + τ R ) ω 2 Q , λ 3 = λ 4 = 3τ C τ R (2τ C + τ R ) (τ C + τ R )(4τ C + τ R ) ω 2 Q ( 10
)
where all eigenvalues have units of s -1 . Already at this stage, the analysis reveals the existence of long-lived modes. Indeed, if we assume that τ R τ C (a common assumption for methyl group dynamics), the relaxation rates are independent of τ C ,

λ 2 = 9 4 ω 2 Q τ R , λ 3 = λ 4 = 3 2 ω 2 Q τ R . ( 11
)
Since the longitudinal Zeeman relaxation rate is

R 1 (D) = τ c ω 2 Q 9 + 24 τ R τ R + τ C + 48 τ R τ R + 4τ C , (12)
we obtain the following ratio:

3λ 2 2R 1 = λ 3 R 1 = λ 4 R 1 = 3τ R (τ R + 2τ C ) 4 4τ 2 C + 21τ C τ R + 9τ 2 R . ( 13
)
If we assume again that τ R τ C ,

λ 2 R 1 = 3τ R 8τ C 1. ( 14
)
Hence, the specified modes relax much slower than zmagnetization. The reason is that upon fast rotation of a CD 3 group, the three quadrupolar interaction tensors are effectively averaged so that the resulting mean quadrupolar interaction is invariant upon rotation. Consequently, crossrelaxation between states belonging to manifolds of different symmetries is suppressed. Since dipolar relaxation is also unable to drive cross-relaxation between such states, one can expect to see long-lived populations modes in CD 3systems. A similar reasoning has already been emphasized in 1979 by Poupko et al. 23 The authors demonstrated that transitions between states of different irreducible representations vanish when the two spins are equivalent and the corresponding interaction tensors are aligned. In other words, any flow of population between states of different symmetry that involves one or more degenerate spins is forbidden to first order. In the case under study, this means that if a CD 3 group rotates so fast that the three deuterons are no longer discernible, one or several LLS due to an SSI can be expected. On the contrary, we do not expect long-lived relaxation modes for CD 3 groups featuring high energy barriers and slow rotation around the C 3v axis (where the assumption τ R τ C breaks down) since the three deuterons are no longer equivalent.

Note that the above equations are based on analytical solutions of the entire product operator space, prior to the reduction of the dimensions, so that all coherences are considered in addition to the populations. Note that zero-quantum coherences spanning nearly degenerate energy states can oscillate with relatively low frequencies and may perturb the dynamic separation of different symmetry manifolds, as discussed in some detail in earlier work. 5,7,16 Second step in the reduction of the dimension. To simplify the problem even further, we note that, first, two modes V 3 and V 4 have identical relaxation rates λ 3 = λ 4 and, second, the difference in the populations of the E1 and E2 states cannot be probed by NMR because these states contribute to the same lines of the 13 C-NMR multiplet in a 13 C-D 3 group. Therefore, the populations of the E1 and E2 states can be lumped together: we do not consider the lumped populations P E1 and P E2 separately but instead introduce the total population of the E-manifold, P E = P E1 + P E2 . Hence, we will work in a reduced 3 × 3 manifold of spin states. In this situation, the relaxation rates are

λ 1 = 0, λ 2 = 9τ C τ R (2τ C + τ R ) 2(τ C + τ R )(4τ C + τ R ) ω 2 Q , λ 3 = 3τ C τ R (2τ C + τ R ) (τ C + τ R )(4τ C + τ R ) ω 2 Q ( 15
)
and the eigenvectors corresponding to the lumped modes of the relaxation matrix are

V 1 = 1 √ 357 10 1 16
, V 2 = 1 √ 6 1 1 -2 , V 3 = 1 √ 42 5 -1 - 4
. ( 16)

The first mode again corresponds to the sum of all 27 populations, which must be a constant of motion. The coefficients reflect the dimensions of the sub-spaces and the fact that in thermal equilibrium, P A = 10 27 , P B = 1 27 , P E = 16 27 . The other two modes correspond to population imbalances that have long lifetimes due to their small auto-relaxation rates λ 2 and λ 3 ,

M 2 = 5P A -P B -4P E , M 3 = P A + P B -2P E . ( 17
)
These are the two observable long-lived modes in CD 3 -groups when the three-fold rotation is fast.

In principle, it is also possible to lump the states in one step, from 27 × 27 to 3 × 3 matrices, by directly lumping together all E states into a single state. However, for the sake of clarity, we broke the reduction down into two consecutive steps via a 4 × 4 matrix.

Detection of long-lived modes. To reveal manifestations of the long-lived modes M 2 and M 3 in the 13 C-multiplet of a 13 CD 3 group, we first express the intensities of the seven NMR lines of the septet, L m (where m = 1,. . .,7), in terms of the populations P A , P B , P E . As explained above, we do not distinguish the populations of individual states within the three manifolds. This means that we neglect differences in intensities between NMR transitions that feature similar symmetry properties, for instance, the difference in intensity between transitions L 1 and L 7 , which both correspond to the A manifold. Thus, we only consider linear combinations that depend on the total populations of the different symmetry manifolds,

         L a = L 1 + L 7 = 2 10 P A , L b = L 2 + L 6 = 2 10 P A + 4 16 P E , 2L b = L 3 + L 5 = 4 10 P A + 8 16 P E , L 4 = L c = 2 10 P A + P B + 4 16 P E . (18)
We readily see that two combinations, (L 2 +L 6 ) and (L 3 +L 5 ), are linearly dependent; hence, it is sufficient to consider their weighted sum [2(L 2 +L 6 )+(L 3 +L 5 )]/2. Now we can express the averaged populations as a function of the line intensities,

P A = 5L a , P B = L c -L b , P E = 4(L b -L a ). ( 19
)
Using these formulas, we can easily express the averaged populations of the long-lived modes, M 2 and M 3 , via the NMR line intensities and extract them from experimental 13 C-NMR multiplets. This allows us to monitor the time dependence of M 2 and M 3 ,

M 2 = 1 2 (82L 1 -30L 2 -15L 3 -2L 4 -15L 5 -30L 6 + 82L 7 ), M 3 = 13L 1 -9L 2 - 9 2 L 3 + L 4 - 9 2 L 5 -9L 6 + 13L 7 .
To probe the existence and lifetime of the two longlived modes M 2 and M 3 , we employed dissolution dynamic nuclear polarization (D-DNP), a method originally devised by Ardenkjaer-Larsen and co-workers in 2003 3 and which has since developed into a powerful technique to investigate nonequilibrium quantum states that would remain unnoticed by means of conventional NMR spectroscopy. 26 In brief, a solid sample of deuterated ethanol-d 6 or DMSO-d 6 , both containing a 13 CD 3 moiety, supplemented with 50 mM of the nitroxide TEMPOL and 50% v/v glycerol-d 8 as cryo-protectant, was irradiated for 3 h by microwaves with a frequency near 188.2 GHz (modulated over a range of 100 MHz with a modulation frequency of 2 kHz, and a power of ca. 35 mW at the locus of the sample) in a prototype dynamic nuclear polarization (DNP) apparatus operating at 1.2 K in a magnetic field of 6.7 T. In this manner, the D as well as 13 C nuclei present in the 13 CD 3 groups are simultaneously hyperpolarized leading to an overpopulation of the lowest or highest energy state, depending on the microwave frequency. The symmetry of the predominantly populated spin state depends on a multitude of factors including the microwave frequency and the methyl group tunneling probability; 13 a detailed discussion of these factors is beyond the scope of this manuscript. For the case at hand, it suffices to note that DNP at low temperatures will overpopulate one of the symmetry manifolds.

After hyperpolarization, the sample it is rapidly dissolved by a burst of 5 mL D 2 O pressurized to 1.05 MPa, squirted onto the sample. The dissolved sample is transported within 4 s to a 9.4 T NMR spectrometer where NMR spectra are taken in a time-resolved manner every second by applying 30 • detection pulses on the 13 C nuclei of the 13 CD 3 groups. The septets are visible for about 2 min, at which time the carbon polarization becomes too small to be detected in a single scan.

It is important to note that the longitudinal deuterium relaxation, as characterized by T 1 (D), is rather fast, typically on the order of a second. We can thus safely assume that the Note that the lines are much narrower in thermal equilibrium than in the hyperpolarized case. This may result from convection and small helium gas bubbles that perturb the homogeneity. The intensity ratios between different lines are not perturbed through this.

Zeeman distribution of the D 3 system is in thermal equilibrium after the dissolution and transfer which take about 4 s. In other words, within each of the symmetry manifolds, A, B, and E, the distribution of populations is in thermal equilibrium during the detection period. All deviations of the septets from their thermal equilibrium intensities of 1:3:5:7:5:3:1 at times longer than T 1 (D) can therefore be attributed to the presence of long-lived SSI.

Typical spectra resulting from hyperpolarization and dissolution are displayed in Fig. 3 for the 13 CD 3 groups of DMSO-d 6 , once 10 s after injection (Fig. 2(a)), after 30 s (Fig. 2(b)), and in thermal equilibrium (Fig. 2(c), averaged over 3584 scans). To determine the intensities of the 7 lines, each one was individually fitted to a Lorentzian (dashed blue lines in Fig. 2). The intensities (integrals) L i of the 7 lines (the numbering is indicated in Fig. 2(a)) were normalized to the central line L 4 . Details on the determination of the integral values are given in Sec. IV. In brief, the spectra were fitted to 7 independent Lorentzians of similar linewidth and split by coupling constants J CD (EtOD) = 19.1 Hz and J CD (DMSO) = 21.2 Hz. The method was adapted from a procedure developed by Miclet et al. 19 Here we used a Nelder algorithm for fitting. All peaks were fitted with a similar linewidth and the different lines are separated by a single J-coupling. For details, see Sec. IV.

Deviations of the intensities of the seven lines from their equilibrium intensity distribution 1:3:6:7:6:3:1 hint toward non-equilibrium deuterium polarizations. For example, the relative intensities L 1 and L 7 are smaller after hyperpolarization than at thermal equilibrium. (As we show below, this is because the E-states are overpopulated.) The very high signal intensities in D-DNP experiments allows us to access such subtle differences.

The carbon nucleus of the 13 CD 3 group is also hyperpolarized after dissolution (Fig. 2(a)). Its enhancement factor was ca. 22 000 corresponding to a 13 C polarization of ca. 18% directly after dissolution and transfer. Similar observations were made for ethanol-d 6 . After hyperpolarization, dissolution, and transfer, the carbon polarization decays toward thermal equilibrium, a process that is dominated by the longitudinal auto-relaxation rate 1/T 1 ( 13 C). This decay is documented in Fig. 3 for both molecules. Each line of the carbon septet is monitored individually in Fig. 3(a) for DMSO-d 6 . The time traces were fitted to mono-exponential decay functions. The corresponding relaxation times are shown in Fig. 3(b), likewise for ethanol-d 6 in Figs. 3(c) and 3(d). The relaxation times of the different lines of the carbon septet span a range between 6 and 12 s, i.e., by a factor up to two. This difference cannot be accounted by the here presented theory as it involves the carbon nucleus of the CD 3 group, but is a consequence of the presence of an SSI as has been shown in our earlier work. [START_REF] Jhajharia | [END_REF] To analyze the SSIs spanning the 13 CD 3 groups, we calculated the values P A , P B , and P E according to Eq. ( 19) as well as the two long-lived modes M 2 and M 3 according to Eqs. ( 17) and ( 19) for both DMSO-d 6 and ethanol-d 6 . The results are shown in Figs. 4 and5. To remove biases due to carbon relaxation from the analysis, we normalized each line to the total intensity summed over the seven lines before applying Eqs. ( 17)-( 19) to the integrated intensities of the seven individual lines. Figures 5(a) and 5(b) show the time dependence after hyperpolarization, dissolution, and transfer to the detection spectrometer of three different populations P A , P B , and P E . Evidently, the time trace for P B is flat, indicating that DNP most likely does not affect the population of the B manifold. On the contrary, the populations P A and P E show a strong time dependence, which is increasing for P A and decreasing for P E . This behavior was observed for both molecules. These observations can be interpreted as an imbalance that starts with an overpopulation of the E manifold and a depleted population of the A manifold. These imbalances decay with time toward their thermal equilibrium values. However, the decay does not appear to be mono-exponential, likely due to cross-relaxation phenomena.

In Figs. 5(a) and 5(b), we calculated the time dependence of the two long-lived modes M 2 and M 3 from the populations P A , P B , and P E . Both modes decay with time, which can be attributed to the imbalance identified above which enters the definition of Eq. ( 17). We find a characteristic relaxation time of 19.0 ± 2.2 s for M 2 and 18.5 ± 2.1 s for M 3 for DMSO-d 6 .

The T 1 ( 2 H) auto-relaxation time in DMSO-d 6 has been determined to be 0.7 s for the 12 CD 3 groups of DMSO-d 6 . This yields a prolongation of the relaxation time of deuterium by the factor of 19.0/0.7 = 27.1. This finding is in good agreement with earlier results. [START_REF] Jhajharia | [END_REF] For ethanol-d 6 , we find a decay time of 12.6 ± 2.4 s for M 2 and 12.7 ± 2.4 s for M 3 , while T 1 ( 2 H) = 2.7 s, which is astonishingly long for deuterons and points toward fast rotation of the 13 CD 3 group and results in a prolongation of 12.6/2.7 = 4.7. Note that in Fig. 5, M 2 and M 3 have been normalized by the integral of the septets. This assures the absence of bias due to 13 C auto-relaxation but causes the "noisy" appearance of the time dependence at longer times.

Our simplified approach cannot account for the observation of significant differences between the intensities L 1 and L 7 , as well as between the pairs L 2 , L 6 , and L 3 , L 5 , i.e., between pairs of lines that appear symmetrically with respect to the center of the septet. Such asymmetries are a consequence of the coupling of deuterium SSIs to the detected carbon-13 nucleus, or of SSIs that cross-relax into multispin order terms, which produce antiphase carbon magnetization with respect to the attached deuterons upon detection (see Kurzbach et al. 16 ). However, these deviations are not considered here since the combinations underlying the modes M 1 and M 2 sum over pairs of symmetrically disposed and therefore ignore such asymmetries.

III. CONCLUSIONS

The analysis of symmetry properties in spin systems such as the CD 3 groups presented here is challenging because of the dimensions of the Liouville space; hence, the corresponding relaxation matrices become quite large. The proposed method here for the reduction of the dimensionality of relaxation matrices helps to identify important lumped modes and relaxation properties. Indeed, with our method, we can identify longlived spin state imbalances in CD 3 groups and analytically calculate their lifetimes avoiding diagonalization of the original 27 × 27 relaxation matrix. Using this method, we find that the quadrupolar relaxation of long-lived modes in CD 3 groups is independent of the overall tumbling of the molecule containing the CD 3 groups, but depends-according to our calculations-only on the rotational correlation time of the methyl groups describing methyl group jumps. This method might help to predict long-lived states in even more complex systems and extend applications of long-lived spin states to NMR spectroscopy and imaging. In this work, we confirm that long-lived modes can be generated and analyzed for spin-1 nuclei, such as 2 H. We anticipate that such longlived states can be utilized for probing dynamics of various molecules and for extending the D-DNP methodology to 2 H nuclei.

IV. EXPERIMENT

Samples were prepared by mixing 50% v/v of glycerol-d 8 with either of two different solutions of 50 mM TEMPOL in 80% DMSO-d6/10% H 2 O or in 80% ethanol-d6/10% H 2 O. Using a Bruker BioSpin prototype D-DNP apparatus, 50 µL of each sample were hyperpolarized at 1.2 K and 6.7 T for ca. 2 h by direct microwave irradiation near 188.38 GHz. The microwave carrier frequency was modulated with a saw-tooth wave over a range of 100 MHz with a modulation frequency of 2 kHz.

The samples were dissolved in 5 mL D 2 O at 180 • C under 1.05 MPa pressure and propelled by helium gas within 3 s via a magnetic tunnel that maintains a minimum field near 0.9 T to a 9.4 T detection NMR spectrometer equipped with a 10 mm broadband "β-barium borate (BBO)" probe for 13 C detection. The temperature was adjusted to 298 K. Radiofrequency pulses with 30 • nutation angles were applied every 1 s for detection. The data were processed, zero-filled, and apodized using NMRPipe, 11 and analyzed with home-written scripts using the SciLab software package. [START_REF] Enterprises | Scilab: Le logiciel open source gratuit de calcul numérique[END_REF] The envelopes I exp of the 7 experimental lines were fitted by minimizing I exp -I fit 2 by a Nelder algorithm to reproduce the shape of the spectrum. A common linewidth was fitted to all seven equidistant peaks, while the individual intensities were fitted independently. Errors were determined via Gaussian propagation of the differences between the experimental and fitted intensities.

FIG. 1 .

 1 FIG. 1. (a)Pictorial representation of the 27 × 27 matrix R describing relaxation of the 27 populations associated with the three deuterons in a 12 CD 3 group. The inclusion of a coupled13 C nucleus would double the size of the matrix. Blue shades indicate negative auto-relaxation rates, while colors from yellow to red stand for positive cross-relaxation rates. The irreducible representations A, E1, E2, and B are indicated. White off-diagonal elements correspond to small but possibly non-vanishing crossrelaxation rates. The matrix corresponds to the case τ R τ C . (b) Energy level diagram of the 27 symmetryadapted spin states of the deuterons in a CD 3 group; explicit eigenfunctions are given by Bernatowicz et al.6

FIG. 2 .

 2 FIG. 2. (a) DNP-hyperpolarized13 C multiplet of the13 CD 3 groups in DMSO-d 6 10 s after injection into a 9.4 T NMR spectrometer. (b) DNP-hyperpolarized13 C multiplet of the 13 CD 3 group in ethanol-d 6 30 s after injection into a 9.4 T NMR spectrometer. (c) Multiplet of the 13 CD 3 group of DMSO-d 6 in thermal equilibrium with an intensity distribution 1:3:5:7:5:3:1. Note that the lines are much narrower in thermal equilibrium than in the hyperpolarized case. This may result from convection and small helium gas bubbles that perturb the homogeneity. The intensity ratios between different lines are not perturbed through this.

FIG. 3 .

 3 FIG. 3. (a) Decays of the seven individual lines of the 13 C septet of DMSO-d 6 (numbered as in Fig. 2) after hyperpolarization, dissolution, and transfer. (b) Similar data for ethanol-d 6 . (c) Characteristic relaxation times of the seven lines of the 13 C septet in DMSO-d 6 . (d) Characteristic relaxation times of the seven transitions in ethanol-d 6 . The relaxation times shown in (c) and (d) have been extracted assuming mono-exponential decays. For peak 7, only the decreasing part of the time trace was considered.

FIG. 4 .

 4 FIG. 4. (a) Time dependence of P A , P B , and P E in DMSO-d 6 after hyperpolarization and dissolution. (b) Similar data for ethanol-d 6 .

ACKNOWLEDGMENTS

The authors thank Bruker BioSpin for the prototype D-DNP equipment. This work was supported by the French CNRS and the European Research Council (contract "Dilute para-water"). K.L.I. acknowledges ENS for a guest professorship and the Russian Science Foundation (Grant No. 14-13-01053).

APPENDIX: SYMMETRY-ADAPTED SPIN STATES

In this Appendix, we briefly discuss symmetry-adapted states. 6 The total spin F of the three S = 1 spins can be equal to 3, 2, 1, or 0. The F = 3 states belong to the first A-manifold of seven states, which remain unchanged upon cyclic permutations of the 3 spins. For instance,

etc. In the notation of symmetry-adapted states, F, M irrep , we define the total spin F, its z-projection M, and the irreducible representation, irrep; in the product states m 1 m 2 m 3 , we specify the z-projections of all three D-spins. The states with F = 2 are the E1 and E2 states, which are invariant upon the pairwise exchange of spins. For instance, the state

is invariant upon the exchange of spins 1 and 2, whereas the state

is anti-symmetric upon the exchange of spins 1 and 2. The definitions of the E1 and E2 states in Eqs. (A2) and (A3) are not unique since one can choose arbitrary linear combinations of these states. This is not essential in this work since the E1 and E2 states can be "lumped together" because the corresponding 13 C transitions overlap. The reader is referred to Refs. 5-7 for a more detailed treatment. The F = 1 states are the A, E1, and E2 states and have similar symmetry properties as the A, E1, and E2 states discussed above. For instance, the state

is invariant to cyclic permutations of the three spins, but the E1 and E2 states remain the same or change the sign when spins 1 and 2 are exchanged,

Finally, there is a single F = 0 state, which is the B-state, This state is invariant to cyclic permutations of the three D-spins.


A A Stanislavsky
email: a.a.stanislavsky@rian.kharkov.ua

A A Konovalenko

• E P Abranin

V V Dorovskyy

• A Lecacheux

• H O Rucker

Ph Zarka

Revisiting the frequency drift rates of decameter type III solar bursts observed in July-August 2002

Keywords: Solar radio bursts, Decameter range, Frequency drift rate

Estimating for the frequency drift rates of type III solar bursts is crucial for characterizing their source development in solar corona. According to Melnik et al. (Solar Phys. 269, 335, 2011), the analysis of powerful decameter type III solar bursts, observed in July-August 2002, found a linear approximation for the drift rate versus frequency. The conclusion contradicts to reliable results of many other well-known solar observations. In this paper we report on the reanalysis of the solar data, using a more advanced method. Our study has shown that decameter type III solar bursts of July-August 2002, as standard type III ones, follow a power law in frequency drift rates. We explain possible reasons for this discrepancy.

Introduction

Solar type III bursts have been observed since the fifties of the last century (Wild, 1950). This is the largest population of bursts, and it would seem that they are the most studied events from observational and theoretical points of view [START_REF] Krüger | Introduction to Solar Radio Astronomy and Radio Physics[END_REF]. Nevertheless, interest in the study of such bursts does not vanish, but even rather vice versa (Reid and Ratcliffe, 2014, and references therein). The manifestation of solar activity is detected in a very wide frequency range, from several GHz down to a few tens of kHz. They are generated by fast electron beams propagating with velocities of about 0.3c (c is the velocity of light) along open magnetic field lines in the solar corona. The fast electrons induce Langmuir waves along the beam propagation path, and the waves are scattered on ions and transformed into radio emission (e.g., [START_REF] Zheleznyakov | Radio Emission of the Sun and Planets[END_REF]. The most detailed and most recent analysis of type III burst properties with their interpretation can be found in the recent review (Reid and Ratcliffe, 2014). The radiation mechanism is nonthermal, and therefore its radiation intensity increases with decreasing frequency. The flux in different solar type III bursts can vary in significant limits, which are even difficult to designate. The strong events are observed with help of both large (such as the UTR-2) and small (e-Callisto, for example) radio telescopes, whereas weak type III bursts are only detected with large instruments. The study of the phenomena does not cease to amaze. For example, several decameter type III bursts can have frequency drift rates with variable sign (Melnik et al., 2015). A possible interpretation of the events is that in some frequency ranges the group velocity of radio emission is smaller than the velocity of beam electrons responsible for this radio emission. Therefore, the burst radiation at a lower frequency is received earlier than at a higher frequency. On the other hand, Stanislavsky et al. (2016) have shown that the solar bursts, observed on 19 August 2012, with high-frequency cut-off are nothing else but the type III radio bursts. This group of solar bursts had frequency drift rates and durations of individual events similar to the features of type III radio bursts at low frequencies. The appropriate interpretation of solar bursts with high-frequency cut-off is that their radiating sources move behind the solar limb relative to an observer on Earth through tunnel-like cavities with low density. Further, one can expect frequency-time distortions of received type III bursts. All this requires a more careful analysis of their measurements.

An important parameter of type III solar bursts is their frequency drift rate. It characterizes the velocity of burst sources (electron beams) in the solar corona. In particular, the value is much higher, for example, than one for type II bursts that originate from shock waves. For many solar type III bursts within 75 kHz to 550 MHz [START_REF] Alvarez | [END_REF] obtained the equation of frequency drift rates in the form df /dt = -(0.01±0.008) f 1.83±0.39 . Then, from observations of a group of solar type III radio bursts on December 27, 1994 by the radiospectrometer (40-800 MHz) together with the WAVES and URAP instruments Mann et al. (1999) have found a similar relation df /dt = -0.0074 f 1.76 . Solar type III radio observations in the range 125 kHz -16 025 kHz from the STEREO/Waves experiment indicate in favor of the frequency drift rate as a function of frequency: df /dt = -0.02 f 1.80±0.03 (Vidojević and Maksimovic, 2009). The Potsdam data from 1997 to 2003 confirm again a power law of frequency drift rates in type III solar bursts (Cairns et al., 2009;Lobzin et al., 2010). In a very recent work Reid and Kontar (2018), using a selection of type III bursts observed in the frequency range 30-70 MHz by LOFAR between April-September 2015, have shown that the drift rate of the bursts can also be fitted by a power-law function of frequency. On the other hand, Melnik et al. (2011) note that the connection between drift rate and frequency for powerful solar type III bursts for each day of observations in July-August 2002 is linear, df /dt = -Af + B, where A and B are constants varying from day to day, and the contribution of B in df /dt attains 33.3 % in 10 MHz and 12.2 % in 30 MHz. It should be noted that this study was carried out at the mean frequencies 11.5, 14, 17.5, 22.5 and 27.5 MHz. In this way 163 bursts of July 2002 and 231 bursts of August 2002 have been analyzed. To cover the frequency range 10-30 MHz, the 60-channel spectrometer was used. The analog multichannel receiver was tuned to selected 60 frequencies with the frequency bandwidth 3 kHz in each frequency channel. However, the frequency channels had non-uniform spacing in frequency, and the frequency gaps between neighbor frequency channels in the spectrometer were from 110 kHz to 1.4 MHz depending on the radio interference environment. Moreover, the selected 60 frequencies changed during the observations due to sporadically occurring interferences, but these changes were not taken into account. We think that it is useful to analyze the observations again, using more advanced tools, modern numerical receivers. It should be mentioned also that the results of many years (1973)(1974)(1975)(1976)(1977)(1978)(1979)(1980)(1981)(1982)(1983)(1984) of observations with the UTR-2 telescope in 12.5-25 MHz have shown that the rate of frequency drift depends little on the phase of the 11-year activity cycle and was proportional to the 1.7 power of the frequency (Abranin et al., 1990).

In this paper we revisit the results of observations in July-August 2002, discussed previously by Melnik et al. (2010) and Boiko et al. (2011a,b). Our analysis is based on the data made with the digital spectral polarimeter, DSP [START_REF] Lecacheux | Proc. SPIE[END_REF][START_REF] Lecacheux | [END_REF]. These measurement were carried out simultaneously with the 60-channel spectrometer records, but the DSP data were not used to determine the frequency drift rate of solar type III bursts before. With help of the additional measurements we are going to confirm or not the law of frequency drift rates in frequency, presented in Melnik et al. (2011).

Observations and Facilities

Features of the summer observational campaign of 2002 with the radio telescope UTR-2 (Kharkov, Ukraine) were described in detail by Melnik et al. (2010). Here we recall only some of them for the convenience of the reader. As for general facilities of the UTR-2, they are presented in detail in the comprehensive review by Konovalenko et al. (2016). To observe solar radio emission, three sections of the radio telescope with a total area of 30 000 m 2 formed a beam with angular sizes 10 • × 13 • , covering the solar corona at decameter wavelengths. The digital receiver DSP recorded the data in the continuous frequency band of 12 MHz (18 -30 MHz) with frequency resolution of 12 kHz and time resolution of 100 ms [START_REF] Lecacheux | Proc. SPIE[END_REF][START_REF] Lecacheux | [END_REF]. In comparison with the 60-channel receiver the number of frequency channels of the DSP is much higher, 1024. This allows us to get noticeably more data points for the analysis of frequency drift rates and other parameters.

During July-August 2002, the Sun made about two rotations. Many active regions (AR) were observed on the solar disk, from AR 10008 to AR 10096. The number of ARs varied from 4 to 13 in July, and from 7 to 14 in August. According to an overview of the strongest solar flares since June 1996 (top 50 solar flares, https://www.spaceweatherlive.com/en/solar-activity/top-50-solar-flares), four events were noticed in July-August 2002 (see Table 1). The greatest manifestation of solar activity was connected with ARs 10017, 10030, 10039 and 10069. Likely, they were responsible for strong solar type III radio bursts observed at that time. The other ARs were not so effective. Using the Their temporal distribution (in days) is shown in Figure 1. In the case when radio time profiles of different type III bursts overlapped so that they could not be distinguished from one another, the overlapping bursts were not taken into account in this analysis, because it was difficult to track their peaks separately.

Data analysis

To explore the frequency drift rates of solar type III radio bursts in July-August 2002, we examined the maximum of the power spectral density time profiles for most of the bursts and at each of the frequency channels, if possible, between 18 and 30 MHz. As usual, frequency channels of the digital records, clogged by radio narrow-band interferences, were ignored. Figure 2 shows the dynamic radio spectrum of such a event recorded with the UTR-2. The frequency drift rate behavior is in direct connection with the mechanisms leading to the emission of type III bursts. The type III bursts have a negative frequency drift rate, because of their sources (streams of fast electrons) originating from the Sun, traveling outward thorough the solar corona, producing radio waves at frequencies equal to the fundamental and to the second harmonic of the local plasma frequency, and the background density of solar plasma decreases as a function of distance from the Sun. As a fitting function, it is convenient to take the form

f (t) = a(t -b) -α , (1)
where a, b and α are parameters giving the best fitting result, if one draws a line through the peak flux value of each type III burst on its dynamic spectrum in each frequency channel (see the bottom panel of Figure 2 as an example). The representation has a clear astrophysical interpretation. If the emission source moves in the solar corona with constant velocity v b and negligible acceleration, it travels a distance equal to r(t) = r 0 +v b (t-t 0 ), where r 0 is the initial position, t 0 being the starting time (Cairns et al., 2009;Lobzin et al., 2010). Generally, the beam velocity is not constant throughout the entire propagation path of electron beams through the solar corona and interplanetary space from the Sun. Nevertheless, the assumption that any change in v b was small on the heights, where the radio emission of solar type III bursts in the frequency range 10-30 MHz was generated, is justified and can therefore be ignored. The background electron density n e (r) at the source location can be characterized by the local plasma frequency f p (r) = C(r/R s -1) -α , where R s is the solar radius. Then, the source produces radio emission with the frequency drift in time, f (t) = mf p (r(t)), according to Equation 1with a = mC(R s /v b ) α and b = t 0 + (R s -r 0 )/v b , where m = 1 and m = 2 for the fundamental and harmonic radio emissions of the given burst, respectively. Consequently, we obtain Equation 1in which constants a and b are dependent on constants C, r 0 , t 0 , v b and m. Next, let us write the frequency drift rate as

df dt = -K f ν , (2)
where K and ν are constants. Note that the values are found directly from the recorded spectra of type III bursts. By the differentiation of Equation 1and the change of variables, we derive the relationship between the constants of Equations 1-2, viz. K = αa -1/α and ν = 1 + 1/α. Based on the processing procedure, we have analyzed frequency drift rates of the solar type III bursts. The fitting errors of K and ν of each burst individually were not taken into account, as they were considered negligible. Our data statistics is shown in the histograms of Figure 3. According to the histograms, the values K and ν of the data set have skewed distributions. They are characterized by the mean, mode and median (Table 2). To avoid the standard deviation being larger than the mean, one can consider K as 10 -x , in a similar way to [START_REF] Alvarez | [END_REF]. In this case we have x = 2.67 ± 1.11. It should be noticed that the power dependence (Equation 2) of frequency drift rate in frequency is not only characteristic of solar type III bursts, but also of solar drift pairs (Stanislavsky et al., 2017), S bursts (McConnell, 1982) and even type II bursts [START_REF] Aguilar-Rodriguez | Proceeding of Solar Wind 11/SOHO 16[END_REF]. Commonly, the duration of type III bursts, using the full-width half-maximum, increases as frequency decreases (Wild, 1950). Elgarøy and Lyngstad (1972) have analyzed the duration of type III bursts in a wide frequency range, from 300 kHz to 500 MHz, obtained by different radio instruments. The best fit between their observed frequency f in MHz and duration τ is described by the following relationship

τ = h f 30 MHz -µ (3)
with h = 6.2, µ = 2/3, and Reid and Kontar (2018) reported τ = (3.7 ± 0.2)/f 0.86±0.11 (per 30 MHz) for 31 selected events. As applied to our data set, the dependence τ (f ) is fitted as τ = (6.07 ± 2.26)/f 0.743±0.42 (per 30 MHz). Fitting errors of individual bursts were small and ignored. When the type III bursts had a tendency to occur in a group, we ignored such events because of their overlapping of radio time profiles. Distributions of h and µ for the data set are shown in Figure 4. They are skewed and have a long tail. Their statistical features are presented in Table 3. Returning to the paper of Melnik et al. (2011), we can find a significantly different conclusion: at high frequencies (near 30 MHz) the 60-channel observational data are closer to the approximation τ = 200/f taken from Wild (1950). However, this result is not confirmed by the DSP data. Our analysis of frequency drift rates and durations in the solar type III bursts, recorded by the DSP in July-August 2002, shows that the solar bursts has clear signs typical for well-known standard type III bursts [START_REF] Alvarez | [END_REF]Lobzin et al., 2010;Reid and Ratcliffe, 2014, etc).

Conclusions

In this paper we have shown that the analysis of solar type III radio bursts, observed in July-August 2002, with help of the DSP data does not confirm a linear character of their frequency drift rates in dependence of frequency that was reported in Melnik et al. (2011). Withal, our results are in very good agreement with [START_REF] Alvarez | [END_REF] who gave an empirical expression for the frequency drift rate as a power function of frequency. It is important because the dependence seems to be valid for both weak and strong bursts. This stresses that in both cases the electron beam speed dependance on distance from the Sun is weak, and it can be extrapolated from model representations. The solar corona is a dynamic structure. The background density gradient is almost unchanged on timescales comparable to the electron beam velocity but will be different from burst to burst. It is well known that active regions are directly responsible for type III radio bursts. Their drift rate increases during the approach of the active region to the central meridian. Therefore, the standard deviations for the exponents ν and µ can be quite large. Note also that some values of ν are exceptionally large, leading to large frequency drift rates. Having a smaller duration, the bursts occurred most often when the active region AR 10030 was near the central meridian. Following Ledenev (2000), if the electron velocity is close to the group velocity of the electromagnetic waves, then the drift rate may be very large. When fluxes of the solar type III bursts were more 1000 s.f.u. (1 s.f.u. = 10 -22 W m -2 Hz -1 ) at low frequencies of the observations and became less at higher frequencies, we did not find any noticeable change in the frequency drift rate law. A possible explanation of discrepancy between our conclusions and ones of Melnik et al. (2011) is that in the latter case the data set for solar type III radio bursts were obtained basically by the analog multichannel receiver with the frequency bandwidth 3 kHz in each frequency channel. Consequently, the data contained notable omissions. In addition, the whole frequency band from 10 to 30 MHz was divided into some frequency sub-bands [10][11][12][13][13][14][15][15][16][17][18][19][20][20][21][22][23][24][25]i. e. the statistical analysis was conducted on the average frequencies of these sub-bands. A simple straight line is not complex enough to accurately capture relationships between the frequency drift rates of solar type III radio bursts and their frequency. In fact, such a model is underfitted. One of the most effective ways to avoid underfitting is making sure that a model is sufficiently complex and the most appropriate for corresponding data set. The same applies to the duration of the solar type III bursts in July-August 2002.

Figure 1 .

 1 Figure 1. Distributions of powerful type III bursts in July and August 2002.

Figure 2 .

 2 Figure 2. Example of applying the fitting procedure to time-frequency traces of type III burst peaks.

Figure 3 .

 3 Figure 3. Histograms of the parameters K and ν, characterizing a power law of frequency drift rates of type III bursts.

Figure 4 .

 4 Figure 4. Histograms of the parameters h and µ, describing a power law of durations of type III bursts.

Table 1 .

 1 The most intensive solar flares observed in July-August 2002.

	Date	Class Region				Start	Max	End
	2002-Jul-15	X3.0		10030					19:59	20:08	20:14
	2002-Jul-23	X4.8		10039					00:18	00:35	00:47
	2002-Jul-20	X3.3		10039					21:04	21:30	21:54
	2002-Aug-24	X3.1		10069					00:49	01:12	01:31
							July of 2002
	1 0	1 1	1 2	1 3	1 4	1 5	1 6	1 7	1 8	1 9	2 0	2 1	2 2	2 5	2 7

Table 2 .

 2 Statistical properties of the solar type III bursts in frequency drift rates for parameters shown in the histograms in Figure3.

		K				ν
	mean	mode	median	std	mean	median	mode	std
	0.0069	0.0051	0.0032	0.0075	2.1	1.85	1.66	0.66

Table 3 .

 3 Statistical properties of the solar type III bursts for parameters shown in the histograms in Figure4.

		h				µ
	mean	mode	median	std	mean	median	mode	std
	6.07	2.15	5.76	2.26	0.743	0.7	0.03	0.42

Acknowledgements This research was partly supported by Research Grant 0118U000563 from the National Academy of Sciences of Ukraine. We thank the anonymous referee for useful comments.

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.