Hydrogen (H 2 ) Spectroscopy[DISCHARGE TUBE] Measurement of the Balmer Lines of the Hydrogen Spectrum

Transcription

1 Physics Department LAB D - 40 References: Hydrogen (H ) Spectroscopy[DISCHARGE TUBE] Measurement of the Balmer Lines of the Hydrogen Spectrum Halliday,Resnick, Walker, Fundamentals of Physics,John Wiley,013. Young and Freedman, University Physics with Modern Physics, Addison Wesley 004 Equipment: Spectrometers, Diffraction gratings, hydrogen spectral tubes, Gas Discharge Tube Power Supply. Introduction: Diffraction Grating. If plane waves of light fall at normal incidence on an opaque wall containing two arrow parallel slits a distance d apart, (Fig. 1); the light spreads out by diffraction upon passing through the slits. On a distant screen the overlapping beams from the two slits undergo interference, to produce a pattern of dark and bright fringes. At C, equidistant from the slits, all wave lengths of the light arrive in phase and interfere constructively to produce a central image or zero-order interference pattern having the same color as the original light. L P L+ nλ Light d θ C Wall Screen P Figure

2 Physics Department LAB D - 40 At some other point P which is at a distance L from one slit and L + nλ from the other (λ is some specific wavelength present in the light beam; n is an integer) there is also constructive interference, and a bright fringe appears with the color pertaining to that specific wavelength. At intermediate points distant L and L+ (n + 1) ( λ ), destructive interference occurs for that wavelength, λ. If the original light beam contains only a number of well separated spectrum lines, each with its own λ (as happens in atomic spectra), the pattern on the screen is a set of lines for each λ, repeated below C and also repeated for numerous values of the integer n. The pattern corresponding to n = 1 is the first-order spectrum (one above and one below point C); n = gives the second-order spectrum, and so on. It can be seen from Fig. 1 that the governing equation is nλ = d sin θ (1) provided d << L o, which is ordinarily true. Therefore measuring the angle θ, counting n from the zero-order pattern, and knowing d in advance, we can calculate λ, the wavelength of the pertinent spectrum line. Doing this for several values of n increases the precision. Greatly increasing the number of slits (to hundreds or thousands), while keeping them all parallel and spaced at regular intervals d, produces a diffraction grating. Original gratings are made by ruling a glass surface with machines of great cost and complexity (each ruling makes an opaque strip on the glass; the slits are the undisturbed parts between rulings). Except in work of the highest precision, replica grating are used. They are mace by depositing a film of collodion, or the like, upon an original grating, stripping off the film, and mounting it on glass. The accuracy of the spacing d is not quite so good on a replica, but ample for a great many purposes. Substituting a grating for the two slits of Fig. 1 still keeping the total size of the grating much less than L 0 does not alter the form of Eq. 1 for the bright fringes, or the meaning of the terms in it. The reason is that light traversing any pair of adjacent slits obeys Eq. 1, and all such patterns are superimposed (provided that L 0 is very great). The availability of a great number of slits, N, means that the bright fringes are far brighter than two slits can produce. Also, each bright fringe becomes much narrower, for at an angle θ differing only very slightly from one that satisfies Eq. 1 the light arrives from the various slits thoroughly out of phase. Therefore a grating far excels the two-slit arrangement, not only in brightness of the pattern but 14050

3 Physics Department LAB D - 40 in resolving power. That is, the bright fringes are so narrow that two of them corresponding to spectrum lines of only slightly different wavelength can be told apart or resolved. This is important, for many atomic spectra are extremely complicated. In this experiment the diffraction of light from a Hydrogen discharge tube by a grating is investigated. To make quantitative measurements of the diffraction patterns a spectrometer is used (see Fig. ). Source Collimator Grating Slit Lens C θ Telescope Fig. Light from the hydrogen discharge tube (from the central portion where the light is red) falls on a slit at the outer end of collimator tubes A lens at the end produces nearly parallel waves of light, to fall on the grating. To get the effect of very large L, spectrum lines are viewed through a telescope focused for objects at infinity. The telescope has a cross-hair, to be set on the spectrum line. The telescope, and in many spectrometers the collimator also, can he swung around in angle, and the angle of setting can be read from a precisely calibrated scale. NOTE: THE HYDROGEN DISCHARGE TUBE IS SHORT LIVED. TURN IT ON ONLY DURING ACTUAL READINGS. Theory: The pattern of wavelengths of light emitted by incandescent (hot and glowing) hydrogen gas is correctly predicted by quantum mechanics to be the following: 1 = 1 1 R λ n 1 n ()

4 Physics Department LAB D - 40 in which n 1 and n are integers. This is equation is derived in many introductory physics textbooks, (we have converted from frequency to wavelength since wavelength is what is actually measured). The energy levels are given by E n 4 mk e 1 n = n= 1,,3, (3) The constant R (called the Rydberg constant) is predicted to have the value 4 mk e 7 1 R = = m (4) 3 4πc In this experiment a diffraction grating is used to measure the four of these wavelengths that fall in the visible range. A diffraction grating is an optical component with periodic structure, which splits and diffracts light into several beams traveling in different directions. The orientations of beams depend on the both spacing of the grating and the wavelength of light. Procedure: Get some guidance from the instructor: the spectrometer is expensive, precise, and somewhat delicate. Most of the necessary adjustment will have been done before you come into the lab. Without changing them, you should only check the adjustments. Adjust the eyepiece of the telescope until the cross-hairs are sharply visible. To see that the telescope has been focused for infinity, view an object the length of the room away. Without changing previous adjustments, swing the telescope until you can see through both it and the collimator, you should now see the collimator slit sharply. Otherwise, ask your lab instructor for help in refocusing the collimator. Then set slit to a nearly closed position (you may have to open it little, later on, to get more light), and clamp the collimator in place. You will find the grating positioned such that the light from the collimator strikes it at normal incidence, Do not change its position. First calibrate the grating with a known wavelength from Hg or He. Generally the replica gratings used in the lab are listed as 300 lines/mm or 600 lines/mm but we can determine the grating distance d more accurately by measuring a known wavelength of a first order spectral

5 Physics Department LAB D - 40 line in our calibration standard. For example the blue mercury line has a wavelength of nm and the green line has a wavelength of nm, by measuring the angles accurately(many spectrometers read angles to 1 arc minute accuracy) at which these lines appear you can determine the grating constant d experimentally from equation 1. Turn on the hydrogen tube, keeping it on the axis of the collimator tube and close to the slit. CAUTION: We apply 5000 volts across the tube. Do not touch the ends of the tube. Measure and record the angular position of the telescope when it is set on the zero-order pattern. Then move the telescope slowly in one angular direction until you recognize the four visible Balmer lines deep violet, blue violet, blue green, and red. There will be many closely spaced faint lines in the background (due to molecular hydrogen and its ion), but the four Balmer lines are not hard to identify in the first-order Spectrum. You may only have time measure only the first-order pattern because of time limitations, if you have extra time you can measure also the second order pattern. To make accurate angle measurements, you will use a vernier scale. In Figure 3, you can see the vernier scale divisions are spaced slightly closer together than main scale divisions. In fact the length of 10 vernier scale divisions is one main scale division shorter than the length of ten main scale divisions. Note: depending on the particular spectrometer available the vernier scale you may be using can differ from the one used in the example below: in many spectrometers the main scale is marked in intervals of 0.5 degrees, and the vernier scale divides this into 30 more parts thus one tickmark of the vernier scale is worth 1/60 of a degree, or 1 arcminute! Figure 3. Using a vernier scale

6 Physics Department LAB D - 40 Rotate the spectrometer telescope, noting that as the pointer moves through one main scale division, each vernier scale mark lines up with a mark on the main scale in succession. As a result, you can read the scale to one tenth of a main scale division. To demonstrate this for yourself do the following: a. Find the main scale mark indicated by the zero or pointer. b. Get the precise subdivision of the interval by finding the vernier scaling that coincides with a main scale mark. What is the vernier of Figure 3 reading? Answer = 18.7 units. Play with the vernier until you can use it consistently. For more info on vernier readings see the lab handout How to read a vernier (available on lab wiki). When measuring the position of a line, place the crosshairs on the fixed edge of the spectral line (see Figure 4). Why should you use the fixed edge? Measure the angles for a spectral line on both sides of zero order. The two angles should be similar. How can you use these two angles to get a more accurate measurement? What error will tend to be corrected by this procedure? Figure 4. Method for measuring position of spectral lines. Take angular readings for all the Balmer lines on each side of the central pattern. For each λ, average the two angles θ on either side of the central zero-order pattern, before substituting in equation (1). Do this multiple times and get a statistical average (See Data and Calculations Section). Finally, compute values of wavelength for all four lines

7 Physics Department LAB D - 40 Data and Calculations: Measured Quantities: N/L (Lines per mm, directly read off label on grating). θ 1, θ : Angular positions of the telescope for which various lines are obtained on left & right. Calculated Quantities: D: Distance between lines of grating (m). θ L, θ R : Angular displacements of the telescope with respect to incident direction on left and right. θ: Mean angular displacement of the telescope (angular diffraction). λ: Wavelength of light observed. N/L= lines per mm. (Read off directly from label on grating). d=1/(n/l) x(0.001) (in meters) RUN # RED BLUE-GREEN VIOLET DEEP VIOLET n θ 1 RUN #1 θ L =180- θ 1 θ θ R = θ -180 θ 1 RUN # θ L =180- θ 1 θ θ R = θ -180 θ sinθ λ=dsinθ (m) 1/ λ (m -1 ) 1/(n ) In above table θ is determined by averaging angular displacements in both left and right directions for both runs. θ=[θ L (RUN1)+ θ R (RUN1)+ θ L (RUN)+ θ R (RUN)]/

8 Physics Department LAB D - 40 Analysis: (1) Calculate your measured wavelength for each of the four Balmer lines, from your angle measurements, using Eq. (). (In your notebook identify each one by its color.) Plot your values on a graph of 1 / λ vs 1 /( n ). According to Eq. (1), this should give a straight line of the form y = a + bx, with slope b = R and intercept a = R / n 1. GIVEN: For all four lines, the whole number n 1 (in Eq. (1)) is the same (unspecified) while n has the values 3, 4, 5 and 6. () Comment on whether your graph confirms the form of Eq. (1) that is, comment on how closely it appears to be a straight line. (Note that our giving the values of n is a shortcut. The values could be determined by trying, for the n values, progressions with different starting values until a straight-line graph results.) (3) From the slope of your graph determine your experimental value for R. Compare this 7 1 with the theoretical value R = m. (4) From the intercept of your graph, determine your value for the constant integer, n 1, in Eq. (1). (5) Predict the wavelength of the next line in the series. (6) At what angle should it occur? (7) Why didn't you see it? Optional Experiments: Other things you might try if you have time: Mystery gas: Your TA will give you some unknown gas in a spectral tube. Try to record and measure enough of its spectra so that you can confidently identify it. Identify the mystery gas by comparing your observed spectra with known spectra. What is the unknown gas? Be sure to record enough information to convince someone who was not present during your observation that you did in fact observe the gas you claim. Try other gratings : Verify that equation 1 is correct for different grating sizes. Several different grating d spacings are available in the lab

9 Physics Department LAB D - 40 Resolve the Sodium doublet : Sodium has a pair of closely spaced lines at nm and nm known as the sodium doublet. It is a challenging experimental exercise to try and resolve them in the second order diffraction pattern with the 300 line/mm or 600 line/mm gratings. Calculate the wavelength splitting. This fine structure splitting is due to a relativistic effect known as spin orbit coupling. Build your own digital spectrometer : Apart from the method of angular readings, you can also build your own digital spectrometer with a digital video camera, diffraction grating, and computer software available in the lab. You can then read the pixels on images and convert them to angles. In this way it should be easier to resolve close spectral lines and also determine the intensity distribution of the spectrum. For more details see the experiment lab handout Making Your Own Digital Spectrometer