Time-Resolved Electronic Spectroscopy To Examine Shock-Wave

Apr 23, 2008 - Shock-wave-induced electronic structure changes in anthracene single crystals ... Shock-wave compression of molecular crystals, unlike ...
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J. Phys. Chem. C 2008, 112, 7761–7766

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Time-Resolved Electronic Spectroscopy To Examine Shock-Wave-Induced Changes in Anthracene Single Crystals Naoki Hemmi, Zbigniew A. Dreger,* and Yogendra M. Gupta Institute for Shock Physics and Physics Department, Washington State UniVersity, Pullman, Washington 99164-2816 ReceiVed: February 26, 2008; ReVised Manuscript ReceiVed: April 4, 2008

Shock-wave-induced electronic structure changes in anthracene single crystals were examined using timeresolved absorption and fluorescence spectroscopy. Crystals were subjected to stepwise loading normal to the (001) crystallographic plane to peak stresses between 2.5 and 6.0 GPa. Absorption experiments revealed (i) a monotonic shift of the absorption edge of the singlet state toward lower energies (red shift) and (ii) formation of a new band on the lower energy side of the red-shifted absorption edge. Furthermore, laser excitation to the new absorption band produced fluorescence with excimer-like characteristics. The combined absorption and fluorescence results indicate the formation of new electronic states as a consequence of shockwave compression and the resulting plastic deformation. The new states are attributed to dimer-type defects formed in shocked anthracene crystals. I. Introduction Shock-wave compression of molecular crystals, unlike other crystalline types, is a relatively unexplored area. Apart from the usefulness of such studies for applications to high explosives (HE), the dynamic response of molecular crystals poses significant scientific challenges. Because of their high compressibility, low threshold for inelastic deformation, and low symmetry, molecular crystals can undergo a variety of physical and chemical changes under shock-wave compression. Limited efforts have been made to identify and understand these changes.1 However, because of the complexity of shock-waveinduced processes and the limited amount of time-resolved data available, the changes remain largely unidentified at the molecular level. In this work, we have used electronic spectroscopy to examine shock-wave-induced deformation in a molecular single crystal. Specifically, time-resolved absorption and fluorescence spectroscopy was utilized to examine the formation of shock-wave-induced local structures (defects). Anthracene was chosen for these studies because it represents a large class of polycyclic hydrocarbons that have served as models for organic molecular crystals. Although the electronic2–12 and crystal13–16 structures of anthracene have been studied to some extent under static high pressures, there are only a few studies under shock-wave loading.17–19 The electronic structure of anthracene is very sensitive to compression,4–7,9 exhibiting significant changes in the fluorescence spectra above ∼1 GPa. New features in the spectra were assigned to fluorescence from excimer states occurring in the compressed crystal upon optical excitation.4–7 Our recent static pressure studies12,20 demonstrated that the new features in the fluorescence spectra were strongly sensitive both to sample morphology and to the compression conditions (e.g., hydrostaticity). To date two studies have been reported on the response of anthracene crystals to shock-wave loading.18,19 In contrast to static high-pressure studies, the shock-wave experiments were primarily focused on stresses beyond 10 GPa. Warnes deter* Corresponding author. E-mail: [email protected].

Figure 1. Schematic diagram of laser-induced fluorescence measurements under shock-wave compression. The pulse laser provides excitation to the anthracene crystal undergoing shock-wave compression due to impact loading. The fluorescence is transmitted into an optical fiber and recorded using an optical system to simultaneously obtain intensity-wavelength-time data. The crystal and windows are not drawn to scale.

mined the Hugoniot of polycrystalline samples up to 55 GPa and noted an abrupt increase in compressibility at 17.6 GPa.18 He suggested that this change is associated with an unspecified intermolecular coupling reaction. Following that work, Engelke and Blais shocked anthracene crystals to various pressures from 9 to 22 GPa and used time-of-flight mass spectroscopy to examine the resulting changes in postshock samples.19 At pressures higher than 18.4 GPa, they observed changes in the molecular weight of detected species, suggesting that complex species were created. They proposed that these species are chemical dimers, resulting from a pressure-driven Diels-Adler reaction.19 In the work presented here, we used real-time absorption and fluorescence spectroscopy to examine shock-wave-induced changes in the electronic structure of anthracene single crystals. Our studies were stimulated, in part, by the recent static highpressure results that found several new features in the electronic spectra. These features, observed under nonhydrostatic conditions, were related to the formation of defect states in the compressed crystal.20 Because shock-wave loading introduces highly nonhydrostatic (anisotropic stress) conditions, it is of interest to determine whether shock compression results in analogous changes and, if so, to examine the time evolution of

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TABLE 1: Summary of Shock Experiments no.

experiment

sample thickness (µm)

projectile velocity (km/s)

1 2 3 4 5

absorption absorption absorption fluorescence fluorescence

201 228 202 221 210

0.505 0.294 0.673 0.670 0.300

these changes. Therefore, we had two specific objectives: (i) to determine the spectral and time dependences of the electronic spectra of shocked anthracene crystals and (ii) to correlate the changes in the spectra with microscopic changes in the crystals. To complement our static20 and previous shock experiments,18,19 we limited these experiments to low pressures, up to 6 GPa. This allowed us to provide new experimental data in this pressure range and to avoid any chemical changes in the shocked crystals. II. Experimental Methods Zone-refined anthracene of 99.9% purity (Aldrich) was used to grow single crystals from a dichloroethane (99.8%, HPLC grade) solvent.21 This method provided large, good-quality crystals with smooth and parallel (001) planes. The thickness of the crystals was controlled by setting the proper cooling conditions. The selected crystals were optically clear and free of any macroscopic imperfections, as examined by an optical microscope with 80 times magnification. For shock experiments, crystals were cut to the appropriate lateral dimensions without further treatment of the surface. The typical crystals used in the experiments were approximately 0.2 mm thick and 10 × 10 mm in lateral dimension and were aligned normal to the (001) crystallographic plane. It is known that two slip systems are operative on (001) plane, namely, [010](001) and [110](001).22 In all experiments, the crystals were sandwiched between two z-cut quartz windows, with liquid glycerol (spectroscopy grade) used to fill the gaps between the sample and the windows. The front and back windows were 3.5 and 9 mm in thickness, respectively. To generate planar shock waves, a quartz crystal (25.4 mm diameter and 12.7 mm thickness), mounted on a projectile, was impacted onto the front quartz window of the sample cell using a light-gas gun,23 as shown schematically in Figure 1. Upon impact, a plane shock wave propagates through the front window and into the anthracene crystal. Reverberation of the shock wave between the front and back windows of the cell results in stepwise loading of the anthracene sample to the peak stress. The peak or final stress was determined accurately from the impact velocity and the known shock response of the impactor and window materials. Absorption/transmission measurements were performed, employing an experimental arrangement described previously.24–26 Briefly, light from a pulsed xenon flash lamp was transmitted through the sample and collected through the back window. An optical fiber delivered the collected light to a spectrometer/streak camera/CCD system. The spectrometer dispersed the light in wavelength, and the streak camera dispersed it in time. The streak camera output (intensity vs time vs wavelength) was recorded by a CCD detector and displayed digitally as a series of transmission spectra separated by ca. 25 ns time intervals. The wavelength was calibrated by mercury-cadmium lines. The transmission spectra were converted to absorption data as described in ref 27.

impactor

windows

quartz quartz quartz quartz quartz

quartz quartz quartz quartz quartz

calcd peak stress (GPa) 4.45 2.53 6.04 6.01 2.59

Laser-induced fluorescence measurements were performed with the configuration indicated in Figure 1. A pulsed dye laser, operating at 514.5 nm wavelength and having a 2.5 µs pulse length, was used as the excitation light source. The 10 mJ excitation pulse, delivered by an optical fiber of 1 mm core diameter was focused to a 2 mm diameter spot on the anthracene sample. The fluorescence resulting from the 514.5 nm excitation was transmitted by another optical fiber of 450 µm core diameter and delivered into a spectrometer where the excitation laser light was rejected by a notch filter. The time-resolved fluorescence signal from the sample was recorded by a system similar to that used for absorption measurements. Spectral sensitivity for the entire detection system was calibrated using a tungstenfilament quartz halogen lamp as a standard light source. Relevant experimental details for the shock absorption and fluorescence experiments are summarized in Table 1. To analyze the time evolution of the absorption and fluorescence spectra, the stress histories of anthracene crystals were calculated using a wave propagation code.28 These calculations made use of the known impact velocity and the known shockwave response of the quartz windows. For anthracene crystals, information for the material model was taken from an equation of state reported by a recent static high-pressure X-ray study.16 III. Results and Analyses A. Absorption. Anthracene crystals are optically transparent in the region of energy below ∼23 × 103 cm-1 at ambient conditions. The absorption band edge located at approximately 24 × 103 cm-1 results from the transition to the lowest excited singlet state: 1La:Ag f B2u.29 The behavior of this absorption band edge was monitored under shock-wave loading.

Figure 2. Time evolution of anthracene absorption spectra shocked to a peak stress of 6 GPa. At 0 ns, the shock wave enters the anthracene crystal. The ∼200-µm-thick crystal reaches the 6 GPa peak stress at approximately 150-175 ns.

Changes in Anthracene Single Crystals

Figure 3. Absorption spectra of anthracene crystals shocked to different peak stresses.

A series of absorption spectra obtained, at 25 ns intervals, during the stepwise loading to 6 GPa, the final stress, are shown in Figure 2. In the plot, time is relative to the instant the shock wave entered the sample, and the final stress of 6 GPa was reached after 150-175 ns. The absorption band profile exhibited significant changes almost immediately after the shock wave entered the anthracene crystal. During the stress ring-up to 6 GPa, the band edge shifted to lower energies (Figure 2). For aromatic molecular crystals (due to the increased dispersion interactions under high pressure), the transition energies are shifted to the lower energies.2,3 Because the ground state of anthracene is nonpolar but the first excited state is polarized, the excited-state energy changes more than that of the ground state, resulting in decreased energy for the optical transition. This red shift was clearly observed for shocked anthracene in our experiments. In addition to the monotonic red shift, the successive spectra in Figure 2 reveal a broad band developed as a shoulder on the red-shifted absorption band edge. This new band occurred around 75-100 ns and grew continually into a distinct band at 175 ns when the sample reached the final stress. The new band has a broad energy distribution, consisting of multiple contributions, extending to energies below 17 × 103 cm-1. To examine the stress dependence of the new band, we shocked anthracene crystals to peak stresses of 2.5 and 4.5 GPa. In Figure 3, we compare the absorption spectra at different peak stresses. The spectra shown were averaged over times from 175 to 225 ns after the shock wave entered the crystal. These results clearly indicate the stress dependence of the new band. The spectrum at 2.5 GPa shows only a red shift of the absorption band edge with a small increase of the baseline absorbance. However, at higher stresses, the spectra include a long tail and a new band as a shoulder on the red-shifted singlet-state absorption. Consequently, the high-pressure absorption spectrum of shocked anthracene consists of three components: (i) a band edge from the lowest singlet-state absorption, (ii) a baseline absorbance, and (iii) a new band. To determine the stress and time dependence of the new absorption band, the baseline absorbance and the singlet-state absorption were subtracted from the absorption spectrum. The baseline was assumed to have a constant value and to be equal to the lowest value of absorbance in the measured spectral range. The contribution of the singlet-state absorption was evaluated

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Figure 4. Analysis of the absorption spectrum of anthracene crystal shocked to 6 GPa. The shape of the new absorption band is extracted from the absorption curve, as discussed in the text.

Figure 5. Effect of pressure on the absorption band shift: (9) absorption edge of singlet states, determined at an absorbance of 0.3 (dynamic range of measurements ∼ 1.4); (b) the new band; (O) results from static high-pressure experiments.20

from high-pressure data of crystalline anthracene.8 The following values were assumed:8 a red-shift rate of 437 cm-1/GPa and a line-broadening rate of 397 cm-1/GPa for the 0-0 transition of the lowest singlet state. An example of the new absorption band extracted from the absorbance curve is shown in Figure 4. It can be seen that the new band has a broad and irregular shape, likely resulting from multicomponent contributions and/ or from oversimplified assumptions made for the pressure dependence of the 0-0 vibronic band. The stress dependences of both the maximum of this band and the band edge of the singlet state from our data are presented in Figure 5. The new absorption band shifts more rapidly with pressure than the singlet-state absorption edge. Furthermore, one can see that the shock results are consistent with the results obtained under nonhydrostatic conditions in the diamond anvil cell.20 B. Laser-Induced Fluorescence. The monotonic red shift observed in the absorption experiments indicates a change in the electronic structure, i.e., a reduction in the energy gap between the ground and first excited states of an anthracene

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Figure 6. Laser-induced fluorescence spectra at ambient conditions and under shock-wave compression. The ambient spectrum was produced using 337 nm (29 673 cm-1) excitation (nitrogen laser), while the spectra under shock were produced using 514.5 nm (19 436 cm-1) excitation. The intensity of the spectrum at ambient pressure is not related to intensities of spectra under shock. The spectra under shock are presented in relative units (solid lines). The dotted line represents the spectrum at 2.5 GPa normalized to the intensity of the spectrum at 6.0 GPa. The excitation laser light (514.5 nm) was rejected by a notch filter. Note: a 514.5 nm line does not produce emission at ambient pressure.

Figure 7. Schematic representation of potential energy curves for the vdW dimer and excimer. Optical excitation of the ground state of the vdW dimer, at energies lower than the singlet-state absorption, directly prepares the excited dimer states. Given that the dimer excited state and the excimer state are strongly coupled, they can yield a similar emission spectrum.

singlet. The appearance of the new absorption band observed above 4.5 GPa indicates additional changes in the electronic structure at high stresses. To examine the origin of the new absorption band, we utilized laser-induced fluorescence spectroscopy with excitation at 514.5 nm (19 436 cm-1). This excitation matched the energy of the new absorption band, but it was below the energy of the singlet-state absorption. In Figure 6, fluorescence spectra are presented for shocks to 2.5 and 6 GPa peak stresses. The intensity of fluorescence at 6 GPa is significantly greater than that at 2.5 GPa, consistent with the strong absorption of the new band at 6 GPa. This finding indicates that laser-induced fluorescence in shocked anthracene originates from excitation to the new absorption band that is produced at higher shock stresses. Examination of the spectral

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Figure 8. Temporal profiles for the 6 GPa peak stress experiment: calculated stress in the center of the anthracene crystal (solid line), new absorption band intensity (O), and laser-induced fluorescence intensity (9). The lines drawn through the points are guides to the eye. Absorption and fluorescence intensities were integrated over the relevant energy ranges and normalized to the highest value in each case.

profile provides additional insight into the electronic transitions. We compared the fluorescence spectrum obtained at 6 GPa with that obtained at ambient conditions for 337 nm (29 673 cm-1) excitation (Figure 6). The ambient fluorescence exhibits a distinctive vibrational progression in the wavelength region from 25 641 to 21 739 cm-1 (from 390 to 460 nm), arising from electronic transitions between the first excited singlet state and the vibrational levels of the ground state. In contrast, the fluorescence spectrum at 6 GPa is structureless and much broader, extending from 20 000 cm-1 to over 15 385 cm-1 (from 500 nm to over 650 nm). These spectral characteristics are analogous to those reported in aromatic molecular crystals under static high pressures where they were related to excimer fluorescence.5–7,9,20 In our experiments, the fluorescence intensity results from excitation to the new band; this implies the same electronic origin for both the shock-wave-induced absorption and excimer-like fluorescence. Next we discuss the possible origins of the new absorption and fluorescence bands and their kinetics. IV. Discussion A. Origin of New Bands. The two main experimental findings of this study are as follows: (i) a new absorption band at stresses higher than 4.5 GPa, with a maximum at ∼21 × 103 cm-1, significantly lower than the energy of the absorption band edge of the singlet excited state (∼24 × 103 cm-1),30 (ii) a structureless fluorescence band, with excimer-like features, from excitation to the new absorption band. The relationship between these new bands and the possible changes in the structure of shock-wave-compressed anthracene is discussed below. It is well-known that the optical properties of molecular crystals are sensitive to structural imperfections.31 In particular, spectroscopic studies of anthracene crystals at ambient conditions revealed “satellite” bands located below the longwavelength edge of intrinsic absorption and/or of intrinsic luminescence of the crystal.31–35 Furthermore, it has been demonstrated that the intensity and spectral distribution of these bands could be altered by mechanical deformation and thermal treatment, indicating clearly the defect-related origin of the lowenergy bands. It was also established that typical defect-related

Changes in Anthracene Single Crystals electronic states occur only a few hundred wavenumbers below the intrinsic excited electronic states of the crystal. They were linked to various types of point defects. Occasionally, electronic states located more than a thousand wavenumbers below the intrinsic excited states have been observed; these were attributed to extended lattice defects. For example, bands located 1.66 × 103 and 2.5 × 103 cm-1 below the excited state were associated with predimer states resulting from the stacking faults or aggregation of dislocations in anthracene crystals.31 Shock-wave compression imposes uniaxial strain on the crystal, and the longitudinal stresses applied in our experiments were well beyond the elastic limit of anthracene,36 resulting in plastic deformation and the formation of defects. Indeed, the appearance of the new band in the absorption spectrum can be associated with shock-wave-induced defects. The energy displacement between this band and the intrinsic electronic state is as large as 3 × 103 cm-1, implying the dislocation character of these defects. Furthermore, on the basis of the lack of structure in the laser-induced fluorescence spectrum, we propose that dislocations can lead to the formation of dimeric structures analogous to those observed under static compression and nonhydrostatic conditions.20 Typically, dimeric structures from aromatic molecules are of the van der Waals (vdW) type and are unbound in the ground state.30 However, recent studies of aromatic vdW clusters demonstrated that two isomers in anthracene clusters (dimers) can exist in the ground state with a parallel-displaced molecular geometry.37,38 The parallel-displaced configuration in anthracene clusters is very similar to that produced by stacking faults in the anthracene crystals that can be formed by partial dislocations along the [110] directions.39,20 As results from analyses of the anthracene crystal structure, the 1/2 [110] dislocation in the (001) plane can lead to the formation of parallel pairs of anthracene molecules along these dislocations.38 It is expected that under pressure these pairs can reach a close distance with enhanced interaction. Therefore, we postulate that dimeric structures produced under shock-wave compression may have a stable ground state, as indicated by formation of the new absorption band. Stabilization of these structures may result from differences in intermolecular interaction near the dimers (local defects) compared to regions far away from these sites. This effect is likely enhanced under high pressure. It is also expected that shock-wave compression could lead to the formation of dimeric structures in some other polycyclic aromatic hydrocarbons. In particular, dimeric structures could be produced in naphthalene, which has a crystal structure and a slip system similar to those of anthracene. The association of dimeric structures with the observed features in the absorption and fluorescence spectra is discussed using the notional energy diagram shown in Figure 7. In the diagram, the potential energy curves represent the ground and excited states of the dimer and excited state of the excimer. Once the ground state of the dimer structure (defect) is formed and stabilized in a manner similar to that of the vdW anthracene clusters,38 excitation of the ground-state dimer leads directly to occupation of the excited dimer state rather than the excimer state. Because of the increased dispersion and resonance interactions upon vdW dimer formation, the dimer transition should occur at lower energies relative to the singlet-state absorption, in agreement with our observations for the new absorption band. Because the excited states of the vdW dimer are easily coupled to the lower-lying excimer states,40 the observed broad and structureless fluorescence spectrum may be

J. Phys. Chem. C, Vol. 112, No. 20, 2008 7765 a combination of emissions from both the excited vdW dimer and the excimer. B. Kinetics of New Bands. To investigate the kinetics of the electronic structure changes related to shock-wave-induced dimeric defects, we examined the time evolution of the absorption and fluorescence spectra. In Figure 8, we compare temporal profiles of intensities of the new absorption band and laser-induced fluorescence with the calculated stress history (in the center of the sample) for the 6 GPa peak stress experiment. The spectroscopic results represent the integrated intensities of absorption and fluorescence over the relevant spectral ranges, normalized to the highest values in each case. With increasing stress, the absorption and fluorescence intensities increase. Furthermore, it can be seen that the two intensity profiles are quite similar but do not directly correspond to the stress history. While the stress increases rapidly at the beginning of the shockwave reverberation, the intensity of the absorption band increases slowly after ∼25 ns from the entry of the shock wave into the crystal. Fluorescence occurs somewhat later than absorption, about 40 ns after the shock enters the crystal. A delay time between fluorescence and absorption intensities remains more or less the same during the reverberation of stress in the crystal. In both cases, the intensities tend to level off as the stress reverberates to the final value. The apparent delay between the entry of the shock wave into the crystal and the appearance of new bands in absorption and fluorescence indicates that a transition time is required to build up the concentration of dimeric defects sufficient for detection. In other words, the concentration of dimeric species created by the [110] dislocations is likely low for the first step of the shock reverberation. The experiments performed at lower stresses reveal that the transition time is related to the input stress value and increases with decreasing input stress. The delay between fluorescence and absorption confirms that fluorescence follows the absorption on the same species. Furthermore, the evident delay of ∼15-20 ns of the onset of fluorescence with respect to absorption can be related to the excited-state lifetime.20 V. Summary To understand the response of molecular crystals to shockwave compression, we examined changes in the electronic structure of anthracene crystals shocked to peak stresses between 2.5 and 6 GPa. Time-resolved absorption and fluorescence spectroscopies were used to detect these changes. The experimental data reveal the following features for anthracene under shock-wave loading: (i) a monotonic shift of the absorption edge of the singlet state toward the lower energies; (ii) the formation of a new absorption band located significantly below the edge of the singlet-state absorption; (iii) the appearance of an excimerlike fluorescence resulting from excitation within the new absorption band. On the basis of these results, we propose that shock-wave compression leads to the formation of dimeric structures in anthracene crystals as a consequence of shockwave-induced dislocational defects along the [110] directions. The formation of dimeric structures requires an incubation time of about 25 ns for shock waves for peak stresses of 6.0 GPa. Our results demonstrate the importance of inelastic deformation for changes in the electronic structure of shocked molecular crystals. Acknowledgment. The authors thank G. Chantler and K. Zimmerman for assistance in the experiments. Dr. J. M. Winey is acknowledged for useful discussions and comments. This work was supported by ONR MURI Grants N00014-01-1-0802 and N00014-06-1-0459 and DOE Grant DEFG0397SF21388.

7766 J. Phys. Chem. C, Vol. 112, No. 20, 2008 References and Notes (1) Dlott, D. D. Annu. ReV. Chem. 1999, 50>, 251. and references cited therein. (2) Wiederhorn, S.; Drickamer, H. G. J. Phys. Chem. Solids 1959, 9, 330. (3) Drickamer, H. G. Solid State Phys. 1965, 17, 1. (4) Jones, P. F.; Nicol, M. J. Chem. Phys. 1965, 43, 3759. (5) Offen, H. W. J. Chem. Phys. 1966, 44, 699. (6) Ohigashi, H.; Shirotani, I.; Inokuchi, H.; Minomura, S. Mol. Cryst. 1966, 1, 463. (7) Jones, P. F.; Nicol, M. J. Chem. Phys. 1968, 48, 5440. (8) Okamoto, B. Y.; Drickamer, H. D. J. Chem. Phys. 1974, 61, 2870. (9) Nicol, M.; Vernon, M.; Woo, J. T. J. Chem. Phys. 1975, 63, 1992. (10) Otto, A.; Keller, R.; Rahman, A. Chem. Phys. Lett. 1977, 49, 145. (11) Mizuno, K.; Matsui, A. J. Phys. Soc. Jpn. 1986, 55, 2427. (12) Dreger, Z. A.; Lucas, H.; Gupta, Y. M. J. Phys. Chem. B 2003, 107, 9268. (13) Leger, J. M.; Aloualiti, H. Solid State Commun. 1991, 79, 901. (14) Oehzelt, M.; Resel, R.; Nakayama, A. Phys. ReV. B 2002, 66, 174104. (15) Oehzelt, M.; Weinmeier, K.; Heimel, G.; Puschnig, P.; Resel, R.; Ambrosch-Draxl, C.; Porsch, F.; Nakayama, A. High Pressure Res. 2002, 22, 343. (16) Oehzelt, M.; Heimel, G.; Resel, R.; Puschnig, P.; Hummer, K.; Ambrosch-Draxl, C.; Takemura, K.; Nakayama, A. J. Chem. Phys. 2003, 119, 1078. (17) (a) Studies were performed in solution: Huston, A. L.; Justus, B. L.; Campillo, A. J. Chem. Phys. Lett. 1985, 118, 267. (b) Yoo, C. S.; Gupta, Y. M. J. Phys. Chem. 1992, 96, 7555. (18) Warnes, R. H. J. Chem. Phys. 1970, 53, 1088. (19) Engelke, R.; Blais, N. C. J. Chem. Phys. 1994, 101, 10961. (20) Dreger, Z. A.; Balasubramaniam, E.; Gupta, Y. M.; Joly, A. G., in preparation.

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