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 ħl. 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 RH 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 = ∞ (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 ni and nf 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 nf=1 are referred to as the Lyman series; those with nf=2, the Balmer series; and those with nf=3, the Paschen series. The lines of a given series will all fall at a wavelength equal to or less than the transition with nf=ni+1, and to longer wavelengths than the convergence limit (the wavelength of a hypothetical transition from a state with ni=∞. 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.
Using equation (2), calculate the wavelengths (nm) of the first seven lines in the Balmer series of hydrogen. Set nf=2 and let ni vary from 3 to 9.
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 Santa Barbara Instruments Group STF-8300 CCD (charge-coupled device) camera for a detector in this experiment. The detector array on the CCD camera is 3352 pixels x 2532 pixels. Each pixel is 5.4 microns square. 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 much smaller than 5.4 micrometers 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 and wider entrance slit width. 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 5.45 software package to control both an Acton Research 0.5 m spectrometer and the STF-8300 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 (0.0054 mm/pixel), we see that (1.7 nm/mm) * 0.0054 mm/pixel = 0.00918 nm/pixel. Since the pixel array is 3352 pixels, we have 3352 pixels * 0.00918 nm/pixel = 30.77 nm. Thus, one "screen shot" with the CCD represents 30.77 nm of the spectrum. For example, if we set the grating on the spectrometer to 485 nm, the resulting spectrum will represent the range 469.61 nm - 500.39 nm. Please make note of this!! When we want to do a 'scan', we'll move the grating from position to position manually and we'll want each 'screen' to overlap by perhaps 2 nm, so that when we 'join' or connect a series of spectra together, we won't inadvertently miss any spectral features at the margins.
Prior to collecting data, the spectrometer must be calibrated using a source of accurately known wavelength. We have Ar-ion lasers at 488.0 nm and 514.5 nm, a He-Ne laser at 632.8 nm, and a solid-state Cube laser at 409 nm. We will use the 488 nm Ar-ion laser for calibration purposes.
To calibrate the spectrometer using KestrelSpec 5.45, do the following:
Start KestrelSpec 5.45 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 Image (Before).
Under the Format menu, select Calibrate X Axis, and select Wavelength (nm).
Next, you must set the grating position to the laser wavelength (488.0 nm). In the Spectrograph Control window, make sure (in the lower left) that wavelength (λ) is selected (not wavenumbers, cm-1 - this is for Raman shift) and type 488.0 in the box at the lower left and click on 'set'. You should hear the grating move into position. The 'center' of the pixel array is now at 488 nm, and the screen with (based on the above discussion) is 488.0-15.39 = 472.61 to 488.0 + 15.39 = 503.39 nm. The actual boundaries on the x-axis on the Kestrel display will be slightly different than this. This difference isn't important - what is important, however, is that you record the range of the x-axis in your lab notebook.
For calibration purposes, an entrance slit width of 10 micrometers works well. Verify that the slit is set to 10 micrometers. Notice that, when the slit is 'closed', it is actually set at 10 microns. This is to prevent damage to the slit edges caused by closing it all the way and having the edges make contact.
If you don't know how to read a vernier scale on a micrometer, here are two tutorials for you to work through:
Turn on the 488 nm laser (CAUTION!). Direct the beam onto a fiber optic collimator. Use the aperture on the laser to partially block the beam or the intense light will result in a very 'washed-out' spectrum. The fiber optic collimator is connected to a fiber optic cable, which is in turn connected to another collimator. The radiation is then focused through a lens onto the entrance slit of the spectrometer.
Here is the setup for the fiber optic collimator for calibration:
Here is the spectrometer/slit/micrometer/CCD camera:
Please make note: this setup is also used for research, and the alignment of the fiber optic/collimator/focusing lens on the entrance slit of the spectrometer is NOT to be disturbed. The only variable in this photo with which you need be concerned is the slit width!
Pull down the Setup menu, select Camera Configuration, and set the exposure time to 0.1 sec. Set the Target Temperature to -8.0 (this is the cooling for the CCD camera - the lower the temperature, the less noise the camera generates.) On the Binning tab, make sure that the Group Size = 1 for both Rows and Columns.
On the spectrograph control menu, hit Acquire.
The spectrum of the laser will appear on the KestrelSpec screen. Ideally, it will look like this:
We want one very well-resolved peak in the spectrum and we want the 'Counts' on the y-axis to be above about 65000. You'll have to experiment with the exposure time and possibly the slit width. Do so until your calibration spectrum looks like the one shown above.
Once you have obtained an acceptable spectrum of the laser, leave the laser on and do the following:
click on the Autocal button on the spectrograph control menu. Kestrelspec will make three or four calibration measurements.
When Kestrelspec has completed the Autocal procedure, Pull down the Setup menu, and select Spectral Configuration. On the CALCurve tab, Set the laser line to 488.0. Click Set up 1-point Cal and be sure that the excitation wavelength is set to the laser line (in this case, 488.0 nm.)
The spectrometer/detector are now calibrated. Turn the laser off.
Move the fiber optic collimator used to collect the laser radition to a position in front of the hydrogen discharge tube. Place a focusing lens between the discharge tube and focus the radiation from the tube onto the collimator. The setup should look like this:
Slide the focusing lens on the optical rail until the radiation from the discharge tube is tightly focused on the fiber optic collimator. Your spectra will be no better than the alignment of your optics in this type of experiment!!
Set the entrance slit width on the spectrometer to 30 micrometers as a starting point.
Enter the longest wavelength (nm) from your Balmer series calculation into the Spectrograph Control panel and press the set button. This wavelength should be approximately 650 nm. The grating will move to this wavelength.
Pull down the setup/Camrea menu and set the exposure time to 10 sec. Acquire a sepctrum of hydrogen. Experiment with exposure time and slit width until you obtain one very well-resolved spectral line.
A note: here is a plot of the quantum efficiency of an SBIG 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! It is not uncommon to have 120-260 sec exposure times in this region.
After you are satisfied with the exposure and resolution of the spectrum, use the peak finder feature and record the wavelength of the spectral line. Name and save the file to a flashdrive.
Move the grating to your next calculated wavelength and repeat the process, recording the wavelength and saving the spectrum each time until you have reached 388 nm.
When you have collected and saved spectra for each line, your instructor or the lab TA will show you how to 'join' these files to make one spectrum which is representative of the Balmer series. Screen capture this spectrum and paste it into a Mathcad doc which will contain your calculations.
Turn down the Variac transformer and turn the discharge tube off. Exit KestrelSpec. When asked if you wish to Disable Temperature Control, click No.
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 (nm) into a vacuum wavenumber (cm-1). Fit your data using a least squares procedure and Eq. (2) above (i.e., plot wavenumber vs 1/ni2), where ni = 3,4,5,6,7,8. 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.