Copper dimer interactions on a thermomechanical superfluid 4He fountain Evgeny Popov and Jussi Eloranta Citation: The Journal of Chemical Physics 142, 204704 (2015); doi: 10.1063/1.4921778 View online: http://dx.doi.org/10.1063/1.4921778 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Second-sound studies of coflow and counterflow of superfluid 4He in channels Phys. Fluids 27, 065101 (2015); 10.1063/1.4921816 Solvation of atomic fluorine in bulk superfluid 4He Low Temp. Phys. 37, 384 (2011); 10.1063/1.3599655 Spectroscopy of the copper dimer in normal fluid, superfluid, and solid H 4 e J. Chem. Phys. 133, 154508 (2010); 10.1063/1.3497643 Applicability of Density Functional Theory to Model Molecular Solvation in Superfluid 4He AIP Conf. Proc. 850, 386 (2006); 10.1063/1.2354749 A time dependent density functional treatment of superfluid dynamics: Equilibration of the electron bubble in superfluid 4 He J. Chem. Phys. 117, 10139 (2002); 10.1063/1.1520139
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THE JOURNAL OF CHEMICAL PHYSICS 142, 204704 (2015)
Copper dimer interactions on a thermomechanical superfluid 4He fountain Evgeny Popov and Jussi Elorantaa) Department of Chemistry and Biochemistry, California State University at Northridge, 18111 Nordhoff St., Northridge, California 91330, USA
(Received 21 April 2015; accepted 14 May 2015; published online 28 May 2015) Laser induced fluorescence imaging and frequency domain excitation spectroscopy of the copper dimer (B1Σg+ ← X1Σu+) in thermomechanical helium fountain at 1.7 K are demonstrated. The dimers penetrate into the fountain provided that their average propagation velocity is ca. 15 m/s. This energy threshold is interpreted in terms of an imperfect fountain liquid-gas interface, which acts as a trap for low velocity dimers. Orsay-Trento density functional theory calculations for superfluid 4He are used to characterize the dynamics of the dimer solvation process into the fountain. The dimers first accelerate towards the fountain surface and once the surface layer is crossed, they penetrate into the liquid and further slow down to Landau critical velocity by creating a vortex ring. Theoretical lineshape calculations support the assignment of the experimentally observed bands to Cu2 solvated in the bulk liquid. The vibronic progressions are decomposed of a zero-phonon line and two types of phonon bands, which correlate with solvent cavity interface compression (t < 200 fs) and expansion (200 < t < 500 fs) driven by the electronic excitation. The presented experimental method allows to perform molecular spectroscopy in bulk superfluid helium where the temperature and pressure can be varied. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921778]
I. INTRODUCTION
Superfluid helium is an ideal weakly interacting solvent for high resolution molecular spectroscopy at low temperatures.1–3 The main interests in these experiments are not just limited to understanding the spectroscopy of the solvated atoms and molecules in this matrix but these “probes” can also be used to interrogate the properties of superfluid helium at atomic and molecular length scales.4–8 Most of the early experiments concentrated on embedding neutral and charged atomic species directly into bulk superfluid helium by laser ablation of the precursor solid either inside the liquid or above the liquid surface.9–15 In the latter case, the ions formed in the plasma could be forced into the liquid by using an external electric field. Regarding the former approach, it was shown recently that laser ablation even at fairly low pulse energies leads to the creation of a long-lasting (millisecond time range) gaseous bubble in the liquid.16 This, in turn, confines the atoms and ions formed effectively inside the hot gas, which leads to rapid clustering. On the other hand, if the spectroscopic measurements are carried out before the clustering takes place, the atomic species would still reside in dense gas within the ablation bubble rather than solvated in the bulk liquid. Furthermore, this technique cannot be applied for molecular species as the high energy ablation process would decompose them instantly. The most successful technique to date for solvating atomic and molecular species in superfluid helium for spectroscopic applications is the helium droplet technique.1–3 After expansion, the droplets cool down by evaporation to a base temperature of 0.38 K and have been demonstrated a)E-mail: [email protected]
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to be in superfluid state depending on their size.3,17 The droplet formation stage is then followed by a molecular pickup process, which allows doping the droplets by guest species. The main limitations of this technique remain the fixed temperature and pressure condition as well as the possible boundary effects due to the finite size of the droplets. However, recent advances in the field allow to perform measurements using larger helium droplets, which can mimic the bulk behavior, and therefore relieve the latter constraint to some extent.18 In order to relax the fixed pressure-temperature constraint present in helium droplets, another approach based on a free-standing superfluid fountain was developed.19,20 In this approach, a thermomechanical fountain pump connected to a bulk liquid reservoir can produce a steady liquid flow up to a few m/s velocity that can easily extend undisturbed for centimeters. This moving liquid column provides a constantly refreshing liquid surface for repeated experimental cycles where the temperature can be controlled by adjusting the helium vapor pressure in the reservoir. For example, laser ablation of metals can be carried out a few millimeters away from the fountain such that thermal perturbations from the hot plasma are minimized while still providing enough kinetic energy for atoms/dimers to reach the liquid. Although the method has not been demonstrated for molecular species other than metal dimers, it may be possible to use the laser desorption method to produce the molecules and subsequently inject them into the fountain. It has been demonstrated previously that copper dimers produced by laser ablation can reach the fountain over a few millimeter distance but it was surprisingly not possible to observe vibrationally well resolved absorption spectrum of Cu2 dimer (B ← X transition) in the fountain.20 Furthermore, the previously published bulk
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© 2015 AIP Publishing LLC
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excitation spectrum of this same transition appears very different from the fountain spectrum.10 At present, it is not clear whether these spectra belong to the dimer solvated in the bulk liquid, dense gas, or at the liquid-gas interface. In this paper, we study the interaction of Cu2 molecule with a thermomechanical superfluid helium fountain at 1.7 K. In the first part, laser induced fluorescence (LIF) imaging is used to study the distribution of Cu2 between the copper target and the fountain, followed by spatially resolved LIF excitation spectroscopy experiments of the B ← X transition in the gas phase as well as in the fountain. In the second part, timedependent bosonic density functional theory (DFT) calculations are performed to model the Cu2 solvation dynamics in the fountain and the optical absorption spectrum of Cu2 B ← X transition in the bulk. Finally, the connection between the experimental findings and the theoretical results is discussed. II. EXPERIMENTAL
The experimental arrangement is similar to that described in our earlier work16 with the main differences between the setups illustrated in Fig. 1. The experiment is carried out inside a modified Oxford Cryogenics triple walled helium Dewar (model Variox with 10 cm inner diameter). Both liquid helium and liquid nitrogen used to cool down the cryostat were supplied by Airgas, Inc. The cryostat provides optical access through a set of suprasil quartz windows along three perpendicular axes, which are used for laser ablation and collecting the fluorescence. The vacuum shroud of the cryostat was evacuated by a turbo molecular pump (Pfeiffer TPH 062; 56 l/s pumping speed) backed by a two-stage mechanical pump (Edwards model E2M2). A needle valve connected to the main helium reservoir can be used to control the liquid level inside the cryostat. The liquid temperature is controlled by adjusting the helium vapor pumping speed using a feedback controlled throttle valve (MKS model 252). The base temperature (1.6 K) was determined by the pumping speed of our single stage mechanical pump (Pompe Per Vuoto Rotant model EU65; 20 l/s pumping speed). The liquid temperature and vapor pressure were measured by a rhodium iron sensor (Oxford model T1-103) and a capacitance manometer (MKS Baratron), respectively. The thermomechanical pump used to produce the liquid fountain flow consisted of a resistively heated metal
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housing, which was filled with Fe3O4 particles (average size 100 µm). The intake side was immersed in the liquid, whereas the metal nozzle capillary was connected to the other end (see Fig. 1; top-down geometry). The heater resistance was 20 Ω and a typical driving voltage was around 1 V. Two different pump nozzle diameters were tested, 0.4 and 1.0 mm but no significant difference between the two was observed. For the 1.0 mm nozzle, the liquid flow velocity was estimated as ca. 0.3 m/s, which is comparable with the previously applied fountain pump in a different geometry (1 m/s).20 As shown in Fig. 1, a metallic copper disk (purity better than 99.9%) was mounted at the end of a stainless steel shaft, which was connected to a gearhead motor located at the top part of the cryostat to minimize the heat load on the liquid. The copper disk was rotated slowly (ca. 3 rounds/min) to provide a fresh metal surface for every laser ablation shot. The motor was equipped with a heater and a temperature sensor to avoid freezing. In contrast to our previous experimental setup,20 the main differences are as follows: (1) the fountain flow is from topdown to avoid the droplet mist from the fountain head, (2) the fountain diameter is 2-10 times larger to increase to fountain capture area, (3) the liquid flow velocity is smaller (0.3 m/s vs. 1.0 m/s), (4) the fountain is placed closer to the target to increase Cu2 penetration, and (5) the ablation laser pulse energy is an order of magnitude smaller than used previously. A third harmonic (355 nm) from a pulsed Nd-YAG laser (Continuum Minilite-II, 9 ns pulse length, 10 Hz repetition rate) was used for laser ablation with pulse energies up to 2 mJ/pulse. The ablation laser beam was focused down to a spot size of approximately 50 µm on the metal target with a quartz plano-convex lens (focal length 100 mm). The copper dimers were detected through B1Σg+ ← X1Σu+ fluorescence, which was excited by a tunable dye laser (Lambda Physik ScanMate Pro; 1 cm−1 linewidth; pumped by Continuum Surelite-II, 9 ns pulse length, 10 Hz repetition rate). Unless otherwise noted, the excitation wavelength was 449.8 nm, which corresponds to the B (v ′ = 2) ← X (v ′′ = 0) vibronic level resonance. The applied LIF excitation laser pulse energy was in the range of some tens of µJ/pulse in order to avoid saturation effects. For LIF excitation experiments, the light was focused onto the entrance slit of Digikröm DK240 monochromator by using the standard two-lens light collection arrangement. The exit plane of the monochromator was
FIG. 1. Overview of the experimental setup. BS denotes a quartz plate, which is used as a beam splitter to guide the beams on top of each other inside the cryostat.
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connected to an intensified charge-coupled device (ICCD; Princeton PI-MAX; 2 ns minimum gate width), which provided 512 × 512 pixel resolution. Vertical binning of the pixels yields the normal wavelength resolved fluorescence spectrum (λ > 450 nm collected). For LIF imaging measurements, the same ICCD camera was equipped with two lenses Meteor 1-5 (17-69 mm, f1.9) and Zeikos Close-up + 10 (77 mm). The gate width of the ICCD was matched with the known radiative lifetime of the B ← X transition (30 ns)21 and the gate position was optimized to minimize the amount of scattered laser light from both the ablation and LIF excitation lasers. The flash lamp, Q-switch, and ICCD timings were provided by two delay generators (SRS DG-535 and BNC565 both with less than 2 ns jitter). Both the delay generators and the ICCD were controlled by IEEE-488 and firewire interfaces through the libmeas scientific instrument interface library.22
III. THEORY
Superfluid 4He was modeled by the Orsay-Trento DFT (OT-DFT) and the interaction with copper dimers was included as an external potential in the Hamiltonian using rotationally averaged (J = 0) pairwise interaction between Cu2 and He.23,24 This assumes that the potential around Cu2 remains sufficiently spherical such that rotational quenching does not take place. The ground state Cu2(X)–He potentials (linear, broadside, and spherically averaged) were obtained at both CCSD(T)/AV5ZDK (coupled cluster with single, double and perturbative triple substitutions) and CCSD/AV5Z-DK levels of theory (see the Appendix),25–28 with the former predicting a slightly larger binding than the previous coupled cluster (CC) based calculations.20 Similarly, the Cu2(B)–He excited state potentials were calculated by equation of motion CCSD-EOM/AV5ZDK method,29 but just like the CCSD/AV5Z-DK calculation for the ground state potentials, this treatment is missing the important triples contributions. For the spectroscopic lineshape calculations, both the ground and excited state potentials were taken from CCSD to provide a balanced description of the states. The main factor limiting the accuracy of these calculations is the rather high value of T1 norm (≈0.03), which exceeds the generally accepted limit for single determinant CC calculations (0.02).30 Presumably for this reason, the inclusion of the triples had an unusually large impact on the calculated binding energies. To eliminate the basis set superposition error, the standard counterpoise correction procedure of Boys and Bernardi was applied.31 All electronic structure calculations were carried out with the Molpro package.32 The dynamics of Cu2 interacting with the fountain was modeled by treating the dimer classically and integrating its equation of motion by the velocity Verlet algorithm33 alongside with time-dependent propagation of OT-DFT for superfluid helium. The force F⃗ acting on the classical particle is given by34,35 ⃗ ⃗ F = −∇r V (|r − r ′|) ρ(r ′,t)d 3r ′ ⃗ r V (|r − r ′|) ρ(r ′,t)d 3r ′, =− ∇ (1)
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where ρ represents the superfluid helium density at time t, V is the Cu2(X)–He spherically averaged pair potential, and the integrals are evaluated over the simulation cube. The initial condition for the calculation is determined by the starting position and velocity for Cu2 as well as the minimum energy liquid helium density distribution. These were chosen such that the starting position is sufficiently far away (>25 bohrs) from the fountain to avoid abrupt liquid dynamics and the initial velocity was fixed to the experimentally observed Cu2 average velocity (15 m/s). The measurements were carried out using a macroscopic size fountain (thickness of several hundred µm), which is not possible to simulate using OT-DFT with the current computational resources. To mimic the fountain in the simulations, a slab of helium with a width of 260 bohrs was used. This is sufficient for modeling the fountain experiment as it can provide large enough liquid helium surface area (avoid boundary effects) and it includes sufficient amount of bulk liquid to allow Cu2 travel over a short period of time. However, it should be noted that the present OT-DFT model cannot handle the finite temperature of the experiments and therefore it is missing, for example, the viscous liquid response as well as the possible thermal excitations on the surface (i.e., ripplons) and in the bulk (i.e., reduced value of critical velocity). The initial liquid distribution for superfluid helium fountain was obtained by propagating the OT-DFT functional in imaginary time using a time step of 10 fs. The discrete spatial grid employed in the calculations was either 128 × 128 × 1024 or 512 × 512 × 512 with a step size of 0.5 bohrs. The real time OT-DFT calculations (∆t = 5 fs) included the time dependent external potential due to the moving classical Cu2 molecule. Predict-correct steps were employed in the Crank-Nicolson propagator where the future external potential was obtained by integrating the classical Cu2 degrees forward in time. The corrective step then used the average of the present and future potentials for the actual propagation. Several different variants of OT-DFT functional were applied: (1) the “basic” version of OT-DFT functional, which did not include the kinetic energy correlation (KC) or the backflow (BF) terms; (2) OT-DFT with the KC term; (3) OT-DFT with the BF term; and (4) full OT-DFT including both the kinetic energy correlation and backflow terms (KC + BF). All these functionals included the high density correction term as the Cu2-fountain impact event may produce high liquid densities near the dimer.36 Each one of these functionals has a different value for Landau critical velocity as the depth of the roton minimum shifts depending on the functional used.37 This is illustrated in Fig. 2 where the calculated dispersion relation for each functional is shown along with the slope determining the critical velocity. Note that the inclusion of the backflow term is most important for reproducing the roton minimum energy. With the full OT-DFT functional and the applied spatial grid resolution, a slightly higher roton minimum energy (9.8 K) is obtained as compared to the calculation with a fine grid (8.8 K) and, consequently, the critical velocity produced by the present calculations is, therefore, also slightly higher. The inclusion of both BF and KC terms required a spatial grid step of 0.5 bohrs or less and KC also placed further restrictions on the longest allowed time step ( 150 fs), the ZPL is present and appears within 2 cm−1 of the corresponding experimental line position. In time domain, the origin of ZPL can be identified by inspecting the first order polarization P(1)(t) shown in Fig. 9. At the level of the present lineshape theory, ZPL arises from the very late stage of the average potential dynamics where a near equilibrium structure is reached. For this reason, the appearance of a ZPL implies sufficiently long dephasing times such that the corresponding portion of the time evolution can be reached. For the present system, starting with the X state equilibrium liquid structure, the average X/B state equilibrium is reached within a couple of picoseconds but the late stage dynamics does not contribute to the spectrum due to dephasing. The gradual appearance of ZPL as a function of increasing dephasing time constant τc is demonstrated in Fig. 8. Although, very low LIF excitation pulse energies were used in the experiments, it is still possible that the ZPL was partially saturated, which could increase the relative contribution of the phonon bands.3 The LIF excitation power level in the present experiments was optimized to provide a good overview of both ZPL and the phonon bands. The band labeled “PW I” in Fig. 8 corresponds to the early time dynamics on the X/B average potential, which takes place within the first 200 fs (see Fig. 9). By inspecting the liquid density evolution during this time period, the underlying motion can be identified as rapid liquid-gas interface profile compression. This can be rationalized using by Jortner’s liquid density profile function,49 ( ) ρ0 1 − [1 + α (r − R)] e−α(r −R) , when r > R ρ(r) = , when r ≤ R 0, (7)
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where ρ0 is the bulk liquid density, R is the approximate solvation cavity radius, and α is the interface width. The two dynamic variables R(t) and α(t) can be assigned to the observed spectral components as follows. The interface profile compression corresponds to a rapid change in α and is the fastest motion present in the evolving bubble. The rapid oscillations present in P(1)(t) during this time period (see Fig. 9) give a strongly blue shifted and broadened band, which is labeled as “PW I” in Fig. 8. The position and width of this band depend on the repulsion present on the excited state and the energy loss during the interface compression dynamics. The phonon band region labeled “PW II” in Figs. 8 and 9 corresponds to the intermediate time regime for the evolving bubble. Within this time period, the compressed gas-liquid interface starts extending towards a larger radius R (i.e., bubble expansion stage; see Eq. (7)). This takes place during 200500 fs and possesses intermediate spectral components from 50 to 250 cm−1. The spectral features produced by this liquid motion are rather broad but two prominent peaks remain, which can also be seen in the experimental spectrum at ca. 160 cm−1 and possibly at 105 cm−1 in Fig. 8. A longer dephasing time would allow more detailed study of these components.
V. CONCLUSIONS
Our experimental results (Figs. 3 and 4) demonstrate that Cu2, formed in the gas phase following laser ablation, are attracted to the helium fountain provided that they have sufficient kinetic energy. The existence of such an energy threshold indicates that the fountain is surrounded by a nonliquid inhomogeneous region, which must be crossed before the bulk part of the fountain can be reached. The OT-DFT calculations at 0 K demonstrate that once the fountain is reached, Cu2 will freely propagate into the core region of the fountain at Landau critical velocity. The LIF excitation spectra recorded in the fountain display an overall broadened vibronic B ← X progression, which contain relatively sharp ZPLs and the accompanying phonon bands. The lineshape simulations are able to reproduce the experimentally observed vibronic band structure qualitatively. This excludes the inhomogeneous background contribution, which we propose to originate from Cu2 trapped near the fountain surface. Real time OT-DFT calculations allow to assign the ZPL and the two phonon bands (PW I and PW II) to specific liquid motions: (1) interface compression (t < 200 fs; PW I), (2) radial bubble expansion (200 < t < 500 fs; PW II), and (3) low amplitude damped oscillations towards the equilibrium structure (t > 500 fs; ZPL). In general, we have demonstrated that the thermomechanical fountain method is a viable approach for studying atomic/molecular spectroscopy in superfluid helium where both temperature and pressure can be varied.
ACKNOWLEDGMENTS
Financial support from the National Science Foundation Grant No. CHE-1262306 is gratefully acknowledged.
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APPENDIX
The calculated Cu2–He pair potentials from CCSD(T)/AV5Z-DK and CCSD-EOM/AV5Z-DK are given in Table I. The basis set superposition error was removed by the counterpoise method.31 The J = 0 spherical average potential VS is obtained from the linear (L) and broadside (T) approach potentials by VS = (VL + 2VT )/3. TABLE I. Raw linear and broadside approach pair-potential data for Cu2(X,B)–He. Two potentials are given for the X state: (1) CCSD(T)/AV5Z-DK and (2) CCSD/AV5Z-DK. The B state data correspond to CCSD-EOM/AV5Z-DK. “T” corresponds to a broadside approach (90◦), whereas “L” is the linear geometry (0◦). R corresponds to the distance between the He atom and the mass center of Cu2 (Bohr) and the corresponding counterpoise corrected energies are given in Hartree.
R 20.0 19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.5 11.0 10.5 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0
1J.
CCSD(T) Linear X
CCSD(T) T-shape X
CCSD Linear X
CCSD T-shape X
EOM-CCSD Linear B
EOM-CCSD T-shape B
−6.6199 × 10−7 −9.7316 × 10−7 −1.3230 × 10−6 −1.9620 × 10−6 −2.9476 × 10−6 −4.5686 × 10−6 −7.2618 × 10−6 −1.1947 × 10−5 −1.9686 × 10−5 −2.5636 × 10−5 −3.3190 × 10−5 −4.2287 × 10−5 −5.1850 × 10−5 −5.8697 × 10−5 −5.4743 × 10−5 −2.2491 × 10−5 7.3131 × 10−5 2.9651 × 10−4 7.5594 × 10−4 1.6192 × 10−3 3.1940 × 10−3 6.6395 × 10−3 1.8398 × 10−2
−4.7340 × 10−7 −6.4688 × 10−7 −9.8462 × 10−7 −1.0628 × 10−6 −2.3063 × 10−6 −2.0082 × 10−6 −4.4701 × 10−6 −6.1984 × 10−6 −1.1325 × 10−5 ... −1.7647 × 10−5 ... −2.7740 × 10−5 ... −3.7990 × 10−5 −3.6513 × 10−5 −1.9071 × 10−5 3.6338 × 10−5 1.7473 × 10−4 4.8529 × 10−4 1.1391 × 10−3 2.4543 × 10−3 5.0077 × 10−3
−5.9020 × 10−7 −8.7209 × 10−7 −1.1728 × 10−6 −1.7485 × 10−6 −2.6183 × 10−6 −4.0583 × 10−6 −6.4270 × 10−6 −1.0502 × 10−5 −1.6940 × 10−5 −2.1789 × 10−5 −2.7715 × 10−5 −3.4362 × 10−5 −4.0264 × 10−5 −4.1718 × 10−5 −2.9930 × 10−5 1.3387 × 10−5 1.2437 × 10−4 3.6980 × 10−4 8.6608 × 10−4 1.8085 × 10−3 3.5832 × 10−3 7.5085 × 10−3 2.0158 × 10−2
−4.3059 × 10−7 −5.8662 × 10−7 −9.0773 × 10−7 −9.1362 × 10−7 −2.1772 × 10−6 −1.6512 × 10−6 −4.0812 × 10−6 −5.4741 × 10−6 −1.0239 × 10−5 ... −1.5489 × 10−5 ... −2.3132 × 10−5 ... −2.7181 × 10−5 −1.9428 × 10−5 8.2123 × 10−6 7.9978 × 10−5 2.4423 × 10−4 5.9465 × 10−4 1.3083 × 10−3 2.7106 × 10−3 5.3873 × 10−3
−1.0583 × 10−6 −1.6406 × 10−6 −2.1444 × 10−6 −3.2439 × 10−6 −4.7462 × 10−6 −6.9023 × 10−6 −9.2464 × 10−6 −1.0026 × 10−5 −8.2241 × 10−8 1.4265 × 10−5 4.2871 × 10−5 9.6985 × 10−5 1.9676 × 10−4 3.7678 × 10−4 6.9761 × 10−4 1.2631 × 10−3 2.2518 × 10−3 3.9655 × 10−3 6.9035 × 10−3 1.1835 × 10−2 1.9780 × 10−3 3.1902 × 10−2 5.3054 × 10−2
−2.8971 × 10−7 −4.7211 × 10−7 −5.2362 × 10−7 −6.4086 × 10−7 −1.5497 × 10−6 −3.0833 × 10−6 −5.2381 × 10−6 −5.2191 × 10−6 −1.3394 × 10−5 ... −1.9592 × 10−5 ... −3.3042 × 10−5 ... −5.7995 × 10−5 −7.4790 × 10−5 −9.2096 × 10−5 −1.0184 × 10−4 −8.3846 × 10−5 6.9846 × 10−6 2.6622 × 10−4 8.8295 × 10−4 2.2254 × 10−3
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Vehmanen, V. Ghazarian, C. Sams, I. Khatchatryan, J. Eloranta, and V. A. Apkarian, J. Phys. Chem. A 115, 7077 (2011). 21M. D. Morse, Chem. Rev. 86, 1049 (1986). 22J. Eloranta, “A simplified C programming interface for research instruments,” http://sourceforge.net/projects/libmeas/ (2015). 23F. Dalfovo, A. Lastri, L. Pricaupenko, S. Stringari, and J. Treiner, Phys. Rev. B 52, 1193 (1995). 24L. Lehtovaara, T. Kiljunen, and J. Eloranta, J. Comput. Phys. 194, 78 (2004). 25C. Hampel, K. Peterson, and H.-J. Werner, Chem. Phys. Lett. 190, 1 (1992). 26M. J. O. Deegan and P. J. Knowles, Chem. Phys. Lett. 227, 321 (1994). 27D. Woon and J. T. H. Dunning, J. Chem. Phys. 100, 2975 (1994). 28N. Balabanov and K. Peterson, J. Chem. Phys. 123, 064107 (2005). 29T. Korona and H.-J. Werner, J. Chem. Phys. 118, 3006 (2003). 30T. J. Lee and P. R. Taylor, Int. J. Quantum Chem. 36, 199 (1989). 31S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 (1970). 32H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz et al., , version 2012.1, a package of ab initio programs, 2012, see http://www. molpro.net. 33M. P. Allen and D. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). 34D. Mateo, A. Leal, A. Hernando, M. Barranco, M. Pi, F. Cargnoni, M. Mella, X. Zhang, and M. Drabbels, J. Chem. Phys. 140, 131101 (2014). 35S. L. Fiedler, D. Mateo, T. Aleksanyan, and J. Eloranta, Phys. Rev. B 86, 144522 (2012). 36F. Ancilotto, M. Barranco, F. Caupin, R. Mayol, and M. Pi, Phys. Rev. B 72, 214522 (2005). 37N. B. Brauer, S. Smolarek, E. Loginov, D. Mateo, A. Hernando, M. Pi, M. Barranco, W. J. Buma, and M. Drabbels, Phys. Rev. Lett. 111, 143002 (2013).
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Physics Vol. 3 (Cambridge University Press, Cambridge, 1991). H. Roberts and J. Grant, J. Phys. A: Gen. Phys. 4, 55 (1971). 47D. Mateo, J. Eloranta, and G. A. Williams, J. Chem. Phys. 142, 064510 (2015). 48F. Dalfovo, M. Guilleumas, A. Lastri, L. Pitaevskii, and S. Stringari, J. Phys.: Condens. Matter 9, L369 (1997). 49K. Hirioke, N. R. Kestner, S. A. Rice, and J. Jortner, J. Chem. Phys. 43, 2625 (1965). 46P.
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THE JOURNAL OF CHEMICAL PHYSICS 142, 204704 (2015)
Copper dimer interactions on a thermomechanical superfluid 4He fountain Evgeny Popov and Jussi Elorantaa) Department of Chemistry and Biochemistry, California State University at Northridge, 18111 Nordhoff St., Northridge, California 91330, USA
(Received 21 April 2015; accepted 14 May 2015; published online 28 May 2015) Laser induced fluorescence imaging and frequency domain excitation spectroscopy of the copper dimer (B1Σg+ ← X1Σu+) in thermomechanical helium fountain at 1.7 K are demonstrated. The dimers penetrate into the fountain provided that their average propagation velocity is ca. 15 m/s. This energy threshold is interpreted in terms of an imperfect fountain liquid-gas interface, which acts as a trap for low velocity dimers. Orsay-Trento density functional theory calculations for superfluid 4He are used to characterize the dynamics of the dimer solvation process into the fountain. The dimers first accelerate towards the fountain surface and once the surface layer is crossed, they penetrate into the liquid and further slow down to Landau critical velocity by creating a vortex ring. Theoretical lineshape calculations support the assignment of the experimentally observed bands to Cu2 solvated in the bulk liquid. The vibronic progressions are decomposed of a zero-phonon line and two types of phonon bands, which correlate with solvent cavity interface compression (t < 200 fs) and expansion (200 < t < 500 fs) driven by the electronic excitation. The presented experimental method allows to perform molecular spectroscopy in bulk superfluid helium where the temperature and pressure can be varied. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921778]
I. INTRODUCTION
Superfluid helium is an ideal weakly interacting solvent for high resolution molecular spectroscopy at low temperatures.1–3 The main interests in these experiments are not just limited to understanding the spectroscopy of the solvated atoms and molecules in this matrix but these “probes” can also be used to interrogate the properties of superfluid helium at atomic and molecular length scales.4–8 Most of the early experiments concentrated on embedding neutral and charged atomic species directly into bulk superfluid helium by laser ablation of the precursor solid either inside the liquid or above the liquid surface.9–15 In the latter case, the ions formed in the plasma could be forced into the liquid by using an external electric field. Regarding the former approach, it was shown recently that laser ablation even at fairly low pulse energies leads to the creation of a long-lasting (millisecond time range) gaseous bubble in the liquid.16 This, in turn, confines the atoms and ions formed effectively inside the hot gas, which leads to rapid clustering. On the other hand, if the spectroscopic measurements are carried out before the clustering takes place, the atomic species would still reside in dense gas within the ablation bubble rather than solvated in the bulk liquid. Furthermore, this technique cannot be applied for molecular species as the high energy ablation process would decompose them instantly. The most successful technique to date for solvating atomic and molecular species in superfluid helium for spectroscopic applications is the helium droplet technique.1–3 After expansion, the droplets cool down by evaporation to a base temperature of 0.38 K and have been demonstrated a)E-mail: [email protected]
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to be in superfluid state depending on their size.3,17 The droplet formation stage is then followed by a molecular pickup process, which allows doping the droplets by guest species. The main limitations of this technique remain the fixed temperature and pressure condition as well as the possible boundary effects due to the finite size of the droplets. However, recent advances in the field allow to perform measurements using larger helium droplets, which can mimic the bulk behavior, and therefore relieve the latter constraint to some extent.18 In order to relax the fixed pressure-temperature constraint present in helium droplets, another approach based on a free-standing superfluid fountain was developed.19,20 In this approach, a thermomechanical fountain pump connected to a bulk liquid reservoir can produce a steady liquid flow up to a few m/s velocity that can easily extend undisturbed for centimeters. This moving liquid column provides a constantly refreshing liquid surface for repeated experimental cycles where the temperature can be controlled by adjusting the helium vapor pressure in the reservoir. For example, laser ablation of metals can be carried out a few millimeters away from the fountain such that thermal perturbations from the hot plasma are minimized while still providing enough kinetic energy for atoms/dimers to reach the liquid. Although the method has not been demonstrated for molecular species other than metal dimers, it may be possible to use the laser desorption method to produce the molecules and subsequently inject them into the fountain. It has been demonstrated previously that copper dimers produced by laser ablation can reach the fountain over a few millimeter distance but it was surprisingly not possible to observe vibrationally well resolved absorption spectrum of Cu2 dimer (B ← X transition) in the fountain.20 Furthermore, the previously published bulk
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© 2015 AIP Publishing LLC
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excitation spectrum of this same transition appears very different from the fountain spectrum.10 At present, it is not clear whether these spectra belong to the dimer solvated in the bulk liquid, dense gas, or at the liquid-gas interface. In this paper, we study the interaction of Cu2 molecule with a thermomechanical superfluid helium fountain at 1.7 K. In the first part, laser induced fluorescence (LIF) imaging is used to study the distribution of Cu2 between the copper target and the fountain, followed by spatially resolved LIF excitation spectroscopy experiments of the B ← X transition in the gas phase as well as in the fountain. In the second part, timedependent bosonic density functional theory (DFT) calculations are performed to model the Cu2 solvation dynamics in the fountain and the optical absorption spectrum of Cu2 B ← X transition in the bulk. Finally, the connection between the experimental findings and the theoretical results is discussed. II. EXPERIMENTAL
The experimental arrangement is similar to that described in our earlier work16 with the main differences between the setups illustrated in Fig. 1. The experiment is carried out inside a modified Oxford Cryogenics triple walled helium Dewar (model Variox with 10 cm inner diameter). Both liquid helium and liquid nitrogen used to cool down the cryostat were supplied by Airgas, Inc. The cryostat provides optical access through a set of suprasil quartz windows along three perpendicular axes, which are used for laser ablation and collecting the fluorescence. The vacuum shroud of the cryostat was evacuated by a turbo molecular pump (Pfeiffer TPH 062; 56 l/s pumping speed) backed by a two-stage mechanical pump (Edwards model E2M2). A needle valve connected to the main helium reservoir can be used to control the liquid level inside the cryostat. The liquid temperature is controlled by adjusting the helium vapor pumping speed using a feedback controlled throttle valve (MKS model 252). The base temperature (1.6 K) was determined by the pumping speed of our single stage mechanical pump (Pompe Per Vuoto Rotant model EU65; 20 l/s pumping speed). The liquid temperature and vapor pressure were measured by a rhodium iron sensor (Oxford model T1-103) and a capacitance manometer (MKS Baratron), respectively. The thermomechanical pump used to produce the liquid fountain flow consisted of a resistively heated metal
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housing, which was filled with Fe3O4 particles (average size 100 µm). The intake side was immersed in the liquid, whereas the metal nozzle capillary was connected to the other end (see Fig. 1; top-down geometry). The heater resistance was 20 Ω and a typical driving voltage was around 1 V. Two different pump nozzle diameters were tested, 0.4 and 1.0 mm but no significant difference between the two was observed. For the 1.0 mm nozzle, the liquid flow velocity was estimated as ca. 0.3 m/s, which is comparable with the previously applied fountain pump in a different geometry (1 m/s).20 As shown in Fig. 1, a metallic copper disk (purity better than 99.9%) was mounted at the end of a stainless steel shaft, which was connected to a gearhead motor located at the top part of the cryostat to minimize the heat load on the liquid. The copper disk was rotated slowly (ca. 3 rounds/min) to provide a fresh metal surface for every laser ablation shot. The motor was equipped with a heater and a temperature sensor to avoid freezing. In contrast to our previous experimental setup,20 the main differences are as follows: (1) the fountain flow is from topdown to avoid the droplet mist from the fountain head, (2) the fountain diameter is 2-10 times larger to increase to fountain capture area, (3) the liquid flow velocity is smaller (0.3 m/s vs. 1.0 m/s), (4) the fountain is placed closer to the target to increase Cu2 penetration, and (5) the ablation laser pulse energy is an order of magnitude smaller than used previously. A third harmonic (355 nm) from a pulsed Nd-YAG laser (Continuum Minilite-II, 9 ns pulse length, 10 Hz repetition rate) was used for laser ablation with pulse energies up to 2 mJ/pulse. The ablation laser beam was focused down to a spot size of approximately 50 µm on the metal target with a quartz plano-convex lens (focal length 100 mm). The copper dimers were detected through B1Σg+ ← X1Σu+ fluorescence, which was excited by a tunable dye laser (Lambda Physik ScanMate Pro; 1 cm−1 linewidth; pumped by Continuum Surelite-II, 9 ns pulse length, 10 Hz repetition rate). Unless otherwise noted, the excitation wavelength was 449.8 nm, which corresponds to the B (v ′ = 2) ← X (v ′′ = 0) vibronic level resonance. The applied LIF excitation laser pulse energy was in the range of some tens of µJ/pulse in order to avoid saturation effects. For LIF excitation experiments, the light was focused onto the entrance slit of Digikröm DK240 monochromator by using the standard two-lens light collection arrangement. The exit plane of the monochromator was
FIG. 1. Overview of the experimental setup. BS denotes a quartz plate, which is used as a beam splitter to guide the beams on top of each other inside the cryostat.
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connected to an intensified charge-coupled device (ICCD; Princeton PI-MAX; 2 ns minimum gate width), which provided 512 × 512 pixel resolution. Vertical binning of the pixels yields the normal wavelength resolved fluorescence spectrum (λ > 450 nm collected). For LIF imaging measurements, the same ICCD camera was equipped with two lenses Meteor 1-5 (17-69 mm, f1.9) and Zeikos Close-up + 10 (77 mm). The gate width of the ICCD was matched with the known radiative lifetime of the B ← X transition (30 ns)21 and the gate position was optimized to minimize the amount of scattered laser light from both the ablation and LIF excitation lasers. The flash lamp, Q-switch, and ICCD timings were provided by two delay generators (SRS DG-535 and BNC565 both with less than 2 ns jitter). Both the delay generators and the ICCD were controlled by IEEE-488 and firewire interfaces through the libmeas scientific instrument interface library.22
III. THEORY
Superfluid 4He was modeled by the Orsay-Trento DFT (OT-DFT) and the interaction with copper dimers was included as an external potential in the Hamiltonian using rotationally averaged (J = 0) pairwise interaction between Cu2 and He.23,24 This assumes that the potential around Cu2 remains sufficiently spherical such that rotational quenching does not take place. The ground state Cu2(X)–He potentials (linear, broadside, and spherically averaged) were obtained at both CCSD(T)/AV5ZDK (coupled cluster with single, double and perturbative triple substitutions) and CCSD/AV5Z-DK levels of theory (see the Appendix),25–28 with the former predicting a slightly larger binding than the previous coupled cluster (CC) based calculations.20 Similarly, the Cu2(B)–He excited state potentials were calculated by equation of motion CCSD-EOM/AV5ZDK method,29 but just like the CCSD/AV5Z-DK calculation for the ground state potentials, this treatment is missing the important triples contributions. For the spectroscopic lineshape calculations, both the ground and excited state potentials were taken from CCSD to provide a balanced description of the states. The main factor limiting the accuracy of these calculations is the rather high value of T1 norm (≈0.03), which exceeds the generally accepted limit for single determinant CC calculations (0.02).30 Presumably for this reason, the inclusion of the triples had an unusually large impact on the calculated binding energies. To eliminate the basis set superposition error, the standard counterpoise correction procedure of Boys and Bernardi was applied.31 All electronic structure calculations were carried out with the Molpro package.32 The dynamics of Cu2 interacting with the fountain was modeled by treating the dimer classically and integrating its equation of motion by the velocity Verlet algorithm33 alongside with time-dependent propagation of OT-DFT for superfluid helium. The force F⃗ acting on the classical particle is given by34,35 ⃗ ⃗ F = −∇r V (|r − r ′|) ρ(r ′,t)d 3r ′ ⃗ r V (|r − r ′|) ρ(r ′,t)d 3r ′, =− ∇ (1)
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where ρ represents the superfluid helium density at time t, V is the Cu2(X)–He spherically averaged pair potential, and the integrals are evaluated over the simulation cube. The initial condition for the calculation is determined by the starting position and velocity for Cu2 as well as the minimum energy liquid helium density distribution. These were chosen such that the starting position is sufficiently far away (>25 bohrs) from the fountain to avoid abrupt liquid dynamics and the initial velocity was fixed to the experimentally observed Cu2 average velocity (15 m/s). The measurements were carried out using a macroscopic size fountain (thickness of several hundred µm), which is not possible to simulate using OT-DFT with the current computational resources. To mimic the fountain in the simulations, a slab of helium with a width of 260 bohrs was used. This is sufficient for modeling the fountain experiment as it can provide large enough liquid helium surface area (avoid boundary effects) and it includes sufficient amount of bulk liquid to allow Cu2 travel over a short period of time. However, it should be noted that the present OT-DFT model cannot handle the finite temperature of the experiments and therefore it is missing, for example, the viscous liquid response as well as the possible thermal excitations on the surface (i.e., ripplons) and in the bulk (i.e., reduced value of critical velocity). The initial liquid distribution for superfluid helium fountain was obtained by propagating the OT-DFT functional in imaginary time using a time step of 10 fs. The discrete spatial grid employed in the calculations was either 128 × 128 × 1024 or 512 × 512 × 512 with a step size of 0.5 bohrs. The real time OT-DFT calculations (∆t = 5 fs) included the time dependent external potential due to the moving classical Cu2 molecule. Predict-correct steps were employed in the Crank-Nicolson propagator where the future external potential was obtained by integrating the classical Cu2 degrees forward in time. The corrective step then used the average of the present and future potentials for the actual propagation. Several different variants of OT-DFT functional were applied: (1) the “basic” version of OT-DFT functional, which did not include the kinetic energy correlation (KC) or the backflow (BF) terms; (2) OT-DFT with the KC term; (3) OT-DFT with the BF term; and (4) full OT-DFT including both the kinetic energy correlation and backflow terms (KC + BF). All these functionals included the high density correction term as the Cu2-fountain impact event may produce high liquid densities near the dimer.36 Each one of these functionals has a different value for Landau critical velocity as the depth of the roton minimum shifts depending on the functional used.37 This is illustrated in Fig. 2 where the calculated dispersion relation for each functional is shown along with the slope determining the critical velocity. Note that the inclusion of the backflow term is most important for reproducing the roton minimum energy. With the full OT-DFT functional and the applied spatial grid resolution, a slightly higher roton minimum energy (9.8 K) is obtained as compared to the calculation with a fine grid (8.8 K) and, consequently, the critical velocity produced by the present calculations is, therefore, also slightly higher. The inclusion of both BF and KC terms required a spatial grid step of 0.5 bohrs or less and KC also placed further restrictions on the longest allowed time step ( 150 fs), the ZPL is present and appears within 2 cm−1 of the corresponding experimental line position. In time domain, the origin of ZPL can be identified by inspecting the first order polarization P(1)(t) shown in Fig. 9. At the level of the present lineshape theory, ZPL arises from the very late stage of the average potential dynamics where a near equilibrium structure is reached. For this reason, the appearance of a ZPL implies sufficiently long dephasing times such that the corresponding portion of the time evolution can be reached. For the present system, starting with the X state equilibrium liquid structure, the average X/B state equilibrium is reached within a couple of picoseconds but the late stage dynamics does not contribute to the spectrum due to dephasing. The gradual appearance of ZPL as a function of increasing dephasing time constant τc is demonstrated in Fig. 8. Although, very low LIF excitation pulse energies were used in the experiments, it is still possible that the ZPL was partially saturated, which could increase the relative contribution of the phonon bands.3 The LIF excitation power level in the present experiments was optimized to provide a good overview of both ZPL and the phonon bands. The band labeled “PW I” in Fig. 8 corresponds to the early time dynamics on the X/B average potential, which takes place within the first 200 fs (see Fig. 9). By inspecting the liquid density evolution during this time period, the underlying motion can be identified as rapid liquid-gas interface profile compression. This can be rationalized using by Jortner’s liquid density profile function,49 ( ) ρ0 1 − [1 + α (r − R)] e−α(r −R) , when r > R ρ(r) = , when r ≤ R 0, (7)
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where ρ0 is the bulk liquid density, R is the approximate solvation cavity radius, and α is the interface width. The two dynamic variables R(t) and α(t) can be assigned to the observed spectral components as follows. The interface profile compression corresponds to a rapid change in α and is the fastest motion present in the evolving bubble. The rapid oscillations present in P(1)(t) during this time period (see Fig. 9) give a strongly blue shifted and broadened band, which is labeled as “PW I” in Fig. 8. The position and width of this band depend on the repulsion present on the excited state and the energy loss during the interface compression dynamics. The phonon band region labeled “PW II” in Figs. 8 and 9 corresponds to the intermediate time regime for the evolving bubble. Within this time period, the compressed gas-liquid interface starts extending towards a larger radius R (i.e., bubble expansion stage; see Eq. (7)). This takes place during 200500 fs and possesses intermediate spectral components from 50 to 250 cm−1. The spectral features produced by this liquid motion are rather broad but two prominent peaks remain, which can also be seen in the experimental spectrum at ca. 160 cm−1 and possibly at 105 cm−1 in Fig. 8. A longer dephasing time would allow more detailed study of these components.
V. CONCLUSIONS
Our experimental results (Figs. 3 and 4) demonstrate that Cu2, formed in the gas phase following laser ablation, are attracted to the helium fountain provided that they have sufficient kinetic energy. The existence of such an energy threshold indicates that the fountain is surrounded by a nonliquid inhomogeneous region, which must be crossed before the bulk part of the fountain can be reached. The OT-DFT calculations at 0 K demonstrate that once the fountain is reached, Cu2 will freely propagate into the core region of the fountain at Landau critical velocity. The LIF excitation spectra recorded in the fountain display an overall broadened vibronic B ← X progression, which contain relatively sharp ZPLs and the accompanying phonon bands. The lineshape simulations are able to reproduce the experimentally observed vibronic band structure qualitatively. This excludes the inhomogeneous background contribution, which we propose to originate from Cu2 trapped near the fountain surface. Real time OT-DFT calculations allow to assign the ZPL and the two phonon bands (PW I and PW II) to specific liquid motions: (1) interface compression (t < 200 fs; PW I), (2) radial bubble expansion (200 < t < 500 fs; PW II), and (3) low amplitude damped oscillations towards the equilibrium structure (t > 500 fs; ZPL). In general, we have demonstrated that the thermomechanical fountain method is a viable approach for studying atomic/molecular spectroscopy in superfluid helium where both temperature and pressure can be varied.
ACKNOWLEDGMENTS
Financial support from the National Science Foundation Grant No. CHE-1262306 is gratefully acknowledged.
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APPENDIX
The calculated Cu2–He pair potentials from CCSD(T)/AV5Z-DK and CCSD-EOM/AV5Z-DK are given in Table I. The basis set superposition error was removed by the counterpoise method.31 The J = 0 spherical average potential VS is obtained from the linear (L) and broadside (T) approach potentials by VS = (VL + 2VT )/3. TABLE I. Raw linear and broadside approach pair-potential data for Cu2(X,B)–He. Two potentials are given for the X state: (1) CCSD(T)/AV5Z-DK and (2) CCSD/AV5Z-DK. The B state data correspond to CCSD-EOM/AV5Z-DK. “T” corresponds to a broadside approach (90◦), whereas “L” is the linear geometry (0◦). R corresponds to the distance between the He atom and the mass center of Cu2 (Bohr) and the corresponding counterpoise corrected energies are given in Hartree.
R 20.0 19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.5 11.0 10.5 10.0 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0
1J.
CCSD(T) Linear X
CCSD(T) T-shape X
CCSD Linear X
CCSD T-shape X
EOM-CCSD Linear B
EOM-CCSD T-shape B
−6.6199 × 10−7 −9.7316 × 10−7 −1.3230 × 10−6 −1.9620 × 10−6 −2.9476 × 10−6 −4.5686 × 10−6 −7.2618 × 10−6 −1.1947 × 10−5 −1.9686 × 10−5 −2.5636 × 10−5 −3.3190 × 10−5 −4.2287 × 10−5 −5.1850 × 10−5 −5.8697 × 10−5 −5.4743 × 10−5 −2.2491 × 10−5 7.3131 × 10−5 2.9651 × 10−4 7.5594 × 10−4 1.6192 × 10−3 3.1940 × 10−3 6.6395 × 10−3 1.8398 × 10−2
−4.7340 × 10−7 −6.4688 × 10−7 −9.8462 × 10−7 −1.0628 × 10−6 −2.3063 × 10−6 −2.0082 × 10−6 −4.4701 × 10−6 −6.1984 × 10−6 −1.1325 × 10−5 ... −1.7647 × 10−5 ... −2.7740 × 10−5 ... −3.7990 × 10−5 −3.6513 × 10−5 −1.9071 × 10−5 3.6338 × 10−5 1.7473 × 10−4 4.8529 × 10−4 1.1391 × 10−3 2.4543 × 10−3 5.0077 × 10−3
−5.9020 × 10−7 −8.7209 × 10−7 −1.1728 × 10−6 −1.7485 × 10−6 −2.6183 × 10−6 −4.0583 × 10−6 −6.4270 × 10−6 −1.0502 × 10−5 −1.6940 × 10−5 −2.1789 × 10−5 −2.7715 × 10−5 −3.4362 × 10−5 −4.0264 × 10−5 −4.1718 × 10−5 −2.9930 × 10−5 1.3387 × 10−5 1.2437 × 10−4 3.6980 × 10−4 8.6608 × 10−4 1.8085 × 10−3 3.5832 × 10−3 7.5085 × 10−3 2.0158 × 10−2
−4.3059 × 10−7 −5.8662 × 10−7 −9.0773 × 10−7 −9.1362 × 10−7 −2.1772 × 10−6 −1.6512 × 10−6 −4.0812 × 10−6 −5.4741 × 10−6 −1.0239 × 10−5 ... −1.5489 × 10−5 ... −2.3132 × 10−5 ... −2.7181 × 10−5 −1.9428 × 10−5 8.2123 × 10−6 7.9978 × 10−5 2.4423 × 10−4 5.9465 × 10−4 1.3083 × 10−3 2.7106 × 10−3 5.3873 × 10−3
−1.0583 × 10−6 −1.6406 × 10−6 −2.1444 × 10−6 −3.2439 × 10−6 −4.7462 × 10−6 −6.9023 × 10−6 −9.2464 × 10−6 −1.0026 × 10−5 −8.2241 × 10−8 1.4265 × 10−5 4.2871 × 10−5 9.6985 × 10−5 1.9676 × 10−4 3.7678 × 10−4 6.9761 × 10−4 1.2631 × 10−3 2.2518 × 10−3 3.9655 × 10−3 6.9035 × 10−3 1.1835 × 10−2 1.9780 × 10−3 3.1902 × 10−2 5.3054 × 10−2
−2.8971 × 10−7 −4.7211 × 10−7 −5.2362 × 10−7 −6.4086 × 10−7 −1.5497 × 10−6 −3.0833 × 10−6 −5.2381 × 10−6 −5.2191 × 10−6 −1.3394 × 10−5 ... −1.9592 × 10−5 ... −3.3042 × 10−5 ... −5.7995 × 10−5 −7.4790 × 10−5 −9.2096 × 10−5 −1.0184 × 10−4 −8.3846 × 10−5 6.9846 × 10−6 2.6622 × 10−4 8.8295 × 10−4 2.2254 × 10−3
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