Background:The diffraction grating provides the simplest and most accurate method for measuring wavelengths of light. It consists of a very large number of fine, equally spaced, parallel slits, usually thousands of lines (slits) per centimeter. Transmission gratings are ruled on glass and the unruled areas transmit the incident light. This experiment uses a transmission-grating replica.
where m is called the order of the spectrum, λ is the wavelength, d is the spacing between grating lines, and θ is the diffraction angle measured with respect to the direction of the light incident on the grating.
When the light from a gas discharge tube is observed with a spectrometer, the colored images of the entrance slit appear as bright lines separated by dark regions: hence, the name line or discrete spectra. Each gas emits a particular set of spectral lines and has its own characteristic spectrum. The discrete lines of a given spectrum depend on the atomic structure of the atoms and are due to electron transitions. The line spectrum of hydrogen was explained by Bohr's theory that describes spectral lines as resulting from electron transitions between energy levels. However, before that, the line spectrum of hydrogen was shown to follow the description of Balmer's empirical formula:
Here, n refers to the principal quantum number of the initial energy level, and R is Rydberg's constant with a value of R = 1.097 x 107 m-1. In this experiment, the hydrogen line spectrum will be observed and the experimental measurements of wavelengths will be compared to those predicted by Balmer's equation.
The apparatus: Make yourself familiar with the parts of the spectrometer. Be sure you understand the purpose and the operation of each part. The light to be analyzed must pass through the grating as a parallel beam so that the direction of all rays will be the same and can be measured. The primary parts of the spectrometer are:
Initial Setup and Adjustments:
Focus the telescope eyepiece on the cross hairs:Remove the telescope from the spectrometer mounting. Adjust the eyepiece until the cross hairs are in sharp focus.
Focus the telescope on an object which is "infinitely" far away:Sight with the telescope at some distant object . Adjust the main telescope tube until the object and the cross hairs are in sharp focus simultaneously. Thereafter, do not make any adjustments on the telescope.
Remount the telescope on the spectrometer. With the central table empty, loosen the set screw which locks the telescope in place. Move the telescope into line with the collimator. Place the mercury light source just behind the collimator entrance slit. While observing the slit through the telescope, slide the collimator slit tube in or out until the vertical slit image and the cross hairs are both in sharp focus. Thereafter, do not move the slit along the tube.
The collimator slit width is adjustable by a small screw near the slit. Use the screw to make the slit narrow enough to define the source position, yet wide enough to allow the passage of light sufficient for easy viewing. A narrow slit gives greater resolution and greater accuracy.
With the cross hairs centered on the slit image, loosen the set screw of the angular scale and adjust it to read 180.0 degrees for this straight through reading.
Rotate the spectrometer table so the groove across the diameter is approximately perpendicular to the light from the collimator and lock the table in this position.
Place the diffraction grating in the table groove such that the grating slits are vertical.
(Relative Intensity versus Wavelength in nm)
Exercise 1: Calibration
Procedures: Measure and record the angular positions of the four major first order (m = 1) lines of the mercury spectrum on each side of the central maximum. Refer to the diagram above for the wavelengths of these mercury lines.
Analysis: Using angle q as half the difference between the readings on each side, apply the grating equation:
to compute the average grating line spacing "d" and thus find the number of lines per
mm for this grating. Compare your value to the given value. These gratings are produced by making replica impressions on collodion
with a master grating. The collodion shrinks by 2 to 3% in the
process. Thus, your calculation may have slightly more lines/mm as a result of the shrinkage.
Then: Record the angular positions on each side of the central maximum for the green and blue lines in the second and third orders. Show that these lines also follow the predictions of the grating equation using m = 2 and m = 3.
Exercise 2: Hydrogen spectra
Procedures: Replace the mercury source with the HYDROGEN light source. The hydrogen spectrum observed is the visible part of the so-called Balmer series in hydrogen. The wavelengths of lines in this series are known to obey the Balmer formula (see Background information). Record the angular positions on both sides of the central maximum for the four prominent lines of hydrogen. The violet line is very weak, but with good technique it can be located. Use the grating spacing "d" determined above to calculate the corresponding wavelength for each of these four lines. The accepted values for these lines are: violet (410 nm), blue (434 nm), aqua (486 nm), and red (656 nm). Compare your values to the given values.
Analysis: Tabulate the reciprocal of your measured wavelengths and the reciprocal of n2 for each line using n = 3,4,5, and 6 for the red, aqua, blue, and violet lines, respectively. Using as large a scale as possible, graph the four 1/l values you measured as ordinates against the corresponding values of 1/n2 as abscissas. Choose the best straight line through the data using linear regression. Find the slope and the intercept for this best line through the data.
Discussion: This plot of your data should yield a straight line. According to the Balmer formula, what slope and y-intercept do you expect? Explain. Compare your experimental results (slope and y-intercept) with the theoretical expectations.