Experiment 11: Atomic spectra Last week, we used a laser to produce light with a sharply defined wavelength. The laser was able to do this by taking advantage of very specific transitions between energy levels in the atoms in the laser cavity (helium and neon). It turns out that very small systems (atoms for example) do not obey the laws of classical mechanics but instead are governed by more complicated laws of quantum mechanics. A fundamental result of quantum mechanics is that electrons that are bound to nuclei of atoms can only occupy discrete states with discrete energy levels. This is in stark contrast with classical mechanics in which an electron orbiting a proton can orbit at any radius and therefore have any energy. This means that light absorbed or emitted by an atom must be exactly the right energy to match the allowed energy levels of the electron. Another important modification to our classical picture is the behavior of light itself. If we were to take a light source and turn the intensity down to a very low level, we would find that light was being emitted in discrete packets called photons. Observation of the emitted light (on a very sensitive screen for example) would show that the light was arriving at definite times with discrete units of energy. The situation is analogous to what happens when a steady stream of water from a faucet is reduced to a drip. In the case of the photons, there is a direct relationship between their frequency (or wavelength) hc and their energy given by E h , where c 2.998 108 m / s is the speed of light, and h = 6.626 x 10-34 Js is Planck’s constant. Part 1: Qualitative examination of emission lines When the transitions in atoms produce photons in the visible spectrum they can be seen by the naked eye. Place a hydrogen-filled spectrum tube in the spectrum tube power supply. Turn on the power and observe the color emitted by the gas. To see the distinct lines, hold the diffraction grating close to your eye and look through the grating at the tube. Describe what you are see, keeping in mind that for a diffraction grating, the bright spots are located at d sin m , where d is the separation between the slits. Sketch a rough picture of the lines you see and indicate their color and values of m on a sheet of paper. This will give you an idea of where the lines are when you use the spectroscope in the next section of the lab to measure the wavelength of these lines. Try looking at the spectra of some of the other tubes, helium, argon, etc… through the diffraction grating. Note that the lines are different due to the different electron states involved. Part 2: Analysis of the spectrum of Hydrogen In this section of the lab, we will measure the energies of photons that are emitted by Hydrogen atoms. Hydrogen consists of a single proton and a single electron, the simplest of atoms. In quantum mechanics, the states of the electron in a Hydrogen atom can be calculated very accurately. These discrete states are found to have energies of E1 En 2 , n 2m 2 e 2 2 where E1 ( ) = - 13.60 eV is a constant in terms of fundamental h 2 4 0 parameters, and n is a positive integer. The fundamental parameters that are needed here are the electron mass, m 9.109 10 31 kg , the charge of an electron, e 1.602 1019 C , Planck’s constant, h 6.626 1034 Js , and the permittivity of space, 0 8.854 10 12 C 2 / Jm . More complex atoms require elaborate approximation methods to find the states, but they are still quantized in an analogous way. Transitions between states emit photons with energy equal to the energy lost by the electron in the transition. Use the above formulas to calculate the wavelength emitted by Hydrogen in the n 2 to n 1 transition. Are the transitions to the n 1 level (called the Lyman series) in the visible part of the spectrum? Repeat your calculation for transitions to the n 2 level (Balmer series). See the figure below for the general picture. 1. Read the introduction in the spectrometer instruction manual to understand how it works. Proceed to setup the apparatus as described in the equipment setup section. (You can skip the leveling of the spectrometer). 2. Study the section describing how to read the vernier scale on the spectroscope. (Note that 1 minute is 1/60 of a degree) 3. Continue on with the instructions, making sure the diffraction grating is aligned properly. 4. Measure the diffraction angles and compute the wavelengths for the lines that you can see. Get the most accurate measurement you can achieve by averaging the angles of diffraction on both sides of the centerline. You can also look at the higher order ( m 1) diffraction peaks to get more data. If done carefully, the experiment should yield errors around the 1- 2 % level. 5. Summarize your results with a comparison of measured values and theoretical values for wavelengths in the Balmer series of Hydrogen.
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