Physics 2107 The Balmer Series of Hydrogen 8 ...incident along the normal of a diffraction grating, then the transmitted light exhibits a diffraction pattern which consists of a number

PY2107 The Balmer Series of Hydrogen Experiment 8 __________________________________________________________________________________

2.1

Physics 2107 The Balmer Series of Hydrogen 8 - Determination of the Rydberg Constant Background Observing that atoms only emit light at definite spectral frequencies demonstrates electrons in atoms only populate distinctive (quantized) energy levels. In the Bohr model of the atom radiation is emitted any time an electron makes a transition from a higher to a lower lying energy level. For the H atom it turns out that for some transitions to the n = 2 level the radiation emitted is in the visible range: transitions to this level generate the so called “Balmer Series” if spectral lines.

Generally, the wavelength λ of a photon created by an electronic transition from a level ni to nf is

1λ= R∞

1nf2 −

1n2i

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where

R∞ =mee

4

8h3ε02c

the Rydberg Constant. The experimental value of the Rydberg constant has to be adjusted to take into account the finite mass of the nucleus: this is done by replacing

the electron mass by a reduced mass mr =memn

me +mn

=18361837

me and hence RH =18361837

R∞

To measure λ we use a transmission line grating. If parallel light of wavelength λ is incident along the normal of a diffraction grating, then the transmitted light exhibits a diffraction pattern which consists of a number of spatially separated intensity maxima at angles θ given by d Sin(θ) = ηλ, where η is defined as the order of the maximum and d is the spacing between the lines on the grating. Hence, knowing d and measuring θ for each order η allows λ to be calculated (see Fig 1).

Fig 1

PY2107 The Balmer Series of Hydrogen Experiment 8 __________________________________________________________________________________

2.2

Experimental Procedure

Measure the wavelength of the lines of the H Balmer series as accurately as possible, using measurements of images on both sides of the straight through position of the spectrometer and in as many orders η as are observable. The Spectrometer

The setup and adjustment of the spectrometer (see Fig 2) are described below. d can be calculated from the number of lines per mm for the grating (e.g. 600 – if in doubt ask the demonstrator).

Fig 2

Eye-piece: Place a piece of well illuminated white paper in front of the telescope and adjust the eye piece until the cross hairs are clearly in sharp focus. Rotate the telescope barrel until one of the cross hairs is vertical. The eye piece should not be re-adjusted during the rest of the experiment.

Telescope: Turn the telescope towards some distant object. Adjust the length of the telescope barrel until the distant object is in sharp focus with no parallax between the image of the cross hairs and the image of the distant object. The telescope is now focussed and should not be re-adjusted.

Collimator: Illuminate the slit of the collimator with light from the hydrogen lamp. It is helpful if the slit is open fairly widely initially. Turn the telescope into line with the collimator and view the image of the slit. Vary the distance between slit and the lens of the collimator until the image of the slit is in sharp focus with no parallax between it and the image of the cross hairs. Adjust the width of the slit to get a narrow image which should be parallel to the vertical crosshair. These adjustments ensure that the light emerges from the collimator in a parallel beam. When the slit is observed directly with the telescope, its image should lie fairly centrally in the field of view. Place the grating on the spectrometer table and observe the diffracted images on both sides of the straight through position. If necessary, adjust the levelling screws on the spectrometer table.

PY2107 The Balmer Series of Hydrogen Experiment 8 __________________________________________________________________________________

2.3

Grating: It is now necessary to set the grating at right angles to the ray from the collimator. Note the straight through position and then turn the telescope to observe a suitable order image on the right hand side and find the angle θr between its position and the straight through position. Doing the same for the same order image on the left hand side with give an angle θl. If θl is equal to θr then the grating is at the right angle to the rays. If not and (for example) θl > θr, then rotate the table through (θl - θr)/2. Then repeat the procedure and the new values of the angles should be much more equal (see Fig 3).

Fig 3

Measurements:

Calculate λ for each value of η of the spectrum and determine the average value of the error. For additional information concerning the use of the vernier scale on the spectrograph, see the appendix.

Plot a graph of 1/λ against 1/ni2 where ni = 3 for the red line, 4 for the blue-green line,

5 for the violet line and 6 for the deep violet line (this one may be difficult to observe). The gradient of this graph gives you a value for the Rydberg Constant. The integer nf can be determined from the intercept in the 1/ni

2 axis – i.e. when 1/λ =0 and 1/ni

2 = 1/nf2

Compare your result for RH to the theoretical value discussed above (using appropriate values for the other physical constants). Use your measured value of RH to calculate the wavelength of the series limit of the H Balmer series (corresponding to ni=∞). The use E = hc/λ to calculate the corresponding transition energy.

PY2107 The Balmer Series of Hydrogen Experiment 8 __________________________________________________________________________________

2.4

Appendix: Reading the Vernier Scale on the Spectrometer

The main scale, which is fixed, runs from 0º - 360º with the smallest separation between lines equal to 0.5º. The Vernier scale, which rotates with the telescope, runs from 0º - 0.5º. The Vernier scale allows the angle of the telescope to be read to an accuracy of 0.01º, as follows.

Firstly read the the angle on the main scale – i.e. read off the angle indicated by the zero line of the Vernier scale. In the image above it lies between 303º and 303.5º. Next, the fractional angle – in this case, the angle between 303º and 303.5º - can be read using the Vernier. Simply follow the lines across on the Vernier scale until you locate the ONE which best lines up with a line on the main scale (see below).

In our example this is ~38 divisions to the left of the zero mark on the Vernier scale, or 0.38º. Finally we add our angle from the main scale of 303º to our Vernier angle of 0.38º to get an angle of 303.38º. Taking this from the image above, the error would be ~0.02º, but you should be able to do better than this from direct measurement. NB. Had the zero line been after the 0.5º marker on the main scale, we would have added 0.38º to 0.5º, which would have given us a final value of 303.88º.