Muhammad Usman Ghani1, M. Imran Jamil1, Afaq Ahmad2,  Saad Tariq2

1Department of Physics and Astronomy, Texas Tech University, Lubbock-79415, USA.

2Department of Physics, University of Management and Technology, Lahore-5459, Pakistan.

2Centre of Excellence in Solid State Physics, University of the Punjab, Lahore-54590, Pakistan.

Received: 28-March-2023 / Revised and Accepted: 09-May-2023 / Published On-Line: 15-May-2023


Abstract: The Photodetachment Microscopy experiment was first carried out in the presence of an electric field by Blondel et al in 1996 for Bromine negative ion. It measures the spatial distribution of ejected electrons on the detector screen which is a direct view of the spatial structure of the wave function of an atomic electron in the form of a ring pattern. From a semi-classical point of view, this ring pattern is formed because of the interference between two electron waves; one is direct while the other is reflected from an electric field. Following Blondel’s photodetachment microscopy experiment, a formula that displays the Newton Rings is derived using a theoretical imaging technique or hydrogen negative ion near a plane interface. The interface means an elastic plane in the vicinity of the source of photoelectrons. The direct and reflected electron waves in this formula generate quantum interference in the form of Newton Rings. It is found that the number of rings increases as we increase the photon energy of the laser light.

This finding is in accordance with the very well-known Einstein photoelectric effect which finally provides help to find the electron affinity of the hydrogen negative ion very accurately.

Keywords: Photodetachment Microscopy; Electric field; Spatial distribution; Ejected electron; Photon energy


A negative ion contains an extra electron and it was first studied by J.J. Thomson as a quantum mechanically weak system [1]. The study of the negative ion is of great interest in the modern era[2]. The process of electron correlation is responsible for a key role of negative ions in the critical test of various theories such as closed orbit theory, Green’s function technique, semi-classical theory quantum mechanical technique and theoretical imaging method [2-6]. By the virtue of characteristic properties of such negative ions, an ideal system for the theoretical framework is built. Various aspects of such weak systems have been under consideration including the binding energy, photodetachment cross section and resonance [7].

A solar spectrum was being studied in early 1939 by Wildt and he pointed out that hydrogen is one of the major components of the Sun’s matter [8]. From the observation of emission spectrum of celestial objects it comes to the knowledge that there must be present both the neutral and positively charged atoms in them [9]. The formation of negative ions or molecules is the result of continuous collisions of free electrons and the neutral atoms[6]. On the same footings the probability of formation of hydrogen negative ions becomes more because of the linking of single free electron with the neutral hydrogen atom. The solar infrared is characterized by the occurrence of black absorption spectrum. In order to make it sure, Wildt explained the infrared absorption and elaborated that hydrogen negative ions were present in the photosphere of the Sun [10]. The work of Chandrasekhar [11] reinforced the work of Wildt, with precise measurement of binding energy of hydrogen negative ions.

In this research work, we studied the process of photodetachment of hydrogen negative ion in the vicinity of a plane hard wall[7] with the help of theoretical imaging method (TIM). We derived an analytical formula for the calculation of electron affinity of ion. The expectations related to plane hard wall are in agreement with both theoretical and experimental studies in micro junctions in past decades [12-15]. A plane hard wall is different from metallic wall. In this study the electrons followed a straight path with the exception of those being reflected from plane wall[16, 17]. A plane hard wall is characterized by the phase loss of π, when a detached electron is reflected from this wall. The detached electron has two components from  for the observer being at large distance[18]. The two components being observed, one propagating directly from  and the other is approaching indirectly after being reflected from plane wall[19]. A relation for detached electron flux was calculated by Afaq et. al.[20] and this relation is used to obtain the interference pattern in the form of ring pattern. This ring pattern helps us to formulate a simple relation between the number of rings and the photon energy. Finally this analytical formula for the calculation electron affinity of  ion is in agreement with experimental result [21].

We organized this paper as follows: in second section of the study, we elaborated the wave function of detached electrons with the help of TIM, proceeded by the process of drawing the results of interference pattern with the help of the relation for electron flux, such interference pattern was also employed for drawing newton rings. In the next section we derived a linear relationship between energy of incident photons and square of newton rings. Conclusions are given in section five.

Photodetachment microscope of  near a plane wall

The study of detachment of hydrogen negative ion near a reflecting surface / wall with the help of theoretical imaging method is considered. This method employs the analytical approach to derive the formulae for total and differential cross sections of the process[20]. A hard wall case is to be considered in this regard. The term hard wall refers to a wall that is responsible for the production of a phase loss of .

Following figure illustrates the process of photodetachment of hydrogen negative ion near a reflecting surface:

Figure I: The schematic diagram for photodetachment microscope of H near a wall.

interference pattern is shown in the form of Newton rings. It can be seen from the figure below that number of Newton rings increases as we increase the incident photon energy.

Figure II: The Newton rings using Eq. (1). We take, d = 100 a.u. L = 1000 a. u. and the value of E taken as (a) E=1eV, (b) E=2eV, (c) E=3eV, (d) E=4eV.

The electron affinity of hydrogen negative ion can be calculated by deriving a relation between number of fringes and photon energy. For this purpose, we use the Eq. (1) for the condition of bright fringe or peak. Bright fringe or

This indicates that when the value of n is zero, photon energy is equal to the binding energy i.e., electron affinity. Following table shows the number of rings for different values of photon energy.

Table I:  Relationship between incident photon energy and Newton rings.

Sr. No. Photon energy “E­ph” (eV) Number of rings “n”
1 1 4
2 2 7
3 3 9
4 4 11

Results and Discussions

The results have been obtained numerically regarding various incident photon energies along with fixing other parameters as constants. It has been observed that the Newton rings pattern obtained are similar to those obtained through experimental setup. With the increase in the incident photon energy the number of fringes also increase as shown in Fig. III below.

We have plotted below the above table-1 between n2 and E­ph which confirms the straight-line equation and extrapolated the graph when n is equal to zero the electron affinity of hydrogen negative ion is found to be 0.754 eV. This is in agreement with the experimental data[21].

  Figure III: Relationship between n2 and  Eph


The method of photo-detachment microscopy has been used for calculating electron affinity of hydrogen negative ion near a hard wall. The effects of hard wall on the detached electron spectra of a hydrogen negative ion have been found with the help of theoretical imaging method, when illuminated by a polarized beam. Such waves are used for the sake of calculation of detached electron flux at a large distance from the hydrogen negative ion which results in the form of Newton rings pattern on the screen used for observation. The concept of photo detachment is similar to that of photo electric effect.

There is a linear relationship between the square of the number of rings and the energy of the photons. If we extend the line backwards to the zero of the vertical-axis the value of electron affinity of is 0.754 eV which is nearly equal to the experimental value [21].


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