Sun Mengtao, Song Peng, Lee Chunxia, Wang Weili, Lee Yongqing, Ma Fengcai (Department of Physics, Liaoning University, Shenyang, 110036, China) Received Nov. 9, 2004; Supported by the National Natural Science Foundation of China (Grant No. 10374040). Abstract Collisonal quantum interference (CQI) was observed in the intramolecular rotational energy transfer in the experiment of the static cell, and the integral interference angles were measured. To observe more precise information, the experiment in the molecular beam should be taken, from which the relationship between the differential interference angle and the scattering angle can be obtained. In this paper, the theoretical model of CQI is described in an atom-diatom system in the condition of the molecular beam, based on the first-Born approximation of time dependent perturbation theory, taking into account the anisotropic Lennard-Jones interaction potentials. The method of observing and measuring correctly the differential interference angle is presented. The changing tendencies of the differential interference angle with scattering angle, including the impact parameter, velocity, and collision partner are discussed. 1 INTRODUCTION Quantum interference (QI) on chemical dynamics is one of main subjects experimentally and theoretically. Experimentally, the QI was studied on rotational energy transfer [1-6], Theoretically, the QI was also studied [7-15] and reviewed [16] on rotational energy transfer. In the open-shell P-states diatomic molecules electronic states, the collisional quantum interference on rotational energy transfer was observed experimentally[1], which results from the splits of open-shell P-states diatomic molecules. Theoretically, models of CQI in this case were also presented [7, 14]. In intramolecuar rotational energy transfer, the evidence of collisional quantum interference (CQI) was obtained by Sha et al. in the CO A1P - e system in collision with He, Ne and other partners [3, 4]. CQI was also observed by Chen et al. in Na2 A - b system in collision with Na (3s) [5]. Based on the time-dependent first order Born approximation, Sha et al. [3, 4] proposed a phenomenon model, which explicitly shows the interference effect by a cross term, (1) where c and d are the mixing coefficients, and and are the energy transfer cross sections for pure singlet and triplet states, respectively, and the last is the interference term, in which measures the degree of coherence. The experiments [3, 4] mentioned above, were taken in a static sample cell, so only the integral cross sections and the integral interference angles having been measured [3, 4]. If experiments can be conducted in molecular beams to measure the differential cross section and the differential interference angle, the collisional quantum interference might be able to be observed more precisely [3], because the integral interference angle is the average effect of the differential interference angle [8-13]. The ion imaging technique, originally developed by Chandler and Houston [17], was significantly improved by Eppink and Parker [18] by using direct velocity mapping technique. Using this technique, Kohguchi [19] measured fully state-resolved differential cross sections for the inelastic scattering of the open –shell NO molecule by Ar. Recently, the direct 3D time sliced ion velocity imaging method [20] was developed to measure the product distribution in crossed molecular beam experiment. Similar to the experiments mentioned above, experiments of the CO (A1 - e) system in collision with He, Ne and other partners might be performed in the crossed molecular beams, using sliced velocity-mapped ion imaging. The theoretical calculations of the CO A1 ~ e system in collision with He, Ne and other partners were done [8-11], and the model of polar diatomic as the partner was also presented [13], but in the theoretical models [8-11], the CQI in the open-shell P-states diatomic molecules electronic states is not considered. To study theoretically proposed experiments of the CO (A1 ~ e) system in collision with He, Ne and other partners in the condition of molecular beams, detected by velocity-mapped ion imaging, a theoretical model is presented, in which the CQI in the open-shell P-states diatomic molecules electronic states is included, based on the time dependent first order Born approximation, taking into account the anisotropic Lennard-Jones interaction potentials, and "straight-line" trajectory approximation. The method of observing and measuring correctly the differential interference angle is presented. The changing tendencies of the differential interference angle with the impact parameter, velocity, and collision partner are discussed. 2 THEORETICAL APPROACH 2.1 Hamiltonian For the atom-diatom system, the Hamiltonian is given as [11], (2) where the collision reduced mass, R the orientation from the atom to the mass center of diatom, the operator for the orbital angular momentum L of the atom-molecule pair, the atom electronic state Hamiltonian, the Hamiltonian of the molecular monomer. can be written as [11], (3) , and are the electronic, vibrational and rotational Hamiltonian of diatom, respectively. represents the interaction potentials of the system, including the interaction potentials between the atom and the two electronic states of the diatom, and the spin-orbit interaction between the two electronic states of the diatom, (4) where S and T represent the singlet and triplet states respectively. The electrostatic interaction potential between the atom and the two electronic states of the diatom in the space frame [27] is, (5) where the Wigner rotational matrix, the Euler angles refer to the space fixed orientation of the diatomic, a Racah spherical harmonic function, the angles , describe the orientation of R in the space frame. In the atom and -states diatomic molecules system, or , which reflects the electronic cloud distribution of C2V group. In the atom and -states diatomic molecules system, . The spin-orbit interaction between the two electronic states of the diatom can be written as, (6) where the spin-orbit coupling constant. The evolution of the interaction potential can be written as, (7) where is the time evolution operator, (8) and where the Hamiltonian of is (9) In this letter, without considering the translational, electronic and vibrational energy transfer, so in Eq. ( 9 ) can be simplified to rotational energy and the rotational kinetic energy of the atom about the diatom, . To simplify the discussion, without considering the rotational kinetic energy of the atom about the diatom [11], the for the -states diatomic molecules can be written as [10], (10) and for the -states diatomic molecules[21], (11) where and or . 2.2 Perturbations between singlet and triplet states The unperturbed wave functions are (12) where the singlet [7] and triplet[22] wave functions are (13) (14) The zeroth order unperturbed energies are defined as and . If is the coupling between these zeroth order states, the perturbed states have energies [28], (15) The perturbed wave functions [29, 11] are (16) Eq.(15) shows that the mutual perturbing states are repulsive each other. The energy level shifts is, (17) where denotes an upward shift and a downward shift. The argument , characterizing the mixing effects, is given by, (18) In the later derivation, according to Eq. (15), we set that the mixing coefficients and , or and depending on the perturbed state is a singlet or a triplet state. From Eq. (15)-(18), the values and the sign of and can be obtained. and will have the same sign if energy level J is shift upwards by perturbation, otherwise, and will have the opposite sign [8,9,12]. If both the initial state and final state of a collision-induced transition are singlet–triplet mixed states, then (19) and (20) 2.3 The transition matrix element and probability According to the first order Born approximation of time dependent perturbation theory, the transition matrix element is, (21) where only electrostatic interaction is involved with no magnetic coupling present [3-5, 11], i.e., a transition between the two states is prohibited, . In Eq. (21), according to the perturbation theory, and [30,31]. Considering the product of three rotation matrix element and spherical harmonic, and the relationship between 3 j symbols and 6 j symbol [21], in Eq. (21), the transitional matrix for the singlet state is [7,21], (22) and the transitional matrix of the triplet state is [22], (23) and the coefficients and are listed in Table 1 [22]. The unpolarized transition probability can be written as [7], (24) Introduce Eq. (21) into Eq. (24), consider the orthogonal relationship of the three J symbol, one can obtain, (25) where , are the pure transition probability of the singlet and triplet states, the third term is the interference term. The interference term in Eq. (25) is (26) Similar to Eq.(1), Eq. (25) also can be written as the form, (27) with the differential interference angle (28) the relationship between the differential interference angle and the integral interference angle are discussed in Refs [7-15]. 3 DISCUSSION 3.1 Obtaining the anisotropic parameter, from the experiments Considering the anisotropic Lennard-Jones interaction potentials, Eq. (5) can be written as, (29) In Eq. (29), is the anisotropic parameter. r is the distance at which , is the depth of the potential well, which is a measure of how strongly the molecules attract each other. Because the ratio of for CO collision with He, Ne and Ar is [32]), and the first term is R-7 for for the long range interaction potential, while the first term is R-6 for the long range interaction potential, so to simply discussion in this paper, we only consider . The integral interference angle can be written as (30) where (31) and and (32) To simplify the discussion, the rigid sphere interaction potential is considered, and the scattering angle for the rigid sphere potential [23] is (33) So in integrating over the impact parameter, for the "straight-line" trajectory approximation is used, which is proposed by Gray and van Kranendonk [24] (b is the impact parameter and is the hard-sphere collision diameter). This approximation requires that the energy transferred to/from the internal degrees of freedom is much less than the energy of relative translational motion. For the static cell experiments of CO collision with partner He, Ne and Ar, the rotational energy transferred () is at , which is much less than the energies of relative translational motion at , respectively. So the "straight-line" trajectory approximation for is a good approximation. For the case of , the integration in Eq. (31) can be integrated by adopting a coordinate system to solve this case where a particle moves along the collisional axis (impact parameter) with velocity v and recoils along the same axis with unchanged velocity [25]. The transition probability can be estimated by interpolation between and at . Since ought to be a smooth function with vanishing slope at , the simplest interpolation function is the parabola [25] (34) The integration of Eq. (31) can be reduced to (35) and the results of the integral for and are listed in Appendix II of Ref [11]. One can fit the ratio of the anisotropic parameter in Eq. (32), through the data [3, 4] of the experiment in the static cell. The parameters for the calculations are listed in Table 1. The fitted results are listed in Table 2. From Table 2, at the same temperature, one can see that with the increase of the reduced mass, the ratio of the anisotropic parameters, increase. Thought the ratio of the anisotropic parameter, , when , and , so the term in Eq. (30) can not be omitted. Table 1 Parameters needed in the theoretical calculation
a. Data from Ref. [8] b. Data
from Ref. [9] c. Data from Ref. [10].d. Experimental Data from Ref. [3,
4] e. Data from Ref. [33]
3. 2. 1 The observable differential interference angle The differential interference angle may be measured from the molecular beam experiment, in which the scattering angle is sensitive with the impact parameter and the potential energy surface. The qualitative analysis can be seen from Figs.1-3, where Fig. 1 is the qualitative potential energy surface of system, and the Figs. 2 is the qualitative scattering angle with the impact parameter b for the system. Fig. 3 is the analysis of the scattering sources. The scattering angle are related to the potential energy surface and the collision energy [23], (36) where is the so-called turning point of the relative motion, which is determined as the solution of the implicit equation [23], (37) if the potential is known, the defection function can be calculated via Eq. (36) at any desired collision energy, then the angular distribution of the scattering can be predicted. From Fig.2 and Fig.3, when , there are three possible sources at one point, the first one is from the repulsive potential () , the second one is from the effect of repulsive plus long-range interaction potential (), and the last one results from the long-range potential (). Also, when , there are two possible sources at one point, one is from the long-range "straight-line", another one is from the long-range induction from . Because of the difficulty of distinguishing the sources, when the scattering angle is in the range of , it can not determine the relationship between the interference angle and the scattering angle. So, to determine the relationship between the interference angle and the scattering angle, the detector should be arranged at the range of the scattering angle . From Fig. 2 and 3, the relationship between the interference angle and the scattering angle can be determined when the impact parameter is , and the detect range of the scattering angle can be seen from Fig. 4. Fig. 1 The qualitative potential energy surface of system Fig. 2 The qualitative scattering angle with the impact parameter b for system Fig. 3 The analysis of the scattering sources to the differential interference angle Fig. 4 The detect range of the scattering angle in experiment of the molecular beam 3. 2. 2 The tendency of the differential
interference angle with the scattering angle
4 CONCLUSION
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