The Balmer Series of Hydrogen

Two quantum numbers, designated n and l, describe the state of the hydrogen atom. The principal quantum number n determines the energy of the state, while l specifies the orbital angular momentum of the electron, which equals hl/2p. Both of these are required to be integers in order for the state to be stationary: n may assume the values 1,2,3..., and l is restricted to the values 0,1,...n. The energy is given to a very good approximation by

Where Z is the charge on the nucleus in units of the electron charge (equal to 1 for hydrogen), c is the speed of light, h is Planck's constant, and R_H is the Rydberg constant for the hydrogen atom. Its numerical value is 13.6 electron volts in energy units or 109,677.581 cm^-1 in wavenumber units. Notice that the energy is defined to be zero when n is infinite (the electron is completely removed from the nucleus) and a minimum when n=1; using this convention, all bound states have negative energy.

Emission or absorption of radiation can occur whenever the hydrogen atom changes its state. When this happens, the energy of the photon emitted or absorbed will be equal to the change in energy of the atom, and the wavenumber of light (the reciprocal of the wavelength) will be proportional to the change in energy according to the formula

Where n_i and n_f are the principal quantum numbers of the initial and final states of the transition. By measuring the wavelength for several transitions, it is possible to determine the energy corresponding to those transitions and, hence, the Rydberg constant for the hydrogen atom.

Transitions that share a common lower state appear to be organized into a series of lines that converge to a limiting wavelength. The transitions with n_f=1 are referred to as the Lyman series; those with n_f=2, the Balmer series; and those with n_f=3, the Paschen series. The lines of a given series will all fall at a wavelength equal to or less than the transition with n_f=n_i+1, and to longer wavelengths than the convergence limit (the wavelength of a hypothetical transition from a state with n_i=(infinity). Thus, the Lyman series will fall entirely between 121.6 and 91.2 nm, while the Balmer series will lie in the visible and near ultraviolet. The convergence limit of the Paschen series is in the red part of the spectrum at about 820 nm, and all other series will appear in the infrared and beyond.

Procedure

Pre-lab calculations

Using equation (2), calculate the wavelengths (nm) of the first seven lines in the Balmer series of hydrogen. Set n_f=2 and let n_i vary from 3 to 9.

Experimental

On the day you collect data, be sure to let the spectrometer warm up for about two hours prior to starting your experiment.

We will be using a CCD (charge-coupled device) camera for a detector in this experiment. The detector array on the CCD camera is 750 pixels x 242 pixels in the 8x1 binning scheme used here; each pixel is 11.5 mm wide. This fixes the effective exit slit width of the spectrometer to the width of a pixel on the detector, and fixes the entrance slit width as well (an entrance slit width smaller than 11.5 mm would cut down on the available light without improving resolution. We will also have to consider the exposure time (the length of time that the pixels on the detector are exposed to the radiation) for the CCD camera; as in all imaging work, a faint light source requires a longer exposure time. It is possible to under- or overexpose the image, and you will need to keep in mind that this is by no means a simple, quick 'point and shoot' experiment!

We will use the KestrelSpec software package to control both the 0.5 m spectrometer and CCD camera. Note that we do not "scan" the spectrum in the usual sense with the CCD as a detector. Based on the dispersion of the spectrometer (1.7 nm/mm) and the width of a pixel (0.0115 mm/pixel), we see that (1.7 nm/mm) * 0.0115 mm/pixel = 0.01955 nm/pixel. Since the pixel array is 750 pixels, we have 750 pixels * 0.01955 nm/pixel = 14.66 nm. Thus, one "screen shot" with the CCD represents 14.7 nm of the spectrum. For example, if we set the grating on the spectrometer to 485 nm, the resulting spectrum will represent the range 477 - 492 nm.

Start KestrelSpec 3.2 by double clicking the icon on the desktop.

From the Window menu select Spectrograph Controls (a control panel will then appear.) The CCD camera will then initialize.

From the Set menu go to the Auto Background Subtraction menu and select Dynamic Background Subtraction.

From the Window menu, select Active Image.

Under the Format menu select Calibrate X axis, and then select Pixels.

Under the Setup menu select Acquire. The Exposure Time should be 5.0 (exposure time is in seconds, but don't type the units). You will need to experiment with the exposure time for the CCD camera to collect the best spectra possible. None of the other parameters under the Acquire menu need be adjusted.

Collecting data

Power-up the hydrogen source lamp. Enter the first wavelength from your Balmer series calculation into the Spectrograph Control panel and press the set button. The grating will move to this wavelength.

Be sure that the fiber optic input into the collimator focused on the spectrometer is connected to the collimator focused on the hydrogen lamp. Press the Acquire button on the Spectrograph Control menu. Watch the upper left corner of the Spectrograph Control Panel; you can see the progress of the data acquisition. Your image will look something like this:

Since this is an imaging spectroscopy experiment, you will need to be able to gauge the quality of your spectrum in terms of the parameters of the CCD camera. The slit width is not a variable in this experiment, but the exposure time for the CCD camera is. In Active Image mode, you are viewing the CCD output by viewing the output of the pixels. The SBIG cameras have 16-bit pixels, so the pixel counts will range from 0 to 2¹⁵=32768. Look at the pixel counts for the image: the best exposure for an image is when the highest pixel count is around 30000 with no pixels being overexposed. The image is underexposed if the counts are no greater than a few thousand.

The pixel count is given by the bar at the right of the image. As you can see from the image, the highest pixel count is about 31000, and no pixels are overexposed. Here's what a badly overexposed image looks like:

Note that the pixel count is reasonable but the image was obviously overexposed. Overexposure is caused by too long an exposure time; each time you make a measurement, view the active image. If it is underexposed, increase the exposure time, and if the image is overexposed, decrease the exposure time. It is preferable to work with the screen showing the active image, and plot the data as a spectrum when the active image from a run is acceptable.

A note: here is a plot of the quantum efficiency of the SBIG ST-6 CCD:

This is from the Rheacorp web site. Note that the quantum efficiency of the CCD is lower around 380-400 nm; if you are working in this region, plan to adjust your exposure time accordingly!

After you are satisfied with the exposure and resolution of the active image from the CCD, pull down the Window menu and select Spectrum Plot. Your image will look something like this:

Notice that the x-axis is given in pixels, and not as a wavelength in nm as you are used to. In this run, the grating was set to 485 nm. Using the Peakfinder feature of KestrelSpec, we would record the pixel position of the above line as 463.

Now we need to calculate the actual wavelength of the line shown in the hydrogen spectrum above. To do this, we will keep the grating in the same position and record the spectrum of an Fe/Ne hollow cathode lamp. A catalog of Fe/Ne spectral lines is available in the lab; we will record pixel positions versus the known wavelengths of the Fe/Ne lines. To do this:

Turn the power down on the hydrogen source

Connect the fiber optic from the Fe-Ne source to the fiber optic collimator focused on the spectrometer slit.

Turn up the power on the Fe-Ne lamp.

Acquire the spectrum of the Fe-Ne lamp (you will have to play with the exposure time.)

The active image from the Fe/Ne lamp with the grating in the same position as was done for the hydrogen spectrum is shown below:

Here are the same data plotted as a spectrum (the sizes are different but you can see how the lines correpsond to the most intense pixel counts.)

The grating was set to 485 nm (this is in the 'middle' of the screen), and know that the spectrum represents the range range 477 - 492 nm. We look in the catalog of known Fe/Ne lines for the wavelength positions of prominent spectral lines in this region. The catalog of Fe/Ne lines contains data presented as wavelengths given with relative intensity of each line. Record the pixel position and the corresponding wavelengths of at least three Fe/Ne lines.

Repeat this procedure (i.e., move the grating to the next target wavelength, record the hydrogen spectrum, switch the fiber optic, record the Fe/Ne spectrum) until you have recorded data for all the lines in the Balmer series. After each successful run (i.e., a hydrogen line and the Fe-Ne run with the grating in the same location), pull down the File menu and select Save Active Curve and save the file on a floppy. You can process your data in the department computer lab at a later time.

You may now make a calibration plot at each of your target wavelengths to determine the wavelength of the hydrogen line. Plot pixel position vs wavelength for each Fe/Ne calibration, fit a line, and from the equation of this line, convert the pixel position of the hydrogen line to a wavelength in nm. If your results are in poor agreement with the known positions of the Balmer series lines, this is probably due to a misassignment of a line in the Fe/Ne spectrum (you will be able to tell immediately from your Fe/Ne calibration plot if you have misassigned a line.)

You now will have a measured wavelength for each hydrogen transition. Because the measured wavelength of your spectrum depends on the speed of light and, hence, the refractive index of the medium in which it is measured, the next step is to correct this wavelength for the refractive index of air. Use the CRC handbook to vacuum correct the wavelength of each of your transitions.

Next, convert your vacuum wavelength into a vacuum wavenumber. Fit your data using a least squares procedure and Eq. (2) above (i.e., plot wavenumber vs 1/n_i²), where n_i = 3,4,5,6,7,8,9. The slope of the line will be the Rydberg constant for the hydrogen atom. Report this quantity at the 95% confidence level. In a Table, compare your experimental wavelengths for the Balmer series of hydrogen (report at the 95% confidence level) to those calculated using Eq. 2.