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Emission Spectrum of Atomic Hydrogen and the Rydberg constant
- 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 Rydberg constant using the atomic spectrum of hydrogen. The emission spectrum of hydrogen is very simple and it is the only spectrum easily linked to theory. By using the grating spectrometer to measure the emission lines, we can verify the Bohr model of the atom and hence the very basis of quantum theory itself. In practice the optical emission spectrum of hydrogen is a mixture of two spectra – that of molecular hydrogen and that of atomic hydrogen. The former is a complex continuous spectrum over all colours, typical of a band spectrum. We shall ignore it but instead look for three/ four fairly clear and sharp lines which represent the only light emitted by the hydrogen atom in the visible range.
- 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
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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 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
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Clamp the table securely in this its operating position.
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
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and use a least squares fitting procedure to obtain a value for the grating constant b. Compare the error in b obtained from this fitting procedure with your own estimate of the accuracy of angle measurements.
(b) The Hydrogen Spectrum
The wavelength λ of emitted radiation from transitions between atomic energy levels are given by:
Em E n hc (^) = −
where Em and En are the energies of the respective levels, of quantum numbers m and n. The Bohr model gives the quantised energy levels as:
2 2
h n E me n
=^ π
where n = 1, 2, 3… are the possible quantum numbers defining the energy levels themselves. Hence, the wavelength of the emitted light may be written as:
= (^2) ⎜⎝⎛^2 − 2 ⎟⎠⎞
h m n
hc π me
It turns out that the only transitions in the visible are those to level m = 2, giving:
= (^3) ⎜⎝⎛^ − 2 ⎟⎠⎞
hc n
π me
where the Rydberg Constant is defined by:
h c R me H (^) 3
2 π^24
Thus:
⎟ ⎠
= ⎛^ −
2
λ R^ H n ………. Eqn (1)
Method
Having optimised the spectrum from the hydrogen discharge tube, identify the three lines of the Balmer series which are red, blue/green and violet (Figure 3a and 3b, Physics RA Serway and JW Jewett).
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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.
Polished Faces
Apex A i r
Collimator Telescope
Figure A.1: Minimun deviation light ray
Schuster's method is based on the fact that the effect of the prism on the divergence of the beam is different on opposite sides of this minimum deviation position. The emergent beam will be less divergent (or more divergent) than the incident beam as the angle of incidence is increased (or decreased) from the minimum deviation value (i.e. as the apex A in figure A.1 is rotated towards, or away from, the telescope). This property of the prism can be used to obtain an accurately collimated beam.
Place the prism on the spectrometer table as shown in the schematic plan view of figure A.1. Ensure that the position of the prism is such that it intercepts the full width of the beam from the
collimator. For your prism the angle of minimum deviation is around 50° so set the telescope at an angle
a few degrees greater than this (~55°). Illuminate the slit of the spectrometer with light from a spectral lamp. Rotate the table holding the prism and, through the telescope, observe the images of the slit pass through the minimum deviation position. Lock the telescope at an angle a few degrees greater than this angle of minimum deviation. Turn the prism table away from its minimum deviation position so that apex A moves towards the telescope and a spectral line is brought into the centre of the field of view of the telescope. Adjust the focus of the telescope until this line image is as sharp as possible. Turn the
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prism table to the other side of the minimum deviation position until the same spectral line is again at the centre of the telescopes field of view. Now adjust the focus of the collimator until a sharp image is once more obtained. Repeat this process until no further adjustment is required. If the same line image is sharply focussed when viewed on either side of the minimum deviation position then the light beam through the prism is properly collimated. If a lamp is used which gives several spectral lines it may be observed that not all these lines can be precisely collimated simultaneously. This is due to chromatic aberration which makes the focal lengths of the lenses in the collimator and telescope slightly dependent on the radiation wavelength. It should not 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
