INSTITUT D OPTIQUE GRADUATE SCHOOL. Examination of Atomic Physics Y. Sortais, V. Josse

Transcription

1 INSTITUT D OPTIQUE GRADUATE SCHOOL Examination of Atomic Physics Y. Sortais, V. Josse Wednesday 5 March 7 Duration : h Authorized documents : One double-sided A4 sheet with personal notes Scientific calculators are allowed Mobile devices and internet connections are NOT allowed. Questions from the course. Bohr model of the hydrogen atom The Bohr model considers the hydrogen atom as a nucleus, with charge q, and an electron of mass m and charge q that orbits around the nucleus with velocity v. The orbit is circular with a radius r, and is governed by the Coulomb interaction (see Annex for a reminder).. Recall the classical equations of motion of the electron, relating the radius r of the orbit to the velocity v and the total energy E of the electron.. Recall Bohr s postulate on the orbital angular momentum of the electron.. Starting from the results of the previous questions, show that the radius of the electronic orbit and the velocity of the electron can only take on integer values, which we denote as r n and v n (n N ). We denote the total energy of the electron in such an orbit as E n. Express r n, v n, and E n as a function of n, of the ionization energy E I, of the Bohr radius a (equal to the radius of the orbit n = ), and of the velocity v. Express E I, a and v as functions of, m et e = q/ 4πε (the reduced charge of the electron). 4. We write α = e c the fine structure constant. Express v as a function of α and show that the movement of the electron is weakly relativistic.

2 Figure Emission spectrum of the hydrogen atom observed by naked eyes through a prism spectroscope.. Stern-Gerlach experiment Briefly recall, with the help of a schematic if necessary, the principle of the Stern-Gerlach experiment, the observed results, and how these results led to the conclusion that electrons have a spin s = /.. Spectrum of the hydrogen atom In this part of the questions, we ignore the nuclear spin. We recall that E I.6 ev.. What is the degeneracy of the n = orbital, taking into account the electronic spin?. What is the degeneracy of the n = orbital, taking into account the electronic spin?. Calculate numerically the wavelength associated with the transition s p? The emission spectrum of the hydrogen atom is classified into series that carry the name of the physicists who discovered and studied them. We call a series a collection of transitions between a given orbital n and the other orbitals n > n. The table below indicates the names associated to the first series. Series n = n n = n n = n Name of the series Lyman Balmer Paschen 4. Calculate the interval of wavelength over which each of these three series of the hydrogen emission spectrum spreads? 5. Figure shows the lines observed by naked eyes through a prism spectroscope. Work out to which series they belong (there could be more than one series) and the wavelengths associated to these lines. Draw a schematic to clearly indicate the lines and their wavelengths.

3 .4 Rubidium atom Rubidium is an element in the first column of the periodic table (given in Annex). Electronic configuration. How many electrons are there in a rubidium atom? Denote this number as Z. We call the distribution of electrons in different orbitals and sub-orbitals of an atom the electronic configuration. For example, the electronic configuration of the atom of Carbon is s s p.. Recall the rule to fill the electrons into the sub-orbitals (the Klechkowsky Rule).. Deduce the electronic configuration of the rubidium atom. Effective atomic number. We use the term of first ionization energy to refer to the minimal energy needed to strip the outermost electron from an atom. Starting from the Bohr model, how much is the first ionization energy for the rubidium atom if the outermost electron is simply subjected to the Coulomb potential of the nucleus? Give this value in ev. We denote it as ɛ 5.. In real life, the first ionization energy of rubidium is measured to be ɛ 5 = 4.8 ev. Explain why ɛ 5 < ɛ 5.. We replace the Coulomb potential of the nucleus by an effective potential V eff = Z eff e r. Deduce the value of Z eff from the measurement of ɛ Compare the values of Z eff and Z, and comment. Exercise : Selection rules of electric dipole transitions in the presence of the fine structure We want to find out if, in the presence of the fine structure, the transition between the states with n = and the states with n = in a hydrogen atom can be excited by light. We take into account the electronic spin, but not the nuclear spin. The electric field is linearly polarized along the direction Oz. The Hamiltonian of the electric dipole coupling is written as Ŵde = qe Ẑ cos(ωt), where q is the electronic charge (its value is given in Annex), E and ω are the amplitude and the frequency of the electric field, and Ẑ is the position operator along the direction Oz. We consider E to be sufficiently weak so that Ŵde is a perturbation.. The atomic energy levels considered here are s / ; s / ; p / and p /. What does this notation mean? Justify that only these states are possible for the orbitals n = and n =.. Two bases of these states are possible, one with states denoted as n; l; s; m l ; m s, and the other with states denoted as n; l; s; j; m j. Which basis is the eigen-basis of the hydrogen atom in the presence of the fine structure? Briefly recall which term in the Hamiltonian of the fine structure gives rise to the coupled basis.

4 . The 4 states given in question have different energies in the presence of the fine structure. What is the degeneracy of each energy level? 4. Recall (without deriving them) the selection rules of electric dipole transitions in the absence of the fine structure, provided the light field is linearly polarized. 5. Calculate n = ; l = ; s = ; m l = ; m s Ŵde n = ; l ; s = ; m l ; m s for the various possible values of l, m l and m s. Express the results in terms of the integrals A and B given in Annex. 6. Decomposition of the coupled basis and the uncoupled basis : (a) Write down the decomposition of a generic vector n; l; s; j; m j in the basis { n; l; s; m l ; m s } ml ;m s by using the Clebsch-Gordan coefficients. These coefficients are denoted as l; s; m l ; m s j; m j. (b) Give the conditions for the Clebsch-Gordan coefficients to be non-zero. (c) With the help of Annex, write down all the eigen-vectors in the coupled basis n; l; s; j; m j for levels n = et n = in terms of the vectors n; l; s; m l ; m s. We recall the expressions of a few useful spherical harmonics in Annex. 7. Transition s / s / : show that this transition is forbidden, with the help of the expression of Y (θ, ϕ). 8. Transition s / p / Calculate n = ; l = ; s = ; j = ; m j Ŵde n = ; l = ; s = ; j = ; m j for all possible values of m j and m j. Give the results in terms of the integrals A and B given in Annex. 9. Transition s / p / Calculate n = ; l = ; s = ; j = ; m j Ŵde n = ; l = ; s = ; j = ; m j for all possible values of m j and m j.. Deduce the selection rules for a transition between the levels n = et n = in the presence of the fine structure. Compare with the selection rules without taking into account the fine structure. Annex : Fundamental constants Charge of an electron : q =.6 9 C. Mass of an electron : m e = 9. kg. Permittivity of free space : ε 8.84 F.m. Planck constant : h = J.s. Reduced Planck constant : = h π. 4

5 Coulomb interaction The Coulomb interaction between two charges q et q separated by a distance r results in the interaction energy : E Coulomb = q q 4πε r. A few useful spherical harmonics Y (θ, ϕ) = 4π. Y (θ, ϕ) = 8π sin θ eiϕ. Y (θ, ϕ) = 8π sin θ e iϕ. A few integrals A = dr r R (r)r (r) = 8 B = π dϕ π dθ sin θ cos θ Y 8 a (a is the Bohr radius). (θ, ϕ) Y (θ, ϕ) =. Clebsch-Gordan coefficients m l, m s j, m j for l = and s = / For simplicity, we omit l and s in the notation of the Clebsch-Gordan coefficients.,, =,, =,, =,, =,, =,, =,, =,, = 5

6 H Hydrogen.8 Li Lithium 6.94 Na Sodium.99 9 K Potassium Rb Rubidium Cs Cesium Fr Francium. 4 Be Beryllium 9. Periodic Table of the Elements B 6 C 7 N Boron Carbon Nitrogen Oxygen Fluorine Neon O 9 F 8 He Helium 4. Ne Magnesium 4.5 Mg Ca Calcium Sc Scandium Ti Titanium V Vanadium Cr Chromium Mn Manganese Fe Iron Co Cobalt Ni 9 Cu Zn Aluminum Silicon Phosphorus Sulfur Chlorine Argon Al Ga 4 Si Ge Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton P As 6 4 S Se 7 5 Cl Br 8 6 Ar Kr 8 Sr Strontium Y Yttrium Zr Zirconium Nb Niobium Mo Molybdenum Tc Technetium Ru Ruthenium.7 45 Rh Rhodium Pd 47 Ag 48 Cd 49 In 5 Sn Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon Sb 5 Te 5 I 54 Xe 56 Ba Barium Lanthanides Hf Hafnium Ta Tantalum W Tungsten Re Rhenium Os Osmium Ir Iridium Pt 79 Au 8 Hg 8 Tl 8 Pb Platinum Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon [8.98] Bi 84 Po 85 At 86 Rn 88 Ra Radium Actinides 4 Rf Rutherfordium [6] 5 Db Dubnium [6] 6 Sg Seaborgium [66] 7 Bh Bohrium [64] 8 Hs Hassium [69] 9 Mt Meitnerium [68] Ds Rg Cn Uut 4 Fl Darmstadtium Roentgenium Copernicium Ununtrium Flerovium Ununpentium Livermorium Ununseptium Ununoctium [69] [7] [77] unknown [89] unknown [98] unknown unknown 5 Uup 6 Lv 7 Uus 8 Uuo Lanthanide Series 57 Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Gadolinium Terbium Dysprosium Holmium Erbium Thulium Ytterbium Lutetium Actinide Series 89 Actinium Thorium.8 9 Protactinium.6 9 Uranium Neptunium Plutonium Americium Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium 5.8 [54] [6] Alkali Metal Alkaline Earth Transition Metal Basic Metal Semimetal Nonmetal Halogen Noble Gas Lanthanide Actinide 5 Todd Helmenstine sciencenotes.org 6