The Grating Spectrometer and Atomic Spectra

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1 PHY 19 Gating Spectomete 1 The Gating Spectomete and Atomic Specta Intoduction In the pevious expeiment diffaction and intefeence wee discussed and at the end a diffaction gating was intoduced. In this lab most of the things leaned in the last expeiment will be applied to measue the wavelengths of the lines in the visible spectum of some elements. You will use a diffaction gating mounted on a spectomete to measue the intefeence maxima fo vaious wavelengths of light. Knowing the gating spacing, a measuement of the angle gives the wavelength. Theoy Gating Spectomete In the diffaction expeiment we showed that the conditions fo constuctive intefeence o maxima in the intensity fo a double slit ae: d sin Θ n = n λ n = 0, 1,, 3... (1) Hee "d" is the distance between the two slits, "Θ" is the angle between the incoming beam and the location whee we obseve the intensity (see Fig. 4 of expeiment 4) and "n" is the ode numbe of the intefeence. In the pevious expeiment no focusing lenses wee used because the finges wee obseved elatively fa away. The spectomete in this expeiment uses collimato lenses as the lage distance appoximations no longe apply and it has the added advantage that the angle "Θ" is measued diectly. The extension fom an N = double slit to a gating whee N is lage (seveal hunded slits/mm) is elatively staight fowad and Eq. 1 emains valid except that "d" is much smalle than it was fo N =. If the wavelength is fixed and n = 1, then the angle "Θ" at which the fist maximum appeas inceases as "d" is deceased (N is inceased). This is exactly what we studied in the last pat of the diffaction expeiment. The highe the gating constant, N, the lage the angle fo the maxima and the bette the wavelength can be measued. Question 1: When the finge patten behind the slits is viewed thee is always diffaction as well as intefeence, as we leaned in the pevious expeiment. The individual slits (lines) of the gating, howeve, ae usually so naow that the cental maximum of diffaction fo each slit sends light into a wide ange of angles. The gatings we use have 600 lines/mm. Assuming that the tanspaent and opaque segments ae of equal widths, calculate the locations of the fist diffaction minimum fo the appoximate limits of visible light (400 nm ~ 700 nm). Atomic Specta When the wavelengths of light emitted by excited atomic gases wee fist studied with pecision spectometes, scientists obseved that most of the light was emitted at a

2 PHY 19 Gating Spectomete few discete wavelengths. In this expeiment, you will obseve the discete emission lines of hydogen and a few othe atomic gases. These discete lines oiginate fom the deexcitation of atoms. That the lines ae discete was initially a geat supise because it implied that the valid enegies of electons in atoms wee also discete. This discovey led to the development of quantum mechanics, which is one of the foundations of moden physics. poton v electon To put this discovey in pespective, conside what one might have expected if Newton s laws govened the enegies of atomic electons. Atomic hydogen consists of a poton and an electon, which is bound to the poton by the attactive Coulomb foce e F C = k, whee k=8.99x10 9 N m /C, e=1.6x10-19 C is the magnitude of the electon and poton chages, and is the distance between them. If the electon is in a cicula obit about the poton, as shown above, Newton s second law, (mass)x(centipetal acceleation)=coulomb foce, implies: v e m = k () (The poton is so much heavie than the electon that one can assume that it emains stationay while the electon obits about it.) The potential enegy U() of the electon can be obtained by integating the Coulomb foce fom ; doing this povides: v U() = d F v C e = k. (3) Thus, the total enegy E=1/mv +U can be obtained fom Equations () and (3) and is given by: 1 1 e E = mv + U =. (4) It is negative because the electon is bound in the attactive Coulomb potential of the poton electic field.

3 PHY 19 Gating Spectomete 3 This deivation, which follows fom Newton s laws, implies that enegy of the electon is negative and has a abitay magnitude that depends only on the adius of the electon obit. This tuns out to be incoect fo small systems like an atom. In fact, the electon in a hydogen atom cannot be in any abitay obit, but only those obits with enegies given by the Boh fomula: 13.6 ev E n = (5) n whee 1 ev= 1.6x10-19 J and n = 1,, 3, 4... When an electon makes a tansition between an obit with a highe enegy (lage n i ) to one with a lowe enegy (smalle n f ), it emits a photon which caies off the enegy lost by the electon. This implies the enegies of photons emitted by a hydogen gas ae given by E 1 1 ph E i E f 13.6 ev n n = =. (6) f i The wavelength of the photon can be obtained fom its enegy by the Planck fomula: 140 ev nm λ = (7) E ph Question : Only tansitions that ae in the Balme seies with n f =, n i =3,4,5,6... ae visible to the human eye. Calculate the photon enegies and wavelengths of the Balme seies fo n i =3,4,5,6. These ae the wavelengths you will be tying to measue. Undestanding the Spectomete A spectomete is a device that allows pecise measuements of the angle θ, between two ays of light. Because the light fom the souce to be studied is often not a naow beam like a lase, it must be gatheed and focused by lenses. The light fom the souce is fist passed though a slit, which should be as naow as possible, consistent with letting enough light though to be visible. The light speads out fom the slit to the collimating lens, which makes it into a paallel beam (i.e. the slit is at the focal point of the lens). The paallel beam of light then stikes the "diffaction" gating (600 ± 4 lines/mm). Beyond the gating, the light tavels in vaious diections due to diffaction and intefeence. If light of a paticula wavelength and ode of intefeence heads in the diection of the telescope, it will be focused down to an image of the slit. If the spectomete is popely adjusted, the image falls on coss hais at the focusing ing. This image is then examined with an eyepiece (a shot focal length conveging lens) to see how closely the image coincides with the coss hais. If the telescope is fist used to locate the 0 th ode of intefeence (staight ahead) and subsequently a fist ode line, then the diffeence in the Venie eadings is the desied angle Θ i.

4 PHY 19 Gating Spectomete 4 Pat 1 In this pat we use the ed line of a hydogen lamp with a known wavelength (656.5 nm) as the souce. We wish to adjust the spectomete and to veify that we can actually measue the wavelength by measuing the position of the fist and second ode maxima. Befoe attempting to move the telescope, be sue that the locking scew is loosened. When you ae fine-setting on a line, the locking scew should be tightened and the final adjustment made with the fine-adjust scew (clockwise diection only). 1. Adjusting the eyepiece. Move the eyepiece in and out until, with you eye set fo distant vision, the coss hais ae in focus.. Adjust the telescope so that thee is no paallax between coss hais and slit image. This can be checked by moving you eye back and foth (fom left to ight). 3. Swing the telescope to the fist ode fo the ed line. Use the fine adjustment to make the line coincide with the coss hai and ead the Venie. Retun to the cental image, set the image on the coss hai and again ead the Venie. The diffeence in the eadings is the angle θ i. Now locate the fist ode image on the othe side of the cental image. How many highe odes can you locate? Estimate the accuacy of you angula eading. 4. Calculate the wavelength fom the position of the maxima and compae it to what you expect. Question 3: By using Eq. 1, deive an expession fo the eo in the measued λ assuming that the only expeimental eos ae in "θ" and "d". Use this expession to detemine which ode is the best one to use in measuing λ. Is it the highest o lowest ode numbe? Question 4: Assuming the eo in the gating constant "d" to be negligible and the eo in the angle measuement fixed (by the Venie scale), is it bette to measue λ with lage gating constant N o with small? Pat Measue in fist and highe odes as many hydogen lines as you can see. If the light is favoable you may be able to see fou (ed, blue-geen, puple and deep puple). The intepetation of these and othe lines led Boh to his model of the hydogen atom which pedicts a discete spectum of light emission athe than continuous. The spectum consists of thee distinct seies called Balme, Lyman and Paschen Seies. Each Seies epesents cetain enegy tansitions fom highe levels to a specific lowe level in the hydogen atom. The specific lowe levels ae n = 1 fo Lyman, n = fo Balme and n = 3 fo Paschen. As discussed peviously, only the Balme Seies tansitions ae in the

5 PHY 19 Gating Spectomete 5 visible ange of the spectum. The Lyman Seies is in the ultaviolet and Paschen Seies is in the infaed. Compae the wavelengths you measue with the expected lines fom the Balme Seies. Calculate the eo in you measuement of the wavelength. Pat 3 Choose anothe light souce fom the ones that ae available and measue the wavelength of a few of the stongest lines. Good examples ae Ne and Hg. Use a handbook (Handbook of Chemisty and Physics o anothe) to identify the pominent lines in the element you picked and compae them to you measuements. In deciding whethe you line coincides with the line of an element, you must have some estimate of the pecision of you measuement. You must also conside only the most intense lines in the tables. Question 5: Which intefeence maximum gives the most accuate esults? Does this agee with you answe to question?

6 PHY 19 Gating Spectomete 6

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the