Atomic Spectra. Eric Reichwein David Steinberg Department of Physics University of California, Santa Cruz. August 30, 2012

Transcription

1 Atomic Spectra Eric Reichwein David Steinberg Department of Physics University of California, Santa Cruz August 30, 0 Abstract To observe helium spectral lines we used a spectrometer. From a table of known wavelengths of the helium spectral lines we were able to determine the spacing between the slits of the diffraction grating(diffraction constant). Knowing the diffraction constant we then analyzed the hydrogen spectral lines of the Balmer series to determine the value of the Rydberg constant. We then used the helium and neon spectral lines and a Helium-Neon LASER (HNL) to determine what wavelengths of light the HNL produced. We concluded that the spectral lines produced by the HNL were in accord with the spectral lines of neon.

2 Contents Introduction The Diffraction Constant 3 3 The Rydberg Constant 5 3. Hydrogen Spectrum Rydberg Constant from Hydrogen Spectrum Rydberg Constant with Error Propagation The Helium-Neon LASER 8 5 Conclusion 0 A Error Analysis A. Propagation of Error: θ sin (θ) A. Propagation of Error: sin(θ) R H Introduction We are using the basic concepts of diffraction grating and introductory quantum theory to determine different aspects of spectroscopy. The main equation being used is the diffraction grating found in most introductory physics textbooks dsinθ = mλ () The spectrometer used (and can be seen in the figure below) was manufactured by Precision Tool and Instrument Company (PTI). Before observing the spectral lines we calibrated and focused the spectrometer by the procedure listed in the laboratory manual [].

3 Figure : The experimental apparatus. A: The diffraction grating. B: The collimator. C: Helium/Hydrogen light source. D: Live feed from camera. E: Telescope. F: Camera adapter to telescope. G: Camera H: Angle Measurement To obtain pictures of the spectral lines we attached a Nikon SLR camera. To attach the camera we used a camera -telescope mount. The mount base supports a camera with attachment screw.75 inches back from telescope connector, however, our camera had attachment screw 3.5 inches back from lens. To overcome this problem we used a piece of wood to broaden the base mount and then attached the camera to the modified base using masking tape. The Diffraction Constant The first step in determining the Rydberg constant is determining the diffraction constant, d. Since the helium spectral lines are assumed to be known and correct we can use them along with equation to determine the diffraction constant. All we need is the angle of diffraction which we can measure using the PTI Spectrometer. There is a vernier scale with precision of a minute ( of a degree) on top of a raw degree scale with precision of half a degree. minute = 60 As we observed the the spectral lines we aligned it first by eye. Then attached the camera and slowly adjusted the exposure, f-stop, and white balance to make the measurements of the diffraction angle as accurate as possible. Then we took a photo of the vernier reading and zoomed in to get as exact minute reading as possible. We concluded that our error on our measurement was at most ±minutes. 3

4 Color λ known φ (Degrees) θ ( 90 φ ) sin(θ) d Blue 447nm m Green 50nm m Orange 588nm m Red 668nm m Blue 447nm m Green 50nm m Orange 588nm m Red 668nm m Table : Using raw data and equation to determine the diffraction constant. We took measurements of the diffraction angle on both sides (i.e. m = ±) and then averaged them using θ = (θ m= + θ m= ) Then propagated the error through equation using the derivative method as seen in appendix A. From table and table we see there is a diffraction constant standard deviation of approximately 5% of the average observed value and a diffraction angle standard deviation of about 8% of the average observed value. λ known θ sin(θ) d σ θ σ d 0.447µm m m 0.50µm m m 0.588µm m m 0.668µm m m Table : Error Propagation for Diffraction Constant The last two columns are the standard deviation which were derived by the differentiation method of error propagation, please refer to appendix A for details. The figure below shows the only visible helium spectrum which is not completely to scale. Figure : Helium spectrum not to scale. The right edge is considered m = 0. From right to left is increasing diffraction angle. 4

5 3 The Rydberg Constant Since we have determined the diffraction constant, d =.75µm, we can deduce what the Rydberg constant is from the unknown hydrogen spectrum. Quantum theory tells us that the energy of electron of a hydrogen atom is given by E n = µe4 8ɛ 0h n Where µ is the reduced mass, e is the fundamental charge, ɛ 0 is the permittivity of free space, h is Planck s constant, and n is the principle quantum number. Since the light we observe is due to the electron transitioning from a higher energy excited state down to a lower energy state we can see that the light has energy given by E n E n = µe4 8ɛ 0h ( n n By using the relation between energy and wavelength, E = hc, we can relate the energy released λ by a electron transitioning between two states to the wavelength of the produced light. It follows that ( E n E n = µe4 ) = () hc 8ɛ 0h 3 c n n λ Where we can group the physical constants into one constant referred to as the Rydberg constant, R H, and is defined as 3. Hydrogen Spectrum R H = µe4 8ɛ 0h 3 c = 0, 973, (55)m (3) Since we have determined the diffraction constant, d, in the previous section and we know the angles of diffraction for m = ± we can find the most probable diffraction angle, θ. Color φ (Degrees) θ ( 90 φ ) θ (Degrees) cyan violet red violet Table 3: Determining average diffraction angle with m = ± cases. By averaging the diffraction angle for the m = case and the m = case we have found the most probable angle of diffraction, θ, just as we did in section. By equation we deduced the wavelengths, and consequently the energy, of the light we observed was. From the energy of 5 )

6 the light we can determine from which two states the electron transitioned to and from. Since a transition from any state to the n = would produce light of frequency higher than the human eye can observe we can automatically exclude these transitions. Any transition to a state higher than n = 3 would produce lower energy light that the human eye can t detect. Figure 3: The hydrogen spectrum. Scale is slightly distorted. Used angle measurements to determine how many degrees a pixel was then moved lines to approximate positions. 3. Rydberg Constant from Hydrogen Spectrum We know that the spectral lines we observe are part of the Balmer Series hence each transition is from some initial state to the n =. To find what energy levels the electron transitions from without using the theoretical Rydberg constant we must use the ratio of wavelengths with n = RH n n λb nb λa a = λa na RH n n λb b Now the ratio is independent of the Rydberg constant and we can now see what ratio of quantum state numbers corresponds to which ratio of wavelengths. From the table below we can match ratios of wavelengths to the corresponding quantum number ratios. For two ratios (one of wavelengths and one of quantum numbers) that match the wavelength of light used in the numerator will have an initial transition state to the quantum number used in the numerator of the corresponding ratio, and similarly with the denominators. Since there is more than four ratios we can confirm that these are the correct quantum state transitions with multiple ratio equivalencies. 6