Experiment 6 – The Study of Atomic Spectra 1Experiment 6Study of Atomic Spectra1 IntroductionIn this experiment, we will employ a plane diffraction grating to createa spectrometer, which we will use to measure a few atomic emissionlines of hydrogen (H), sodium (Na) and mercury (Hg). We will use theHg lines to calibrate the spectrometer. The goals of this experimentare (1) to use the H spectrum to estimate the Rydberg constant and(2) to determine the energy level diagram of the low lying levels of Naand the fine structure splitting of the 3p configuration.2 Background- see Hecht, Chap. 102.1 Diffraction GratingIn this experiment, we will use a plane diffraction grating, the largeN limit of multiple slits. Diffraction gratings can be made by ruling alarge number of equidistant lines on a glass substrate. Most gratingstoday, however, are replicas (molds of ruled gratings).The maxima produced by a diffraction grating are very sharp. Forparallel light incident on a diffraction grating with angle θi, the posi-tions of the maxima are given bypλ = d (sin θp+ sin θi) , p = ±1, ±2, ±3, . . . (1)where p, d, θpand θiare defined in Fig. 1 with p designating the orderof the spectrum.It is clear from Eq. (1) that for fixed p, the angle θpwill be a functionof the wavelength of the light. Thus, if we illuminate the grating withlight composed of several wavelengths, each wavelength will emergewith a different angle. Note, however, dispersion only occurs for |p| ≥ 1.At zero order, θp= 0, we will see all the wavelengths.Experiment 6 – The Study of Atomic Spectra 2Gratingp=-2p=-1p=0p=1p=2d!iCollimated light incidenton grating travels alongdotted path!pFigure 1: Schematic illustration of diffraction of light by a diffractiongrating. Various orders of the spectrum are shown.The diffraction grating is a particularly simple instrument to use forstudying the components of light produced by an excited atom. Whenthe electrons in an atom are excited (have their energy increased), theyreturn to lower states by emitting light of specific wavelengths knownas spec tral lines. The spectrum of an atom is one of its basic signatures.The existence of many elements in astronomical studies is often estab-lished by measurements of their spectra. Atomic spectroscopy has alsoplayed a fundamental role in the development of quantum mechanics.Hence, the great theoretical and experimental importance of atomicspectroscopy is well established.2.2 Hydrogen AtomIn 1889, Rydberg proposed the following formula to describe the fre-quency of the light emitted whe n a Hydrogen atom makes a transitionbetween energy levels n and n0,˜νn0n=En0− Enhc= R1n2−1n02. (2)In Eq. (2), ˜ν(≡ ν/c) is called the wavenumber – the frequency of thelight divided by the speed of light and expressed in units of cm−1– Enis energy of level n, hc is Planck’s constant times the speed of light andthe constant R is the Rydberg constant.For H, the energy of the nthlevel given byEn= −12e24πεon2ao, (3)Experiment 6 – The Study of Atomic Spectra 3where aois the Bohr radius. From Eqs. (2) and (3) it follows that theRydberg constant for an infinitely massive nucleus is given byR∞=12hce24πεoao=mee44πc~3, (4)while the Rydberg constant for an atom of mass M is given byRM= R∞1 +meMN−1. (5)The energy unit of R is most often cm−1. Combining Eqs. (2) and (4)leads to the following simple result for the H spectrum:En= −hcRH1n2. (6)2.3 Sodium AtomThe structure of Na is more complicated than that of H because ofthe presence of 10 additional electrons. Although these electrons aretightly bound in the 1s, 2s and 2p levels, they produce a significantaffect on the energy of the outer electron in the 3p level. The predom-inant lines in the Na spectrum are from transitions from the excited 3plevel to the 3s ground state. For the 3p level of Na, n = 3, l = 1 ands = 1/2 leading to total angular momentum j = 1/2 and j = 3/2. Theenergy difference of these two levels is known as fine structure, and isdue to the spin-orbit coupling, the interaction of the orbital angular mo-mentum with the spin angular momentum. These yellow lines are thesame as those measured previously with the Michelson interferometer.2.4 Mercury AtomThe structure and spectra of the Hg is very complicated, due to thelarge number of electrons (Z = 80). Mercury is added to fluorescentlights because the large number of different wavelength lines in thespectra yields an overall color close to white. We will use mercury linesto calibrate the spectrometer.3 Experiment3.1 Overview of the SpectrometerThe apparatus used in this experiment is schematically shown in Fig. 2.It consists of a plane transmission diffraction grating (∼ 600 lines/mm)mounted on the rotating table of a spectrometer. The spectrometer alsoconsists of a telescope and a collimator, both of which can be rotatedindependently about the axis of the table. A description of each of thenumbered parts will now be given.Experiment 6 – The Study of Atomic Spectra 4234781465Figure 2: The experimental apparatus used to study atomic spectra.1. Table. Used for mounting the grating.2. Eyepiece fo cus. At the end of the telescope, this eyepiece canbe gently pulled or pushed to get a sharp focus on the cross-hairs without moving their position. This allows the apparatusto allow for different individual viewing preferences without up-setting the focus of the image being viewed.3. Telescope focus. The chrome flange can be rotated to focusthe telescope on the image.4. Leveling Screws. These are used to align the collimator andtelescope vertically . One screw is a clamp and the other is usedfor leveling.5. Collimator stop. Adjusts the length of the collimator. Thecollimator should be set to the correct length when the stop ispushed in against the outer shell.6. Collimator slit Allows adjustment in w idth and orientation ofthe slit.7. Telescope arm clamping and tangent screws. The twoscrews on the side of the spectrometer just below the telescopeeach serve a distinct purpose. The screw projecting radiallyoutward is the clamping screw which should always be loosenedbefore attempting to move the telescope. After the telescopeis nearly in the desired position, the clamping screw should begently tightened. The perpendicular tangent screw then allowsfor very fine adjustment of the telescope position.Experiment 6 – The Study of Atomic Spectra 58. Object table clamping and tangent screws. These adjustthe