Interference Angle on Quantum Rotational Energy Transfer in Na+Na 2 (A 1 + v = 8 b 3 Π 0u, v = 14) Molecular Collision System

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1 Commun. Theor. Phys. (Beijing, China 52 (29 pp c Chinese Physical ociety and IOP Publishing Ltd Vol. 52, No. 6, December 15, 29 Interference Angle on Quantum Rotational Energy Transfer in Na+Na 2 (A 1 + u, v = 8 b 3 Π u, v = 14 Molecular Collision ystem WANG Wei-Li, 1, MIAO Gang, 1 LI Jian, 2 and MA Feng-Cai 3 1 Department of Basic Course, Liaoning Technical University, Huludao 12515, China 2 Department of cience, henyang Institute of Aeronautical Engineering, henyang 1131, China 3 Department of Physics, Liaoning University, henyang 1136, China (Received January 14, 29; Revised June 5, 29 Abstract In order to study the collisional quantum interference (CQI on rotational energy transfer in atom-diatom system, we have studied the relation of the integral interference angle and differential interference angle in Na+Na 2 (A 1 + u, v = 8 b3 Π u, v = 14 collision system. In this paper, based on the first-born approximation of timedependent perturbation theory and taking into accounts the anisotropic effect of Lennard Jones interaction potentials, we present a theoretical model of collisional quantum interference in intramolecular rotational energy transfer, and a relationship between differential and integral interference angles. PAC numbers: Qg, 34.5.Ez Key words: rotation energy transfer, integral interference angle, differential interference angle, collision diameter 1 Introduction Quantum interference (QI on chemical dynamics is one of main subjects experimentally and theoretically. There have been various studies of QI on rotational energy transfer. [1 4] In the open-shell Π-states diatomic molecules electronic states, the collisional quantum interference on rotational energy transfer was observed experimentally, which results from the Λ splits of open-shellstates diatomic molecules. In intramolecular rotational energy transfer, the evidence of collisional quantum interference (CQI was obtained by ha et al. in the CO A 1 Π(v = e 3 Σ (v = 1 system in collision with He, Ne, and other partners. [5 7] CQI was also observed by chen [8] et al. in Na+Na 2 (A 1 + u, v = 8 b3 Π u, v = 14 collision system by optical-optical double resonance fluorescence spectroscopy. In the article, our purpose is to compute the differential interference angle (cosθt D and the integral interference angle (cosθt I, for the singlet-triplet mixed states of Na 2 (A 1 + u, v = 8 b3 Π u, v = 14 system in collision with Na. It is meaningful and necessary to calculate the values of cosθt D, since the magnitude can give us a quantitative and accurate description of CQI. Through our model we also get the conclusion that the integral interference angle is the average effect of the differential interference angle. In order to study the effect-factors that has effect on CQI in Na 2 (A 1 Σ + u, v = 8 b3 Π u, v = 14 in collision with Na system at 75 K in condition of molecular beams, using velocity-mapped ion imaging, a theoretical model is presented, basing on the time dependent first order Born approximation, taking into account the anisotropic Lennard Jones interaction potentials and straight-line trajectory approximation. We obtained the relation of the integral interference angle and differential interference angle. 2 Theoretical Approach in Calculating Interference Angle In Hund s case (a the wave function for a diatomic molecular is written as [9] JM ΩνΛΣ = JM Ω νλσ, (1 where J is the angular momentum quantum number, M is the projection of J along a space-fixed Z-axis, Ω is the projection of total angular momentum on the molecular frame. Λ and Σ are the projections of orbit angular momentum L of molecule and spin momentum on the molecular frame, and ν is the vibration quantum number. The rotational wave function JM Ω which appears in Eq. (1 can be written as a rotational matrix element, [1] namely JMΩ = [(2J + 1/8π 2 ] 1/2 D J MΩ (αβγ. (2 The wave function of Na 2 (b 3 Π u, v = 14 can be written as b 3 Π u v = 14 JMΩε, which can be subsequently coupled to the orbital angular momentum L of the atommolecule pair to form the total angular momentum, upported by the National Natural cience Foundation of China under Grant No and the Foundation of the Educational Department of Liaoning Province under Grant Nos and Corresponding author, ww79223@163.com

2 No. 6 Interference Angle on Quantum Rotational Energy Transfer in 163 b 3 Π u v = 14 JΩεIζ = b 3 Π u v = 14 M JM L JMΩε, LM L Iζ JMΩε LM L, (3 A 1 Σ + u v = 8 JΩIζ = A 1 Σ + u v = 8 M JM L JMΩ, LM L Iζ JMΩ LM L, (4 where LM L is a spherical harmonic. [11] LM L = Y L M L (θ, φ = [(2L + 1/4π] 1/2 D L M L, (θ, φ,. (5 If both the initial state i and final state f of a collision-induced transition are singlet-triplet mixed states, then i = C J A 1 Σ + u v = 8 JΩIζ + d J b 3 Π u v = 14 JΩεIζ, (6 f = C J A 1 Σ + u v = 8 J Ω Iζ + d J b 3 Π u v = 14 J Ω ε Iζ. (7 For an atom-diatom system, the Hamiltonian is given as, H = H + V. (8 Based on Born Oppenhamer approximation, L 2 H = 1 2µ dr 2 L2 2µR 2 + H at + H mol, (9 where µ is the collision reduced mass, R is the orientation from the atom to the mass center of diatom, L is the operator for the orbital angular momentum L of the atommolecule pair, H at the atom electronic state Hamiltonian, and H mol the atom electronic state Hamiltonian, H mol = H e + H vib + H rot (1 where H e, H vib, and H rot are the electronic, vibrational, and rotational Hamiltonian of diatom, respectively. The evolutions of the interaction potential can be written as V (t = U + (t, V U(t,, (11 where U(t, is the time evolution operator, U(t, = e iht/. (12 In this paper, without considering the translational, electronic, and vibrational energy transfer, H in Eq. (12 can be simplified to rotational energy H = L 2 /2µR 2 +H rot. Eq. (9 can be written as H = H rot = ω = hcbj(j + 1, (13 where B is a rotational constantand J is the rotational quantum number. V Represents the interaction potential 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, V = V s + V T + V T. (14 The electric static interaction potential between the atom and the two electronic states of the diatom in the space frame [12] is, V,T (R, θ, φ = V l,k (RD l, m,k (ΩCl m(θ, φ, (15 l,m,k where R is the orientation from the atom to the mass center of diatom, D l, m,k (Ω is the Wigner D rotational matrix, the Euler angles Ω = α, β, γ refer to the space fixed orientation of the diatomic, Cm l (θ, φ is the Racah spherical harmonic function, the angles θ, φ describe the orientation of R in the space frame. In the atom and Π-states diatomic molecular system, k = or k = ±2, [11 12] which reflects the Π electronic cloud distribution of C 2 V group. In the atom and Σ-states diatomic molecules system, k =. [13] V,T (R, θ, φ = V l,k (RD l, m,k (ΩCl m(θ, φ, (16 l,m,k because of v v = 1. Based on the first order Born approximation of time dependent perturbation theory, the transition matrix element of the potential is f V (t i = C J C J J Ω Iζ V (t JΩIζ + d J d J J Ω ε Iζ V T (t JΩεIζ. (17 In Eq. (15, the transitional matrix for the singlet state is J Ω Iς V s (t JΩIς = δ Ω Ω( 1 J+J +l [(2J + 1(2J J + 1(2L + 1(2L + 1] 1/2 ( L FJ,J l { }( l L J L I J L J e i J J t Vl, [R(t], (18 l l where FJ,J l = (1/2[1 + εε ( 1 J+J +l ], and the transitional matrix of the triplet state is J Ω ε Iς V T (t JΩεIς = ( 1 J+J +I Ω [(2J + 1(2J + 1(2L + 1(2L + 1] 1/2 ( L FJε,J l { }{( l L J L I J ε L J l Ω Ω l

3 164 WANG Wei-Li, MIAO Gang, LI Jian, and MA Feng-Cai Vol. 52 ( e i Ω= J J J t Vl, T } [R(t] + I(Ω, Ω ε Ω e i Ω=2 J 2 Ω J t Vl,2 T [R(t], (19 with FJ,J l = F l Jε,J ε and I(Ω, Ω where [14] { 1, if Ω = Ω = 1; Ω =, Ω = 2; or Ω = 2, Ω =, I(Ω, Ω =, otherwise. The I(Ω, Ω term reflects the fact that for purely electronic coupling the spin Ω = 1 projection quantum number can not change. From Eq. (2, the coupling will always vanish between the rotational level and the Ω = 1 or Ω = 1 rotational levels. Thus collision induced transitions between the 3 Π 1 and 3 Π or 3 Π 2 rotational manifolds will be rigorously forbidden. In the case of transitions within the 3 Π manifold Ω = Ω =, the second term in Eq. (19 will not contribute, due to the symmetry of the 3j symbol, the first term will contribute only if J + l + J is even. This implies, from Eq. (19 that the coupling will vanish unless ε = ε. The unpolarized transition probability can be written as [15] 1 P J J = 1 f V (t i 2. (21 (2J + 1 i Inserting Eq. (17 into Eq. (21, one can obtain the transition probability, LL P J J = c 2 Jc 2 J P J J + d 2 Jd 2 J P T J J + 2c J c Jd J d JP Inter T, (22 where PJ J, P J T J are the pure transition probability of the singlet and triplet states, and the third term is the interference term, PJ J (A1 Σ + u = 2J (2L L L[(2L + 1] ( L (FJ,J l l L 2 { } J L I 2 ( J 2 2 L J l l e i J J t Vl,[R(t]dt 2, (23 P T J J ( Ω = (b3 Π u = 2J P Inter,T + 1(2L L L[(2L + 1] l 2 ( J (A + 1, v = 8 b 3 Π u, v = 14 u Equation (22 also can be written in the form with the differential interference angle (F ε,j ε 2 ( L l L 2 { J L } I 2 L J l (2 e i Ω= J t J Vl, T [R(t]dt 2, (24 = 1 2J (2L L L[(2L + 1] l ( L l L 2 { J L I L J l { ( e iω J J t Vl, [R(t]dt F,J F ε,j ε } 2 ( J 2 e i Ω= } J t J Vl, T [R(t]dt + h.c.. (25 P J J = c 2 J c2 J P J J + d2 J d2 J P T J J + 2c Jc J d Jd J (P J J P T J J 1/2 cosθ D, (26 The energy transfer cross section can be expressed as, [16] P Inter cosθ D = [PJ J P J T. (27 J ]1/2 P J J = 2π By introducing Eq. (22 into Eq. (28, the integral cross-section can be obtained, P J J(bbdb. (28 σ J J = c 2 J c2 J σ J J + d2 J d2 J σt J J + 2c Jc J d Jd J (σ J J σt J J 1/2 cosθ I, (29 with cosine of the integral interference angle, P cosθt D Inter bdb = (P J J (bbdb1/2 (PJ T. (3 J (bbdb1/2

4 No. 6 Interference Angle on Quantum Rotational Energy Transfer in 165 It can also be obtained by introducing Eq. (27 into Eq. (3 (P cosθt I = JJ (bp JJ T (b1/2 (P J J (bbdb1/2 (PJ T. (31 J (bbdb1/2 Equation (31 gives out the relation between the differential and integral interference angle. The integral interference angle is the average effect of the differential interference angle. [17] The experiment of Na 2 (A 1 + u, v = 8 b3 Π u, v = 14 system in collision with Na(3s has been done in a static sample cell with Na(3s the bulk temperature 75 K. However, Eq. (31 for the integral interference angle, which has been derived for the partners with uniform collision velocity, is not appropriate to deal with the cell experiments in which the gas species have the Maxwell-Boltzmann velocity distribution. In this case, Braithwaite et al. have derived a formula to get velocity-averaged probability, [18] / P AV = P(νν 3 e µν2 /2KT ( µ 2 dv P(νν 3 e µν2 /2KT dv = 2 P(νν 3 e µν2 /2KT dv. (32 KT Now, the velocity-averaged integral interference angle can be obtained by first introducing Eq. (26 into Eq. (32 to get P JJ (AV and then the P JJ (AV into Eq. (31, the integral interference angle is obtained. P cosθt I Inter (AV = ν 3 e µν2 /2KT bdbdv ( PJJ ν3 e µν2 /2KT bdbdv 1/2 ( PJJ T. (33 ν3 e µν2 /2KT bdbdv 1/2 3 Discussion Considering the anisotropic Lennard Jones interaction potentials, Eq. (15 can be written as, [ ρ 12 ] V T (R, θ, φ = 4ε R(T 12 ρ6 R(T 6 Σ m,k a,t 2,k D2, m,k (α, β, γc2 m(θ, ϕ. (34 In Eq. (34, a 2k, ρ, and ε are the anisotropic parameters, the distance at which V (R =, and the depth of the potential well respectively. In the case of transitions within the 3 Π manifold Ω = Ω =, the second term in Eq. (19 will not contribute. Due to the symmetry of the 3j symbol, a 2,±2 is not considered, and Eq. (31 can be written as cosθt I Is I Ω= T ν 3 e µν2 /2KT bdbdv = ( I 2 ν 3 e µν2 /2KT bdbdv 1/2 (, (35 I 2 ν 3 e µν2 /2KT bdbdv 1/2 where I T = + e t( ρ 12 iω,t J J R(T 12 ρ6 R(T 6 dt. (36 collision diameter. Fig. 2 The differential interference angle with velocity for b ρ, 3 < v < 8. Fig. 1 The differential interference angle with velocity for b ρ,1 < v < 3. Equation (36 may be integrated with straight-line trajectory approximation. [19] The infective factors of the angle are collision velocity and collision diameter. o we get the relation of the angle with collision velocity and When b ρ, the differential interference angle with velocity and impact parameter are Figs. 1 and 2. When b ρ, the differential interference angle with velocity and impact parameter are Figs. 3 and 4. We choose relative velocity from 1 m/s to 8 m/s, due to the fact that the velocity distributing probability drop to zero when the velocity is v < 1 m/s or v > 8 m/s, the biggest collision parameter is 8Å, and

5 166 WANG Wei-Li, MIAO Gang, LI Jian, and MA Feng-Cai Vol. 52 the distance (b > 8Å has overstepped the collision partners effective distance. functions is weak, and the interference degree becomes weaker as well. Fig. 3 The differential interference angle with velocity for b ρ,1 < v < 3. From the figures, we can see that the differential interference angle is between 2 8 degree. We also obtained the integral interference in Na+Na 2 (A 1 + u, v = 8 b 3 Π u, v = 14 system to be [2] which is the value is average value of differential interference angles. The largest differential angle is found to be b = 2 1/6, with is the collision diameter. This is because that the repulsive potential energy is equal to long-range potential in the distance, so the mixing degree of corresponding wave Fig. 4 The differential interference angle with velocity for b ρ, 3 < v < 8. 4 Conclusion The collision-induced quantum interference on rotational energy transfer in Na+Na 2 (A 1 + u, v = 8 b3 Π u, v = 14 system is studied in this paper and a relationship between differential and the integral interference angles is obtained. The tendency of the differential with impact parameter and impact velocity is also analyzed. References [1] W.M. Gelbart and K.F. Freed, Chem. Phys. Lett. 18 ( [2] M.H. Alexande, J. Chem. Phys. 76 ( [3] J. Boissels, C. Boulet, D. Robert, and. Green, J. Chem. Phys. 9 ( [4] K.T. Lorenz, D.W. Chandler, and J.W. Barr, cience 293 ( [5] M.T. un, J. Liu, and W.Z. un, Chem. Phys. Lett. 365 ( [6] G.H. ha, J.B. He, B. Jiang, and C.H. Zhang, J. Chem. Phys. 12 ( [7] X.L. Chen, G.H. ha, B. Jiang, J.B. He, and C.H. Zhang, J. Chem. Phys. 15 ( [8] X.L. Chen, H.M. Chen, J. Li, et al., Chem. Phys. Lett. 318 (2 17. [9] H.M. Tian, M.T. un, and G.H. ha, Phys. Chem. Chem. Phys. 4 ( [1] R.N. Zare, A.L. chmeltekopf, W.J. Harrop, and D.L. Albritton, J. Mol. pectrosc. 46 ( [11] M.H. Alexander, Chem. Phys. 92 ( [12] M.H. Alexander, J. Chem. Phys. 76 ( [13] M.H. Alexander, J. Chem. Phys. 76 ( [14] M.H. Alexander, J. Chem. Phys. 79 ( [15] M.T. un, F.C. Ma, and G.H. ha, Chem. Phys. Lett. 374 (23 2. [16] P. Brumer, A. Abrashkevich, and M. hapiro, Faraday Discuss 374 ( [17] T.J. Wang, W.C. twalley, L. Li, and A.M. Lyyra, J. Chem. Phys. 97 ( [18] C.G. Gray and J.V. Kranendonkan, Can. J. Phys. 44 ( [19] W.L. Wang, P. ong, Y.Q. Li, G. Miao, and F.C. Ma, Commun. Theor. Phys. 46 ( [2] M.T. un, W.L. Wang, P. ong, and F.C. Ma, Chem. Phys. Lett. 386 (24 43.