Phase Space Calculations Research Articles

Role of Intersystem Crossing in the Dynamics of the O(3P) + C2H5I Reaction

Reactive scattering of O(3P) atoms with C2H5I molecules has been studied at initial translational energies E ∼ 51 and 16 kJ mol-1 using a supersonic beam of O atoms seeded in He and Ne buffer gas generated from a microwave discharge source. A mildly forward- and backward-peaked angular distribution of IO product observed at lower initial translational energy becomes backward-scattered at higher initial translational energy. An isotropic angular distribution is observed for HOI product at lower initial translational energy which becomes forward- and backward-peaked, slightly favoring the backward direction at higher initial translational energy. Extended phase space calculations indicate that reaction leading to IO product proceeds via intersystem crossing from the lowest triplet 3A‘‘ potential energy surface to a bound OIC2H5 intermediate on the singlet 1A‘ potential energy surface in small impact parameter collisions at low initial translational energy. However, the emergence of backward-scattered IO product indicates the onset of direct reaction over the triplet 3A‘‘ potential energy surface at higher initial translational energy. The formation of HOI reaction product arises from the singlet OIC2H5 complex via a five-membered ring transition state where the H atom is abstracted from the terminal CH3 group at the top of a barrier leading to vibrational excitation of HOI and repulsion between the reaction products in this strongly exoergic exit valley of the potential energy surface.

Role of intersystem crossing in the dynamics of the O(3P)+ICl reaction

Reactive scattering of O(3P) atoms with ICl molecules has been studied at an initial translational energy E ∼ 98 kJ mol-1 using a supersonic beam of O atoms seeded in He buffer gas generated from a high temperature radiofrequency discharge source. A mildly peaked angular distribution of IO reactive scattering slightly favouring the forward direction is observed with a product translational energy distribution which peaks at a very low fraction f′ ∼ 0·04 of the total available energy with a long tail extending out to higher energy. This compares with a more sharply peaked angular distribution at lower initial translational energy E ∼ 45 kJ mol-1. In both cases, the product translational energy distributions are shifted to higher energy for the forward and backward scattering compared with sideways scattering. Extended phase space calculations indicate that reaction leading to IO reaction product is confined to very small impact parameters b m ∼ 1·2 A at high initial translational energy increasing to b m ∼...

The role of nuclear incompressibility in the production of hard photons in heavy-ion collisions

We summarize the properties of hard photons emitted in heavy-ion collisions. Dynamical phase-space calculations are used to describe the initial phase of the collision. Guided by the predictions, the photon spectra and the two-photon interference pattern are interpreted as a result of the collision dynamics and the photon spectra are viewed as an image of the phase space occupancy of participating nucleons. The data demonstrate the existence of two spatially and temporally distinct photon sources. It is shown how their relative intensity provides us with a measure of the incompressibility modulus of infinite nuclear matter.

Quantum phase space theory for the calculation of v⋅j vector correlations

The quantum state-counting phase space theory commonly used to describe ‘‘barrierless’’ dissociation is recast in a helicity basis to calculate photofragment v⋅j correlations. Counting pairs of fragment states with specific angular momentum projection numbers on the relative velocity provides a simple connection between angular momentum conservation and the v⋅j correlation, which is not so evident in the conventional basis for phase space state counts. The upper bound on the orbital angular momentum, l, imposed by the centrifugal barrier cannot be included simply in the helicity basis, where l is not a good quantum number. Two approaches for an exact calculation of the v⋅j correlation including the centrifugal barrier are described to address this point, although the simpler helicity state count with no centrifugal barrier correction is remarkably good in many cases. An application to the photodissociation of NCCN is consistent with recent classical phase space calculations of Klippenstein and Cline. The experimentally observed vector correlation exceeds the phase space theory prediction. We take this as evidence of incomplete mixing of the K states of the linear parent molecule at the transition state, corresponding to an evolution of the body-fixed projection number K into the total helicity of the fragment pair state. The average over a thermal distribution of parent angular momentum in the special case of a linear molecule does not significantly reduce the v⋅j correlation below that computed for total J=0. Predictions of the v⋅j correlations for the unimolecular dissociation of NCNO and CH2CO are also provided.

Subthreshold K? production in pion-nucleus reactions

The inclusive subthresholdK − mesons production in charged pion-induced reactions is analyzed with respect to the one-step production processes on the basis of an appropriate first collision model, which allows one to take into account the various forms of an internal nucleon momentum distribution. Free elementary production processes are described by the phase space calculations normalized to the corresponding total experimental cross sections. Simple parametrizations for the total cross sections of theK − production in pion-nucleon collisions are given. Primary-pion energy dependence of the inclusive cross sections for K− production inπ + C− collisions is calculated. The influence of the uncertainties in the elementaryK − production cross sections close to the production threshold on theK − yield is explored.

Chaotic dynamics of single particles in axially symmetric nuclear shapes

Abstract The spectral statistics of single particle motion in deformed cavities with axial symmetry are presented. Considering cavities with Legendre-polynomial deformations P 2 to P 6 , a systematic study of the fluctuation measures of the energy spectra is performed. For each deformation type the evolution of the fluctuation measures with increasing deformation indicating an order-to-chaos transition is observed. The nature of such transitions is found to be in agreement with earlier classical phase space calculations.

Resonances and fluctuations in the unimolecular reaction of NO2

Fluctuations and oscillations in the unimolecular reaction of NO2, and their manifestations in photofragment yield (PHOFRY) spectra, NO rotational state distributions and decomposition rates are examined. Comparisons between experimental and simulated PHOFRY spectra show that extraction of rates from linewidths in state-selected spectra is unjustified in the regime of overlapping resonances. Measurements of the alignment parameter of the NO product in the excess energy range EE= 200–500 cm–1 evidence the existence of fluctuations in the decay rate. Changes in the patterns of fluctuations and oscillations in the NO rotational state distributions reveal the progressive tightening of the transition state (TS) as the excess energy increases and the importance of exit-channel interactions beyond the TS. Distributions well fit on the average by phase-space calculations can be obtained even when the transition state is tight.

Calculations of interstellar radiative association rates including tunnelling: the strange case of C2H + 2 + H2

Both the radiative association reaction and the ion–molecule reaction between the reactants C2H+2 and H2 occur appreciably at temperatures under 100 K and must be included in models of interstellar and circumstellar chemistry. In this paper, we consider the mechanism of the association reaction to form C2H+4. Ab initio quantum chemical calculations have been undertaken to determine the pathway of minimum potential energy. This pathway is distinct from another pathway we have calculated for the ion–molecule reaction, which forms C2H+3+ H. For the association channel, our ab initio calculations show that there is a small entrance channel barrier under which low-temperature reactants must tunnel to access the deep potential well associated with C2H+4. Other than redissociation into reactants, the C2H+4 reaction complex can dissociate only to the classical (higher energy) form of C2H+3+ H; this channel is endothermic and does not occur at low temperatures, enhancing the prospect that the C2H+4 complex can be stabilized by collision with a third body or via emission of a photon. Phase space calculations extended to include tunnelling for both the formation and redissociation of the C2H+4 complex do not reproduce low-temperature experimental measurements of the rate of the association channel. Rather, a phase space calculation in which it is assumed that the entrance channel barrier does not exist is in agreement with most of the experimental data. It is likely, however, that a small barrier does exist. Accurate radiative association rate coefficients, largely unaffected by the existence of such a small barrier, are tabulated in the temperature range 2–80 K.

Photodissociation dynamics of (C6H6)2+

Photodissociation of (C6H6)2+ is studied with photon energies between 1.29 and 2.81 eV. Mass-selected ion beam of (C6H6)2+ is photodissociated by a pulsed laser beam in the field-free region of a reflectron-type time-of-flight mass spectrometer. The average relative translational energy, 〈εt〉, and the angular distributions of the photofragments (C6H6+ and C6H6) are measured as a function of photon energy. With a photon energy of 2.81 eV, the (C6H6)2+ ions are promoted to a bound upper state correlated to C6H6+(ππ)+C6H6(X). Only a small fraction (≊3%) of the available energy is partitioned into the translational energy of the fragments and the product angular distribution is isotropic. Absorption of a photon in the range of 1.29–2.14 eV induces a charge resonance transition to a repulsive upper state which correlates to C6H6+(X)+C6H6(X). The observed values for 〈εt〉 are at most 10% of the available energy, although the statistical phase space calculation shows that the complete randomization of the available energy is not achieved. As the photoexcited (C6H6)2+ moves apart on the dissociative potential surface, ≊90% of the available energy flows into the intramolecular modes of the fragments. The energy partitioning becomes almost statistical regardless of the photoexcitation to a dissociative state.

The effect of vibrational excitation (3≤v″≤19) on the reaction Na2 (v″)+Cl→NaCl+Na*

The variation of the total integral cross section for the chemiluminescent channel with the vibrational excitation 3≤v″≤19 is investigated using laser-induced fluorescence or coherent population transfer, a new method employing delayed interaction with two cw lasers to populate vibrationally excited levels. The cross section increases with v″ by about 0.76% per vibrational level. The results are in agreement with the results of phase-space calculations, as well as with those from simple curve crossing models.

Fluctuations of the one-body distribution function.

A general method to evaluate fluctuations of the semiclassical one-body Wigner function is presented both in the equilibrium and nonequilibrium framework. The model is valid far away from critical points, where the Boltzmann-Nordheim-Vlasov equation represents a good approximation of the average Wigner function. A first application of the model in momentum space is shown in comparison with different analytical results. An extension to realistic phase space calculations of heavy ion collisions is straightforward and numerically feasible. In this case, together with fluctations coming from the scattering processes, mean field fluctuations are automatically included. The role of Landau damping is discussed.

Light particle emission in Boltzmann master equation theory of pre-equilibrium reactions.

A method is proposed to account for the emission of light particles in heavy-ion fusion reactions using the Boltzmann master equation approach. The method consists in a simple phase space calculation which allows one to estimate the formation probability of a cluster for a given energy distribution of the nucleons. The emission probability of the clusters at each stage of the equilibration cascade is calculated and the changes of the occupation probabilities of the nucleon energy states due to these emissions is accounted for by a new depletion term in the set of master equations. The calculated spectra of deuterons, tritons, and \ensuremath{\alpha} particles reproduce satisfactorily the exclusive spectra measured in $^{16}\mathrm{\ensuremath{-}}$ and $^{32}\mathrm{Ag}$ fusion at 30 MeV/nucleon incident energy. The same model is used to predict the emission of clusters in nucleon-induced reactions with satisfactory results.

Calculations on the rate of the ion–molecule reaction C2H+2+H2→C2H+3+H

The minimum energy pathway for the ion–molecule reaction C2H+2+H2→C2H+3+H has been calculated via ab initio quantum chemical techniques. The reactants first form a weakly bound entrance channel complex with H2 in a bridged position perpendicular to the C2H2 linear structure. A transition state is then encountered with an asymmetric structure in which H2 favors one of the two carbon atoms as it begins to come apart. After proceeding through a weakly bound exit channel complex, once again of bridged structure, the system produces the product ion C2H+3 in the nonclassical (bridged) position. The reactants do not access the deep potential well of C2H+4. At our highest level of calculation, the reaction is determined to be endothermic by approximately 2 kcal mol−1, in good agreement with thermochemical values. However, phase space calculations on the reaction dynamics, including tunneling, using the calculated potential energy surface are not in good agreement with measurements of the rate coefficient at both 300 and in the vicinity of 2 K. A revised potential energy surface, in which the products and intermediate stationary points are lowered in energy by 2 kcal mol−1, making the reaction thermoneutral and reducing the barrier over the transition state, leads to better agreement with measured rate coefficients. With the reduced energies, the calculated rate coefficient shows a temperature dependence similar to that obtained previously for the reaction between NH+3 and H2.

Sequential reactions of SiD+2 with SiD4

The thermal (300 K) reaction of SiD+2 with SiD4 proceeds at greater than the Langevin collision rate (21±3×10−10 cm3/molecule s ). The reaction products SiD+3, Si2D+2, and Si2D+4 are produced in a 54:7:39 ratio. Both silicon isotope exchange and adduct (Si2D+6) formation are <1% of the collision rate at silane pressures of 1–4×10−7 Torr. The branching ratio for SiD+3 formation increases with increasing internal energy of SiD+2. Sequential reactions of SiD+3 and Si2D+2 with SiD4 have been previously found to produce terminal species containing five silicon atoms. Si2D+4 reacts with SiD4 only by silicon isotope exchange at 2.0±0.7% of the collision rate (0.20±0.07×10−10 cm3/molecule s ) with no evidence of other reactions (<0.5% of the collision rate). Reaction of SiD+2 with SiD4 does not lead to unconstrained clustering and particle formation in silane plasmas. High level ab initio calculations on this system are reported by Raghavachari in his companion paper. Energies of the critical intermediates and transition states along the reaction surface are compared quantitatively to the experimental results via phase space calculations. The energies agree to within 6 kcal/mol.

Measurement of the 12C(p, pi 0)13N reaction by recoil detection.

Differential cross sections for the neutral pion production reaction $^{12}\mathrm{C}$(p,${\mathrm{\ensuremath{\pi}}}^{0}$${)}^{13}$${\mathrm{N}}_{\mathrm{g}.\mathrm{s}.}$ have been measured at 153.5, 166.1, 186.0, and 204.0 MeV bombarding energy using recoil detection. The shape of the angular distribution agrees well with that of the mirror reaction $^{12}\mathrm{C}$(p,${\mathrm{\ensuremath{\pi}}}^{+}$${)}^{13}$${\mathrm{C}}_{\mathrm{g}.\mathrm{s}.}$. Isospin invariance predicts for the ratio of the cross sections \ensuremath{\sigma}(p,${\mathrm{\ensuremath{\pi}}}^{+}$)/\ensuremath{\sigma}(p,${\mathrm{\ensuremath{\pi}}}^{0}$)=2. At 153.5, 166.1, and 186.0, MeV, where (p,${\mathrm{\ensuremath{\pi}}}^{+}$) cross sections have been reported previously by other groups at the same reduced pion momentum \ensuremath{\eta}=${\mathit{p}}_{\mathrm{\ensuremath{\pi}}}$/${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}$c=0.34,0.55,0.78, the ratio of the total cross sections is 2.02\ifmmode\pm\else\textpm\fi{}0.14, 3.14\ifmmode\pm\else\textpm\fi{}0.12, and 2.12\ifmmode\pm\else\textpm\fi{}0.16, respectively. The (p,${\mathrm{\ensuremath{\pi}}}^{0}$) cross sections at 166 MeV is therefore 1.57\ifmmode\pm\else\textpm\fi{}0.06 times larger than expected. For ${\mathit{T}}_{\mathit{p}}$160 MeV, the experimental cross sections are larger than predicted by a phase space and Coulomb barrier penetration calculation that fits the higher-energy data. Differential cross sections were also obtained for the $^{12}\mathrm{C}$(p,${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$${)}^{13}$O reaction, and upper limits set for differential cross sections for radiative capture to the ground state of $^{13}\mathrm{N}$.

Kinetic-energy dependence of the reactions of C+(2P)+N2 from threshold to 28 eV CM

Guided-ion-beam mass spectrometry is used to examine the kinetic-energy dependence of the reaction of ground-state atomic carbon ion with molecular nitrogen. Integral reaction cross sections are measured for the three possible product channels: N2+ + C, CN+ + N, and N+ + CN, all strongly endothermic. Thresholds for all three channels are found to be consistent with literature thermochemistry, although formation of N2+ is inefficient near threshold. The observed behavior suggests that the Π3 state of CN+ is either the ground state or within 0.14 eV of the ground state, and that N2+ is formed primarily in its A 2Πu first excited state rather than the X 2Σg+ ground state. The thermochemistry measured here finds ΔfH00(CN)=102±3 kcal/mol, D00(CN)=7.82±0.14 eV, and IE(CN)=14.02±0.14 eV. The reaction mechanism for this system is explored by examination of the electronic state correlations and by phase-space calculations.

Phase-space calculation of the three-particle decay rate of dtμ and the sudden approximation theory of sticking in muon catalyzed fusion

Recently Chatterjee questioned the validity of the standard sudden approximation (SA) theory of the initial muon sticking probability ω 0 s by performing a relativistic phase-space calculation and finding a 20% reduction of sticking relative to the standard SA result. We show that at the nonrelativistic level of approximation the phase-space formula of Chatterjee reduces exactly to the SA expression in its complete form, as given by Bogdanova et al. This expression computed by us in the Born-Oppenheimer approximation leads to a reduction of the sticking probability by only 0.28% compared to the standard SA result — in agreement with the qualitative estimate of Bogdanova et al. The accurate relativistic calculation of the relevant phase-space integrals, also performed by us, leads to 0.26% reduction of the sticking probability relative to the standard nonrelativistic SA result.

Mechanistic studies of processes involving carbon-carbon bond cleavage in gas-phase organometallic reactions using product kinetic energy release distributions: Co+ reacting with cyclopentane

A reverse-geometry mass spectrometer is used to obtain product ion kinetic energy release distributions to probe the energetics and mechanisms of several gas-phase organometallic reactions. In particular, we examine the mechanism for C{sub 2}H{sub 4} and C{sub 3}H{sub 6} elimination from Co(cyclopentane){sup +}. The kinetic energy release distribution associated with these processes can be modeled by using phase space calculations assuming, for the C{sub 3}H{sub 6} elimination process, propene rather than cyclopropane is being eliminated as the product neutral, and for the C{sub 2}H{sub 4} elimination process, Co(propene){sup +} rather than (cobaltacyclobutane){sup +} is being formed as the product ion. In addition, we obtain a heat of formation for the cobalt ethylene ion of 255 kcal/mol at 0 K, corresponding to a bond dissociation energy of 42 kcal/mol at 0 K, by fitting the theoretical results to the experimental distribution.

Theoretical studies of clustering reactions. Sequential reactions of SiH+3 with silane

The mechanisms for the sequential clustering reactions of SiH+3 with SiH4 have been studied by means of accurate quantum chemical techniques using polarized basis sets and including zero-point corrections and the effects of electron correlation. Detailed reaction pathways including the necessary transition structures have been investigated to understand the structures and stabilities of the reaction intermediates and products. In accordance with the experimental observations of Mandich, Reents, and Kolenbrander, the reactions lead to the formation of Si2H+5 and Si3H+7 without significant overall barriers. In addition, bimolecular attachment complexes (Si2H+7,Si3H+9 and Si4H+11) bound by bridging hydrogen atoms are involved as reaction intermediates and have also been observed by Mandich et al. as collisionally stabilized products. Unusual hydrogen bridged complexes play a key role in the termination of the growth sequence. The first bottleneck in the sequence of growth reactions is due to the formation of the particularly stable hydrogen bridged complex Si3H+7 containing a four-membered Si–Si–Si–H ring. Further growth beyond Si3H+7 other than bimolecular attachment involves transition states which have significant barriers and could only happen in the presence of excess energy. Isotopic exchange reactions which have been seen experimentally are also rationalized by this mechanism. Quantitative comparisons are made between our calculated transition state barriers and complexation energies and those derived by Mandich et al. from phase space calculations using the experimental reaction rates. The mean absolute deviation between the two results for five different energetic quantities is only ≂3 kcal/mol.

Temperature dependence of termolecular association reactions N2+ + 2N2 .fwdarw. N4+ + N2 and O2+ + 2O2 .fwdarw. O4+ + O2 occurring in free jet expansions below 20 K

A new experimental technique that combines supersonic expansions and laser ionization mass spectrometry has been used to measure the termolecular association rate coefficients, k{sub 3}, for the gas-phase reactions N{sub 2}{sup +} + 2N{sub 2} {yields} N{sub 4}{sup +} + N{sub 2} and O{sub 2}{sup +} + 2O{sub 2} {yields} O{sub 4}{sup +} + O{sub 2} below 20 K. The continuity of the proposed inverse temperature law of k{sub 3} = CT{sup -n} was observed down to 4 K for both reactions. Statistical analysis of our results along with the higher temperature data obtained by other authors gives the values of 1.92 {plus minus} 0.15 and 1.86 {plus minus} 0.15 for n for the reactions of N{sub 2}{sup +} and O{sub 2}{sup +}, respectively. The results of phase space calculations are presented for these reactions and excellent agreement between theory and experiment is obtained.