Laboratory #29: Spectrometer

Transcription

1 INDIANA UNIVERSITY, DEPARTMENT OF PHYSICS, P309 LABORATORY Laboratory #29: Spectrometer Goal: Learn to adjust an optical spectrometer, use a transmission grating to measure known spectral lines of mercury, calibrate the grating spacing a. Measure the unknown lines of hydrogen, compare with the Balmer formula. Determine the index of refraction and the dispersion of a prism. Equipment: Spectrometer, grating, prism, and spectral discharge tubes. A. Getting to know the spectrometer, adjustment procedure A.. Principle The spectrometer consists of o light source, an telescope adjustable slit, and the collimator that produces a collimator parallel beam of light. This beam interacts with the grating or the prism that is mounted on the stage stage. The outgoing beam is observed with the telescope. The telescope rotates around a vertical axis; its angle can be read with a precision of 30 (arc seconds). A.2. Adjustment of the instrument Before you start any measurement, you need to adjust the instrument. Follow all steps in Appendix: Spectrometer Adjustment., and make sure that you understand the possible adjustments and the criteria that are used to make them. B. Calibration of the transmission diffraction grating Refresh your knowledge of diffraction on a grating: why does a grating disperse light into its spectral components? (See any textbook on introductory physics or on optics, e.g., [HEC02]). For a transmission grating at normal incidence, the deflection angle θ and wavelength λ are related by the equation m λ = a sinθ, () where a is the grating spacing and m is an integer, called the order (thus, each line in a spectrum makes multiple appearances for m =,2 etc.). The deflection angle is obtained from the current telescope angle minus the reference angle θ 0. The wavelengths of the four most prominent lines in mercury are listed in table. Measure, as accurately as possible, these lines in the st and 2 nd order diffraction spectrum, both on the left and the right of the collimator axis. Deduce the grating line separation, a i, from each of these 6 measurements. Obtain the grating spacing a and its error. 29 -

2 Color Purple Green Yellow Yellow 2 Wavelength nm nm nm nm Table : the four most intense, visible lines in the mercury spectrum. Listed are the wavelengths in dry air. C. The Balmer series in hydrogen The Bohr model predicts that the electron in the hydrogen atom exists in discrete energy states (see, e.g., [PRE9]). Since the electron is bound, these states have negative energy: n E n = hcr. (2) 2 Here, R = m - is called the Rydberg constant and n is the quantum number of the level. The lowest energy, E = -3.6 ev, corresponds to the ground state. Spectral lines correspond to transitions between these energy levels. The energy of a photon is given by E=hf, where f is the frequency of the radiation. The speed of light is related to frequency and wavelength λ by c=fλ. Thus, the wavelength is given by (/λ)=e/hc. With this, we find for the wavelength of a transition from level n i to level n f λ if = R 2 2 n f ni. (3) The lines in the hydrogen spectrum form groups, depending on the final level. The group with n f =, n i =2,3, is called the Lyman series, and the one with n f =3 the Paschen series. Here, we are concerned with transitions with n f =2, n i =3,4,, called the Balmer series. These transitions lead to radiation in the visible part of the spectrum. Mount the hydrogen spectral lamp in front of the collimator. Measure the deflection angles for all visible lines and deduce their wavelengths (with their errors). Plot (/λ i2 ) against (¼ /n i 2 ), starting with n i =3 (see eq.3). Fit a straight line through your data and determine the value of the Rydberg constant. D. Index of refraction of a prism Remove the diffraction grating and mount the equilateral triangle prism on the stage. Unlock the ring {3.} and raise the stage until the midplane of the prism is at beam height. Lock ring {3.}. Repeat step 4 of the adjustment procedure (see Appendix) to level the stage. The following measurements require the knowledge of the diffracting angle α of the prism. We can use the spectrometer to measure this angle. To this aim, place the prism on the stage, with its diffracting edge at the center of the stage. Then determine the angles Θ L and Θ R at which the image of the slit is reflected from the faces of the prism. The difference Θ L Θ R is twice the diffracting angle α. Θ L α Θ R 29-2

3 α α δ R δ L To study diffraction by the prism, shift the prism such that its center is at the center of the stage. Use the mercury spectral lamp. Observe the deflection angle δ for the green line. Rotate the stage until the deflection angle is a minimum. This corresponds to symmetric beam traversal (the angles at the entrance and exit faces of the prism are the same). In this case, the index of refraction n(λ) is given by sin ( α + δ ( λ)) 2 n ( λ) = (4) sin α 2 Carry out a measurement on both sides of the collimator axis: the deflection angle δ is then half the difference δ L δ R. Repeat this procedure for all visible lines in the mercury and hydrogen spectra, as well as for the yellow line in the helium spectrum. Make a plot of n(λ) versus λ. The dispersive power of a material is usually expressed in terms of the Abbė number V d. This number can be obtained from a measurement of the index of refraction at three different wavelengths. V d n = n F. (4) n The subscripts refer to prominent spectral lines. These lines are among the so-called Fraunhofer lines (see table 2). d C Fraunhofer name λ (nm) color origin element d yellow He F blue H C red H Table 2: wavelengths of some Fraunhofer lines E. References [PRE9] D.W. Preston and E.R. Dietz, The Art of Experimental Physics, John Wiley, New York 99. [HEC02] E. Hecht, Optics, Addison-Wesley, San Francisco, 2002 table 3: colors in the visible spectrum Color: violet blue green yellow orange red Wavelength (nm) :

4 Appendix: Spectrometer Adjustment. The first goal is to set up a parallel beam through the instrument. Refer to fig., Spectrometer Adjustments. To receive a parallel beam the telescope must be focused to infinity. Open a window to view a distant object. Release lock screw {.4} and focus the telescope to infinity with knurled ring {.3}. Turning the eyepiece assembly until the cross hairs are vertical. Re-lock screw {.4}. Focus the cross hairs with ring {.5}. To test for parallax move your eye sideways: the cross hairs should not move relative to the distant object. 2. Aim the telescope into the collimator {2}. Place a lamp in front of the slit {2.5} and close the slit jaws almost completely with screw {2.}. You should see a narrow vertical line. Release lock screw {2.4} and slide and rotate the slit assembly {2.5} until it is in focus and aligned with the vertical cross hair. Tighten screw {2.4}. The slit is now at the focal point of the collimator lens. Align the slit image with the center cross hair. a b Next we level telescope, collimator and stage. The figure on the left shows the stage seen from above with the three leveling screws {3.3}. Place the auto-collimation mirror (ACM, transparent glass plate) in position a on the stage. Release (counterclockwise) the ring {3.}, which locks the stage post. Slide the stage vertically until the center of the ACM is at the same height as the telescope axis. Re-tighten ring {3.}. Center the ACM on the stage Turn on the power to the telescope (switch {4.}). This illuminates the little cross that protrudes from the left into the field of view of the telescope. This cross is projected out of the telescope. When it is reflected from a plane that is normal to the telescope axis, it produces an image that appears at the secondary cross hair on the right (see figure on the right). Unlock the stage rotation (screw {3.4}) and the telescope leveling (screw {.2}). Rotate the stage until you see the cross image. Rotate the stage by 80 o until you see Slit image cross image the cross image again. If the two images are not at the same height, adjust the two leveling screws {3.3} on the right in the figure above. If the two images are not centered horizontally, adjust {.} to level the telescope. Repeat until the cross image appears at the right cross hair in both orientations of the ACM. Rotate the ACM by 90 o to the position b. Repeat the above procedure, but this time you only need to adjust the one remaining leveling screw {3.3}. When done, lock screw {.2}. Unlock {2.3} and level the collimator with screw {2.2} until the slit image is centered up-down and fix this adjustment with screw {2.3}. Telescope and collimator are now in line. Also, this line and the plane of the stage are orthogonal to the main axis of the spectrometer. 5. Place a mercury lamp in front of the slit and adjust the lamp position sideways until the image is brightest. Place the transmission grating in the center of the stage {3}. Rotate the telescope until the slit image is aligned with the center cross hair. Rotate the stage until the reflected cross is centered at the right cross hair (see figure). Lock the stage rotation {3.4}. The grating is now perpendicular to the incident light beam (the collimator axis), as well as the telescope axis. 29-4

5 The angle of the telescope in this position is the reference angle θ 0 relative to which all subsequent measurements are taken. Read this angle carefully. Angle readings are taken as follows. 6. How to read angles: There are two ports with little covers {4.2} on opposite sides of the main body {4}. Looking down into one of them, you see two scales. The outer scale is part of a ring that goes all the way around the table from 0 to The major tick marks are 0 apart. They are separated by two minor tick marks, 20 (arc minutes) apart. The inner scale is the vernier. First, note the angle of the minor tick mark on the outer scale that is just to the right of the zero (right end) of the inner scale (say, ). Now look at the inner scale. It goes from 0 to 20 in steps of 0.5 (or 30 ). Somewhere along that scale you will see a bright line that connects it to the outer scale. Note the position of that bright line on the inner scale (say,.5 ). The present angle of the telescope is the sum of the two values ( ). 29-5

6 Fig..: Spectrometer Adjustments