Physics 2020 Lab: Atom and Hydrogen Spectrum page 1 of 9Lab: The Atom and the Hydrogen SpectrumINTRODUCTION & BACKGROUND:This lab will allow us to look inside an atom. Using tools we have developed over thesemester and a simple model of the atom, we will be able to determine which atomsemit which colors of light. Each atom has a unique optical fingerprint and we will usethat concept to study the hydrogen atom, and then determine the identity of amystery atom. This approach is similar to how astronomers determine what stars aremade of.The goals of this lab are to learn how to use a diffraction grating, to understand theBohr model of the atom and electron orbital structure, energy levels in an atom, andto learn how the orbital structure relates to the spectrum of emitted light.Diffraction grating review: Recall our last lab working with two-slit interference.We observed that the separation of interference maxima on a screen depended onthe wavelength of light and the distance d between the slits. Now instead of usingjust two slits, we will use thousands. The effect is the same as with two slits, exceptthe peaks get much sharper and further apart on the screen (due to smaller slitseparation d). So we observe sharp bands of light corresponding to specificwavelengths.A diffraction grating is simply a piece of glass or plastic which has a series of veryfine scratches or grooves cut in its surface. The grooves are perfectly straight andparallel and are equally spaced so that there are a fixed number of grooves permillimeter, typically around 500 grooves/mm. A grating behaves essentially like a multi-slit aperture, that is, a mask with manyclosely spaced slits. If the number of grooves per length is n (n grooves per cm),then the separation between adjacent slits is d = 1/n (cm per line or simply, cm).Consider what happens when a beam of monochromatic (single-wavelength) lightstrikes a grating at normal incidence, as shown below. Each groove or slit scattersthe light in all forward directions. However, in only certain directions will the lightscattered from different grooves interfere constructively, producing a strong beam. University of Colorado at Boulder, Department of Physicsdincident beamscattered lightgratingdd sin wavelength to observerPhysics 2020 Lab: Atom and Hydrogen Spectrum page 2 of 9The diagram on the right shows two light rays emerging from adjacent slits in thegrating and heading toward an observer (or a point on a screen) at an angle fromthe normal (perpendicular) direction. In traveling to the observer, the ray from thelower slit has to travel an extra path distance; this path difference is p.d. = d sin().The two rays will interfere constructively only if the path difference is an integernumber of wavelengths:(1) d msin , (m = 0, 1, 2, 3, …) Constructive interferencewhere is the wavelength of the light and m is any integer. At only these specialangles, corresponding to integer m's, will the rays from all the slits interfereconstructively, producing a bright beam in that direction. In any other direction, therays from the various slits interfere destructively and produce no light intensity. Theinteger m is called the order of the diffraction.An incident light beam made of a several distinct wavelengths will be split by thegrating into its component wavelengths, with each separate wavelength heading indifferent directions, determined by the condition d msin . In this way, the variouswavelengths can be determined by measuring the angles.Bohr model of the hydrogen atom: In the 19th century, it was known that hydrogengas, when made to glow in an electrical discharge tube, emitted light at fourparticular visible wavelengths. In 1885, a Swiss high school teacher named Balmerdiscovered that the four wavelengths, here labeled i (where i = 1,2,3,4) preciselyobeyed a curious mathematical relation:(2)1 1212 2iiRn where R is a constant, and ni = 3, 4, 5, 6. The four wavelengths (or "lines") werehenceforth called the Balmer lines of hydrogen. Why hydrogen emitted only thosevisible wavelengths and why the wavelengths obeyed the Balmer formula was acomplete mystery. The mystery was solved in 1913 by the Danish physicist Niels Bohr. According to theBohr model, the electron orbiting the proton in a hydrogen atom can only exist incertain orbital states labeled with a quantum number n (n=1, 2, 3, 4...). When theelectron is in orbit n, the total energy of the hydrogen atom is given by the formula:(3) n2 21 13.6eVE R h cn n= - ��� = -,University of Colorado at Boulder, Department of PhysicsE(eV)n = 1n = 2n = 3n = 465–13.6–3.4–1.5–0.85Physics 2020 Lab: Atom and Hydrogen Spectrum page 3 of 9where c is the speed of light, h is a constant (Plank's constant), and R is a numberpredicted by the Bohr model to be R = 1.09737 107 m-1. The different energies Encorrespond to different orbital states of the electron. Smaller-radius orbitscorrespond to lower values of n and lower, more negative, energies. The n=1 stateis the lowest possible energy state and is called the ground state. When an electron makes a transition from an initial state of higher energy Ei to afinal state of lower energy Ef , the atom emits a photon of energy (4)E hf hcE Ei f . Useful number: hc = 1240 eVnmHere we have used the expression for the energy of a single photon: E = hf, where his Planck's constant and f is the frequency of the light. From equations (3) and (4),the wavelength of the emitted photon is related to the initial and final quantumnumbers like so:(5)hcE E R hcn nRn ni fi f f i 1 1 1 1 12 2 2 2,. This is none other than Balmer'sformula! Transitions between any pairof states such that ni > nf produces aphoton; however, only those transitionswith nf = 2 and ni = 3, 4, 5, or 6,happen to produce photons in thevisible range of wavelengths.University of Colorado at Boulder, Department of PhysicsEnergy level diagram for Hydrogen. Longer linesindicate transitions that emit higher energy photons.NOTE: this figure is not a picture of the electron orbits of the Bohr model atom; rather it
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