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Study on the Energy Level Splitting of the Francium Atom (Fr) in an External Magnetic Field

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Study on the Energy Level Splitting of the Francium Atom (Fr) in an External Magnetic Field International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 3 Issue 5, August 2019 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470 Study on the Energy Level Splitting of the Francium Atom (Fr) in an External Magnetic Field Win Moe Thant Lecturer, Department of Physics, Shwebo University, Shwebo, Myanmar How to cite this paper: Win Moe Thant ABSTRACT "Study on the Energy Level Splitting of the In This paper the normal Francium atom is considered with the use of non- Francium Atom (Fr) in an External relativistic quantum mechanical approach and the Bohr's atomic model. We Magnetic Field" study the energy shift values of a Francium atom by the external magnetic Published in fields effect; we considered the two cases; that effect without external International magnetic fields and those of with the external magnetic fields. Journal of Trend in KEYWORDS: energy shift, energy level splitting, magnetic field effect on the atom Scientific Research and Development INTRODUCTION (ijtsrd), ISSN: 2456- IJTSRD26730 Atom is a Greek word which means “indivisible”. The Greek believed that matter 6470, Volume-3 | can be broken down into very small invisible particles called atom. Greek Issue-5, August 2019, pp.1714-1720, philosophers such as Democritus and John Dalton put forward the concept of https://doi.org/10.31142/ijtsrd26730 the atom, Democritus explained the nature of matter; He also proposed that all substances are made up of matter. He stated that atoms are constantly moving Copyright © 2019 by author(s) and invisible, minuscule particles that are different in shape, size, and temperature International Journal of Trend in Scientific and cannot be destroyed. Research and Development Journal. This Later in the year 1808, John Dalton proposed the atomic theory and explained is an Open Access the law of chemical combination. By the end of the 18th and the early 20th article distributed centuries, many scientists such as J. J Thomson, Gold Stein, Rutherford, Bohr under the terms of among other developed and proposed several concept on the “atom” the Creative Commons Attribution License (CC BY 4.0) J. J Thomson was the first and one of the many scientists who proposed models (http://creativecommons.org/licenses/by for the structure of an atom. J. J Thomson discovered negatively charged /4.0) particles by cathode ray tube experiment in the year 1897. The particles were named “electrons”. In the year 1904, J. J Thomson suggested a model of the atom projection mj where along mj J, J 1, J 2,..., J as a sphere of positive matter in which electrons as the z-axis. positioned by electrostatic force. J. J. Thomson believed When an atom with angular momentum J is in the absence of electrons to be two-thousand times lighter than a proton. He external fields and when the atom is not interacting with assumed that an atom is composed of a cloud of negative other particles, all the (2J+1) sub-levels are completely charge in a sphere of positive charge. The positive and degenerate with respect to energy. When the external negative charge is equal in magnitude and therefore an atom magnetic field is applied, the energy levels are split into the has no charge as a whole and is electrically neutral. possible numbers of (2J+1) and this is called the Zeeman In each element, the ground state of the atom contains a Effect. fixed and equal number of protons and electrons. An atom characterized by an atomic number Z has a nucleus which The Ground State Energy of Single-Electron Hydrogen carries almost the entire mass of the atom and a positive Atom electric charge (Ze) surrounded by Z electrons each of which In 1913, Bohr proposed his quantized shell model of the carries a negative charge (-e). The orbital motion of the atom to explain how electrons can have stable orbits around electron is characterized by an orbital momentum quantum the nucleus. The motion of the electrons in the Rutherford number l and its projection mll , 1, l 2,..., l model was unstable because according to classical j mechanics and electromagnetic theory; thus, the electron with reference to the axis of quantization, which is usually would lose energy and spiral into the nucleus. The energy of chosen as the z-axis. The electron has also an intrinsic spin an electron depends on the size of the orbit and is lower for quantum numbers 1/ 2 , its projection ms 1/ 2, 1/ 2 smaller orbits. The atom will be completely stable in the with reference to the same axis. Thus the total number of state with smallest orbit, since there is no orbit of lower electrons with the same angular momentum l are 2 energy into which the electron can jump. Bohr assumed that an electron in an atom moves in an orbit about the nucleus (2 l +1) which are filled, following the Pauli's exclusion under the influence of the electrostatic attraction of the principle. The orbital and spin angular momentum of the nucleus. Circular or elliptical orbits are allowed by classical electron add up quantum mechanically to yield a mechanics, and Bohr selected to consider circular orbits for totalangular momentum quantum number J and its simplicity. He then, postulated that instead of the infinity of orbits which are possible in classical mechanics, only a @ IJTSRD | Unique Paper ID – IJTSRD26730 | Volume – 3 | Issue – 5 | July - August 2019 Page 1714 International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 certain set of stable orbits, which he called stationary states are allowed. As a result, atom can only exist in certain Consider an electron of charge e and mass m, it will be taken allowed energy levels with energies Ea,Eb, Ec, . to be moving with a velocity v in a circular orbit of radius r in If the orbits are quantized, the amount of energy absorbed or an atom. emitted is also quantized, producing discrete spectra. Photon The Coulomb force between a stationary nucleus with the absorption and emission are among the primary methods of nucleus of charge +Ze and orbiting electron with charge ‘e’ is transferring energy into and out of the atoms. The energies Ze2 F K (1) of the photons are quantized, and their energy is explained 2 r as being equal to the change in energy of electron when it moves from one orbit to another. From the Newton’s second law of motion, the electron mass 2 Bohr was clever enough to find a way, to calculate the m times the centripetal acceleration v called centripetal electron orbital energies in hydrogen atom. This was an r important first step that has been improved upon, but it is mv2 well worth repeating here, because it does correctly describe force Fc , It can be expressed as follow; many characteristic of hydrogen atom in the next portion. r Ze2 mv 2 The simplest example of the Bohr model is for the hydrogen K atom (Z=1) or for a hydrogen-like ion (Z >1), in which a 2 r r negatively-charged electron orbits a small positively-charged Ze2 nucleus; electromagnetic energy will be absorbed or emitted v K (2) if an electron moves from one orbit to another. Only certain mvr electron orbits are permitted. N. Bohr proposed that the angular momentum (L) of the Bohr explained how electrons could jump from one orbit to h another only by emitting or absorbing energy. When the allowed orbit takes the integral values of h . electron jumps to a larger orbit, it must absorb a equal in 2 energy to the difference in orbits. From Bohr’s postulate (3), the angular momentum (m v r) of the allowed orbits takes the integral values of h/2π. Hence, Electrons in an atom are held their orbits by the forces of Bohr's postulate that the angular momentum is given by electrical attraction acting between them and the nucleus, the higher is its energy, although with an increase in this L= m v r = n ħ where n=1, 2, 3, . (3) distance its bound with the nucleus naturally, weakens, However, no two electrons can be in the same energy state By using the above equations (2) and (3), we get the value of and therefore electrons are arranged in layer in the shell. v and r as There may be only strictly limited number of electrons in 2 2 2 K Ze each level, 2 in the first innermost shell, 8 in the second and v (4) nh so on these shells are denoted by the symbols K, L, M, N respectively. Substituting equation (4) into the equation (3), we get the Each shell can contain only a fixed number of electrons. The radius of the orbit first shell can hold up to two electrons, the second shell can (nh )2 hold up to eight (2+6) electrons, the third shell can hold up r . mKZe2 to 18 (2+6+10) and so on. The general formula is that the nth shell can in principle hold up to 2n2 electrons. (h )2 o where, 0.529A called Bohr’s radius. Km e2 For the case of the one electron atom, Bohr's was able to modify the classical model to obtain the quantization of 0.529 n2 o For nth state, r A .(n=1, 2, 3, 4, …) (5) energy levels by making the additional postulate that the n Z angular momentum of the electron moving in circular orbit nh can only take one of the value L nh , when n is a From which the total energy E0 of the electron in the 2 Hydrogen atom is positive integer, n= 1, 2, 3, 4,… and the common occurring E TV h/2π is conventionally denoted by ħ.
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