Ultrafast Shock-Induced Chemistry in Carbon Disulfide Probed with Dynamic. Kathryn E. Brown, Cynthia A. Bolme, Shawn D. McGrane, David S.

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1 Ultrafast Shock-Induced Chemistry in Carbon Disulfide Probed with Dynamic Ellipsometry and Transient Absorption Spectroscopy Kathryn E. Brown, Cynthia A. Bolme, Shawn D. McGrane, David S. Moore Shock and Detonation Physics Group, Los Alamos National Laboratory, Los Alamos, New Mexico United States Abstract We used transient visible/near-infrared absorption spectroscopy and ultrafast dynamic ellipsometry to characterize carbon disulfide (CS 2 ) shocked with an ultrafast laser pulse. We found a volume-decreasing reaction, characterized by the deviation of the shock and particle velocity (u s and u p ) points from the unreacted Hugoniot, above u p =1.5 km/s. This result contrasts with literature plate-impact data, which found the reaction-induced deviation from the unreacted Hugoniot to occur at u p =1.2 km/s. We attribute this disparity to the difference in timescale between plate-impact experiments (ns to μs) and our ultrafast experiments (sub-ns), as our ultrafast experiments require higher shock pressures and temperatures for an observable reaction. The volume-decreasing reaction was accompanied by a large increase in absorption of the reaction products, necessitating the use of impedance matching techniques to characterize the u s -u p points above the reaction cusp. Using transient absorption spectroscopy, we discovered a change in the absorption spectrum for shock strengths below and above the volume-decreasing reaction, suggesting there are multiple chemical reactions in CS 2 shocked to above 7.4 GPa in 300 ps. 1

2 I. Introduction Carbon disulfide (CS 2 ) has been widely studied under extreme pressures and temperatures. In the middle of the 20 th century, Bridgman was the first to discover a black polymer 1 formed by subjecting the liquid CS 2 to high pressures and temperatures in a diamond anvil cell (DAC). This work spawned decades of research to try to characterize the CS 2 static phase diagram 2-4 and shock equation of state (EOS), 5-12 as well as the high pressure/temperature (P/T) chemistry of CS The EOS for unreacted CS 2 has been shown to closely follow the universal liquid Hugoniot (ULH), 8,26 an empirical relationship that is valid for several liquids, based on the ambient speed of sound (c 0 ) in that liquid. However, there is evidence of a volumedecreasing reaction that takes place at 5.1 GPa under explosively-driven plate-impact shocks on the μs-ns timescale, where the data deviate from the ULH. 6-8,10,11 Likewise, increases in reflectivity and absorption under similar conditions indicate a chemical reaction occurring in shocked CS ,19-24,27,28 The nature of this chemical reaction is a subject of debate. It was originally proposed that this reaction was the formation of Bridgman s black polymer, 10,11 but this hypothesis was later discounted as the CS 2 shock Hugoniot does not access the P/T region where Bridgman s polymer is formed. 8,29 Another proposed reaction was simply the decomposition of CS 2 into elemental carbon and sulfur. 28 Yoo et al. postulated that a direct dissociative reaction was unlikely, favoring instead a scheme in which the CS 2 molecules are first rotated to a state parallel to the shock front, then polymerized, and finally dissociated. 19,22 More recently, Engelke et al. observed explosively-driven CS 2 shock reaction products using mass spectrometry, and the products consisted of many multi-atom carbon-sulfur compounds. 14 2

3 In this study, we used dynamic ellipsometry to obtain shock and particle velocity data for CS 2 on the ultrafast (300 ps) timescale. We have previously shown that our techniques can access early reaction chemistry of reactive materials. 30,31 Additionally, we used transient visible/near-infrared absorption spectroscopy to correlate shocked CS 2 product absorption with the volume-decreasing reaction seen in the EOS. II. Experimental All experimental techniques, including the liquid sample cell, ultrafast laser-driven shock drive, ultrafast dynamic ellipsometry (UDE), and transient absorption spectroscopy have been described in detail in previous studies by the authors, 30,32-35 and so will only be briefly described here. All experiments utilized a single, amplified Ti:Sapphire laser to generate both the pump and probe beams. A schematic of the sample cell with the pump and probe beams is shown in Fig. 1. 3

4 FIG. 1. Schematic of sample cell with pump and probe beams (not to scale). HA: high angle, LA: low angle. A) Liquid Sample Cell: The liquid sample cell consisted of four layers: 0.5 mm sapphire, 2 μm vapor-coated Al, CS 2 inside of a 6-12 μm spacer, and 2 mm CaF 2 (Fig. 1). The CS 2 was obtained from Sigma-Aldrich and used without further purification. The sample was tested under ambient conditions without degassing, but microscopic observation verified the absence of gas bubbles. The cell was 1 inch in diameter, allowing for several tens to hundreds of shots per sample. Care was taken to ensure that the shocked regions were clear of debris from previous shots by stepping mm between shots and microscopically viewing the region before the shot. B) Shock Drive: 35 A portion of our Ti:Sapphire laser was chirped to ~300 ps and shaped with a fast rise. Part of this beam was used to generate the shock drive. The laser pulse was incident on the sample cell through the sapphire side (Fig. 1) 4

5 and was absorbed in the Al drive layer, creating an expanding plasma that drove a shock wave through the remainder of the Al and into the CS 2. The diameter of the shock region was approximately 65 μm full width at half maximum (FWHM). CS 2 was shocked at several different input energies between 0.2 and 2.5 mj/pulse, adjusted using a λ/2 waveplate and polarizer. The shot-to-shot relative intensity of the shock drive was monitored with a photodiode, and for each input energy was found to be consistent to within 3% (standard deviation). C) UDE: 32 The remainder of the chirped beam was split into two interferometers, high angle and low angle, each comprised of a signal and reference beam. In some experiments, the high angle was 65 from normal and the low angle was 27 from normal (Fig. 1); other experiments had high angle of 64 and low angle was normal incidence (0 ). The signal and reference beams for each angle were recombined, producing interference fringes that were imaged onto spectrometers, and split into s- and p-polarization images at the 2-D charge-coupled device (CCD) detectors using Wollaston prisms. We converted wavelength to time using a cross-correlation frequency-resolved optical gating (XFROG) 36 measurement of the pulse. Fourier transform analysis of the interferograms yielded the timedependent optical phase shift. Lineouts from the center ~5 μm of the interferograms were fit to thin-film equations, resulting in fits for shocked real and imaginary refractive indices (n s and k s ), shock velocity (u s ), and particle velocity (u p ). The equations and detailed methodology of the fitting procedure can be found in Refs. [32,34] and in the supplementary material of this manuscript. 37 5

6 D) Impedance Matching: For u s -u p data taken at energies above the volumedecreasing reaction, the high absorption of the reaction products strongly reduced the intensity of the reflection off of the Al surface so that thin-film analysis through the shocked layer was not possible. Instead, u p was determined using impedance matching, and u s was determined from interference of the optical reflection from the shock front in CS 2. For each laser intensity used, 10 shots were first taken on bare Al and probed with the UDE interferometers. The free surface velocity (u fs ) of Al was calculated by u fs = λ φ 4πn cos θ Δt (1) where λ is the interferometer wavelength (nominally 800 nm), n is the refractive index of the medium (for bare Al shots, the refractive index for air, n air =1 was used; for CS 2 shots, n= at the 800 nm probe wavelength) 38, ϴ is the incident angle of the interferometer probe onto the Al, and φ is the slope of the t obtained time-dependent phase shift. To determine the impedance matched input u p of CS 2, the Al Hugoniot was reflected about the calculated u fs /2. Where the reflected Al Hugoniot and the ULH for CS 2 intersected was the impedance matched CS 2 u p. Corresponding values for u s at each laser-driven shock energy were determined using the time-dependent phase shifts from the UDE interferometers for 5-10 shots on CS 2 samples. Because the data resulted only from the reflection off of the shock front, Eq. 1 was used to determine u s. E) Transient Absorption Spectroscopy: The remainder of the Ti:Sapphire laser beam was compressed to 100 fs. The compressed pulse was sent through a 12 mm thick CaF 2 window, generating a ~ 3 ps pulse of white light (~ nm). The beam 6

7 was split into a signal and reference leg, and the signal beam was focused onto the central ~30 μm of the shocked region (Fig. 1). Absorbance was calculated by A = log ( I R I ) (2) 0 R0 where I, R, and I 0, R 0 are the signal and reference intensities during and before the shock, respectively. The delay between the shock drive and the white light probe was varied in 50 ps increments to obtain a transient absorption history from shock arrival at the Al-CS 2 interface (0 ps) to 300 ps, the time over which the shock wave in the material is supported. The amount of shocked material linearly increased with time, while a decreasing thickness of the film remained unshocked during these measurements. F) Pressure and Volume Estimation: The pressure and volume of CS 2 were calculated using the Rankine-Hugoniot jump conditions: P = ρ 0 u s u p (3) V V 0 = u s u p u s (4) where P is the shock pressure, ρ 0 is the initial density, g/cm 3, 14 and V and V 0 are the specific volume of the compressed and uncompressed material, respectively. It is worth noting that the pressure and volume calculations at the volume-decreasing reaction cusp and for the reaction products are estimations, as the jump conditions are only valid for the unreacted material. III. Results A) Shock and Particle Velocity Measurements: Shock and particle velocity measurements for CS 2 from both UDE (red circles) and impedance matching 7

8 (blue diamonds) experiments are shown in Fig. 2, along with literature plate impact data (black squares), 5-7,9,10,12 the ULH (solid line) for CS 2 using c 0 =1.157, 8,11,26 and Dick s empirical products EOS (dashed line). 10 Also shown in Fig. 2 are points from two ultrafast experiments performed by Bolme in her thesis (yellow triangles). 39 Further explanation regarding these points will be provided in Section IV-A. Each UDE point represents one shot; each impedance-matched point represents the average u s and u p values for one laser shock-drive energy, and the error bars represent one standard deviation from 5-10 shots. The scatter of our unreacted points under u p =1 km/s is one standard deviation of Δu s =0.14 km/s from the ULH. The first point that lies outside one standard deviation below the ULH is at u s =4.0 km/s; extrapolated back to the ULH, the cusp is at u p =1.5 km/s and u s =4.0 km/s, approximately 7.4 GPa. The literature cusp location is at 5.1 GPa, corresponding to u p =1.2 km/s. 7 Both literature and our data show a volume decrease of about 26%. 10,14 It is reasonable and expected for our results to indicate a higher cusp onset u p than in the literature, as higher shock pressure is needed to expedite the chemical reaction so that it is observable on our ultrafast timescale. There is an increased amount of scatter in the data both above and below the u p of the volume-decreasing reaction cusp, centered around u s =4.0 km/s. We have previously shown that the cusp region of our data is subject to imprecision, as the UDE and chemical reaction timescales are comparable in that region. 30 As yet, we do not have a suitable method for analyzing UDE data for multiple, reactive waves. Table S1 in the Supplemental Material lists the u s, u p, and n s points for our UDE experiments from this study, Table S2 lists values of u s, 8

9 u p, and n s from Bolme, 39 and Table S3 in the Supplemental Material lists the average u s, u p, and associated standard deviations for our impedance matching experiments. 37 FIG. 2. Shock and particle velocity data for CS 2. Literature plate-impact data are shown as black squares, 5-7,9,10,12 UDE data from this study are shown as red circles, and impedance matched data are shown as blue diamonds. The yellow triangles are UDE data obtained by Bolme. 39 The solid line is the ULH for CS 2, and the dashed line is Dick s empirically determined products EOS. 10 B) Transient Absorption Spectroscopy: Transient absorption results are shown in Fig. 3 for CS 2 shocked at four different laser conditions. We were unable to take transient absorption spectra and u s -u p data concurrently, so the u s -u p data for each set of transient absorption data are assumed to be similar to UDE or impedance matched data for which the laser shock drive energy was the same. The conditions for our transient absorption data are thus estimated to be at u p =1.3 km/s, 2.2 km/s, 2.8 km/s, and 3.4 km/s, approximately 6, 11, 16, and 23 GPa input pressures, respectively. 9

10 FIG. 3. Transient absorption of CS 2 shocked to a) u p =1.3 km/s, b) u p =2.2 km/s, c) u p =2.8 km/s, and d) u p =3.4 km/s. On our timescale, the reaction products of CS 2 shocked to u p =1.3 km/s are highly absorbing, with an apparent optical density (OD) of 0.23 at 480 nm after 300 ps. The apparent OD increases with increasing input energy to OD=0.57 when shocked to u p =3.4 km/s. Corrections for reflectivity of the shock front will be discussed in Section IV-A. The spectral evolution is different between the reaction products of CS 2 shocked to a state characteristic of the reactant as seen on the Hugoniot of Fig. 2 (u p =1.3 km/s; Fig. 3a) and a product state (u p =3.4 km/s; Fig. 3d), and spectral lineouts from those experiments are shown in Fig. 4a and 4b, respectively. 10

11 FIG. 4. Spectral lineouts of CS 2 shocked to a) u p =1.3 km/s and b) u p =3.4 km/s. From bottom to top, 0, 50, 100, 150, 200, 250, 300 ps after shock arrival. The dashed red lines are guides to the eyes. In the lower-energy case of CS 2 shocked to u p =1.3 km/s, there is an increase in absorption primarily at 480 nm, shown in Fig. 4a. In the high-energy case, however, the reaction product or products exhibit transient absorption that shifts to longer wavelengths after 50 ps, with the maximum absorption at approximately 550 nm, as shown in Fig. 4b. IV. Discussion A) Absorption of CS 2 Reaction Products and Implications for UDE Analysis Above the volume-decreasing reaction cusp, the time-dependent phase shifts could not be fit using our traditional UDE thin-film analysis. Selected low-angle lineouts from various shock conditions are shown in Fig. 5. Typical materials, with the strongest optical reflection from an interface moving at the particle velocity, have phase shifts of less than 20 radians (cf. Fig. 2 of Ref. [33] for an example); however, above our reaction cusp, we observed apparent optical phase shifts of nearly 40 radians. We attribute this phenomenon to the material behind the shock front becoming nearly opaque, which 11

12 causes the interference fringes to be formed from the reflection off of the shock front. The small oscillations seen in the lineouts in Fig. 5 can be attributed to the thin film interference caused by the CaF 2 window rather than the reflection off of the shocked Al surface. FIG. 5. Optical phase shifts from selected experiments. From bottom to top, u s =3.95 km/s, u p =1.24 km/s (fit with UDE); u s =3.97 km/s, u s =4.24 km/s, u s =4.58 km/s, u s =5.08 km/s. (determined by the phase shift vs. time slope). Each plot is vertically offset in increments of 3 radians for clarity. We attempted to quantify the degree to which the products absorbed the UDE probe and to which the shocked/unshocked CS 2 interface reflected the probe light. This quantification was accomplished using the thin film equations on our layered sample: a CaF 2 window, unshocked CS 2, and shocked CS 2, necessitating the determination of the refractive index of shocked CS 2. We determined k, the imaginary part of the shocked refractive index, from our transient absorption data, using: I = I 0 e αx (5) k = αλ 4π (6) 12

13 where I and I 0 are the intensities before and during the shock (Eq. 2), α is the absorption coefficient, x is the path length of the shocked region, and λ is the wavelength. Values obtained for k are shown in Table 1. As we were unable to use the thin-film equations to fit the high-energy data, we were unable to calculate n s for higher u p experiments. In her doctoral thesis, Bolme made similar UDE measurements on CS 2 ; 39 the data shown in Fig. 2 represent two of her shots. Her experimental setup differed only slightly from ours, primarily in the liquid cell. She used a 330 μm sapphire substrate (versus our 500 μm sapphire substrate), a 0.5 mm tilted fused silica window (versus our 12 mm perpendicular CaF 2 window), and the liquid sample in the chamber was 3 mm thick, compared to our 6-12 μm thick liquid sample. We further examined her unpublished data, which show the same phenomenon of the shocked CS 2 becoming opaque as we saw in our experiments. However, she was able to use thin-film equations on one shot above the reaction threshold. Rather than choosing the center-most region for analysis as we currently do, her analysis used the spatial variation of the shock drive to obtain several points from a single shot. Table 1. Approximate shock and particle velocity, extrapolated shocked index of refraction (at 800 nm), reflectivity, and measured and reflectivity-corrected OD corresponding to the transient absorption data shown in Fig. 3. At maximum absorption a At 740 nm measured corrected measured corrected u p (km/s) u s (km/s) n s k s OD max R max OD max k s OD 740 R 740 OD a Maximum absorption wavelength for u p =1.3 km/s (Fig. 3a) was 480 nm and 550 nm for higher energy conditions (Fig. 3b-d). 13

14 Fig. 6 shows refractive index data from our experiments and Bolme s experiments. Our data and Bolme s data appear to connect. A linear fit to our and Bolme s combined data corresponding to the unreactive CS 2 below u p =1.5 km/s gave n s =0.39u p We used Bolme s values that correspond to data on the products EOS to extrapolate values of n s for our experiments above the reaction threshold. A linear fit to Bolme s values of n s between u p =2.1 and 2.6 km/s gave n s =0.72u p Extrapolated values for n s corresponding to our transient absorption data shown in Fig. 3 are given in Table 1. FIG. 6. u s -u p points for CS 2 from this study (open circles) and Bolme 39 (open triangles) compared to u p -n s points for CS 2 from this study (red circles) and Bolme (yellow triangles) in the reactants (a), cusp (b), and products (c) regions. The thin black lines are the CS 2 ULH 8,26 in the reactants region (a), Dick s empirically determined products EOS 10 (c), and a guide to the eye in the cusp region (b). The thick lines are the linear fits to the shocked refractive indices, n s, in regions (a) and (c) and a connecting line in the cusp region (b). Using the determined values of n s and k, we calculated the percentage of reflected light off of our samples at 740 nm (close to the 800 nm wavelength UDE probe), 480 nm for the u p =1.3 km/s (Fig. 2a) experiment and 550 nm for the experiments where u p =2.2 14

15 km/s, 2.8 km/s, and 3.4 km/s (Fig. 3b-d). Table 1 shows the values of n s, k, reflectivity (R), and the measured and corrected OD at two wavelengths for the transient absorption data shown in Fig. 2. At the u p =2.2 km/s condition, just above the volume-decreasing reaction cusp, the CS 2 reaction products become too opaque to successfully use the thin film equations to characterize the shock state. The corrected OD for that condition at 740 nm is 0.36, which means that the CS 2 products absorb approximately 50% of the incident UDE probe. Bear in mind that our transient absorption probe sees an average over the center 30 μm of the 65 μm shocked region, whereas the UDE probe sees only the center 5 μm of the shocked region. Because our shock drive pulse is not spatially flat, the centermost area of the shocked region experiences the greatest shock strength. 32 Therefore, the OD of the center 5 μm is likely higher than what we have corrected for, explaining how the UDE probe reflects only off of the shock front at conditions higher than the volume-decreasing cusp. B) Evidence of Multiple Chemical Reactions Both the u s -u p measurements and transient absorption data indicate that there is a chemical reaction occurring in shocked CS 2 that produces one or more strongly absorbing species. The transient absorption spectral lineshapes are different at the pressure regimes below the volume-decreasing reaction cusp (Fig. 4a) and above it (Fig. 4b). The correspondence of the change in spectral shape to the volume-decreasing reaction cusp indicates a chemical reaction to one or more highly absorbing products. However, despite significant absorption at u p =1.3 km/s, the volume-decreasing reaction cusp on our timescale does not occur until u p =1.5 km/s. The significant 15

16 absorption below the volume-decreasing cusp may indicate a chemical reaction or electronic interaction that does not cause any significant change in volume that can be observed by u s -u p measurements. There is a great deal of scatter above the ULH at u p =1.3 km/s, which may also be indicative of a chemical reaction on our timescale that we are not able to resolve using UDE. Yoo and Gupta performed absorption experiments on 1-μm-thin samples of CS 2, comparable to the thickness of our shocked regions. 22 Their shock compression was achieved through step wave loading, where the shock was reverberated between the windows confining the CS 2 until peak pressure was reached, nominally in a few ns. Due to the nature of their experiments, their peak temperatures were much lower than in our single-shot experiments. For peak pressures less than 10.5 GPa, they postulated a partial chemical reaction or interaction, based on the partial recovery of an absorption spectrum under those conditions. Despite the differences in time and temperature, our evidence of an early chemical reaction is consistent with their results. A change in the slope of the shocked refractive index with particle velocity may also indicate a chemical reaction. Fig. 6 shows our calculated refractive indices with particle velocity as well as u s -u p data. At the volume-decreasing cusp, there is a clear change in the slope of the refractive index, indicating a chemical reaction at a u p consistent with our u p -u s measurements. Yoo and Gupta also saw evidence of an irreversible chemical reaction of a thin film of CS 2 shocked to peak pressures between 11 and 12 GPa, with an estimated temperature of 820 K. 22 Their evidence of an irreversible chemical reaction was broad absorption across nm; likewise, our transient absorption measurements at 11 16

17 GPa and above show broad absorption over our spectral window, between 450 and 750 nm. The difference in timescale and temperature between their measurements and ours makes it difficult to directly compare our results, but our absorption spectral shapes are consistent with their experiments. One plausible reaction pathway has been proposed by Yoo and coworkers, 19,22 where the initial chemical interaction is shock-induced parallel alignment of the CS 2 molecules. Following alignment, there is an associative reaction, resulting in CS 2 polymers, which eventually dissociate into carbon and sulfur. The notion that the first reaction is associative rather than dissociative is also supported by the experiments of Engelke et al. 14 An associative reaction is consistent with a volume-decreasing reaction as we see in our experiments and the literature EOS, 5-7,9,10,12 and early electronic alignment could explain our early absorption. B) Decomposition Timescale Based on the phase shift data shown in Fig. 5, it is apparent that under our highest strength shock conditions, we form an opaque product in under 10 ps. However, we do not reach a maximum absorbance per unit length in our transient absorption measurements at similar input conditions until 250 ps. This discrepancy may be caused by the large diameter of our transient absorption white light probe compared to the small area probed by UDE. Using the Beer-Lambert law, A = εbc (7) 17

18 where ε is the molar absorptivity, b is the path length, and c is the concentration of the absorbing species, we can estimate the relative increase in concentration of the absorbing CS 2 reaction products. Path length was calculated as 300 ps multiplied by the shock velocity, and we assumed a constant value for ε. Looking only at the maximum corrected absorbance at 550 nm for CS 2 shocked at conditions above the volume-decreasing reaction, and if we assume that the maximum absorbance from our highest strength shock represents a complete reaction (product concentration fraction=1), we find that the relative concentration of the absorbing species at u p =2.2 km/s was 0.67 and 0.87 at u p =2.8 km/s. The non-unity concentration values indicate that a complete reaction is not reached in 300 ps, at least under u p =3.4 km/s. However, if the u s -u p data is on the products EOS, we assume a full reaction to products, which contradicts the transient absorption data. One explanation for this phenomenon is that our transient absorption measurements are much more sensitive to chemical reactions than our u s -u p measurements. It is also worth noting that our assumption of a constant ε at our different shock conditions is flawed; if more than one step is involved in CS 2 shock reaction chemistry, products of each step may also have different ε. Additionally, studies have shown that ε is sensitive to temperature for several compounds The temperatures for our products were estimated by Sheffield s proposed equation of state, 8 and, for our transient absorption data on the products curve, the temperatures range from 3600 K 6500 K, which likely has a large impact on the value of ε, and thus our calculated concentrations of the absorbing product. V. Summary and Conclusion 18

19 We have used UDE and impedance matching techniques as well as transient absorption spectroscopy to characterize the behavior of CS 2 shocked on the sub-ns timescale. Our u s -u p data indicate a volume-decreasing reaction occurring above u p =1.5 km/s, higher than the literature cusp at u p =1.2 km/s. This change was expected, since higher shock pressures accelerate the chemical kinetics. We found that CS 2 shocked to conditions above the cusp becomes opaque and show broadband absorption centered at 550 nm. However, below the cusp, the shocked CS 2 absorption peaks at 480 nm, possibly indicating multiple reaction steps that can be detected on our sub-ns timescale. VI. Acknowledgements Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA The authors thank Laura Wetzel and Thomas Hoyt for their assistance conducting experiments. The authors gratefully acknowledge the support of this study by Rick Martineau through Science Campaign 2: HE Science. References 1 P. W. Bridgman, Proc. Am. Acad. Arts Sci. (Daedalus) 74, 399 (1942). 2 E. G. Butcher, M. Alsop, J. A. Weston, and H. A. Gebbie, Nature 199, 756 (1963). 3 S. F. Agnew, R. E. Mischke, and B. I. Swanson, J. Phys. Chem. 92, 4201 (1988). 4 P. F. Yuan and Z. J. Ding, J. Phys. Chem. Solids 68, 1841 (2007). 5 J. M. Walsh and M. H. Rice, J. Chem. Phys. 26, 815 (1957). 19

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