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CONCAVE DIFFRACTION GRATING SPECTROGRAPH
Revised May 16, 2006
Diffraction gratings are widely used for high resolution spectral studies. The grating available in the lab is a holographically-produced, concave grating of 600 lines/mm and radius R = 1075 mm. It is placed in a Rowland mounting. In such a mount, the grating is constrained to be on a circle of diameter equal to the radius of curvature of the grating (the Rowland circle) on which the entrance slit (the slit through which light is incident on the grating) is also located. If the entrance slit and the grating are on the circle, the slit image will be in focus on the same circle. Concave gratings are often used in the ultraviolet portion of the spectrum where their focusing properties eliminate the need for UV-transmitting lenses. In this experiment, strong lines from the visible part of the spectrum of the elements mercury and cadmium are used to calibrate the spectrograph. The spectrograph is then used to measure the wavelengths of the lines in the visible spectrum of hydrogen (Balmer series) from which a value of the Rydberg constant for hydrogen can be obtained. The lamp used also contains the isotope of hydogen, deuterium, so the spectrum of deuterium can be obtained as well; for this part, we use a CCD array detector which allows a more accurate measurement to be made of the separation betwen similar lines of the two isotopes, and also allows the experimenter to make a quantitative measurement of the resolving power of the spectrograph.
REFERENCES
- Hecht, Optics (4th ed.), pp. 476–481 (general properties).
- Thorne, Anne P.,Spectrophysics, 2nd ed., Chap. 6, pp. 144–170 (types of diffraction gratings and mounts, focusing property of concave gratings).
- Davis, Sumner P., Diffraction Grating Spectrographs (practical information on the alignment and aux- iliary optics for diffraction gratings).
- Tipler, Modern Physics, pp. 144-151 (reduced mass correction).
APPARATUS
The apparatus is pictured in the Fig. 1 with the Rowland Circle shown as a dashed line. The radius S of the grating is 1075 mm (± 2 mm), the width of the grating is 25 mm, and the grating constant is 600 lines/mm (error unknown). The grating is blazed to give maximum intensity in the first order. The grating mounting has been carefully adjusted and should not be disturbed. A movable carriage holds a viewing microscope (and CCD array), which can be focused on the slit image. A set of cross-hairs in the microscope allows for precise location of the spectral line under observation.
The distance x from the slit to the slit image can be measured on a wooden meter stick + associated vernier to a precision of about 0.1 mm. For a Rowland mounting with the angle of reflection (i.e., the angle between the grating normal and the axis of the detector) θr = 0◦, the grating formula becomes d sin θ = d x S = mλ, where d is the line-to-line separation on the grating, θ is the angle of incidence of the light on the grating, x is the slit to image distance, S is the grating-to-image distance, m is the order of the spectrum (m = 1, 2 , etc.), and λ is the wavelength of the spectral line.
CALIBRATION OF THE SPECTROGRAPH
The wavelength of a spectral line can be determined from a measurement of x and use of the grating equation, but this method has significant limitations. The grating equation is only as precise as the values of d and S, and it depends on positioning the entrance slit exactly above the zero of the meter stick scale, a result difficult to achieve in practice.
Figure 1: The concave grating spectrograph. The carriages with the grating and detector (microscope or CCD array) move perpendicular to each other along tracks, so that the distance between them S stays constant, even though the angle θ changes. The image of the slit is focused by the grating, which is slightly concave, on the detector. Because S stays constant, the spectrograph stays in focus over the range of angles θ. The grating, detector and entrance slit lie on an imaginary circle of fixed diameter called the “Rowland circle”.
- Compute the positions of the calibration lines in the first order (i.e., m = 1) from the grating equation. These computed positions need not be especially precise, but they will help in identifying the correct calibration lines.
- Carefully record the positions of the calibration lines in first order. Plot each point carefully on the graph paper provided after you measure a line. Plot x (in cm) on the vertical and λ (in nm) on the horizontal. Choose the scales for x and λ so that the data nearly fill the page. Visually verify that a spectral line of the right color exists at that value of x. Use a straight edge to draw a line through the data points. If you have correctly identified each line and measured its position carefully, the plotted points will lie on a straight line as closely as you can plot them. If not, remeasure the lines that are off.
When you have finished recording the calibration lines, look at several of them (green & yellow) in the second order. Notice how much dimmer these lines are than their first order counterparts. This is the result of the grating being blazed so that maximum intensity is reflected in the first order. After looking at the second order lines, turn off the mercury/cadmium lamp.
Later, when you write your report, also do the following:
- Use EXCEL (or another curve fitting program such as KaleidaGraph or Mathematica) to fit a least- squares straight line to the calibration data. Obtain the slope s, the standard deviation in the slope σs, the intercept x 0 , the standard deviation in the intercept σx 0 , and the standard deviation in an individual measurement of x, σx. Comment on the value of the intercept you obtained. Is it reasonable? Why might the intercept be nonzero?
- From the slope and the radius R calculate the lines per mm in the grating and the standard deviation in this number. Compare this to the nominal value of 600 mm−^1 , and comment on how any difference might arise.
BALMER SERIES AND THE RYDBERG CONSTANTS
The visible spectrum of hydrogen consists of a series of lines that result from transitions terminating on the energy level with principal quantum number n = 2 and originating from higher-lying energy states for which n = 3, 4 , 5 , etc. The series of lines is named after Johann Balmer who, in 1855, published a numerical relation accurately describing their wavelengths. Starting with the longest (red) wavelength, the lines are labeled Hα, Hβ , Hγ , etc. The Balmer series wavelengths for hydrogen are described by the Balmer formula
1 λ
= RH
n^2
, n = 3, 4 , 5 ,... (1)
The constant is called the Rydberg constant for hydrogen. A similar expression applies for deuterium, but the Rydberg constant for deuterium, RD , is slightly larger than that for hydrogen. The difference arises from the effect of the greater mass of the deuterium nucleus. A larger nuclear mass results in a larger value for μe, the electron reduced mass. The reduced mass of the electron arises from the fact that the two-body problem of an electron and a nucleus can be recast into an effective one-body problem involving the reduced mass of an electron only. The formula for the reduced mass is
μe =
me 1 + me/M
where me is the mass of an electron and M is the mass of the nucleus. See the reference by Tipler and the material on the class website for the derivation of this formula. The Bohr (and Schr¨odinger) theories of the hydrogen spectra give a formula for the Rydberg constant:
R = μee^4 8 ε^20 h^3 c
where e is the charge of an electron, ε 0 is the permittivity of free space, h is Planck’s constant, and c is the speed of light. The Rydberg constant for an infinitely heavy nucleus R∞, which is the same as Eq. (3) with μe = me, is one of the best known constants in physics, R∞ = 10 973 731.568 549(83) m−^1 [CODATA recommended values of the fundamental constants 1998]. Note that this is larger (by a small amount) than the value of the Rydberg constant for hydrogen.
In the next exercise, you will take data in order to determine RH.
Move the large carriage supporting the mercury/cadmium lamp toward the lens carriage and clamp it a cm or so from the lens carriage. Move the small carriage and mercury/cadmium lamp housing toward the wall, out of the light path, and clamp it in position. Move the carriage supporting the hydrogen/deuterium lamp assembly forward until the small piece of white tape on the left edge of the carriage is above the 515 mm mark on the metal rail, and clamp the carriage with the silver clamping knob. With the lamp and lens in these positions, the lamp image is focused on the slit with 1:1 magnification (to see the image hold a white card next to the slit).
You may need to make some adjustments of the lens in order to optimize the position and the focus of the image of the hydrogen/deuterium lamp.
The hydrogen/deuterium lamp gets very hot after being on for a period of time, and as its temperature increases, the intensity of the emitted light decreases. To help keep the lamp cooler and the intensity higher, a small fan is positioned at the base of the black tubing, which protects the lamp. Turn on the power supply for the fan, and then turn on the power supply for the lamp.
For x ≈ 42 cm, the red (Balmer α) line should be visible. If the line is not uniformly bright, it may be necessary to adjust the transverse position of the lens and/or the longitudinal position of of the hydro- gen/deuterium lamp.
The hydrogen/deuterium lamp draws 20 milliamperes (mA) of current at 5000 volts potential difference across the tube. 20 mA can be a lethal current so it is important to handle the hydrogen/deuterium lamp assembly carefully and touch it only at grounded surfaces.
To change the position of the lamp, slightly loosen the knob marked with blue tape, and rotate the lamp holder by grasping the short horizontal rod marked with orange tape. Do not touch the black tubing protecting the lamp or the white caps holding the tube in place. When you have achieved maximum brightness by adjusting the lens and lamp positions, clamp the lamp in position by tightening the knob marked with blue tape.
Once you have found the Balmer α line, reduce the slit width in order to see the separation between the hydrogen and deuterium lines. Which line corresponds with which element? Hint: study Eqs. (1) through (2). When you have figured out which lines are the hydrogen lines, focus the cross-hairs of the microscope on those lines.
Measure the position of the three visible Balmer lines, (α, β and γ) for hydrogen in first order. (The other lines in the Balmer series fall outside the visible range.) Note: there are a number of lines in the spectrum of the lamp that come from impurities rather than from hydrogen or deuterium. These tend to be notably dimmer than the hydrogen lines, but you can use your calibration from the mercury/cadmium lamp to make sure you are observing the correct lines.
- Determine the value of RH from measurements of the Balmer lines of hydrogen. Extract the Rydberg constant for hydrogen RH from the parameters of a straight-line fit (done with EXCEL or other program) to an appropriately chosen plot of your data. Hint: The Balmer formula can be reduced to a linear equation by substituting the proper functions of λ and n. See Eq. (1). Calculate an uncertainty in your value of RH , and compare your result with the accepted value of R∞ adjusted for the reduced mass of hydrogen.
Figure 2: Scope traces showing the CCD array output. (a) The Balmer α lines with the entrance slit too wide to resolve the difference between Hα and Dα. Note the ≈ 2 .5 V readout cycle start spike directly below the trigger marker (“T”). (b) The α lines resolved into the Hα and Dα components. Note that the carriage has been moved so that the lines are near the start cycle spike, and the horizontal and vertical scales have been greatly increased so that the steps corresponding to individual pixels are visible.
Once you can see the signal of the α lines, reduce the slit width enough to split the large bump into two smaller bumps. Then loosen the silver knob on the side of the post holder that supports the CCD card assembly and, while watching the scope, slowly rotate the card assembly back and forth to find the position which makes the peaks the highest and narrowest possible. This optimizes the position of the CCD detector so that it is at the focus point of the grating.
Next, slide the carriage slowly along the track to position the peaks near the beginning of the CCD readout cycle, which is marked by the short “spike” or “blip” of about 2 volts amplitude (See Fig. 2a). The vertical position of the trace on the scope is determined by two knobs: the vertical “Position” knob on the scope panel sets the position of zero volts, as denoted by the channel marker on the left side of the display, and the “Offset” knob on the amplifier box, which applies a DC offset to the CCD signal. It is helpful to position the channel marker about two large divisions below the centerline of the display, and then adjust the “Offset” knob to park the trace at this spot. This is shown in Fig. 2a; note the marker for channel 1 at the left.
Increase the sweep speed to about 400 μS/div in order to zoom in on the peaks. Then, make successive small adjustments of the CCD card position, carriage position, slit width, and oscilloscope controls in order to make the peaks on the scope as narrow and as distinct as possible. You should be able to obtain an image similar to that shown in Fig. 2b. Note the significant change in both the vertical sensitivity (“CH 1”) and horizontal sweep speed (“M”) between Figs. 2a and 2b. When you have zoomed in on the peaks, the trace will be quite jumpy because of electrical noise. You can quiet this noise by using the averaging feature of the scope. Press the MENU button in the ACQUIRE column, then press the Average button to the right of the scope screen next to the Average selection. An average of 32 sweeps will make for a steady display. To change the number of sweeps in the average, rotate the knob with the ⊕ on it at the upper right of the screen. Trace averaging was done in Fig. 2b.
At this point, the Hα and Dα lines should be clearly resolved. The readout from individual pixels should now be visible on the scope: each pixel shows up as a flat section with a faint spike at the edge.
A very nice hard copy of the CCD readout can be obtained with the Tektronix TDS 3012 digital scope. If there are waveforms displayed other than Channel 1 (yellow), you can remove them from the display by pushing the appropriately colored button (white for REF, red for MATH), and then pushing the OFF
button. To print the scope display, press the button at the lower left of the screen with the printer icon on it; retrieve your print from the printer below the scope.
By counting pixels (the “stair-steps” in Fig. 2b), you can measure the separation between the two peaks and estimate the full-width at half maximum (FWHM) of the peaks. But to relate the pixel counts to the change in wavelength, you need a conversion factor, which you can determine by using the CCD array and the alpha lines as follows.
Set the sweep speed back to 10 ms/div, so that you can see an entire readout cycle of the CCD array. Move the carriage to the left so that the very top of the largest peak is just at the left edge of the CCD readout. Note the x reading (call it “x 1 ”) on the meter stick. Now move the carriage to the right so that the very top of the same peak is just at the right edge of the CCD readout, and again note the x reading (x 2 ) on the meter stick. Hint: if you understand how this system works, you can get a more accurate result by moving the horizontal position marker, or “trigger point” to the middle of the screen, and then increasing the sweep speed sufficient to see individual pixels. Then at one extreme, the peak will be on the right side of the cycle-start spike, and at the other extreme, the peak will be on the left side of the cycle-start spike.
From your original calibration you can derive the wavelength associated with x 1 and x 2 , and thereby the wavelength difference ∆λ = λ(x 2 ) − λ(x 1 ) across the 1024 pixels. The change in wavelength per pixel will be ∆λ/1024 = ∆λpixel.
After you have obtained your pixel calibration data and printout for the α lines, repeat the procedure to measure the Hβ and Dβ lines. You will need to increase the vertical sensitivity on the scope, since these lines are less intense, and the CCD is less sensitive to this wavelength. Make sure to turn off the Averaging feature when first locating the β lines. After locating the lines, and parking them close to the cycle-start point, increase the sweep speed and vertical sensitivity to see the lines in more detail. Turn on the Averaging feature to get rid of the bounce in the display.
Since the β lines produce a much smaller signal than the α lines, the background pixel-to-pixel variation of the CCD readout will visibly affect the line shapes. To eliminate this background variation, the REFERENCE and MATH features of the scope can be utilized. After finding the β lines and getting them displayed on the digital scope, proceed as follows.
- Cover the slit assembly with the black paper (or slide the carriage a wee bit to one side so that the lines move off screen).
- Press the REF button, turn Ref 1 on (press the button below the Ref 1 selection on the screen), and store Channel 1 to Reference 1 by pressing the “Save/Recall” button to the right of the knob marked ⊕, and chosing the appropriate softkeys on the screen.
- Remove the black paper covering the slit, (or move the carriage back to its original position).
- Now press the MATH button, and select the subtraction operation (-), with the first source set to Channel 1, and the second source set to Reference 1.
The MATH trace (red one) will now display the difference between the signal and background, with the pixel-to-pixel variation largely eliminated and the line shapes displayed much more cleanly. This effect is shown in Fig. 3.
Make a print out of the β lines for later analysis. If you have time, you may wish to explore whether you need to calibrate the pixel width versus wavelength for the β line measurements. Can you think of a reason why it might differ from that of the α lines?
Copies of the scope display for both the Hα/Dα and Hβ /Dβ lines are available in the lab, so you can see what sort of data you should be able to obtain.
- Determine the mass ratio of deuterium to hydrogen from the difference in wavelength of the α and β lines. Use your printouts and calibration data to determine the difference in wavelength ∆λHD ≡ λH − λD between the Hα and Dα lines and the difference in wavelength between the Hβ and Dβ lines. Then calculate ∆λHD /λH for each pair of lines, along with an uncertainty in this ratio. Use Eqs. (1), (2) and (3) to prove the following relationship:
∆λHD λH
1 − MH /MD
1 + MH /me
where MH and MD are the masses of the hydrogen and deuterium nuclei. Finally, use the result of Eq. (6) along with your measurements to calculate the ratio of the of the masses of the hydrogen and deuterium nuclei, MH /MD , along with its uncertainty. You will need to use the proton/electron mass ratio mp/me = 1836.15/1. If the nuclear mass ratio you calculate is not close to 1/2, try again.
