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Byline: Sadia Ashiq and Afaq Ahmad

ABSTRACT: We have investigated the classical limit in the photo-detachment of homo-nuclear diatomic negative ion using two-center-model. The molecular axis of the system is considered parallel to the laser polarized light. In the classical regime the sum of probability densities |ps1|2+|ps2|2 of the detached-electron from individual atoms is equal to the probability density |ps|2from the diatomic molecule. The classical limit calculations for interatomic distance between two coherent sources depend on detached-electron energy as well as photon energy.


The interference pattern in Young's double slit experiment gave an idea of the wave nature of light. But Feynman [9] observed the same effect in double slit experiment with electrons and demonstrated the dual nature of quantum objects. Demkov et el.[10] suggested that negative ions act as localized source of electron waves in the photo-detachment cross section. Near threshold photo-detachment cross section of H[?] displays oscillations in the presence of a static electric field [6] and in the absence of external field they disappear. Afaq and Du [4] have obtained similar oscillations in the photo-detachment of H[?] near a reflecting wall using theoretical imaging method.

The oscillations in photo-detachment cross section for homo-nuclear diatomic anion H2[?] in the absence of external fields with different laser orientations have been well explained using two- center-model [1-3]. H2[?] photo-detachment was also discussed in the presence of external fields [7]. Wang presented an analytical formula for the electron flux distribution of a hetero-nuclear diatomic anion HF[?] using two-center-model [5]. From photo-detached-spectrum, the bond-length of a diatomic anion can be calculated in the thermal equilibrium. By increasing the bond-length of a diatomic anion in a theoretical experiment, the amplitude of oscillations in the photo-detachment cross section changes and goes to zero at some particular distance called classical limit. In this paper, we investigate this classical limit in the photo-detachment H2[?] by taking results from Ref. [2]. The probability densities of detached-electron from the two individual centers of the system and from the system as a whole are calculated.

To vanish interferences, the distance between two atoms is increased hypothetically for fixed photon energy and classical limit is calculated.


In two-center-model of H2[?], each atom act as a coherent source of electron waves and separated by small distance d called bond-length or equilibrium distance. Similar to H[?], it is assumed that there is only one active electron. In the photo-detachment process, these detached-electron p-waves originate from each coherent source and interact with one another at large distance and on the observation plane we observe interferences [8]. In spherical polar coordinates, detached-electron outgoing waves ps1(r1, th1) and ps2(r2, th2) from two coherent sources after the photo-detachment process are given as [2],

Where B = 0.31552 is the normalization constant [7], k is related with detached-electron energy and is given by E = k2/2, kb is related with the binding energy Eb as Eb= kb2/2 and both of these energies are related with the photon energy by the relation Eph = E + Eb

The schematic diagram of H2[?] for the case in which the molecular axis and the laser polarization are along z-axis. r1, r2and r are the distances from center 1, center 2 and origin respectively. The d is the distance between center 1 and center 2 and these two centers act as two coherent sources of detached-electron waves emanating from the system after photo detachment. These electron waves interfere at any point p(x,y) on the screen placed at large distance L.

We use large distance approximation to simplify the problem that is the distance between observation plane and the system is very large as compared to the bond-length of the system. These approximations are


The outgoing detached-electron wave from the diatomic system is the superposition of ps1and ps2 and is given by


The probability densities from the individual atoms and from the diatomic system using Eq. (2) and Eq. (3) can be calculated as

To see interferences on an observation plane, the distance between two coherent sources should be of the order of wavelength of laser light used. If this distance is changed keeping photon energy fixed then the interference effect changes and will vanish at a particular distance. At this particular distance, then probability density of detached electron wave arriving on the screen is equal to the sum of probability densities of detached-electron waves from the two sources [9]. So we can write as

where dc is the distance where the interference in the photo-detachment cross section ceases and is called classical distance or classical limit beyond which a diatomic molecular negative ion dissociates into a neutral and a negative atomic ion and there will be no quantum interference. We consider the simplest case, where x = 0 and y = 0, then

Where dc is the classical distance in atomic unit (a.u.). This final relation shows that it depends on the incident photon energy such that by increasing photon energy it decreases and vice versa. When distance

between two atoms of diatomic anion increases for particular k value of detached-electron energy then interferences begin to disappear. This is according to X-ray diffraction principle in which the wavelength of the X-rays should be of the order of the lattice spacing to see diffraction pattern. It can be seen that dc is inversely proportional to electron energy. At high energies the wavelength of the electron waves decreases and hence the classical distance also decreases. Thus for particular photon energy, the classical distance dc is the maximum distance between the two centers to show interferences. For larger values of n with fixed electron energies, dc increases hence oscillations vanish showing classical behavior which is according to correspondence principle.

So it is not possible to determine exact value of n where the quantum system is completely changed to classical system. Above equation also demonstrate the classical behavior even if the distance is not changed. This can be done with photon energy which has wavelength either very small or very large as compared to the equilibrium distance d0. At very high energy, electron waves are so tiny that the interference pattern become very smooth and it became difficult to distinguish the maxima and minima peaks and hence we see a kind of average, which is the classical curve. This is similar to the Feynmen's double slit experiment with bullets [9].


In conclusions, we have investigated the classical limit in the photo-detachment of homo-nuclear diatomic negative ion using two-center-model calculations and by simply equating probability densities from two coherent sources to the probability density from the diatomic molecule negative ion. This limit depends inversely on the photon energy. This theoretical study is indeed helpful in view of the experiments on photo-detachment microscopy. It may also provide an help to understand quantum chaos experiments.


[1] A. Afaq and M.L. Du, Interferences in photodetachment of a negative molecular ion,Commun.Theor.Phys. 46, 119-122(2006).

[2] A. Afaq and M.L. Du, Interferences in photodetachment of a negative molecular ion model, Commun. Theor. Phys. 50, 1401-1406(2008).

[3] A. Afaq and M.L. Du, Oscillations in the photodetachment cross section of a two center model, J.Phys.B:At.Mol.Opt.Phys 42, (2009).105101

[4] A. Afaq and M. L. Du, A theoretical imaging method for photodetachment of H[?] near a reflecting surface, J.Phys.B:At.Mol.Opt.Phys 40, 1309-1321(2007).

[5] D. H. Wang, Electric flux distribution in photodetachment of heteronuclear diatomic molecular negative ion, Chin. Phys. B 19, 020306(2010).

[6] H. C. Bryant, A. Mohaghaghi and J.E. Stewart, Observations of motional-field-induced ripples in the photodetachment cross section of H [?], Phys. Rev. Lett. 58, 2412(1987).

[7] M.L. Du and J. B. Delos, Photodetachment Of H [?] in an electric field, Phys. Rev. A 38, 5609(1988).

[8] M.L. Du, Oscillations of electron flux in photodetachment of H [?] in an electric field, Phys. Rev. A 40, 4983(1989).

[9] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Addison-Wasley Vol.III, (1964).

[10] Y. N. Demkov, V. D. Kondratovich, and V. N. Ostrovskii, Interference of electrons resulting from photodissociation of an atom in electric field, JETP Lett.34, 403-405(1981).
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Author:Ashiq, Sadia; Ahmad, Afaq
Publication:Science International
Article Type:Report
Geographic Code:9PAKI
Date:Jun 30, 2012

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