Re-examining Photodetachment of H near a Spherical Surface using Closed-Orbit Theory

Transcription

1 Commun. Theor. Phys Vol. 65, No. 3, March 1, 216 Re-examining Photodetachment of H near a Spherical Surface using Closed-Orbit Theory Xiao-Peng You and Meng-Li Du Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 119, China Received December 18, 215 Abstract Photodeachment of H near a reflective spherical surface was studied by Haneef et al. [J. Phys. B: At. Mol. Opt. Phys ] using a theoretical imaging method. The total cross section displays interesting oscillations. Here we re-examine the total photodetachment cross section of this system by directly applying the standard closed-orbit theory. Our result for the total cross section differs from the result obtained by Haneef et al. The difference between the two results vanishes in the limit of large radius of the reflective sphere. We argue that the theoretical imaging method developed originally for photodetachment near a flat surface can not be directly applied to the present system. PACS numbers: 32.8.Gc, i, xg Key words: photodetachment, hydrogen negative ion 1 Introduction Photodetachment of negative ions in different external fields has been studied during the past decades. [1 11] The most interesting aspect of these studies is to understand and characterise the induced oscillations in various systems by the additional external fields. Different methods have also been introduced to calculate the photodetachment cross sections. As a prelude to the present system, the photodetachment cross section for a negative ion H near a reflective wall was first introduced and studied [12] using closed orbit theory. At almost the same time, Afaq and Du [13] applied a theoretical imaging method to study the same problem. More recently, Haneef et al. [14] studied the photodetachment of H near a spherical surface by using the theoretical imaging method. [13] They found oscillations in the total photodetachment cross sections. In this paper, we re-examine the total cross section of H near a spherical surface by directly applying the closed-orbit theory. When we compare our result and that by Haneef et al., significant difference between the two results is found. We also find the difference vanishes in the limit of large radius of the reflective sphere. We argue the discrepancy between the two results is caused by the incorrect application of the theoretical imaging method developed originally for photodetachment near a flat surface. This article is organized as follows. In Sec. 2, the system is briefly described. In Sec. 3, the total photodetachment cross section is calculated using the standard closed-orbit theory. [15] In Sec. 4, we calculate and compare our photodetachment cross sections with those by Haneef et al. [14] for various values of parameters such as the radius of the sphere and the distance between the sphere and the negative ion. Conclusions are given in Sec. 5. Atomic units are used throughout this paper unless otherwise noted. 2 The System Foonvenience, here we first briefly describe the system considered by Haneef et al. [14] As shown schematically in Fig. 1 we study the photodetachment of H near a spherical surface. The system is three-dimensional and Fig. 1 represents only the intersection of the system with the x z plane, assuming z-axis points upward. The reflective spherical surface is represented by the higheircle centered at O 1 with a radius. Outside the sphere is a negative ion H located at O. The distance between the negative ion and the center of the spherical surface O 1 is assumed to be d +. For the photodetachment process, the initial bound state of H can be regarded as a one-electron system. [7 8] In such state, the electron moves in a short-range potential of the hydrogen atom. In the one active electron approximation for the system, the initial bound state Ψ i = B e br /r, where B is the normalization constant and equals b is the momentum related to the binding energy E b of H. E b is approximately.754 ev. When a laser is turned on, the ion may absorb a photon and the electron then moves under the fields of combined short-range potential of the hydrogen atom and the reflective sphere. The photodetachment cross section is a measure of the amount of excited electron cloud after the absorption of a photon by the negative ion. The cross section depends sensitively on the parameters of the system Supported by National Natural Science Foundation of China under Grant Nos and duml@itp.ac.cn c 216 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 355 including the radius of the reflective sphere and photon energy. The cross section also reveals much about the motion of the excited electron after a photon has been absorbed. Fig. 1 The schematic of photodetachment of H near a reflective spherical surface. The radius of the reflective spherical surface centered at O 1 is. H is outside the sphere. The distance between H and the closest tangent plane to it is d. The origin of coordinate is set at the negative ion located at O. The z axis points upward. A trajectory from O to M and from M to B is also shown. 3 Closed-Orbit Theory for Photodetachment Cross Section Let us assume the laser polarization is linear and parallel to the z axis. According to closed-orbit theory, [15] the active electron goes into outgoing p z waves after a photon is absorbed. The outgoing waves emerging from O propagate away from the hydrogen atom in all directions. Sufficiently far from the residual atom, the wave propagates according to semiclassical mechanics. Part of these waves propagating upward toward the sphere gets reflected by the sphere. The waves then reverse its direction near the surface of the sphere and propagate downward toward the negative ion. The interference of the outgoing electron wave and the returning wave in the region of the negative ion leads to the oscillation in the photodetachment cross section. The outgoing and returning wave follows a closed-orbit. In Fig. 1, the closed-orbit goes along the z-axis from O first to C and then goes bac to O. This is the only closed-orbit that goes out from and returns to the point O in this system. According to closed-orbit theory, [15] the photodetachment cross section can be written as σe = σ E + σ ret E, 1 where σ E = 16 2B 2 π 2 E 3/2 /3cE b + E 3[9] represents the photodetachment cross section of H in free space, again B = is related to the normalization of the initial bound state Ψ i of H, c is the speed of light and its value is approximately 137 a.u., E denotes the energy of the escaping electron, the photon energy is E p = E + E b. As mentioned earlier, E b is the binding energy of the negative ion of H. σ ret E is the oscillatory term which is identified with the closed orbit, σ ret E = 4E p c Im DΨ i Ψ ret, 2 E p is the incident photon energy and D is the dipole operator, Ψ i denotes the initial bound state wave function of H. In the closed orbit theory, it is crucial to get Ψ ret, which is a returning wave obtained by propagating an outgoing wave along the closed-orbit O C O when it comes bac to the negative ion region. In the case of linear polarization parallel to the z axis, D is simply z. The outgoing wave is defined as the outgoing solution of an inhomogeneous Schrödinger equation, E H Ψ out r, θ, φ = DΨ i, where H represents the Hamiltonian of the detached electron moving in a short-range potential excluding the spherical sphere after absorbing a photon. H = p 2 /2 + V p r, where V p r represents a short-range potential for the detached electron. The outgoing wave has been obtained [8] and has the form Ψ out r, θ, φ = 4B2 i b h1 1 rcosθ, 3 where h 1 1 r is an outgoing Bessel function. The outgoing wave in Eq. 3 describes the outgoing photodetached electron with an angular factoosθ. Far away from the source region, for example, when r is larger than 2 a.u. from the negative ion, the outgoing wave is well correlated with classical trajectories. As mentioned before, there is only one closed-orbit for this system. After a photon is absorbed, the detached electron as a wave first travels towards the sphere centered at O 1. Then, it is reflected bac to the region of the hydrogen atom at O by the spherical surface. Consider a small sphere centered at the negative ion. Its radius is R. For the moment, R has a value around 2 a.u. However, the final result will be independent of R as we will see. For r R the asymptotic approximation h 1 1 r = eir π /an be used. Then the direct outgoing wave for the detached electron on the surface of sphere with a radius of R centered at O has the form Ψ out R, θ, φ = 4B2 i eir π b 2 + cosθ 2 2 R. 4 The returning wave function is constructed using the standard semiclassical method. [15] Ψ ret r, θ, φ = Ae is µπ/2 Ψ out R, θ =, φ, 5 where A and S are respectively the amplitude and the action along the closed orbit, µ represents the Maslov index and indicates the phase loss caused by the reflective spherical surface centered at O 1. Assume the reflective surface is hard, [13] then µ equals two. In the region of the negative ion, the returning wave function can be well represented by a plane wave propagating downward, [9] Ψ ret = N e iz. Therefore, N = Ae is π Ψ out R, θ =, φ.

3 356 Communications in Theoretical Physics Vol. 65 When the outgoing wave propagates out from the source region along the closed orbit, its amplitude and phase change accordingly. The electron travels in free space until it meets the spherical surface centered at O 1, it then travels toward and finally returns bac to the region of hydrogen atom. The action S along the full closed orbit O C O is easy to obtain, S = 2d = 2 2Ed. 6 The amplitude A is calculated with [15] A = detj2 ρ, z, 1/2 ρ detj 2 ρ, z, T ρt 1/2, 7 where T is the time of the full closed orbit and J 2 ρ, z, t is the two dimension Jacobian. ρ ρ detj 2 ρ, z, t = t z z. 8 t Note the elements of the Jacobian matrix are calculated with respect to the closed orbit. As shown schematically in Fig. 1, when the trajectory first travels from O vertically to the sphere surface centered at O 1, it is reflected at C and travels bac to O, forming the closedorbit. In order to calculate the amplitude in Eqs. 7 and 8, we now launch a trajectory from O slightly away from the vertical upward direction. It first travels along OM, and then along MB. The angle between OC and OM is assumed small. The distance between the initial point O and the reflection point M is denoted as l. For the triangle OMO 1, sinθ 1 = 2 sin θ cosθ 1 using the law of cosines and noting O 1 M =, we have r 2 c = l2 + d + 2 2ld + cos θ. Solving the above equation for l, we get l = d + cosθ d + 2 cos 2 θ d 2 2 d. 9 Assume the line BM is extended and it intersects the vertical line OO 1 at F. Similarly the line O 1 M is extended to M 1 and outside the reflective surface. To simplify the notations, we denote OFM = θ 1 and OMM 1 = θ 2. In the triangle OMO 1, one may use the law of sines sinπ θ 2 /OO 1 = sin θ/o 1 M to obtain sin θ 2 = d + sin θ. 1 Then because sin 2 θ 2 + cos 2 θ 2 = 1, it leads to 2 cosθ 2 = 1 sin 2 θ. 11 In the triangle OFM of Fig. 1, one has the relationship θ 1 = π θ π 2θ 2 = 2θ 2 θ. We can use the above equation to rewrite the sine and cosine function of θ 1 : cosθ 1 = cos2θ 2 θ = cos2θ 2 cosθ + sin 2θ 2 sin θ similarly, = 2 cos 2 θ 2 1cosθ + 2 sinθ 2 cosθ 2 sin θ, 12 sin θ 1 = 2 sinθ 2 cosθ 2 cosθ sin θ2 cos 2 θ Substitute Eqs. 1 and 11 to the above two expressions, one can obtain 2 sin 2 θ d + cosθ 1 = 1 2 sin 2 rc θ cosθ + 2 sin 2 θ 1 2 sin 2 θ sin θ, 14 2 sin 2 θ. 15 Now we are ready to evaluate the amplitude A in Eqs. 7 and 8. When t is small and the electron is not far away from the artificial surface of sphere center at O, the trajectories are For t =, ρθ, t = R + tsin θ, 16 zθ, t = R + tcosθ. 17 ρθ, = R sin θ. 18 The elements of Jacobian at the initial point are obtained, = R, θ=,t= 19 t =, θ=,t= 2 t =, θ=,t= 21 =. θ=,t= 22 Substituting the above four elements to Eq. 8, we get detj 2 ρ, z, = R. 23 For t larger than l R/, the trajectories have been reflected by the sphere centered at O 1. The returning trajectories near the region of the negative ion are given by ρθ, t = sin θ sin θ 1 l + zθ, t = sin2θ 2 sin θ 1 l t l R sin θ l + sin θ 1 sin θ 1, 24 t l R cosθ From Eq. 24 one has immediately ρ/ t θ= =. So only two elements are needed to determine the Jacobian in Eq. 8 for the returning trajectory. From Eqs. 24 and 25 we have = 2d + r c rc l + t l R, 26 θ=,t>l R/ 2d + =. 27 t θ=,t>l R/ For the closed orbit, we have also l = d and t = 2d R/.

4 No. 3 Communications in Theoretical Physics 357 Denoting the transit time of the closed-orbit by T, we have = 2dd +, 28 θ=,t=t =. 29 t θ=,t=t The Jacobian for the closed orbit at the returning time T is then detj 2 ρ, z, T = 2dd +. 3 For small θ, Eq. 24 also gives the following approximation ρθ, T = dsin θ + sin θ Now combine it with the expression of ρθ, obtained earlier, one can show ρθ, lim θ ρθ, T = lim R θ d1 + sin θ 1 / = R 2dd When Eqs. 23, 3, and 32 are used in Eq. 7, we finally obtain the amplitude R A = 2dd +, 33 for the closed orbit in the present system. Since the amplitude A primarily determines the amplitude of the oscillation in the photodetachment cross section corresponding to the closed-orbit, its importance can ρ i ρ f = R tan θ cosθ 1 l sin 2θ 2 = R1 2d + / 2 sin not be over-estimated. As a chec, we now demonstrate explicitly that the expression for A in Eq. 33 satisfies the conservation of the electron flux. Denoting the short arc segment AA 1 by ρ i and the short horizontal line OB by ρ f. That part of the detached electron crossing AA 1 must also cross OB. When θ is small and when the cylindrical symmetry of the system is taen into consideration, the electron flux crossing AA 1 is approximately v i A 2 i πρ i 2 and the electron flux crossing OB is approximately v f A 2 f πρ f 2, where we have used A i and A f to represent the amplitudes of the detached electron wave when crossing AA 1 and OB. Since the initial speed of the electron v i equals to the final returning speed v f. The conservation of electron flux requires From Fig. 1 we have Therefore A f A i = ρ i ρ f. 34 ρ i = R tanθ, 35 ρ f = OFtanθ 1 = l sin2θ 2 cosθ θcos θ + 2d + / sin 2 θ 1 d + / 2 sin 2 θ 2ld + / 1 d + / 2 sin 2 θ cosθ. 37 When θ approaches, ρ i. R = ρ f 2dd + 1 θ d2 + 2d 2r 2 c Combining Eqs. 34 and 38, we get A f lim = θ A i + oθ R 2dd A is equal to A f by assuming A i = 1. This provides confirmation on the correctness of the result in Eq. 33. The returning wave function near the source region is now written as 2iB Ψ ret = b d + d exp i2 2Ed z. 4 Substitute this wave function in Eq. 2, the overlap integral DΨ i Ψ ret is written as with DΨ i Ψ ret = 2iB2 exp i2 2Ed b I, 41 d + d I = π 2π r 2 sin θdrdθdφexp iz z r exp br = 2π dr exp b rr dτ exp irττ = 2π dr exp b rr 2 τ exp irτ ir r 2 = 2π dr exp b rr 2 cosr 2i 1 r 1 = 4iπ 2 exp b r sin rdr 1 r exp b osrdr sinr 2 r 2 = 8iπ 2 b + 2, 42 where we have used exp b r sin rdr = / 2 b + 2, r exp b osrdr = 2 b 2 / 2 b Consequently we have DΨ i Ψ ret = 16πB b d + d exp i2 2Ed. 43 Afteombining the above equation and Eq. 2, we ob-

5 358 Communications in Theoretical Physics Vol. 65 tain the oscillatory part of the cross section 8π 2 B 2 E σ ret E = cdd + E b + E 3 sin 2 2Ed. 44 The smooth bacground is the total photodetachment cross section without any external fields. It has the expression 16 2π 2 B 2 E 3/2 /3cE b + E 3. [9] Our total photodetachment cross section using closed-orbit theory is obtained as σ COT E = 16 2π 2 B 2 E 3/2 3cE b + E sin 2 2Ed 2 + d 2Ed Comparisons and Discussions We now show the results based on closed-orbit theory in Eq. 45 and compare it with the formula obtained by Haneef et al. [14] using an imaging method. For the convenience of comparison, their formula is re-written below, σ I E = σ E [ sindh + 6 cosdh ] dh dh 2 6sindh dh 3, 46 where h = 1 + 1/1 + 2d/. In Fig. 2 the results of total cross sections from both Eqs. 45 and 46 as a function of photon energy E p is plotted for d = 25 a.u. and = 5 a.u. The dashed lines represent the smooth cross section σ E in free space. The thic solid line is the cross section of the closed-orbit theory calculated from Eq. 45, and the thin solid line is the cross section calculated using Eq. 46. As mentioned earlier, the oscillation in the cross section is explained as an interference between the direct outgoing wave and reflected wave in closed orbit theory. With the increase of the photon energy, the oscillation amplitude decreases. We note the difference between our result and the result of Haneef et al. [14] is very significant, especially in the phase of the oscillation. Fig. 2 The thic solid line represents the total photodetachment cross section of H near a reflective spherical surface obtained using closed-orbit theory in Eq. 45 with d = 25 a.u. and = 5 a.u. and the thin solid line represents that obtained by Haneef et al. [14] in Eq. 46. The dashed lines represent the cross section in free space. The difference in the oscillations is significant and is discussed in the text. Fig. 3 Similar to Fig. 1 but here we fix d = 25 a.u. and compare the cross sections for several values of. It is noted that for larger, the present cross sections in Eq. 45 and that obtained by Haneef et al. [14] in Eq. 46 are much closer.

6 No. 3 Communications in Theoretical Physics 359 If the radius of the reflective spherical surface changes, the oscillation changes accordingly. In Fig. 3 we fix the distance d indicated by OC in Fig. 1 between the negative ion and the surface of the sphere and compare the total photodetachment cross sections for the radius of the sphere going from = 5 a.u. to = 5 a.u. We note that in general the results from the two theories differ. But for large, the difference between the results of the two theories becomes smaller, and when = 5 a.u. the two results are almost identical as shown in Fig. 3d. In fact, one can readily show that in the limit of large radius the expression in Eq. 45 is dominated by the first two terms and it approaches σe = σ E + σ ret E 16 2π 2 B 2 E 3/2 3cE b + E sin2 2Ed 2d The formula in Eq. 46 also approaches this limit.. 47 The oscillation in the cross section also depends on the distance d between the negative ion and the spherical surface OC in Fig. 1. In Fig. 4 the photodetachment cross sections are displayed for several values of d with a fixed = 5 a.u. One can see the amplitude of the oscillation decreases and the cross section approaches the cross section in free space σ E when d increases. On the other hand, the difference between the results from the two theories become smaller for smaller d. Fig. 4 Similar to Fig. 1 but here we fix = 5 a.u. and compare the cross sections for several values of d the distance between the negative ion and the surface or OC in Fig. 1. It is noted that for smaller d, the oscillation in Eq. 45 obtained based on closed orbit theory in the present wor and that in Eq. 46 by Haneef et al. [14] agree better. 5 Conclusions We have studied the photodetachment of H near a reflective spherical surface using the standard closed-orbit theory. [15] Closed-orbit theory has been obtained for various systems over the decades with success. [16 19] The present system of H photodetachment near a reflective spherical surface was first proposed by Haneef et al. [14] They applied an imaging method to study the cross section of the present system. The imaging method was originally developed for the photodetachment near a flat reflective surface. [13] Because the problem is interesting, we have studied the photodetachment cross section of it based on the closed orbit theory. We have shown the cross section in Eq. 45 based on standard closed-orbit theory differs from the cross section in Eq. 46 based on an imaging method. However, we have also observed that the difference decreases when either the radius of the reflective surface increases or when the distance between the negative ion and the reflective surface d OC in Fig. 1 decreases. Larger or smaller d correspond to a more flat reflective surface relative to the negative ion. Inspection of the formula in Eq. 46 suggests both the phase and the amplitude are incorrect in the first dominating oscillation term. It is our belief that the imaging method originally

7 36 Communications in Theoretical Physics Vol. 65 developed for photodetachment near a flat reflective surface can not be directly applied to the photodetachment near a reflective surface of sphere as is done. [14] Further theoretical study or experiment of this system should clarify the situation and seems necessary. References [1] Yu.N. Demov, V.D. Kondratovich, and V.N. Ostrovsii, JETP Lett [2] I.I. Fabriant, Sov. Phys. JETP [3] H.C. Bryant, A. Mohagheghi, J.E. Stewart, et al., Phys. Rev. Lett [4] J.E. Stewart, H.C. Bryant, P.G. Harris, et al., Phys. Rev. A [5] A.R.P. Rau and Hin-Yiu Wong, Phys. Rev. A [6] A.R.P. Rau and Chitra Rangan, Phys. Rev. A [7] M.L. Du and J.B. Delos, Phys. Rev. A [8] M.L. Du, Phys. Rev. A [9] M.L. Du, Phys. Rev. A [1] N.D. Gibson, B.J. Davies, and D.J. Larson, Phys. Rev. A [11] N.D. Gibson, B.J. Davies, and D.J. Larson, Phys. Rev. A [12] G. Yang, Y. Zheng, and X. Chi, Phys. Rev. A [13] A. Afaq and M.L. Du, J. Phys. B: At. Mol. Opt. Phys [14] M. Haneef, Iftihar Ahmad, A. Afaq, and A. Rahman, J. Phys. B: At. Mol. Opt. Phys [15] M.L. Du and J.B. Delos, Phys. Rev. Lett ; Phys. Rev. A ; [16] H.J. Zhao and M.L. Du, Phys. Rev. A [17] De-hua Wang, Yi-hao Wang, and Jian-Wei Li, Braz. J. Phys [18] B.C. Yang, J.B. Delos, and M.L. Du, Phys. Rev. A [19] B.C. Yang, J.B. Delos, and M.L. Du, Phys. Rev. A and references therein.