Interferences in Photodetachment of a Negative Molecular Ion Model

Commun. Theor. Phys. (Beijing, China) 50 (2008) pp. 1401 1406 c Chinese Physical Society Vol. 50, No. 6, December 15, 2008 Interferences in Photodetachment of a Negative Molecular Ion Model A. Afaq 1, and DU Meng-Li 2, 1 Department of Physics, COMSATS Institute of Information Technology Lahore, Pakistan 2 Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China (Received May 19, 2008) Abstract By employing a two-center model, the total and differential cross sections in the photodetachment of a negative molecular ion are studied theoretically and obtained for the case of light polarization parallel to the molecular axis. We find that in contrast to the smooth behavior of the total cross section for perpendicular polarized light, the cross section for parallel polarized light shows an interesting oscillatory structure. The oscillations in the total cross section may provide a method to determine the distance between the two centers. We explain the oscillation in the total cross section as an interference effect using closed-orbit theory. We also calculated the detached-electron flux distributions on a screen placed at a large distance from the negative molecular ion. The distributions display multiple-ring-like interference patterns. Such interference patterns are similar to those in the photodetachment microscopy experiments. PACS numbers: 32.80.Gc, 32.80.Qk Key words: quantum interference, two-center model, photodetachment 1 Introduction Interference phenomena are very important in quantum mechanics. Analogies of the Young double-slit experiment played a fundamental role in the description and comprehension of the dual nature of quantum objects such as an electron. Richerard Feynman in his famous electron gun thought experiment [1] explored what might happen if an electron travels simultaneously along two trajectories, and interferes with itself. He felt that all of the strangeness of quantum behavior can be seen in one simple and elegant experiment called the two-slit experiment. In the 1980s, Demkov et al. [2] suggested a version of Feynman s idea: photodetachment of negative ions provides a localized source of electrons with fixed energy, and an externally applied homogenous electric field acts to create the double-slit effect in the cross sections. [3 9] Recently, Afaq and Du have studied the photodetachment of a negative molecular ion using a two-center model for the case of perpendicular polarized laser light, and they showed that the total cross section is smooth and it approaches one and two times of the cross section for onecenter system in the high and low photon energy limits respectively. [10] Here we discuss the photodetachment of a doublecenter system using the same model, [10] but for parallel polarized laser light case, that is, when the laser light is parallel to the axis of double-center system. For convenience, we call this double-center system a negative molecular ion even when the distance parameter between the two-centers is set to values larger than the values of usual molecules. The two centers of the system in the photodetachment process play the role of slits as in the Young double-slit experiment, so that interference of detached electrons from two coherent centers can be observed on a screen placed at large distance from the system. This problem is inspired by the interferences in strong laser field [11] and in the scattering of D 2 molecule by fast electron. [12] For one-center photodetachment of H in the presence of a static electric field, similar interference patterns have been observed experimentally [3,4,7 9] as well as theoretically. [5,6,13,14] In contrast to the perpendicular case, [10] the total cross section for the present parallel case shows strong oscillations. The oscillations in the total cross section may provide us a useful method to determine the distance of the two centers in the negative molecular ion. The origin of the oscillations can be explained using closed-orbit theory [15] developed for atoms in external fields. The oscillations in the differential cross sections show interference patterns similar to these in photodetachment microscopy. [7,9] For an observer at large distance from the system, there are two coherent detached-electron waves going from each center of the system to the observer. Superposition of these two coherent detachedelectron waves gives the total outgoing wave from the system. Detached-electron flux calculated from this outgoing wave produces ring like interference patterns on a screen. By integrating the detached-electron flux for all outgoing angles, we are able to derive analytic formula for the total photodetachment cross sections for the parallel polarized laser light case. This article is organized in the following way. In Sec. 2, we first describe the two-center model of the system and the wave function of detached-electron at large The project supported by National Natural Science Foundation of China under Grant No. 90403028 E-mail: aafaq@ciitlahore.edu.pk E-mail: duml@itp.ac.cn

1402 A. Afaq and DU Meng-Li Vol. 50 distances from our system, the differential and the total photodetachment cross section are then derived. The total cross section for the parallel case is studied in comparison with the perpendicular case. [10] In Sec. 3, we calculate and demonstrate the ring-like interference patterns of detached-electron flux on a screen placed at a large distance from the system. In Sec. 4, we re-examine the oscillations in the total cross sections using closed-orbit theory. The oscillations are explained as an interference effect. Conclusions are given in Sec. 5. Atomic units are used unless otherwise noted. 2 Total Photodetachment Cross Section Assume the laser polarization is in the z-direction, and axis of the double-center system is parallel to the polarization of light and is also in z-direction. The schematic diagram of the system is shown in Fig. 1. 1 and 2 on the z-axis represent the two centers of the system. The origin of the coordinates system is chosen to be the middle point of the line connecting the two centers. The screen perpendicular to z-axis is placed at a distance L from the two centers. Let d be the distance between the two centers of the system, d is a parameter and its value can be from a few to several hundred Bohr radii. L is much greater than d and usually equal to several thousand Bohr radii in the experiment. [7,9] Two outgoing electron waves Ψ 1 and Ψ 2 are shown. The double-circle with a plus and a minus at 1 and 2 represents the angular dependence of the outgoing wave from each center (see Eq. (4)). Fig. 1 The schematic diagram of our system. 1 and 2 on the z-axis represent the two centers in the model negative molecular ion. They are separated by d, which is a few to several hundred Bohr radii. The double-circles with a plus and a minus at each center reflect the angular dependence of the outgoing wave amplitudes in Eq. (4). Similar to the H model, [6] we assume that there is only one active electron in the system. Furthermore, we assume the normalized wave function for the active electron in the negative molecular ion can be written as a superposition of the H -like bound state at the two centers Φ = (Φ 1 + Φ 2 )/ 2, where Φ 1 and Φ 2 are the normalized wave function for H but centered at the first center and second center in Fig. 1. The photodetachment process can be regarded as a two-step process: [15,16] in the first step, the double-center system of negative ion absorbs one photon energy E ph and generates an outgoing electron wave; while in the second step, this outgoing wave propagates to large distances, parts of the outgoing wave may return to the source region to interfere and produce oscillations in the total cross section. In the H case, an outgoing wave of detached-electron centered at the nucleus is generated in the photodetachment process. [17] In the present case, the detachedelectron wave function is a superposition of detachedelectron waves generated from the two centers. These waves are coherent in nature and can be obtained from the results in the photodetachment of H in the absence of an electric field. [17] Let Ψ 1 and Ψ 2 be the waves produced from centers 1 and 2 respectively, then the outgoing detached-electron wave Ψ from the double-center system is given by Ψ = 1 2 (Ψ 1 + Ψ 2 ). (1) Using (r 1, θ 1, φ 1 ) and (r 2, θ 2, φ 2 ) as the spherical coordinates of the detached-electron relative to each center, we have [17] Ψ 1 (r 1, θ 1, φ 1 ) = U(k, θ 1, φ 1 ) exp(ikr 1) kr 1, Ψ 2 (r 2, θ 2, φ 2 ) = U(k, θ 2, φ 2 ) exp(ikr 2) kr 2. (2) The factors U(k, θ 1, φ 1 ) and U(k, θ 2, φ 2 ) for the laser polarization parallel to z-axis can be written as U(k, θ 1, φ 1 ) = U(k, θ 2, φ 1 ) = (k 2 b + k2 ) 2 cos θ 1, (k 2 b + k2 ) 2 cos θ 2, (3) where k b is related to the binding energy E b of H by E b = kb 2 /2, B is a constant.[6] The detached-electron wave from the double-center system Ψ(r, θ, φ) can be written as Ψ(r, θ, φ) = 1 4k 2 Bi [ exp(ikr 1 ) 2 (kb 2 + k2 ) 2 cos θ 1 kr 1 exp(ikr 2 ) ] + cosθ 2. (4) kr 2 Using this detached-electron wave function, both the differential and total photodetachment cross section for parallel polarized laser light case can be derived. Let (r, θ, φ) be the spherical coordinates of the detached-electron relative to the origin at the middle of the two centers. Assume r is much greater than d. The outgoing wave function in Eq. (4) can be simplified further. For phase terms, we use the approximations r 1 r (1/2)d cos θ, r 2 r+(1/2)d cos θ, and in all other places in Eq. (4), we use r 1 r 2 r, and θ 1 θ 2 θ.

No. 6 Interferences in Photodetachment of a Negative Molecular Ion Model 1403 With these approximations, Eq. (4) becomes Ψ(r, θ, φ) = 2 ( kdcos(θ) ) (kb 2 + k2 ) 2 cos(θ) cos 2 ( exp(ikr) ). (5) kr Now we calculate the electron flux [10,17] as j(r, θ, φ) = i 2 (Ψ Ψ Ψ Ψ). (6) Using the expression for Ψ(r, θ, φ) in Eq. (5) in the above formula, the detached-electron flux normal to the surface of a huge sphere of radius r, j r (r, θ, φ) = j 0 (r, θ)[1 + cos(kdcos θ)], (7) where j 0 (r, θ) = [ 16k 3 B 2 /(kb 2 + k2 ) 4 r 2] cos 2 θ. We now calculate the total photodetachment cross section of the system. Let us imagine a large surface Γ such as the surface of a sphere enclosing the source region, a generalized differential cross section dσ(q)/ds may be defined on the surface from the electron flux crossing the surface, [10] dσ(q) = 2πE ph j r ˆn, (8) ds c where c is the speed of light and is approximately equal to 137 a.u., q is the coordinate on the surface Γ, ˆn is the exterior norm vector at q, ds = r 2 sin(θ)dθdφ is the differential area on the spherical surface. The total cross section may be obtained by integrating the differential cross section over the surface, dσ(q) σ(q) = ds. (9) Γ ds The result for the cross section in the parallel polarization can be written as σ (E) = σ 0 (E)A ( 2Ed)a 2 0, (10) where σ 0 (E) is the photodetachment cross section of H in the absence of electric field, [6] the argument 2Ed = kd of A in Eq. (10) is the action of the detached-electron from one center to another center along the straight line. A is the modulation function in the parallel polarized laser light case and is defined by A (u) = 3 2 π 0 cos 2 θ sin θ[1 + cos(u cos θ)]dθ. (11) By substituting v = u cos θ, Eq. (11) is reduced to the following elementary integral, and its result is A (u) = 3 2u 3 u u v 2 (1 sin v)dv, A (u) = 1 + 3 sin(u) + 6 cos(u) u u 2 6 sin(u) u 3. (12) The asymptotic form of the modulation function is given by A asy (u) = 1 + 3 sin(u). (13) u In Fig. 2, we show the total photodetachment cross sections in Eqs. (10) and (12) for several values of d. Oscillations are strong in the photodetachment cross sections in the present parallel case. In contrast, the cross sections in the perpendicular case for the double-center system are smooth and non-oscillatory. [10] The oscillation in the total cross section is controlled by the second term of Eq. (12). The amplitude of the oscillation decreases as the distance d increases, but the frequency of the oscillation increases as the distance d increases. We will explain the origin of the oscillation in the total cross section using closed-orbit theory in Sec. 4. Fig. 2 The total photodetachment cross sections (thick solid line) of the two-center negative molecular ion for laser polarized parallel to the molecular axis. The parameter d is the distance between the two atoms in the molecule ions. The dashed lines represent the result of one-center system for comparison. a 0 is the Bohr radius.

1404 A. Afaq and DU Meng-Li Vol. 50 3 Interference Patterns of Detached-Electron Figure 3 is a schematic diagram describing the geometry for calculating the spatial distributions of the detached-electron. A screen perpendicular to z-axis is placed at z = L from the system, and L is much greater than the distance d between the two centers of the system. L is usually equal to thousands atomic units in the experiments. [7,9] The flux distributions on the screen is cylindrically symmetric. The intensity of detachedelectron depends on the distance between any point (x, y) on the screen and the z-axis ρ = x 2 + y 2. By projecting the radial flux in Eq. (7), the detached-electron flux distribution crossing the screen for the parallel polarization case as a function of ρ can be written as 16k 3 B 2 L 3 [ ( kdl j z (ρ) = (kb 2 + 1+cos )].(14) k2 ) 4 (ρ 2 + L 2 ) 5/2 ρ2 + L 2 In Fig. 4, we show the detached-electron flux distribution calculated using Eq. (14) with d = 4 Bohr radii and L = 1000 Bohr radii for nine different values of photon energy E ph. The corresponding interference fringes on the screen are presented in Fig. 5. Such interference patterns are similar to the ones observed in the photodetachment microscopy experiments. [7,8] Figures 4 and 5 show that when kd is an integer multiple of 2π, a large peak appears at the center of the screen because of constructive interference of the two coherent detached-electron waves, and when kd is a half integral multiple of 2π, a valley appears at the center of the screen because of destructive interference of the two coherent detached-electron waves. Fig. 3 The schematic diagram showing the detachedelectron waves from the negative molecular ion and the flux patterns on the observation screen. The screen is perpendicular to the z-axis and is placed at z = L. Fig. 4 The detached-electron flux distributions on the screen calculated using Eq. (14) are shown for nine values of photon energy E ph (solid lines): (a) E ph = 303 ev; (b) E ph = 329 ev; (c) E ph = 355 ev; (d) E ph = 383 ev; (e) E ph = 412 ev; (f) E ph = 442 ev; (g) E ph = 473 ev; (h) E ph = 505 ev; (i) E ph = 538 ev. The laser is assumed to be polarized in the z-direction parallel to the molecular axis. In these calculations, we set d = 4 bohr radii and L = 1000 Bohr radii. The dashed lines are the results of one-center system.

No. 6 Interferences in Photodetachment of a Negative Molecular Ion Model 1405 Fig. 5 The interference fringes on the screen corresponding to Figure 4. 4 Cross Section Oscillations and Closed-Orbit Theory The oscillations in the total photodetachment cross section of the double-center system can be explained using closed-orbit theory, [15] which was developed originally to describe the oscillations in the photoionization cross section of atoms in external electric and magnetic fields. The physical picture for the present problem as follows. When a photon is absorbed, a coherent wave describing the detached-electron from the double-center system is produced. The total wave is a sum of two waves. The first one is produced by center 1 and the second wave is produced at center 2. When the detached-electron wave produced in one center reaches another center, it interferes with the source there and results in an oscillation in the total cross section. Figure 6 is a schematic diagram illustrating the situation. Fig. 6 The schematic diagram illustrating the physical origin for the oscillation in the total cross section of the double-center system. To be more specific, let Φ 1 (r) = B e k br 1 /r 1, Φ 2 (r) = B e k br 2 /r 2 be the localized wave functions [6] centered at 1 and 2 of the double-center system respectively, and Ψ 1 (r) = (k 2 b + k2 ) 2 cos(θ 1) exp(ikr 1) kr 1, Ψ 2 (r) = (kb 2 + k2 ) 2 cos(θ 2) exp(ikr 2) kr 2 be the outgoing detached-electron wave functions from center 1 and 2 respectively. Consider z-polarized laser light here. The oscillator-strength density can be calculated by using the formula, [13,14] Df(E) = 2E ph Im DΦ(r) Ψ(r), (15) π where Φ(r) = (1/ 2)[Φ 1 (r) + Φ 2 (r)] is the wave function of the double-center system, and Ψ(r) = (1/ 2)[Ψ 1 (r) + Ψ 2 (r)] is the outgoing wave function of the system. D(r) = z is the projection of electron coordinate on the direction of laser polarization. Using the expressions of Φ(r) and Ψ(r), Eq. (15) can be written as Df(E) = E ph π Im[ DΦ 1 Ψ 1 + DΦ 2 Ψ 2 + DΦ 2 Ψ 1 + DΦ 1 Ψ 2 ]. (16) Because of the symmetry of the two centers, in Eq. (16), the value of the first term is equal to the second, and the third is equal to the fourth. The cross section formula in Eq. (16) can be written as Df(E) = 2E ph π Im[ DΦ 1 Ψ 1 + DΦ 2 Ψ 1 ]. (17) The following result can be inferred from the problem of H in the presence of a static electric field, [13] 2E ph π Im[ DΦ 1 Ψ 1 ] = 8 2B 2 E 2/3 3(E b + E) 3. (18)

1406 A. Afaq and DU Meng-Li Vol. 50 We now evaluate the second term in Eq. (17). The overlap integral DΦ 2 Ψ 1 represents the part of the detached-electron wave initially produced at center 1 arriving at center 2 and interfering with the source at center 2. In a small region around center 2 the detached-electron wave produced at the first center Ψ 1 (r) can be approximated by a plane wave as where f can be calculated from Ψ 1 (r), Ψ 1 = f e ikz, (19) f = [4k 2 Bi/kd(k 2 b + k 2 ) 2 ] e ikd. (20) The overlap integral in the second term of Eq. (17) can be worked out, 2E ph π Im DΦ 2 Ψ 1 = 2E ph π = Im zb e k br r [ 8 2B 2 E 2/3 ][ 3 sin(kd) 3(E b + E) 3 kd 4k 2 Bi kd(kb 2 + k2 ) 2 eikd e ikr cos θ r 2 sin θdrdθdφ ]. (21) The total oscillator-strength density of the doublecenter system derived using closed-orbit theory is [ Df(E) = Df 0 (E) 1 + 3 sin(kd) ], (22) kd where Df 0 (E) = 8 2B 2 E 2/3 /3(E b + E) 3 is the smooth oscillator-strength density for H. The total photodetachment cross section for our double-center system derived using closed-orbit theory is [ σ(e) = σ 0 (E) 1 + 3 sin(kd) kd ], (23) where σ 0 (E) is the cross section of H. [6] The total cross section in Eq. (23) derived using closedorbit theory is identical to Eq. (10) with the asymptotic modulation function in Eq. (13). The derivation using closed-orbit theory clearly shows that the origin of the oscillation in the total cross section is the interference effect: the detached-electron produced from one center propagates to another center and interferes with the source there. 5 Conclusions In summary, we have studied the photodetachment of a negative molecular ion using a double-center model when laser polarization is parallel to the molecular axis. The detached-electron flux on a screen shows interesting interference patterns. Such interference patterns may be possible to observe as in the experiments on photodetachment microscopy. [7,8] The total cross section in the present parallel polarization case displays a strong oscillation, the oscillation is absent when the laser polarization is perpendicular to the molecular axis [10]. Both the amplitude and the frequency of the oscillation depend on the distance between two centers of the molecular ion. The double-center photodetachment cross section may be used to determine the distance between the two atoms in molecular ions. We have also applied closed-orbit theory [15] to this double-center problem and re-derived the total cross section. The derivation shows that the oscillation in the total cross section is the result of interference between the detached-electron wave produced at one center interfering with the source at another center. The quantity controlling the oscillation in the total cross section is the action for the detached-electron going from one center to another center, which is kd. References [1] R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, MA (1964) Vol. III. [2] Y.N. Demkov, V.D. Kondratovich, and V.N. Ostrovskii, JETP Lett. 34 (1981) 425. [3] H.C. Bryant, et al., Phys. Rev. Lett. 58 (1987) 2412. [4] J.E. Stewart, et al., Phys. Rev. A 38 (1988) 5628. [5] A.R.P. Rau and H. Wong, Phys. Rev. A 37 (1988) 632. [6] M.L. Du and J.B. Delos, Phys. Rev. A 38 (1988) 5609. [7] C. Blondel, C. Delsart, and F. Dulieu, Phys. Rev. Lett. 77 (1996) 3755. [8] C. Blondel, C. Delsart, and F. Dulieu, et al., Eur. Phys. J. D 5 (1999) 207. [9] C. Blondel, W. Chaibi, C. Delsart, et al., Eur. Phys. J. D 33 (2005) 335. [10] A. Afaq and M.L. Du, Commun. Theor. Phys. 46 (2006) 119. [11] S.X. Hu and L.A. Collins, Phys. Rev. Lett. 94 (2005) 073004. [12] O. Kamalou, et al., Phys. Rev. A 71 (2005) 010702(R). [13] M.L. Du, Phys. Rev. A 70 (2004) 055402. [14] M.L. Du, Eur. Phys. J. D 38 (2006) 533. [15] M.L. Du and J.B. Delos, Phys. Rev. Lett. 58 (1987) 1731; M.L. Du and J.B. Delos, Phys. Rev. A 38 (1988) 1896; M.L. Du and J.B. Delos, Phys. Rev. A 38 (1988) 1913. [16] C. Bracher, J.B. Delos, V. Kanellopoulos, M. Kleber, and T. Kramer, Phys. Lett. A 347 (2005) 62. [17] M.L. Du, Phys. Rev. A 40 (1989) 4983.