The Effects of Spherical Surface and Laser Polarization on the Photodetachment Cross Section of H

Transcription

1 Commun. Theor. Phys. 59 (2013) Vol. 59, No. 3, March 15, 2013 The Effects of Spherical Surface and Laser Polarization on the Photodetachment Cross Section of H Muhammad Haneef, 1, Suneela Arif, 1 Jehan Akbar, 1 Nasrullah Shah, 2 Muhammad Zahir, 3 Aneela Shamim, 1 and Hameed Ullah 4 1 Lab of Atom and Quantum Interference, Department of Physics, Hazara University, Garden Campus, Mansehra, Pakistan 2 Department of Chemistry, Abdul Wali Khan University, Mardan, Pakistan 3 Department of Statistics, University of Peshawar, Pakistan 4 Department of Chemistry, Hazara University, Garden Campus, Mansehra, Pakistan (Received October 9, 2012; revised manuscript received December 31, 2012) Abstract We report the combined effects of laser polarization and curvature of the spherical surface on the detached electron spectra from H. The Theoretical imaging method is used as a tool of investigation. The photodetachment cross sections for various polarization angles, radii of curvatures and inter ion surface distances are displayed. The analysis of the spectra reveals that the laser polarization angle θ L, curvature of the surface r c and inter ion surface distance d strongly affect oscillations in the spectra. Therefore, a fine control on the laser polarization and that of curvature in the surface can be used to control oscillations in the photodetachment of negative ions. PACS numbers: Gc, f, Ee Key words: negative ions, photodetachment, spherical surface 1 Introduction Negative ions are weakly bound quantum mechanical systems, which have attracted prime interest of researchers in the recent years. [1] As the electrons are bound and they have correlations, therefore the negative ions and their dynamics proved to be a challenge for various theories. [2] The distinctive properties of these ions make them ideal systems for theoretical studies. Binding energies, [3] resonances [4 5] and photodetachment cross sections [6] of such feebly systems (negative ions) have been investigated experimentally and studied theoretically. Overall, the negative ions are attractive quantum mechanical systems with applications in different fields of science. The physics of the photodetachment in the presence of external fields, [7 8] surfaces, [9] cavities, [10] interfaces [11] or potential walls is more important for the understanding of surface dynamics and negative ion structures. [12 14] The occurrence of quantum interference in quantum particles has been of critical significance since the birth of quantum mechanics. Resemblance of quantum interference with that of the young s double slit experiment helps in the interpretation and plays a vital role in the dual nature of quantum objects i.e. photons, electrons, neutrons etc. The photodetachment cross section of hydrogen negative ion (H ) in the vicinity of metallic surface, endures transition from a staircase structure to a smooth structure for high energetic incident photon. [15] It has also been observed that two mirror metallic micro cavities have a stronger enhancement effect on the detached electron spectra of H. Recently, we have investigated the effects of different surfaces on the spectra of mono-atomic and diatomic negative ions. [12 14,16 17] The advancement of photodetachment microscopy has made it possible to observe the spatial distributions of detached electrons on an observing screen. However, there is a need for study of the laser polarization effects on the photodetachement of negative ions near surfaces. Photodetachment of negative ions in externally applied electric field has been experimentally examined. The photodetachment yield is maximum, if the static electric field direction and polarization direction of the laser is collinear. [18 19] However, the oscillations go to zero for mutually perpendicular electric and polarization direction of the laser. [20 21] The photodetachment of negative ions by a laser beam polarized in arbitrary direction were examined in an isolated and/or in the vicinity of electric field, with different models [22 23] (two and three center models) and closed orbit theory. [20] Inspired from the work of Du, [20] instead of electric field and closed orbit theory, here we use spherical surface and theoretical imaging method. We present the combined effects of laser polarization and curvature of the spherical surface on the detached electron spectra from H. Therefore, a fine control on the laser polarization Corresponding author, hanifsaqi85@gmail.com c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 357 and that of curvature in the surface, can be used to control oscillation in the photodetachment of mono-atomic negative ions. To the best of our knowledge, no one has paid any attention to this problem yet. This study is anticipated to be helpful in the experimental investigations using the understudy technique. 2 Theory/Calculation Light interacts with atomic species through some fundamental processes, for example photodetachment of negative ions and photo ionization of neutral or positive ions. These processes occur all the time in nature, especially in plasma, stellar, interstellar nebulae and planetary atmosphere etc. Among the atomic species negative ions have a vital importance because of the correlations/associations of the electrons in the valance orbit. The strong correlations are responsible to bind the additional electron in the negative ions. In the photodetachment of atomic negative ions, these loosely bound electrons in their respective outer shells are detached by the laser photons. For photodetachment of H, the mathematical expression is given as: H + ω = H + e. (1) surface is considered as an ideal reflecting surface, which reflects all electron waves with a phase shift of π, or in other words a surface with an infinite potential barrier as described in Ref. [24]. The surface is oriented in such a fashion that its principal axis is parallel to z-axis. In front of the surface, the H ion is placed at a distance d on z-axis. The surface like-spherical mirror produces image of H behind the surface. Let r c represents the radius of curvature and d is the distance of the image from the surface. In the given schematic d < d. A laser beam polarized in an arbitrary direction (θ L, φ L ) focussed on H. Quantum mechanically, H acts as a source of detached electron waves, propagating away from the source (S) in all directions. Out of all these waves, only two component waves are assumed. The component wave, ψ 1 represents the direct wave, propagating along the trajectory r 1, which makes angle θ 1 (less than 90 ) with z-axis. The component wave, ψ 2 (represents the reflected waves) is supposed to propagate towards the surface with an angle π θ 2. The reflected component (ψ 2 ) appears as it originated from an image (I) behind the surface. Those waves for which, d sin θ 2 > r c, will not strike the surface and therefore not reflected. At this stage, only those spheres are treated for which d sin θ 2 r c. The total wave function of the detached electron at the observation screen is obtained by the linear superposition of ψ 1 and ψ 2. ψ = 1 (ψ 1 + ψ 2 ). (2) 2 The expression for the direct wave of one center system is worked out in Ref. [20]. ψ 1 (r 1, θ 1, φ 1 ) = 4ikB (kb 2 + k2 ) 2 f(θ 1, φ 1 ; θ L, φ L ) exp(ikr 1), (3) r 1 Fig. 1 Schematic picture of the photodetachment of H near a spherical surface is shown. The principal axis of the surface is parallel to z-axis. The H, which acts as a source (S) of the detached electron waves is placed at a distance (d) from the pole of the surface. The surface like-mirror forms image of the H behind the surface at a distance d. A laser polarized in arbitrary direction is focussed on the system, which makes angle θ L with z-axis. The two component waves are indicated by ψ 1 and ψ 2. The component wave ψ 1 is the direct outgoing wave and the component wave ψ 2 appears as produced at image (I). The observing screen is assumed to be placed at a large distance from the system. In Fig. 1 the schematic picture of the photodetachment of H ion, near a spherical surface is shown. The spherical where, k = 2E, E is the energy of detached electron, k b is related to the binding energy E b of H by k b = 2E b and B is normalization constant having value [21] The angular factor f(θ 1, φ 1 ; θ L, φ L ) represents the dependence of the detached-electron wave on the outgoing direction (θ 1, φ 1 ), which is given in Ref. [20]. f(θ 1, φ 1 ; θ L, φ L ) = cosθ 1 cosθ L + sinθ 1 sinθ L cos(φ φ L ). (4) The presence of the spherical surface produces an additional phase shift of π [24] in the reflected waves and can be written as: ψ 2 (r 2, θ 2, φ 2 ) = 4ikB (kb 2 + k2 ) 2 f(π θ 2, φ 2 ; θ L, φ L ) exp(i(kr 2 π)) r 2. (5) The expression for f(π θ 2, φ 2 ; θ L, φ L ) can be obtained from Eq. (3), if θ 1 is replaced by π θ 2. Let (r, θ, φ) be the spherical coordinates of the detached electron wave relative to the origin (i.e. pole of the spherical surface) and

3 358 Communications in Theoretical Physics Vol. 59 C = 4ikB/(kb 2 + k2 ) 2, then the total detached electron wave function can be written as: ψ(r, θ, φ) = C [ f(θ 1, φ 1 ; θ L, φ L ) exp(ikr 1) 2 kr 1 f(π θ 2, φ 2 ; θ L, φ L ) exp(ikr ] 2). (6) kr 2 The total detached electron wave function can be simplified by large distance approximation, and substituting r 1 r d cosθ, r 2 r + d cosθ, in the phase terms and θ 1 θ 2 θ and r 1 r 2 r at all other places. To observe the effects of curvature, spherical mirror equation, d /d = r c /(2d + r c ) is used. The total detached electron wave function with these substitutions reduce to: ψ(r, θ, φ) = C [ ( ikdhcos cosθ cosθ L 1 + e θ) 2 ikdhcos + sinθ sin θ L cos(φ φ L ) (1 e θ)] eik(r d cos θ), (7) r where h = 1 + r c /(2d + r c ) defines effects of curvature present in the surface. The function h depends on inter ion surface distance d and radius of the surface r c. For r c d, h = 1 whereas for r c d, h = 2 and for moderate values of r c and d, h can take any fractional values in the range 1 2. The information about any physical quantity related to the system can be evaluated by treating this wave function accordingly. An analytical expression for the detached electron flux can be derived by substituting Eq. (7) in the following relation. j = i 2 [ψ ψ + ψ ψ]. (8) By substituting the detached electron wave-function in the above relation, we get: j r (r, θ, φ) = kc2 r 2 [cos2 θ cos 2 θ L (1 + cos(kdh cosθ)) + sin 2 θ sin 2 θ L cos 2 (φ φ L ) (1 cos(kdh cosθ))]. (9) A large imaginary spherical hollow shell (Γ), enclosing the system is assumed to investigate the behavior of the total photodetachment cross section. Let the electron flux traverses surface of the hollow shell, then the generalized differential cross section dσ(q)/ ds is defined in the following equation. [25] dσ(q) ds = 2πE ph c j r ˆn, (10) where q is the coordinate on the surface, ˆn is the exterior normal unit vector of the differential area (ds = r 2 sin θdθdφ) at coordinate q and c is the speed of light in a.u. The total cross section of the system can be derived by integration of the differential cross section over the entire surface, σ(q) = (dσ(q)/ds)ds. Substituting Γ the detached electron flux from Eq. (9) in Eq. (10) and integrating, we get a more simplified result. σ(e, θ L, h, d) = σ 0 (E)H(θ L, h, k, d), (11) H(θ L, h, k, d) = [cos 2 θ L (1 + Q(kdh)) Q(kdh) = 3 sin(kdh) (kdh) P(kdh) = 3 cos(kdh) (kdh) 2 + sin 2 θ L (1 + P(kdh))], (12) + 6 cos(kdh) (kdh) 2 6 sin(kdh) (kdh) 3, (13) 3 sin(kdh) (kdh) 3, (14) where σ 0 (E) is non oscillating photodetchment cross section of H in free space. [21] The function, H(θ L, h, k, d) given in Eq. (12) is a generalized modulation function, which modulates the photodetachment cross section. The function H(θ L, h, k, d) defines polarization effects and spherical effects on the photodetachment cross section. It is also evident from Eqs. (11) (14) that cross section depends on the radius of curvature r c, laser polarization angle θ L, photon energy, k = 2(E ph E b ) and inter-ion surface distance, d. For very high photon energy or for very large inter ion surface distance d, Eq. (12) reduces to unity, then Eq. (11) yields cross section of H in the free space. For θ L = 0, Eq. (11) recovers the result of Ref. [12], whereas for θ L = 90 Eq. (11) reduces to the result of Ref. [16]. 3 Results and Discussion A typical spectrum of the photodetachment cross section given in Eq. (11) is presented in Fig. 2. The graph is plotted for fixed values of inter ion surface distance d = 20 a.u. and radius of curvature r c = 100 a.u. For these values of d and r c, the value of h is In order to see the effect of the laser polarization angle, the graph is plotted for several θ L values i.e. θ L = 0, θ L = 54 and θ L = 90. The figure shows that θ L strongly affects oscillations in the cross section. The amplitude of oscillation varies with an increase in θ L. The strongest oscillating structure in the cross section is observed for θ L = 0, which is what reported in Ref. [12]. The peaks of the oscillation fall down for increasing values of θ L. When the laser polarization is perpendicular to the principal axis of the surface i.e. θ L = 90 then Eq. (11) predicts that the oscillation in the cross section disappears, which is confirmed in the figure. The Photodetachment cross section for the laser polarized perpendicular to the principal axis of the surface, obtained earlier [12] is just a special case of Eq. (11) with θ L = 90. For any other polarization angle, the amplitude of the oscillation lies between the above two limiting cases and it can be calculated readily using Eq. (11). It is quite simple to predict, the peaks of the oscillation should fall down by 50% when θ L is turned from 0 to 45. Hence oscillation

4 No. 3 Communications in Theoretical Physics 359 in the photodetachment cross section of H near a spherical surface can easily be reduced to invisible oscillation by changing θ L from 0 to 90. In Fig. 3, the influence of r c on the photodetachment cross section for fixed values of d = 80 a.u. and θ L = 54 is shown. Using the given values of d for r c = 100 a.u., h = 1.57 for r c = 150 a.u., h = 1.48 and for r c = 200 a.u., h = The cross section apparently displays oscillating structures. This can be clearly understood from the figure, as r c varies, the oscillating structure also varies. the figure, for increasing values of d, the amplitude of oscillations fall down, whereas the frequency of oscillations increases. For very large distance (d ), the oscillations completely vanish. This photodetachment cross section reduces to that of H in free space, which indicates the vanishing effect of the surface at large distance. The comparison of Figs. 2 and 3 suggests that θ L affects amplitude of the oscillation more than r c. From the careful analysis of Figs. 2 and 4, it is noted that θ L greatly affects only the amplitude of the oscillations, whereas d simultaneously affects both the frequency and the amplitude of the oscillations. Fig. 2 The photodetachment cross section given in Eq. (11) is plotted for θ L = 0, θ L = 54 and θ L = 90. The radius r c = 100 a.u. and distance d = 20 a.u. are fixed. Solid thick line (θ L = 0 ) represents the cross section of Ref. [12]. Dashed line (θ L = 90 ) represents the cross section of Ref. [16]. Fig. 4 The photodetachment cross section given in Eq. (11) is plotted for several d values. The radius r c and θ L are fixed. The figure shows that oscillations in the photodetachment cross section varies with distance d. Fig. 3 The photodetachment cross section given in Eq. (11) is plotted for several values of radius r c, whereas θ L = 54 and d = 80 a.u. are fixed. The graph shows that the curvature strongly affects the photodetachment cross section. Finally, the photodetachment cross section is displayed in Fig. 4 for several values of inter ion surface distance d, where as r c = 50 a.u. and θ L = 54 is fixed. It is presented that how the inter ion surface distance d affects the cross section. From the figure, it is clear that when the ion is close to the surface, it greatly affects the cross section and induces strong oscillations. It can also be seen in 4 Conclusions In conclusion, theoretical imaging method is used to explore the combined effects of laser polarization and radius of curvature on the photodetachment cross section of H near a spherical surface. A more general expression in terms of θ L and r c is derived for the photodetachment cross section of H. An oscillating structure, for all values of r c and θ L (except for θ L = 90 ), is observed in the photodetachment cross section. The results of Refs. [12] and [16] are the limiting cases of our result, which can be obtained from Eq. (11) for θ L = 0 and θ L = 90, respectively. It should be noted that the amplitude of oscillation varies with angle θ L, however both the amplitude and frequency change with the increase in inter ion-surface distance. Our thought experiment clearly demonstrates that laser polarization angle, curvature and inter ion surface distance, play an essential role in the control of oscillation induced in the photodetachment of negative ions. Moreover, it could broaden the road to new investigations of photodetachment dynamics in the presence of both surfaces and fields.

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