Demonstration Experiments: The Grating Spectrometer 27 September 2012

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2 The Grating Spectrometer Introduction: A spectrometer is an instrument used to study the spectrum of light. The light source can be anything from a cryogenically cooled crystal to a superhot plasma or a star, and the light might be anything from the longest infrared all the way down to γ-rays. In the case of light the spectrum generally takes one of two forms: hot solids, such as a tungsten lamp filament, emit spectra where the intensity is a slowly varying function of wavelength- a continuous spectra. Hot or electrically excited gasses emit spectra in which the emission is concentrated at a few well defined wavelengths- called emission line spectra, from the appearance of the slit image of the spectrometer. In such a spectrum the ratio of intensities of spectral lines can tell you the temperature of the gas, while the line widths can give you the density of the emitting gas or, in an electrical discharge, the density of the free electrons in the discharge. Spectroscopy is a noninvasive technique and is one of the most powerful tools available to physicists studying plasmas, stars, flames, semiconductors, etc. In its simplest form, a spectrometer uses a lens or mirror to produces a collimated beam of light and disperses it with a grating or a prism. The dispersed spectrum is then focused and viewed with an eyepiece (or recorded with an electronic detector). This experiment has two parts. In the first you will measure the spectrum of hydrogen and use your data to deduce the Rydberg constant. In the second, you will investigate the resolving power of a grating spectrometer. this is a measure of the smallest wavelength difference that can be resolved in a spectrum. These experimental tasks are open endedyou might want to put more effort into exploring systematic effects in the Rydberg measurement at the expense of determining the resolving power, or vice versa. Setting up the spectrometer Figure 1 shows the important parts of the spectrometer. The following sequence of operations should get the instrument properly aligned and focused. 1. Focus the eyepiece on the cross wire. ideally this should be done so that the image is at a convenient viewing distance, around 30cm. The aim is to find a setting that is free of eyestrain. It is better done with both eyes open.. Focus the camera lens so that a distant object is focused on the cross wire. Distant in the context means 5 to 10 meters; in focus means no parallax between the image and the cross wire. 3. Set the telescope (camera lens and eyepiece) so that it is aligned with the axis of the collimator. Position the hydrogen lamp close to the slit, so that it is illuminated. Without changing the camera focus adjust the focus of the collimator until a sharp image of the slit is focused at the cross wire. You might need to adjust the slit to be near its minimum width while carrying out this step. The cross wires are easier to use if set as an rather than as a +. 83

3 Figure 1. The spectrometer. Light from the source passes through a narrow slit and is made parallel by the collimating lens. It strikes the diffraction grating at normal incidence and the diffracted light is focused by the camera lens onto the cross wire X. A magnified image is viewed using the eyepiece. The inset shows the path length difference d sinθ. 4. Check out how the clamp screws and the slow motion drives work, then align the camera and collimator so the cross wires are set on the slit image. ote the reading of the two scales. The main scale is divided into thirds of a degree, the vernier thus reads in arc minutes, 1/60 th of a degree. Check with your demonstrator that you are reading the scales correctly. 5. Move the telescope through precisely 90º and clamp it in position. Put a grating in the holder, being careful not to touch the (fragile!) surface and to get the rulings approximately vertical. 6. Turn the grating table until the grating surface is at 45º to the incident beam from the collimator. you will see a reflected image of the entrance slit in the eyepiece. Adjust the height of this image using the adjustment screws under the grating table. You may need to use a shortened slit. Use the grating drive to set the grating so the slit image is precisely at the cross wire. 7. ote the reading. Rotate the grating through precisely 45º, using the scales, so that it is accurately set at 0º (normal) incidence. 8. Move the telescope out to the way and look into the grating. This should give you an approximate idea of the positions of the spectral lines. Move the telescope around to find the diffracted image of the slit- choosing one at a large angle of diffraction. If the grating rulings are not parallel to the rotation axis of the telescope, this image will be above or below the plane defined by the cross wire. Use the grating adjustment screws (think carefully about which one or ones to use) to rotate the grating in the plane of its surface until the image is at the right height. Check by going to large angle on the other side of the grating normal. The spectrometer is now set up to make accurate, repeatable measurements. It is probably the most accurate instrument you will meet in the lab. In careful hands and using the average of a dozen or so measurements, it is capable of a precision of under three parts in ten thousand, or about ±0.nm for yellow-orange light. 84

4 Spectrometer theory The grating can be considered as a large number of equally spaced very narrow slits, with slit separation d. For light of wavelength λ a principal maximum occurs when the phase difference between neighboring rays is an integer multiple m of π: π phase difference = path difference = π m. λ The path difference is d sinθ, thus π d sinθ = π m, λ which means d sin θ = mλ. (1) m is called the order number. This path difference is shown in the inset in figure 1, figure shows a fringe pattern for =4. Fig.. The diffraction pattern for =4 slits, showing the principal maxima and minima. Resolving power: With slits illuminated, the first minimum occurs at a phase angle π φ = π m +. We find the resolving power by considering the simple criteria that the minimum of one diffraction pattern occurs at the same position as the maximum of another pattern, as shown in figure 3. Light of a second wavelength λ = λ Δλ will produce its m th principal maximum at a slightly different angle θ given by d sinθ = mλ. It will have a minimum at angle θ if 85

5 Fig. 3. Resolution. The principal maximum of one pattern occurs at the same position as the first minimum of the other. π π d θ π m λ sin = +, that is, d sinθ = mλ +. () λ Taking eqs. 1) and ), we find λ mλ = mλ +. We have defined λ λ = Δλ, where Δ λ << λ. This means λ = m. (3) Δλ At this separation the intensity at a point half way between two equal lines is about 0% less than the peak and the two lines can easily be seen to be separate. Experiment 1: The Rydberg constant Transition wavelengths in hydrogen are given by the remarkably simple formula = R, (4) λ n p where n and p are integers. R is the Rydberg constant, which has the measured value m -1. It can be expressed in terms of more fundamental constants as α mec R =, h where m e is the mass of the electron, c the speed of light, h Plank s constant and α the fine structure constant, e α =. 4πε 0 c ote that 1/α It is dimensionless. * * see 86

6 The Balmer series in hydrogen is given by eq. 4 with p = and n = 3, 4, 5 etc. Balmer transition wavelengths are in the visible part of the spectrum. The aim of this part of the experiment is to use the spectrometer to measure the wavelengths via the grating equation (1) and to thereby deduce R, and its uncertainty. Fig 4. Schematic diagram showing two Balmer transitions in a hydrogen atom, p=, n=3 and p=, n=4 For each transition wavelength measure θ as a function of order m, considering both positive and negative values. Before recording data determine the optimal slit size for resolution and intensity. Don t change the slit size once you start taking data. You and your partner should alternate readings of the same line to get a value for the uncertainty in θ. Eq. 1 implies a plot of sin θ vs. m will yield a straight line. You will probably want to use a computer to determine the slope and its uncertainty. You might also want to consider other ways to analyze your data. Recall that d( sin θ) = cosθ dθ. With practice you can make wavelength measurements quite quickly and thus build up a large database. (You will probably want to computerize your analysis.) This means your statistical uncertainty should be good, but there are numerous possibilities for systematic error in this experiment. Some things to consider are the effects of spectrometer misalignment, defocus, lamp position, grating period, air pressure and humidity, background light etc. Some of these you can test experimentally. You might want to reread the discussion at the end of the speed of light script. 87

7 Experiment : Spectrometer resolution Use the spectrometer to test eq. 3. With the sodium source, find an order where you can just resolve the two lines of the yellow doublet. The yellow (or orange, as color can be subjective) sodium wavelengths are 589.0nm and 589.6nm. Gradually reduce the effective number of grating lines by sliding pieces of card just in front of the grating, parallel to the lines. Don t touch the grating with your fingers! Try to determine the number of lines at which you just fail to resolve the doublet. Repeat this measurement several times and calculate the mean and standard error. Compare your result for Δλ derived from eq. 3 to the known value. If you have time measure the separation of the pair of green lines at 569nm or red lines at 616nm. Is the separation equal to that of the orange lines in wavelength units and/or in frequency units? What does this tell you about common levels for these transitions? At 589nm, Δν = GHz. 88

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