I. **The Spectrometer and the Balmer Series**

Wavelength measurements are among the most accurate measurements in physics. The tool of measurement in this experiment is the optical spectrometer, principally a device for the accurate measurement of the angle of deflection of light, whether by reflection, refraction, or diffraction. The appearance of sharp lines characteristic of the individual elements challenged classical physicists, and Balmer's discovery of a mathematical relationship which fit the optical spectrum of hydrogen was an important contribution to the development of the Bohr model of the atom.

The diffraction grating we shall use is an aluminized glass surface with precisely cut parallel grooves - or it may be a plastic replica of an original glass grating. The number of reflecting/transmitting surfaces per cm may be as great as 10,000. You will determine the number for your grating)

**NEVER TOUCH OR RUB THE SURFACE OF THE GRATING. TO DO SO WILL SEVERELY DAMAGE THE GRATING**, regardless of whether it is a glass original or plastic replica.

Each surface (or groove) acts as a source of spherical waves (Huygens’ principle). A pair of grooves acts much like a double slit interference (Young’s) apparatus. Due to the angle made with the normal to the grating, there will be a path difference between the two rays as they travel from the grating to the detector. The path difference will be

** x=d sin** q
.

If x= **ml
**, where m =0,1,2, ..., constructive interference occurs.. (See Tipler, section 33-11)

Alignment of Spectrometer

You need to adjust the spectrometer and light so that the grating is __perpendicular__ to the __parallel__ beam of the incident light from the collimator.

- While looking through the telescope, slide the eyepiece in and out until the cross-hairs of the graticule come into sharp focus. Rotate the graticule alignment ring (should be the ring nearest the telescope holder) until the crosshairs form a X (not a
**+**). Refocus if necessary. - Focus the telescope at infinity. This would be best done by focusing out a window, were that possible. You can sight done the length of the hall. Having focused the telescope, DON'T touch the telescope focus again!
- Check that the slit is partially open and align the telescope directly opposite the collimator.
- Looking through the telescope, adjust the focus of the collimator and , if necessary, the rotation of the telescope until the slit comes into sharp focus. (The telescope focused at infinity means that if the collimator is also focused at infinity, the waves emanating from the slit as a cylindrical wave are converted to plane waves (parallel rays) by the optics of the collimator and the then parallel rays will be focused back to the image of the slit by the telescope. )
- Tighten the telescope rotation lock-screw and use the fine adjust knob to align the center of the cross of the graticule with the
**fixed**edge of the slit. If the slit is not vertical, loosen the slit lock ring, realign the slit, and retighten the lock ring. Adjust the slit width for a clear, bright image. Measurements of the diffraction angle are always made with the graticule line aligned along the fixed edge of the slit, so a very narrow slit is not necessarily advantageous. On the other hand, too wide a slit will lead to a "graying" of the field you observe, and make it difficult to observe faint lines.

When properly aligned and focused, the slit should be sharply focused in the center of the field of view of the telescope and one cross-hair of the graticule should be parallel and aligned with the fixed edge of the slit. If proper alignment cannot be achieved as described, talk to your instructor.

Note the reading of the vernier indicating telescope rotation. - Move the sodium lamp near the slit and adjust it to be aligned with the slit, about 1 cm away.
- Insert the diffraction grating into the clips of the mount. Check to be sure the light is being dispersed horizontally. You wish to align the grating so that it is exactly perpendicular to the incident light path. You may be able to do so by rotating the grating slightly so as to super impose the undiffracted image of the grating and one or more multiple reflection "ghosts".
- Rotate the telescope away from its aligned position to find a bright slit image that allows easy alignment of the graticule with the fixed edge of the slit image. Align the graticule and carefully measure the angle of diffraction.
- The diffraction grating diffracts the incident light into identical spectra on either side of the line of the undiffracted beam. Rotate the telescope back, past the zero diffraction angle, to find the corresponding slit image Carefully measure the angle of diffraction for this slit image.
- If the grating is perfectly aligned, the diffraction angles for the corresponding slit images will be identical. If not, you may use the fine adjust knob to rotate the table base to compensate for the difference (i.e., to align the grating exactly perpendicular to the collimator beam so that these angles of diffraction will be exactly equal. ). You will be able to determine the amount to rotate from the difference of the diffraction angles. If the difference is small, you may choose to always take readings on both sides, and find the average.
- Once the grating is aligned, do not rotate the rotating diffraction grating table or its base again. Rotate the telescope to re-align the graticule with the undiffracted slit image and record the new zero point. All angles of deflection should be measured with respect to the undeflected reading.

Wavelengths are determined according to the formula

l
= (d sin q
)/n (1)

where **d** is the spacing of the lines on the grating, q
is the angle of diffraction, and **n** is the order of the diffraction spectrum. Be sure you know where this formula comes from. Check, now, to see that you get wavelengths that are what you expect. And this is the point at which you should observe the sensitivity of your results to changes in the measured values. You should make an estimate of your experimental uncertainty and dominant source of uncertainty at this point - not after laboratory! (How accurately do feel you can measure angles reproducibly? How much difference does it make in your results if your measurements vary by the maximum amount you have estimated to be possible? This is the sort of consideration you should make in each laboratory to estimate the uncertainty of your results.)

Calibration of Grating

__Qualitatively__observe the sodium spectrum in first order for m=+1, identifying as many lines as possible. To observe fainter lines, it may be necessary to open the slit a little more, but not so much that the scattered light causes the background to become too light. You will also want to use the black cloth to reduce the scattered light entering the telescope or interfering with your vision.- Now, carefully measure the angular position for m=+1 of the 5890.0 (actually, 5889.953) line. Take an average, with both partners contributing readings. Note that you should always move the crosshairs onto the slit image from the same direction, to avoid error due to backlash. Calculate your standard deviation. (How does it compare with your estimated uncertainty, above? Why?) (It is probably best to continue to take readings of each line on both sides of zero, even though if you correctly adjust the grating, the readings should be the same.)

Knowing this wavelength and the angular position of this line, calibrate your grating by calculating**d**and the uncertainty in**d**: - Without spending too much time with it, measure the wavelengths of the other sodium lines you see, and then compare with the table of wavelengths given below; note also their color, and that they are "doublets."
- Now, just to see how accurate your measurements were, measure the wavelengths for the two yellow lines; that is, the two lines of the primary doublet. What is their separation? Comparing this difference with the known value of 5.9 Angstroms will give you further indication of the accuracy of your measurements.

d = nl /sin q .

Also work out the number of lines per mm on the grating, and compare with the approximate figure given on the grating.

This is the value you will use in the rest of this laboratory, so make sure you can get the same grating next time.

(Note, the values of the grating spacing vary from grating to grating by a small amount. This has to do with stretching or shrinkage of the plastic replica, after it has been removed from the glass original.)

*Note*: in your notebook, you should include a column for __wavelength__, and calculate these as you go. The outcome of these exercise is the measurements of the wavelengths of spectral lines. Other columns should record color, the accepted value, the percentage discrepancy, and the apparent grating-spacing - that is, the grating lines/mm which would give you agreement.

Hopefully, you will be able to enter this data into a spreadsheet program which will allow some of the calculations to be done automatically. However, as is true anytime you use the computer to calculate a physical answer, you should check the accuracy of the calculation by doing at least one point by hand.

It is the average of these apparent spacing that you will use to measure the Balmer spectra. See below.

Note: You should get to here by the end of the first lab.

Sodium lines :

Wavelength in Angstroms (10^{-10}m).

6154.3 - 6160.7 (red)

5889.95 - 5895.92 (the bright orange doublet)

5682.7 - 5688.2 yellow

5149.1 - 5153.6 blue-green

4978.0 - 4982.9 blue

4748.0 - 4751.9 "

4664.9 - 4668.6 "

4494.3 - 4497.7 blue–violet

Balmer Series

After you have calibrated and learned to use your spectrometer, you will be given a discharge tube for hydrogen. The purpose here is to observe the Balmer series of lines, and see that they may be expressed as

1/l
=R(1/4-1/n^{2}).

(It was the discovery of this fact that was instrumental in leading to the proposal of the Bohr model of the atom. ) You should be able to measure at least three, and possibly four, lines of the Balmer spectrum. Determine R, and the uncertainty of R, by fitting your data to a linear plot. Discuss the uncertainty of the fit and the uncertainty of your measurements.

The given value of R is found in your text or the Handbook. The values for the lines may be readily calculated from R and the Balmer formula, (were it so easy for the spectra of other elements!) Determine your discrepancy. State how accurately you can determine spectral wavelength with your apparatus? (Imagine your boss asks you how acurately he should state your resutlts to a client: what would you answer?)

Equipment:

Pasco Precision Student Spectrometer

Diffraction grating

Sodium and Hydrogen (Deuterium) light sources.

Various other spectral light sources (if you have time)

5895.923 Angstroms.