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Keele University Physics/Astrophysics Laboratory 31
The Optical Spectrometer
- Introduction
The optical spectrometer is a precision instrument capable of an accuracy of measurement far greater than that found in most other areas of physical measurement. The function of the spectrometer is to disperse light into its various component wavelengths (or frequencies) and to determine the wavelength (or frequency) of each resolved component. The dispersive element is usually a diffraction grating but it could also be a glass prism. In this experiment the line spectrum of an element (He, Hg or Cd), obtained from a gas discharge lamp, is used to verify the fundamental equation for a diffraction grating (the grating equation) and hence also to obtain a value for the grating constant. This information is then used to determine the wavelength separation of the yellow doublet in the sodium spectrum (the sodium “D” lines), a measurement which is close to the resolution limit of this instrument.
- The Instrument The optical spectrometer has three essential components (see fig. 1), a collimator, a dispersive element and a telescope.
Slit
Collimator
adjustment
screw Table clamp and adjustment screws
Eyepiece
Telescope adjustment screw
Grating
Vernier scale
Fig. 1 Plan view of the optical spectrometer The collimator is the fixed arm of the instrument. It consists of a vertical slit of adjustable width in the focal plane of the collimator lens. The collimator accepts light passing through the slit and produces an emergent light beam having a precisely defined direction in the horizontal plane. The narrower the slit the more precisely collimated is the emergent beam. The telescope receives this collimated light beam and focuses it to produce a real image of the slit. This real image should be in the same plane as the cross wires so that when the moveable eyepiece is
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used the image of the slit and the cross wires are both seen together in sharp focus. The telescope and the table which holds the dispersive element can both be rotated independently and both have clamping screws and fine rotational adjustment screws. The latter only operate when the respective clamping screw is fixed. Two vernier scales are provided for measuring the angular displacements of the telescope and the table. During your measurements readings from both verniers should be taken and an average value used because this will reduce the effect of errors in the alignment of the axis of rotation and the centre of the graduate circle. The vernier scales should allow angular measurements to within 1' of arc (or 1/100° on some instruments). A magnifying glass and a reading lamp are provided to assist in reading the vernier scales.
- Adjusting The Optical Spectrometer Firstly familiarise yourself with the various features described in section 2. Locate and use the clamping and fine adjustment screws of the table and the telescope. Note that the table height is adjustable (and is locked by its own clamping screw) and that the table has three levelling screws. a) The table height should be set so that the full width of the collimated beam falls on the dispersive element, without obstruction by the table or grating mounts. The collimated beam can be viewed directly on a white paper screen if a reasonably wide slit is used and illuminated by a spectral lamp with the room lights off. Observe the collimated beam reflected off such a paper screen and move the lamp about across the slit. Ensure that the lamp is always used in a position such that the collimated beam is a uniformly illuminated circle. This ensures that the collimator lens is fully illuminated and that the image of the slit seen in the telescope will be uniformly bright. b) With suitable illumination (e.g. indirect light from the room lights or reading lamp) an image of the cross wires should be seen through the eyepiece of the telescope. To obtain a sharp image move the eyepiece in or out. In order to focus the telescope parallel light rays are required and this in turn requires a properly adjusted collimator. Therefore for this reason the adjustment of the telescope and the collimator are usually done together. In appendix A the procedure devised by Schuster for this adjustment is given. As this procedure is followed verify that the slit is vertical and that its image is in the centre of the field of view. If it is not then rotate the slit mounting as necessary. When the Schuster procedure is complete the telescope and collimator should not need further adjustment. If the adjustment screws are now left undisturbed the only adjustment needed to accommodate individual eyesight variation is that of the eyepiece. The eyepiece will require constant repositioning as each member of a team takes their turn to make observations through the telescope. Resist the
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Telescope
Collimator
Grating
Figure 2. The diffraction angle θ and
the orientation of the diffraction grating
- Measurements (a) The Grating Constant
For a transmission grating used as in figure 2 the diffraction angle , θ, for light of wavelength, λ, is
given by the grating equation.
n λ= d sin θ (1)
where d is the distance between lines of the grating and n is the order of the diffraction (n = 1, 2, 3 ....). The grating constant is b = 1/d, the number of lines per unit length. Use one of the spectral lamps which gives several brightly visible spectral lines (e.g. He, Hg, Cd) of known wavelength (values are given in Appendix B.). For each of say 4 to 6 lines covering a wide
range of wavelengths measure the diffraction angle θ, in the first order spectrum (n=1). For the highest
accuracy take readings from both verniers. The diffraction angles both to the left and the right should be equal but errors in the alignment of the grating may give rise to measurable differences. These errors will
be largely cancelled if an average of left and right diffraction angles is used. Plot a graph of λ against sin θ
and use a least squares fitting procedure to obtain a value for the grating constant b. Compare the error of b obtained from this fitting procedure with your own estimate of the accuracy of angle measurements.
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(b) The Sodium “D” Lines Use a sodium lamp to illuminate the slit and observe that the yellow light characteristic of sodium has a doublet structure. Observe that the resolution (i.e. angular separation) of the doublet is greater in the second order spectrum (n=2) than in the first order, as expected from the diffraction equation.
Measure the diffraction angles θ 1 , and θ 2 of the two components of the “D” lines in the second order
spectrum. Use both verniers and both left and right diffracted beams as before. Repeat these measurements a few times to assess the reproducibility of your angular measurements and telescope settings (Each person in turn could set and reset the telescope while the other reads the verniers without sharing the results until the sequence of observations is complete). Show for yourself from the diffraction equation(1) that
where
Obtain a value for Δλ , the wavelength separation of the “D” lines. In view of the accuracy and
reproducibility of the angle measurement what is your estimated error in Δλ?
Appendix A : Schusters method for the focussing of a spectrometer As a ray of light passes through a glass prism it suffers refraction at two surfaces. For one particular angle of incidence i the angle of deviation of the ray r will have a minimum value. This corresponds to a ray which passes symmetrically through the prism as shown in figure A.1.
n Δ λ = 2d cos ( θ m ) sin ⎛⎝⎜ Δ 2 θ⎞⎠⎟ (2)
θ m =^ θ^ + θ
1 2
1 2 1 2
⎝⎜^
(3.a)
(3. b) (3.c)
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be necessary to reset the collimator for each spectral line individually. Where necessary a simple focussing readjustment of the telescope alone is now sufficient.
Appendix B: Wavelengths for spectral lines The textbooks report the following wavelengths in nm (nanometres = 10-9 m) for the strong spectral lines given off from Cadmium, Mercury and Helium gas discharge lamps at room temperature and atmospheric pressure. Note there may be other weak lines as well and some care is needed in making the correct identifications. Cadmium Mercury Helium Red 643.85nm Yellow 579.07nm Red 706.52nm Green 508.58nm Yellow 576.96nm Red 667.81nm Blue 479.99nm Green 546.07nm Yellow 587.56nm Blue 467.81nm Blue/Green 491.60nm Green 501.57nm Purple 442.00nm Violet 435.84nm Blue/Green 492.19nm Violet 407.78nm Blue 471.31nm Violet 404.66nm Purple 447.15nm Violet 438.79nm
