Collisional quantum interference on rotational energy transfer: The relationship between the differential interference angle and the scattering angle

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 C_2V 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

Collision System	Reduced Mass (a.m.u)	s(Å) c	q(0) d		Rotational constants () d
			253K	470K	CO A1)	(CO e)
CO-He	3.5a	2.56	610	650	1.6105	1.2836
CO-Ne	11.75b	2.75	710	690
CO-Ar	17.5b	3.41	710	720

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]

Table 2 The ratio of the anisotropic parameter for CO (A¹) - He, Ne and Ar

3. 2 The relationship of the differential interference angle with the scattering angle
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            
To simplify the discussion, we consider that the interaction potential is the rigid sphere potential, and the scattering angle for the rigid sphere potential is for , and we only consider the scattering angle from the "straight line" trajectory when for , without considering the other two cases. Then one can obtain the relationship of the differential interference with the impact parameter and the relative velocity, where the relative velocity can be controlled by choosing the collision angle of the two sources of the crossed molecular beam, . For , the transition probability can be estimated by interpolation between and of Eq. (34). Figs 5 and 6 are the relationships of the differential interference angle with the relative velocity impact parameter for and for , where only the source of the “straight line” trajectory is considered for . From Figs 5 and 6, one finds the long-range interaction potential plays an important role for the static cell, because the experimental result in the static cell is 61⁰, which is the average effect to velocity and impact parameter, so some data should be larger than 61⁰ and some data should be less than 61⁰ in the differential interference angle, while in case of the short-interaction range of Fig. 5, all the differential interference angle are less than it. So, in the experiment of the static cell, the experimental result mainly reflects the influence of the long-range interaction potential. With the increase of the reduced masses, the repulsive potential begins contributing to the interference angle inchmeal ^[10]. To observe the influence of the short range interaction potential, the experiment in the molecular beam should be done. According to the discussion in Section 3. 2. 1, the observable differential interference angle should be measured at the range of , Figs. 7 and 8 are the relationship of the differential interference angle with the relative velocity and impact parameter for the system of CO-He, for , at T=253 K and 470K. If one compares Figs. 7 and 8, one will find that with the increase of the experimental temperatures, the differential interference angles increase, i. e., the interference decrease. Figs. 9 and 10 are the relationship of the differential interference angle with the relative velocity and impact parameter for , for the system of CO-Ne and Ar at 470K. Comparing Figs. 8-10, one can find that with the increase of the reduced mass, the differential interference angle increase, i. e., the interference decrease. The changing tendencies of the differential interference angle with the relative velocity and impact parameter in the molecular beam experiment are consistent with the experimental results in the static cell. With the increases of the experimental temperatures and the reduced mass, CQI is of the tendency that being washed out ^[3,4,26]. The lower temperature and smaller reduced mass, the more clearly the CQI can be observed.

Fig.5 The differential interference angle with velocity and impact parameter for CO-He at T=253K for	Fig.6 The differential interference angle with velocity  and impact parameter for CO-He at T=253K for

Fig.7 The differential interference angle with velocity and impact parameter for CO-He at T=253K for	Fig.8 The differential interference angle with velocity and impact parameter for CO-He at T=470K for

Fig.9 The differential interference angle with velocity and impact parameter for CO-Ne at T=470K for	Fig.10 The differential interference angle with velocity and impact parameter for CO-Ar at T=470K for

4 CONCLUSION
The collision-induced quantum interference on rotational energy transfer is studied, and the differential, the integral interference angle and the relationship between them are obtained. The relationship between the differential interference angle and the scattering angle in the experiment of the molecular beam at the observed range is discussed.

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转动传能中的碰撞量子干涉效应：微分干涉角和散射角之间的关系
孙萌涛^#，宋朋，李春霞，王伟丽，李永庆，马凤才^*
(辽宁大学物理系，沈阳，110036)
摘要  在分子内部转动传能的静态池实验中观察到碰撞量子干涉效应，并且测得积分干涉角。为了获得更加精确的信息，要采用分子束实验，通过此实验可获得微分干涉角和散射角之间的关系。本文考虑各项异性L-J相互作用势，应用含时微扰理论的一级波恩近似，在分子束实验的条件下，建立在原子-双原子分子体系中碰撞量子干涉的理论模型。采用正确的观测途径，讨论了微分干涉角随散射角(包括碰撞参数、速率以及碰撞伴)的变化趋势。
关键词：量子干涉效应，微分干涉角，散射角。