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Foglio1 Pagina 1 4/28/2015 12:58:39 PM Wavelength (nm) Absorbance 190 674 191 657 192 383 193 311 194 410 195 668 196 873 197 779 198 527 199 241 200 371 201 606 202 597 203 542 204 494 205 674 206 865 207 786 208 626 209 552 210 420 211 425 212 620 213 719 214 709 215 423 216 467 217 387 218 650 219 755 220 769 221 764 222 510 223 520 224 475 225 690 226 647 227 582 228 496 229 354 230 445 231 478 232 459 233 478 234 450 235 530 236 531 237 493 238 376 239 284 Foglio1 Pagina 2 240 322 241 283 242 362 243 352 244 331 245 427 246 475 247 524 248 471 249 386 250 489 251 452 252 525 253 454 254 488 255 499 256 344 257 372 258 350 259 353 260 364 261 352 262 362 263 362 264 348 265 370 266 410 267 393 268 333 269 381 270 369 271 441 272 477 273 512 274 526 275 547 276 524 277 454 278 418 279 401 280 395 281 514 282 560 283 560 284 666 285 807 286 1010 287 1134 288 1185 289 1257 290 1316 291 1302 292 1336 Foglio1 Pagina 3 293 1414 294 1589 295 1782 296 1948 297 2096 298 2165 299 2179 300 2150 301 2176 302 2118 303 2283 304 2362 305 2415 306 2370 307 2232 308 2307 309 2313 310 2446 311 2462 312 2454 313 2425 314 2336 315 2356 316 2301 317 2291 318 2301 319 2295 320 2306 321 2236 322 2201 323 2228 324 2223 325 2265 326 2286 327 2359 328 2446 329 2385 330 2326 331 2287 332 2250 333 2257 334 2260 335 2323 336 2352 337 2307 338 2242 339 2198 340 2190 341 2272 342 2316 343 2355 344 2310 345 2326 Foglio1 Pagina 4 346 2325 347 2279 348 2222 349 2163 350 2197 351 2155 352 2122 353 2076 354 2048 355 2006 356 1979 357 1946 358 1880 359 1812 360 1750 361 1703 362 1645 363 1579 364 1496 365 1429 366 1368 367 1314 368 1262 369 1201 370 1144 371 1090 372 1037 373 986 374 939 375 893 376 850 377 808 378 767 379 730 380 695 381 659 382 624 383 593 384 561 385 533 386 505 387 479 388 455 389 431 390 408 391 386 392 366 393 346 394 328 395 313 396 299 397 286 398 273 Foglio1 Pagina 5 399 260 400 248 401 237 402 225 403 215 404 205 405 197 406 190 407 184 408 177 409 171 410 165 411 162 412 158 413 155 414 150 415 147 416 146 417 146 418 146 419 145 420 145 421 146 422 146 423 148 424 149 425 151 426 153 427 155 428 158 429 161 430 165 431 168 432 173 433 179 434 185 435 190 436 195 437 202 438 209 439 217 440 225 441 234 442 242 443 250 444 257 445 267 446 277 447 287 448 298 449 306 450 316 451 328 Foglio1 Pagina 6 452 338 453 349 454 360 455 373 456 386 457 396 458 408 459 419 460 430 461 441 462 453 463 466 464 479 465 492 466 503 467 514 468 524 469 537 470 549 471 561 472 572 473 581 474 591 475 601 476 612 477 621 478 631 479 640 480 649 481 659 482 666 483 671 484 676 485 682 486 690 487 696 488 701 489 708 490 716 491 722 492 723 493 725 494 730 495 733 496 736 497 737 498 739 499 738 500 738 501 737 502 738 503 738 504 737 Foglio1 Pagina 7 505 733 506 731 507 729 508 724 509 720 510 716 511 711 512 705 513 699 514 695 515 690 516 684 517 677 518 671 519 664 520 656 521 649 522 642 523 634 524 626 525 616 526 609 527 599 528 .

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Foglio1 Pagina 1 4/28/2015 12:58:39 PM Wavelength (nm) Absorbance 190 674 191 657 192 383 193 311 194 410 195 668 196 873 197 779 198 527 199 241 200 371 201 606 202 597 203 542 204 494 205 674 206 865 207 786 208 626 209 552 210 420 211 425 212 620 213 719 214 709
215 423 216 467 217 387 218 650 219 755 220 769 221 764 222 510 223 520 224 475 225 690 226 647 227 582 228 496 229 354 230 445 231 478 232 459 233 478 234 450 235 530 236 531 237 493 238 376 239 284 Foglio1 Pagina 2 240 322 241 283 242 362 243 352
244 331 245 427 246 475 247 524 248 471 249 386 250 489 251 452 252 525 253 454 254 488 255 499 256 344 257 372 258 350 259 353 260 364 261 352 262 362 263 362 264 348 265 370 266 410 267 393 268 333 269 381 270 369 271 441 272 477 273 512 274 526 275 547 276 524 277 454 278 418 279 401
280 395 281 514 282 560 283 560 284 666 285 807 286 1010 287 1134 288 1185 289 1257 290 1316 291 1302 292 1336 Foglio1 Pagina 3 293 1414 294 1589 295 1782 296 1948 297 2096 298 2165 299 2179 300 2150 301 2176 302 2118 303 2283 304 2362 305 2415 306 2370 307 2232 308 2307
309 2313 310 2446 311 2462 312 2454 313 2425 314 2336 315 2356 316 2301 317 2291 318 2301 319 2295 320 2306 321 2236 322 2201 323 2228 324 2223 325 2265 326 2286 327 2359 328 2446 329 2385 330 2326 331 2287 332 2250 333 2257 334 2260 335 2323 336 2352 337 2307 338 2242 339 2198 340 2190 341 2272 342 2316 343 2355 344 2310
345 2326 Foglio1 Pagina 4 346 2325 347 2279 348 2222 349 2163 350 2197 351 2155 352 2122 353 2076 354 2048 355 2006 356 1979 357 1946 358 1880 359 1812 360 1750 361 1703 362 1645 363 1579 364 1496 365 1429 366 1368 367 1314 368 1262 369 1201 370 1144 371 1090 372 1037 373 986
374 939 375 893 376 850 377 808 378 767 379 730 380 695 381 659 382 624 383 593 384 561 385 533 386 505 387 479 388 455 389 431 390 408 391 386 392 366 393 346 394 328 395 313 396 299 397 286 398 273 Foglio1 Pagina 5 399 260 400 248 401 237 402 225
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1 student 4/28/2015 12:58:39 PM Foglio1 In the Laboratory 336 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org The electronic absorption–emission–fluorescence spec- trum of I2 in the gas phase has long been a staple in physical chemistry laboratory instruction on spectroscopy (1–7). It is relatively easy to record the I2 absorption spectrum with reso- lution sufficient to measure many band heads in the B ← X electronic spectrum, and thereby to characterize the vibrational structure in the excited B state. By extrapolating to the point where the B-state vibrational intervals ∆Gυ� vanish (Birge– Sponer extrapolation), students can estimate the photodisso- ciation limit of the B state and thence the dissociation energy of the ground X state. Apart from the latter indirectly ob- tained quantity, the absorption spectrum provides much less information about the ground state, since normally bands can be assigned only up to ground-state vibrational levels υ� = 2 or 3, higher levels being insufficiently populated at typical cell temperatures due to small Boltzmann factors. The I2 B ↔ X fluorescence spectrum nicely comple- ments the absorption spectrum and is thus often included in the teaching instruction on this transition. The ideal fluo- rescence spectrum is produced by exciting a single (υ�, J�) level in the B state and consists of the P( J� + 1) and R( J� − 1) rotational lines (which, by convention, are labeled with the lower rotational level J �) going to a large range of vibrational levels in the X state (6, 7). Fluorescence thus offers not only
information about the vibrational structure in the latter but also, with sufficient resolution, the rotational structure. Un- fortunately, the needs for suitable excitation sources and for a high-resolution spectrometer to record the spectrum make this experiment problematic in many teaching labs. The earliest descriptions employed 546.1-nm excitation from low-pressure Hg lamps, with photographic recording of the spectrum (1, 3); but such procedures can be tedious and time-demanding. The 514.5-nm line from an Ar-ion la- ser excites strong fluorescence; but unfortunately this source produces two main excited levels, (43, 12) and (43, 16), and the resulting fluorescence appears as narrow triplets, requir- ing high resolution for quantitative analysis. The widely avail- able He–Ne laser excites two well-characterized fluorescence series, from upper levels (6, 32) and (11, 127) (6, 7). The latter of these is particularly appealing, because the rotational doublet splitting (roughly proportional to J ) is large. How- ever, the absorption involves highly excited levels (υ� = 3 and 5, respectively), so with the readily available low-power He– Ne lasers, fluorescence intensities tend to be weak. Also, both upper levels may be excited together, depending on the op- erating characteristics of the laser (7). Such complications can make it difficult to complete the experiment “on schedule” in the teaching lab. For these reasons, I have generally treated this experi- ment as a “dry-lab”, done in conjunction with a demonstra- tion of laser-induced fluorescence excited by the Ar-ion laser and observed by students through a hand spectroscope. The data they then analyze are for the (6, 32) fluorescence, as re- corded at ∼1-cm resolution using a Raman spectrometer (6). In the latest edition of their laboratory text, Garland et al. (8) mention some of these laser sources and also suggest
an appealing alternative—the doubled Nd�YAG laser, which is now readily available at ∼5 mW power in the form of the green laser pointer (GLP). When I used such a pointer re- cently to demonstrate I2 fluorescence, I observed a remark- able behavior, which bodes ill for the use of GLPs to excite fluorescence spectra for quantitative analysis. At the same time, this behavior does offer a “flashy” and compelling dem- onstration of some important properties of both laser-induced fluorescence (LIF) and lasers. It nicely complements and re- inforces the instructional value of a demonstration of LIF using the Ar-ion laser with I2. In following paragraphs, I de- scribe this behavior and explain how it arises from temporal variations in the output wavelength of the GLP. This expla- nation is corroborated with high-resolution spectra taken as a function of time for several different pointers. I also de- scribe the procedures used to photograph fluorescence spec- tra at low resolution, in color, using a digital camera in combination with a hand spectroscope. Some of these spec- tra are included in this article; however they are much better appreciated in color, to which end I have provided supple- mentary material for online viewing, in the form of a Power- Point document. The Supplemental MaterialW also includes four QuickTime movies of the behavior in question and of related phenomena involving the fluorescence and emission spectra of I2 in the gas phase. What Happens and Why? When the beam from a green laser pointer is directed through a cell containing I2 vapor at low pressure, the beam transiting the cell is alternately bright and vanishingly dim. This behavior was observed for four different GLP models (the only ones examined, details below), so it is clearly a prop- erty of the typical GLP. For none of these did the intensity of the laser itself seem to vary much, ruling out that obvious explanation. The correct interpretation is twofold: (1) the
wavelength of the laser is varying with time and (2) at any time, the spectral purity of the laser is high compared with the inherent I2 linewidths (∼0.03 cm �1, dominated by Dop- pler broadening and molecular hyperfine structure; ref 9). Thus, as the wavelength of the laser changes with time, it alternately tunes itself into and out of resonance with lines in the absorption spectrum of I2. 1 Garland et al. (8) have included a figure (p 429) dis- playing the gain profile for a research-level doubled Nd�YAG laser, with prominent absorption lines identified. A much Laser-Induced Fluorescence in Gaseous I2 Excited with a Green Laser Pointer W Joel Tellinghuisen Department of Chemistry, Vanderbilt University, Nashville, TN 37235 Advanced Chemistry Classroom and Laboratory edited by Joseph J. BelBruno Dartmouth College Hanover, NH 03755 http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm
l In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 337 more extensive characterization of the I2 absorption in this region has been given by Forkey et al. (10), who have listed all the detectable lines in absorption from υ� = 0, 2, and 3 in a ∼2-cm–1 interval.2 In fact, I have succeeded in photo- graphing spectra that are clearly attributable to these three υ� levels. Results are illustrated in Figure 1. The confirma- tion of excitation from the two excited levels (υ� = 2 and 3) is the observation of two and three anti-Stokes lines, respec- tively (anti-Stokes meaning lines falling at wavelengths shorter than that of the exciting line). Spectra of green laser pointers confirm that the output varies progressively rather than randomly with time. It seems likely that this behavior relates to thermal changes in com- ponents of the optical cavity or power supply as the laser warms up under power. As one indication of this, if the laser is allowed to operate for several minutes and is then turned off and on again quickly, it appears to cycle rapidly through its previous tuning cycle, producing two or three fluorescence “flashes” before it stabilizes again near its performance when turned off. In one particularly dramatic instance of such be- havior, through the spectroscope I observed in repeated fash- ion spectra like those illustrated in Figures 1B and 1C, with a cycle time of ∼2 s.3 The logic for high spectral purity is probabilistic in na- ture. The I2 B ← X absorption spectrum is so dense in this region that excitation at random wavelengths will produce
strong fluorescence about half the time. For example, in early attempts to measure the continuous absorption between the lines of the B ← X system using atomic line sources, discrete absorption was evident for all but three of a dozen Ar atomic lines (11). In a later, more extensive study, almost all mea- surements had to be corrected for discrete absorption (9). When the present fluorescence “turns off ”, the beam can com- pletely disappear in a dark room; from observations of the strong fluorescence through neutral density filters, this means a drop in fluorescence intensity by a factor of at least 104. If the GLP emitted light in broad lines or in too many different modes, such “off ” behavior would be improbably rare. Green laser pointers operate on a doubled transition of Nd doped in a host crystal. The latter is often YVO4 (yttrium orthovanadate) instead of YAG (yttrium aluminum garnet, Y3Al5O12), because YVO4 is said to better couple optically with the 808-nm radiation from the laser diode pump (12). The IR laser output at 1064 nm is doubled in a crystal of KTP (potassium titanyl phosphate, KTiOPO4), which is contained within the laser optical cavity, where it is exposed to the high optical powers needed to achieve frequency doubling. The IR output is blocked with a filter, leaving only 532-nm radia- tion in the output beam.4 The lasers are said to operate in the single transverse optical mode TEM00. Longitudinal modes occur at frequency intervals of c �2d, where c is the speed of light and d is the length of the optical cavity (13). For the typical cavity lengths of ∼5 mm (12), this translates into a wavenumber spacing of ∼1 cm�1 in the IR fundamental, 2 cm�1 in the doubled visible output. Since the Nd gain profile is only several cm�1 wide, it is reasonable for the GLP to oper- ate on only one or two modes, as required probabilistically
for the observed “off ” behavior of the fluorescence. Green laser pointers may operate either pulsed or CW (continuous wave). Pulsed operation can be demonstrated by sweeping the beam across a wall rapidly, giving a “dashed- line” display. Spectra of Green Laser Pointers Spectra were recorded at high dispersion (0.04 nm�mm) for three of the pointers (numbers 1, 2, and 4), revealing a surprising range of behaviors, with each displaying a spec- trum clearly distinct from those of the others. (Lasers are iden- tified in the Materials section.) Spectra were recorded with short exposures (0.3–1 s), slits almost closed (∼2 µm), and neutral density filters to further attenuate the light so as not to saturate the CCD array detector. For all three GLPs, some spectra were recorded in time sequence (< 1 min apart). Typi- cal results are illustrated in Figures 2 and 3. Figure 2 shows a behavior more complex than antici- pated. The strong modes are spaced ∼2.7 cm�1 apart, imply- ing a cavity length of 3.6 mm. However, there are other peaks, usually weak, that do not fit this pattern. The extra peaks do not appear to be due to different transverse modes; these Figure 1. Calibration and I2 fluorescence spectra for excitation with a green laser pointer: (A) reference spectra from Hg [bright green (546.1 nm) and yellow (577.0 and 579 nm)], Ne (red, from He– Ne laser at 632.8 nm), and from the green laser pointer (532 nm). Spectra (B–D) are I2 fluorescence spectra recorded at different times. (B) and (C) are dominated by fluorescence excited mainly from
υ� = 0 and 2, respectively; (D) includes a detectable component ex- cited from υ� = 3 (note “blue” line at top). Figure 2. Representative spectra for GLP 4, as recorded for differ- ent “on” times and battery powers. These spectra were calibrated with Ne lines at 533.0778, 534.1094, and 534.3283 nm (stan- dard air), giving 532.05(1) nm or 18790.0(4) cm�1 for the zero point on the wavenumber scale. http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-000.jpg&w=215&h=120 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-001.jpg&w=234&h=131 In the Laboratory 338 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org would be expected to vary spatially in the laser spot, but no differences were observed in spectra obtained by directing different regions of the spot onto the spectrometer slit. Time sequence spectra for this GLP showed a smooth progression of changes, with modes drifting to lower wavenumber ac- companied by slow intensity changes. With fresh batteries, the longtime pattern was typically one strong mode and ei- ther one somewhat weaker mode (∼50% of main) or two much weaker modes (each ∼5% of the main mode) on ei- ther side.
Figure 3 shows the much different output of the other two GLPs. GLP 1 displayed multiple peaks displaced con- siderably to the blue (spectrum C) when operated with weak batteries. On the other hand, with fresh batteries, only one of the many recorded spectra (D) showed more than one peak, and that an order of magnitude weaker than the main one. It is hard to interpret the peak spacing in spectra C and D in terms of longitudinal modes. Similarly, the mode pat- tern in the spectra of GLP 2 (A and B) is not simple; and in this case the pattern is similar for weak and fresh batteries and shows many peaks of significant intensity. Not surpris- ingly, the fluorescence beam excited by GLP 2 never weak- ened to the vanishing point in a dark room. It is noteworthy that GLP 1 operated CW while the other two were pulsed. To check on the possibility that the single-mode character was tied to this property, I recorded spectra for another CW GLP (not used in the fluorescence experiments) but found its spectrum more like that of GLP 4 in Figure 2. LIF in the Gas Phase: What Else Is Required? The green-laser-induced fluorescence dramatically illus- trates the prime condition necessary for LIF: the radiation must be absorbed by the fluorescing species. That this con- dition is necessary but not sufficient is nicely illustrated by directing the same laser beam through a cell containing a sig- nificant pressure of another gas in addition to the I2, as shown in Figure 4. I2 in its B electronic state is known to be de- stroyed by collisional predissociation (14), I + I + I2I2(B) + I2 (1)
and I + I + Ar I2(B) + Ar (2) These processes represent radiationless decay of the B state, so the fluorescence is said to be quenched. Actually, the beam can be perceived weakly in a high-pressure cell like that shown in Figure 4, since the quenching is not 100% efficient. And interestingly, this weak beam seems less subject to the tem- poral variations of the exciting laser. This behavior, which has not yet been quantified, could be a consequence of pres- sure broadening of the I2 absorption lines, leading to a re- duced specificity in the excitation process, due to more extensive overlap of the absorption lines. So I2 visible fluorescence requires absorption into the B state at low pressures, in the absence of quenching gases. Is that sufficient? The answer is “No,” as is illustrated by Fig- ure 5, which shows several beams from an Ar-ion laser oper- ated multiline, directed through the same low-pressure I2 cell. Only the longest wavelength line, at 514.5 nm, excites fluo- rescence (spectrum in Figure 6). With a power meter, one can confirm that the other wavelengths are absorbed (9). However, all but the “green” line excite I2 above the B-state photodissociation limit, resulting in direct dissociation (hence no fluorescence), I + I*I2(X) + hν (3) where I represents (as before) the ground state (2P3/2) of the atom and I* the spin-orbit excited state (2P1/2). 5 Figure 6 includes the spectra produced by exciting from υ� = 0 at two different wavelengths, from which it is clear
that the wavelength range of the fluorescence is determined firstly by the excitation wavelength of the source.6 This seems Figure 3. Spectra recorded for GLP 2 (A and B) and 1 (C–E). The wavenumber calibration is the same as in Figure 2. Figure 4. A GLP beam traverses (right to left) two cells containing I2 vapor at room temperature. The cell on the right contains only I2, while that on the left has ∼500 torr Ar in addition. The laser was positioned ∼7 m from the cell to give a less focused beam. http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-002.jpg&w=229&h=123 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-003.jpg&w=191&h=143 In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 339 at odds with comments made in a recent discussion of fluo- rescence and scattering in this Journal (15), in which the fluo- rescence wavelength was stated to be independent of exciting wavelength. The difference is that the present gas-phase fluo- rescence spectra are obtained under essentially collision-free conditions in the gas, while the fluorescence spectra discussed in ref 15 were observed in solution, where internal relaxation occurs in the excited electronic state prior to emission. In
that sense the present fluorescence spectra are more like the resonance Raman spectra described in ref 15; in fact workers with a primary background in Raman spectroscopy have of- ten referred to such resonance fluorescence spectra in exactly that way, as resonance Raman spectra.7 The intensity patterns in Figures 1 and 6 reflect the vary- ing Franck–Condon factors (FCFs) for the transitions, since the intensities are proportional to the FCFs (6). Figure 7 shows these FCFs for υ� = 43 and 32, the latter being the lowest υ� level excited at 532 nm (10). Note especially the strong alternation in FCFs for the latter in the region υ� = 7–15—a behavior that is clearly evident in the spectra of Fig- ures 1B and 6B. Other Spectroscopic Processes in Gaseous I2 The phenomena discussed above all involve the B–X transition in I2, which occurs conveniently in the visible spec- tral region. However, I2 has a rich emission–fluorescence spec- trum at shorter wavelengths, involving a number of other electronic transitions, some of which can be illustrated easily as a part of a demonstration of LIF in the B–X system. Thus, for example, if a Tesla coil is discharged in contact with the two cells shown in Figure 4, the resulting optical emissions are radically different (see the Supplemental MaterialW): The low-pressure cell gives a weak, yellowish-white glow, which on spectral inspection can be seen to display many of the same B–X bands that are observed in absorption (16). On the other hand, the high-pressure cell now displays an in- tense blue emission. Both cells show strong emission in the UV, but again with radically different spectra. The low-pres- sure spectrum is dominated by an extensive series of diffuse bands known historically as the McLennan bands (17, 18), while the high-pressure cell shows several discrete band sys- tems (19). Similar (but weaker) spectra can be excited by a
low-pressure mercury discharge lamp like that used to ob- tain the calibration spectra in Figures 1 and 6. The states responsible for the optical emissions in the UV are of ion-pair character and lie at much higher energies than the valence B state (20). Their behavior in the presence of inert gases (rare gases, N2) is in stark contrast with that of the B state. The low-lying ion-pair states are practically im- mune to quenching, permitting the excited molecules to re- lax into near-thermal υ and J population distributions prior to emission. The resulting fluorescence–emission spectra are relatively insensitive to excitation process, making this case more like that described for fluorescence in (15). This be- havior—immunity to collisional quenching—is also the hall- mark of the rare-gas halide (RgX) excimer lasers, which operate on similar charge-transfer electronic transitions. In- deed, the strongest transition in the high-pressure I2 emis- sion spectrum, known as D� → A�, was long ago made to lase in experiments like those which led to the development of the RgX lasers. Figure 5. Several beams from an Ar-ion laser operated multiline transit the low-P I2 cell. The beams are dispersed by a prism lo- cated ∼25 cm to the right of the cell, the window of which is viewed from the inside, looking through the cylindrical wall. Fluorescence is excited by only the beam at 514.5 nm. The spots on the en- trance window to the left of the fluorescence beam mark the points of entry for beams at 501.7 nm, 496.5 nm, 488.0 nm, and 476.5 nm, from right to left. The lower spots are from reflected beams, and the faint green and blue lines are “ghost” images. Figure 7. Franck–Condon factors for the lowest υ� level excited
at 532 nm and for the υ� = 43 level excited at 514.5 nm. Figure 6. I2 fluorescence spectra excited by the Ar-ion laser at 514.5 nm (A) and, for comparison, spectrum (B) from Figure 1. Reference spectra are from Hg and (in A) from the laser, at wavelengths noted in Figure 3 (except missing the last). http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-004.png&w=215&h=138 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-005.jpg&w=191&h=143 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-006.jpg&w=232&h=188 In the Laboratory 340 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org Observing and Photographing the Fluorescence Spectra The reader may have noticed that the fluorescence beam in the low-pressure cell in Figure 4 appears as an elongated cone, with intensity clearly diminishing with distance from the entrance window. The laser light is absorbed so strongly
that it is significantly attenuated by the time it exits the cell. For absorption near the centers of lines originating in υ� = 0, this attenuation is much greater than the ∼25% one might predict from the average molar absorptivity of ∼800 L mol�1 cm�1 (9), for a 10-cm cell at P = 0.3 torr (the approximate vapor pressure of I2 at room temperature). On the other hand, for observing the fluorescence spectrum, it is good to use a disperse laser beam (as in Figure 4) rather than a focused one, and then to view the fluorescence near the entrance window, where it is most intense. Lasers that operate single mode with high spectral purity can interact with only small velocity groups of molecules, so a focused beam can effectively de- plete the population of absorbers as it “burns” its way through the cell. The result is reduced absorption and fluorescence (and violation of Beer’s Law, which assumes constant con- centration of the absorbing species). In essence, this phenom- enon is the basis of one method of sub-Doppler spectroscopy, known as hole-burning. The gross attenuation of the beam seen in Figure 4 is also a dramatic illustration of the origins of nonlinear behavior in analytical fluorimetry: when the fluo- rescence intensity is observed some distance from the entrance window, it will first rise with increasing pressure of absorber, then decline as the absorption attenuates the exciting light before it reaches the observation region. Spectra like those shown in Figures 1 and 6 are readily observed “live” using a hand spectroscope; however, record- ing them required much trial-and-error work with a digital camera. This task provides a golden opportunity to get ac- quainted with the more subtle options in the camera’s menu. A tripod is necessary, since exposures of 1–8 s may be needed. It is helpful to use a bright source to achieve proper align- ment of the source, spectroscope, and camera. The fluores- cence excited by the Ar-ion laser can be bright enough for this purpose (depending on the laser power), but that pro-
duced by the green laser pointer probably will not be. I used a mercury Pen-Ray lamp, the strong green and yellow lines from which could be seen easily on the camera’s LCD moni- tor at lowest magnification. I could then zoom in on these lines, adjusting the alignment to maintain good images. Manual focus is needed for this operation; about 60 cm was the optimal stated distance in my case, though the lens of the camera was actually only about 1 cm away from the eye- piece of the spectroscope. Manual settings for both the aper- ture and shutter speed are also required, and I found it necessary to operate at largest aperture (F2.8) to obtain good image quality. For the longer exposures it is also necessary to invoke the “noise reduction” option. Alternatively one may use the “night scene” setting, which automatically employs noise reduction. Finally, I had to tweak the white-balance to render the yellow Hg doublet as yellow; an enhancement of the red component sufficed. However, I was never able to record a full spectrum with the nuanced differences that are obvious to the eye, for example, in the several laser lines to shorter wavelength from the green line in the Ar-ion laser output. In part, this is likely a problem of color/intensity in- terdependence, which the eye can handle much better than the camera. Materials I observed the on–off behavior for all four of the laser pointers I tried: (1) from Sean and Stephan Corp. (Taiwan); (2) Model GP4 from Limate Corp. (Taiwan); (3) AltasNova (Mead, WA and Taiwan); and (4) Beta Electronics (Colum- bus, OH). All of these pointers are stated to be less than 5 mW, but some evidently push this limit harder than others, judging from differences in the observed fluorescence inten- sities. Pointer 1 appeared to be best overall. Pointers 1 and 3 appeared to operate CW, while pointers 2 and 4 were pulsed,
with a much faster repetition rate for pointer 2. For pointer 1 the stated wavelength was 532 ± 10 nm, while that for pointer 2 was 500–550 nm! However, as shown above, all operate near the stated doubled Nd wavelength of 532 nm (12). The spectra illustrated in Figures 2 and 3 were obtained using a Jobin–Yvon 1.5-m spectrometer equipped with a 3600-groove�mm holographic grating and a CCD array (Photometrics CCD9000, 27-mm pixel width) as detector. The reciprocal dispersion in this region was obtained by re- cording several Ne lines (noted in Figure 2) from a discharge source. The spectroscope contains a prism as dispersing element and has an adjustable slit, which was set as narrow as pos- sible for recording the spectra. When focused, with narrow slits, this instrument easily resolves the Hg yellow lines, which are separated by 2.0 nm. The specific manufacturer’s infor- mation has been lost, but a very similar spectroscope is sold by Edmund Industrial Optics for less than $300. Cenco (Sargent–Welch) offers a prism model with adjustable slit and focus for less than $60, but the resolution is not stated. An Olympus Camedia model C-4000 (3× optical, ∼10× total zoom) was used to record the spectra. Procedures and settings have already been described, and similar procedures should work for other digital camera models. The mercury discharge source used for alignment and for reference–cali- bration was a Pen Ray lamp (Ultra-Violet Products, Inc.). The Ar-ion laser was a model Innova 300 from Coherent (15 W maximum power all lines). The Tesla coil was a model BD-20 from Electro-Technic Products. The cells were silica and were charged with I2 (and Ar, for the high-P cell) on a vacuum line and sealed off with a torch. Hazards
It is never good to stare directly into any laser, includ- ing the Class IIIa devices that are approved for laser point- ers. However, numerous studies have shown that momentary exposure to such light is no sight hazard, and certainly not at the distances involved in purported cases of airplane “spot- ting” that made news not long ago (21). Ar-ion lasers, on the other hand, can operate at powers that represent great sight hazard, so any use of the 514.5 nm line from such a laser should be carefully planned so that observers are not exposed to either direct or reflected beam light. The Hg Pen- Ray lamp produces strong UV light at 253.7 nm, which can damage the cornea of the eye. Both glass and plastic safety glasses block this radiation, but side exposure should be avoided, also. Operation of the source at a distance of 1 m http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 341 or more for only short times (1–2 min) should result in mini- mal exposure, with or without safety glasses. The Tesla coil output is very high voltage (up to 50 kV) but also very high frequency, so the spark is not dangerous (22). Instructors can “wow” students by boldly taking a ∼0.5- inch spark from the tip, though this is more comfortably done if the spark is directed to a coin held between thumb and forefinger.
Both the Tesla coil and the Hg discharge lamp produce ozone at detectable levels (by smell), so should be left on only while needed. Summary A green laser pointer can be used in a “flashy” demon- stration of laser-induced fluorescence in the gas phase, by di- recting the beam of the laser through a cell containing I2 at its room temperature vapor pressure. The demonstration is a good one to provoke discussion, and the explanation of the on–off behavior provides valuable insight into the require- ments for LIF and the properties of lasers. On the other hand, if this source is to be used to record fluorescence spectra for quantitative analysis, it will be necessary to use an array-type detector and to be aware that the time-integrated spectrum may contain contributions from numerous (υ�, J�) levels. Under the assumption that it is changing cavity tem- perature that is responsible for the varying wavelength of the laser, it is conceivable that one can modify such a pointer to permit control of its temperature and hence its wavelength. I am looking into this possibility.8 WSupplemental Material Color versions of Figures 1, 4, 5, and 6 and four QuickTime movies illustrating the temporal behavior of the fluorescence and of the discharges obtained by bringing the low- and high-pressure cells into contact with an operating Tesla coil are available in this issue of JCE Online. Acknowledgment I thank Tim Hanusa for providing two of the green la- ser pointers and Mike Bowers and Laura Swafford for help
setting up and operating the Ar-ion laser. Notes 1. This behavior is so astounding to observers that it readily elicits hypothetical explanations. In my experience on demonstrat- ing the phenomenon to a number of students and faculty, few of these are close to reality. 2. This wavelength region corresponds to a Franck–Condon “hole” in the absorption from υ� = 1, so no lines from this level are listed. 3. After about two minutes of operation, Pointer 1 produced a particularly intense spectrum that was dominated by excitation from υ� = 0. Then in the next minute it slowly drifted off this ex- citation (or excitations) and onto one involving predominantly υ� = 2. When the laser was clicked off and on again, it cycled back to the latter spectrum in about two seconds, displaying the former spectrum for about 1 second in the process. 4. Reference 12 includes many very instructive photographs and diagrams of green laser pointers and their components, in ad- dition to the diagram cited in the reference. 5. There is also a weaker continuous absorption in this same spectral region, designated 1Πu ← X, which yields two ground- state I atoms via photodissociation (9, 11). However, this transition never yields fluorescence and is in no way sensitive to the B-state
photo- dissociation limit. 6. It is sheer coincidence that the 532-nm wavelength so nearly coincides with the third line in the fluorescence spectrum excited at 514.5 nm: the energy difference in the two wavelengths just happens to match closely the energy difference between υ� levels 0 and 3 in the X state. This is perhaps unfortunate, as it suggests a “symmetry” in nature that is not there. 7. The weak fluorescence that persists at high pressures shows less dependence on exciting wavelength, as the surviving B- state molecules have mostly relaxed into low υ� levels. This situation is thus closer to that described for fluorescence in ref 15. 8. Additional spectra recorded at lower dispersion show that GLP 1, which appears to operate single mode in Figure 3, actually operates on one or two additional modes spaced 2 nm (70 cm�1) apart most of the time. Such broad spacing is hard to explain from expectations for both the gain profile and the cavity length. Literature Cited 1. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw–Hill: New York, 1996; pp 425–434. 2. Stafford, F. E. J. Chem. Educ. 1962, 39, 626–629.
3. Steinfeld, J. I. J. Chem. Educ. 1965, 42, 85–87. 4. Hollenberg, J. L. J. Chem. Educ. 1970, 47, 2–8. 5. D’alterio, R.; Mattson, R.; Harris, R. J. Chem. Educ. 1974, 51, 282–284. 6. Tellinghuisen, J. J. Chem. Educ. 1981, 58, 438–441. 7. Muenter, J. S. J. Chem. Educ. 1996, 73, 576–580. 8. Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 7th ed.; McGraw–Hill: New York, 2003; pp 423–432. 9. Tellinghuisen, J. J. Chem. Phys. 1982, 76, 4736–4744. 10. Forkey, J. N.; Lempert, W. R.; Miles, R. B. Appl. Opt. 1997, 36, 6729–6738. 11. Tellinghuisen, J. J. Chem. Phys. 1973, 58, 2821–2834. 12. Goldwasser, S. M. Sam’s Laser FAQ. http://repairfaq.ece. drexel.edu/sam/laserssl.htm#sslafgl. See especially http://repairfaq. ece.drexel.edu/sam/dpss1.gif for a diagram of the optical cavity (both accessed Nov 2006). 13. Svelto, O. Principles of Lasers; Plenum: New York, 1976. 14. Masiello, T.; Vulpanovici, N.; Nibler, J. W. J. Chem. Educ. 2003, 80, 914–917. 15. Clarke, R. J.; Oprysa, A. J. Chem. Educ. 2004, 81, 705–707. 16. Singh, S. M.; Tellinghuisen, J. J. Mol. Spectrosc. 1973, 47, 409–419. 17. McLennan, J. C. Proc. Roy. Soc. 1913, A88, 289–291. 18. Tellinghuisen, J. Chem. Phys. Lett. 1974, 29, 359–363. 19. Guy, A. L.; Viswanathan, K. S.; Sur, A.; Tellinghuisen, J. Chem.
Phys. Lett. 1980, 73, 582–588. 20. Mulliken, R. S. J. Chem. Phys. 1971, 55, 288-309. 21. The Register. http://www.theregister.co.uk/2005/01/06/ laser_man_letters/ (accessed Nov 2006). 22. McKeever, M. R.; Sur, A.; Hui, A. K.; Tellinghuisen, J. Rev. Sci. Instrum. 1979, 50, 1136–1140. http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://www.jce.divched.org/Journal/Issues/1996/Jun/abs576.htm l http://repairfaq.ece.drexel.edu/sam/laserssl.htm#sslafgl http://repairfaq.ece.drexel.edu/sam/laserssl.htm#sslafgl http://repairfaq.ece.drexel.edu/sam/dpss1.gif http://repairfaq.ece.drexel.edu/sam/dpss1.gif http://www.jce.divched.org/Journal/Issues/2003/Aug/abs914.ht ml http://www.jce.divched.org/Journal/Issues/2004/May/abs705.ht ml http://www.theregister.co.uk/2005/01/06/laser_man_letters/ http://www.theregister.co.uk/2005/01/06/laser_man_letters/ Morse Oscillators, Birge-Sponer Extrapolation, and the Electronic Absorption Spectrum of l2 Leslie Lessinger Barnard College, New York. NY 10027
The Absorption Spectrum of 12 The visible spectrum of gaseous I2 affords a most inter- esting and instructive experiment for advanced under- graduates. Analysis of the electronic absorption spectrum, measured at intermediate resolution (vibrational progres- sions resolved; rotational fine structure unresolved), was described in this Journal by Stafford (I), improving the ini- tial presentation of Davies (2). (Assignment of vibrational quantum numbers u' to the bands of the excited B elec- tronic state of 12 was corrected in two key papers, by Stein- feld et al. (3) and by Brown and James (41.) Improvements both in experimental techniques and in data analysis ap- plied to the 12 absorption spectrum continued to be pre- sented in this Journal (5-9). Instructions for the I2 experi- ment also appear i n several physical chemistry laboratory manuals (10-14). Steinfeld's excellent spectroscopy text- book (15) often reproduces parts of the actual spectrum of 12 as examples. Birge-Sponer extrapolation is one method often used to analyze the visible absorption spectrum of 12. All measured differences AG between adjacent vibrational energy levels u' + 1 and u' are plotted against vibrational quantum num- ber u'. The differences between all adjacent vibronic band head energies observed in the usual undergraduate labo- ratory experiment on 12 give a nicely linear Birge-Sponer plot. Thus, these vibrational energy levels (which do not include 20 to 30 levels a t very high values of u') of the ex- cited B electronic state can be accurately described as those of a n anharmonic oscillator with a single anharmo- nic term. This, in turn, implies that the potential well gov- erning the vibrations up to the maximum u' observed in these experiments is closely approximated by a Morse po- tential function.
What is the Correct Procedure for BirgeSponer Extrapolat~on? Unfortunately, many sources, including some in the ref- erences, give incomplete, ambiguous, or self-contradictory descriptions of the Birge-Sponer extrapolation method. I N T E R N U C L E A R DISTANCE + Figure 1. Morse potential curve and vibrational energy levels. Equations for determining spectroscopic parameters pre- sented in different sources are not consistent with each other. Several erroneous presentations are common. This paper aims to give a complete, correct exposition of Birge- Sponer extrapolation for the special case in which the plot derived from the observed data is taken as exactly linear. If extrapolation beyond the observable differences is as- sumed to be strictly linear in both directions, and particu- larly toward the dissociation limit, then a Morse function would exactly describe the vibrational potential for the di- atomic oscillator. For real molecules this is not strictly cor- 388 Journal of Chemical Education rect, but the Morse potential is oRen a good appmxima- tion. Also. the analvsis of actual molecular s ~ e c t r a l data in terms of t ~ s exactiy soluble problem in quakum mechan- ics should be clear and unambirmous. .. A schematic diagram of a Morse potential, with its quan- tized vibrational enerw levels. and ~ictorial definitions of various terms used inyhe following isc cuss ion, is shown in
Figure 1. Morse Oscillators In 1929, Morse (16) introduced a convenient two-param- eter analytical function to approximate the shape of the anharmonic potential energy curve for a diatomic molecu- lar oscillator: The Sehrodinger equation for a particle of reduced mass p i n this potential can be solved exactly. The energy levels are given by With the conventional spectroscopic units of wavenum- ber (cm-'1, these quantized levels are oRen written in the following form. There are no higher order terms. The energy levels of a Morse oscillator are given by a harmonic oscillator term plus a single anharmonic correction term, which is pre- cisely what is required for the linear Birge-Sponer extrap- olation procedure to be valid. D. is t h e depth of t h e vibrational potential well; P governs the curvature a t Re, and thusthe force constant k,. k. = wep2 ( 4 ) D. and p determine the fundamental vibration frequency, v. = cw., and the anharmonicity, v&, = ewe: Once we and we are found from analysis of the spec- trum, the well depth D. for the Morse oscillator can be de- termined exactly: The zero-point energy E(, = ,,, of the Morse oscillator is given by
Subtracting E(, . o, from D. gives the bond dissociation energy DO: Do (0.- 0&J2 - = hc 4w& (9) A key feature of the vibrational energy levels in a Morse potential is that the number of bound states is finite; the integer vibrational quantum numbers u for the bound states have a maximum possible value u,, governed by the following inequality Moreover, in order to correspond to a finite, normalizable wave function, the highest quantized vibrational energy level must be less than the well depth; E(v,,) cannot be exactly equal to D.. The difference AED between the disso- ciation limit D. and the highest quantized level E(v,,) must obey a related inequality: Both inequalities result from the condition put on all phys- ically meaningful bound states: the vibrational wave func- tion must vanish as the internuelear distance R eoes to - infinity ( I 7). As a Morse oscillator approaches dissociation, the den- sity of states does not increase without limit, and the en- ergy level spacing does not approach zero, in contrast to the pattern in certain other bound systems, for example, the hydrogen atom. Finally, there is no necessary relation- ship between the values of the two parameters D. and P defining any Morse potential. Thus, w. and a&. can also have any arbitrary ratio greater than 1. BlrgeSponer Extrapolation Both we and w g e are often obtained from spectroscopic
data by a graphical method introduced by Birge and Spo- ner (18). For any two adjacent vibrational energy levels of a Morse oscillator, AG(u) = G(u + 1 ) - G ( u ) = m e - 0 g e ( 2 u + 2 ) or equivalently, The spacings between adjacent vibrational enerw levels decrease as a-linear function of the quantum number v or, alternatively, of the variable v + 'h. The second differences are constant: All the Morse potential parameters can be found analyt- ically, of course, using equally well the line defined by ei- ther eq 12 or eq 13. However, a closer look a t the geometri- cal properties of the plot and its commonly presented interpretation shows why it is far better, particularly for pedagogical purposes, to make a Birge-Sponer extrapola- tion from a plot of AG vs. v + U rather than AG vs. u. A proper Birge-Sponer plot of AG vs. v + U, corresponding to the energy levels of Figure 1, is shown in Figure 2. The discrete values AG define a line (eq 13). From its slope, -2m&., the anharmonicity parameter w&. is found. This value is then used with the ordinate intercept of the line, w. - w&,, to determine the wavenumber w. corre- sponding to the fundamental vibration frequency v, in the h a m o n k oscillator approximation. All the of Volume 71 Number 5 May 1994 389 Figure 2. Linear BirgeSponer plot of A G v s . v + th.
the Morse potential can now be determined. In particular, eqs 7 and 9 can be used to calculate D. and Do. Typical Sources of Error I n the Birg-Sponer plot (Fig. 2) the area of the triangle under the line between the ordinate intercept and the ab- scissa intercept is exactly equal to Ddhc (compare with eq 9): This is so because that area corresponds, as Figure 1 makes clear, to the following sum, by which the value ofDo can be expressed: The Contributions The last term in eq 15 is almost always either omitted, because a n unnecessary approximation is made, or erron- eously included within the sum, which is incorrectly writ- ten to run to u = urn.,. Clearly, Ddhc, the area of the large triangle in Figure 2, equals the sum of the areas of all the rectangular strips, each with height AG(u + h) and unit width, plus the area of the little shaded triangle. I will comment on each contribution in turn. If the plot were of AG vs. u rather than AG vs. u + h , the area under the line would not equal Ddhc. Then just a bit more than half the value of &GI, =ol, the area of the first and largest strip in the correct plot, would be left out, and DO would be seriously underestimated. The error in Ddhc would in general be For the B state of 12 this is = 65 em-'. If students are told to calculate the area of the triangle in the BirgeSponer plot to find Ddhc, the plot must be made correctly.
The area of the little shaded triangle in Figure 2 is equal to (D, - E(u,))lhc. It corresponds exactly to the energy difference AED, shown in Figure 1, between the well depth D. and the highest quantized vibrational energy level E(u,,). Recall that AED bears no necessary relation to the pattern of quantized E values, or the corresponding AG values, for integral u. For the special case when D. acciden- tally equals exactly E(u, + 11, the numerical value of AEdhc is expressed by cyr. = AG(urn,). The highest bound state in this case, however, is still the one with u = u,,; there is no Morse oscillator wave function corresponding to E(u, + 1) = D.. Therefore, no rectangular strip is drawn for AEdhc in Figure 2, and i t is treated as a distinct sepa- rate term in eq 15. The highest vibrational quantum number urn, is the largest integer less than (we - w&.)l2w&, the value of the abscissa intercept. To what value does the abscissa inter- cept (which is generally nonintegral) itself correspond? Extrapolation in Incorrect Terms Birge-Sponer extrapolation is often incorrectly expli- cated in terms of the intercept a t AG = 0 and the supposed significance of the corresponding uib~=o,, with u treated as a continuous variable: when or equivalently, when The value U(AG= 0, does not correspond to urn, except ac- cidentally for the special case when
m d h c = 0 ~ J 4 In general, uiAc=o, is a noninteger in the range Moreover, uiA~=o,, treating u as a continuous variable, does not correspond to UD, defined as the value of u a t the dissociation limit, because In short, the quantity uiAo=o, has no physical significance a t all. The correct value UD corresponding exactly to the disso- ciation limit, at which GI,=,, = DJhc, is This is also clearly shown by the proper BirgeSponer plot (Fig. 2) of AG vs. u + h , where uo exactly equals the value of the abscissa intercept. In this plot, each strip is bounded left and right by successive integer values of u. The abscissa intercept is simply the upper bound of a hy- pothetical last strip, withinteger lower bound u,, and an area AEdhc equal to the area of the shaded triangle. Gen- erally, U D is nonintegral, except for the special case when AEdhc = wo, its maximum ~ossible value. The disti%on between u = l, and u n can also be under- stood analvticallv. Eauations 12 and 13 fur AG are bv defi- - - nition expressions for finite differences Au = 1. 'This, the 390 Journal of Chemical Education condition AG = 0 does not correspond exactly to D d k , the maximum value of G. Then uD is defined as that value of u, treated as a continuous variable. for which G(u1 = D J k , the maximum in the parabolic function G(u). Thus, vD is
correctlv found bv setting the derivative of G with respect to u equal to zero; and s o h g for u = UD: This gives for u~ the value in eq 19, precisely equal to the abscissa intercept on a Birge-Sponer plot of AG vs. u + U . Application of BirgeSponer Extrapolation to Real Data To apply the method to actual spectroscopic data, a least- squares line should be fit to a plot of the measured AG vs. u + 49. If "hot" bands are seen (so that AG data come from several vibronic progressions, originating in different vi- brational levels of the ground electronic state and going to vibrational levels of the same excited electronic state) then all the data should be used in the same Birge-Sponer plot to find the parameters of the upper state. For good results, it is important to calibrate the wavelength scale of the spectrophotometer. For I z , the band head positions (i.e., the data measured in the usual undergraduate experi- ment) are very good approximations to the positions of the band o r i ~ n s (6,151. ~tudeGts in the advanced laboratory course a t Barnard College apply linear Birge-Sponer extrapolation to band head data from the three overlapping vibrational progres- sions they see in the visible absorption spectrum arising from the B c X electronic transition of 12. Over the past 15 years typical student data bas yielded results for the B state in the ranges (cn-'1 shown in the table from mea- surements on the following transitions. McNaught warns against too facile comparison of stu- dent results with literature values (6). Older values may have been revised or reinterpreted. Newer data analyses use fitting methods more sophisticated than Birge-Sponer plots, so not all parameters are comparable. Student re- sults also depend on instrument calibration, resolution,
and extent of data. We believe the most useful reference values for com~arison are those shown in the table. Accnrate high resolution spectroscopy shows that values of D, estimated by simple linear Birge-Sponer extrapola- tion are systematically incorrect. The discrepancy arises principally from the departure of successive term differ- ences from linearity. a t high u, approaching the dissocia- tion limit. The Morse potential does not adequately repre- sent the long-range attractive forces between the two atoms of a diatomic molecule at larrre separations. For the - - excited B state of I z , however, this inadequacy becomes ap- parent only a t very high values of u'. These are usually difficult or impossible to observe because these transitions have such low intensities. Correct Extrapolation a s v Approaches the Dissociation Limit More accurate methods for large u that take into account the actual long-range potentials are discussed in Steinfeld Typical Student Data Compared to Literature Values Student Data me 127-135 me% 0.94-1.05 Do/hC 4 1 7 1 4 2 2 D$hc 42364490 Lierature Values
0s 132.1 cm-' We& 1 .05 cm" (fit of band head data to a Morse potential (6)) Ddhc 4391 cm-' (estimate of actual well depth (3) (15). In an elegant application to 12, Le Roy and Bernstein (19) used a n extrapolation appropriate to a potential V ( R ) = D. - CR5 at large internuclear separation R. This is the correct potential between the J = l/z and J = 3 h atoms into which 12 in the B state dissociates. The fit to the 1 2 data a t very high values of u' is excellent. Thus, this ex- trapolation provides a n accurate determination of the en- ergy a t the dissociation limit, which lies just above u- = 8 7 ( ~ i n e a r BirgeSponer extrapolation U & ~ A G up to u' = 50 underestimates the dissociation limit of this state by 140 cm-' (201.1 An even more elaborate analysis, applying a power series in Rm, of the longe-range potential curves of the excited B electronic states in the series 12, Br,, and C12 was given by Le Roy (21). Literature Cited 1. Sfaffmd,F.E. J. Chem. Edue. 1 9 6 2 . 3 9 . 6 2 6 6 2 9 . 2. Danes, M . J . Chem. Educ. 1 9 5 1 , 2 8 , 4 1 4 4 7 7 . 3 . Steinfeld, J. I.; Zare, R.N.: Jones, L.;Lcek,M.;l(lemperer,W. J. Chem. Phys. 1866, 4 2 . 2 S 3 3 . 4 . Bmwn,R. L.; 3ames.T. C. J. Chem. Phys. 1 9 6 5 , 4 2 , 3 M 5 . 5 . D'altedo, R.; Matteon, R; Har6a.R. J . Chem. Edue.
1974,51,282-284. 6 . M a a u g h t , I. J. J. Chem Educ 1980.57.101-105. I . C a m a h t , H. M. J. Chem. Edue. 19&3,60,606607. 8 . Snadden, R.B. J . Chem.Educ. 1987, €4,919-921. 9 . Armanlous, M.: Shaja, M. J . Chem. Edue. 1%36,63,621- 628. 10. Bdm, A. G . I" Erpon"U"b in Physical chemistry, 2nd ed.: Wilson, J. M.; New- mmbe, R. J.; Densm, A. R.; Riekett, R. M. W , Eds.; P q a m n : O l f d , 1968; pp 303-306. 11. Salzbeq, H. W.; M m w , J. I.; Cohen, S. R.: Green, M. E. Phyrieol Chemisfm:A Modem Lobomfory Caum; Academic: New York, 1969, pp 2 3 6 2 4 9 : pp 443-445. 12. Hofacker, U. A. Chemiml Erperimtation; Free-: San Francisco, 1972; pp 4 2 51; p p 5 1 4 9 . 13. Findlqv'sPmcfimlPhysimlChomisfq,9thed.; Levitt, B. P., Ed.;hngnan: Londm. 1973; p 183. 14. White,J.M.PhysidChemialqLobomtoryE~p~iiitt:Pmtie~Hall:Engl ewood Cliffs, NJ, 1975: pp 3 8 1 3 8 8 . 15. Steinfe1d.J. I . MoleculesondRodlotion, 2nd ad.: M I T Cambddge, MA, 1985: C h a p ters 2 6 : e~peelally pp 128-134: pp 145-161. 16. M m e , P . MPhvs. Rou. 1 9 2 9 , 3 4 , 5 7 - M .
17. F M w , S. Pmetieol Quantum Mdmlulies; SpringerVerlag: New York, 1914: Vol. I. pp 1 8 2 1 8 6 . 18. Birge,R. T.; Sponer, H. Phys. Rau. 1928.28.259-283. 19. Ie Roy, R. J.; Bemstein, R. B. J. Md. Spoctmscopy 1911.37, 109-130. 20. Stelnfeld, J. I . ; Campbell, 3. D.: Weisa, N.A. J. Mol. Spetrosrnpy 1989,29,204-215. 21. Ie Ro5 R. J. Con. J. Phys. 1 9 7 4 , 5 2 , 2 4 6 2 5 6 . Volume 71 Number 5 May 1994 391 When experimental d a t a are obtained from expensive e t al. suspect a t lower values of the vihrational quantum equipment i t is desirable to extract as much information as number as i t is clear t h a t the (u', 0) hands for u' less than 14 possible from those data. I t is also desirable t o make the are swamped by the intensity of the (u', 1) and (u', 2) bands. analysis as self-contained as possible by minimizing the need Therefore, particular care must be taken in analyzing this to use literature values for quantities unobtainable by the region. Fortunately, the existence of these hot hands makes experimental design. i t possible, as will be shown below, to obtain much more in- The electronic spectrum of iodine has played a central role formation than is usually done. Ian J. McNaught University of Rhodesia P.O. Box MP 167
Salisbury. Rhodesia in testing the cons&ency of quantum mechanics and observed spectroscopic fine structure. T h e results have been summa- rized in a masterly fashion by Mulliken ( I ). The analysis of the low resolution electronic spectrum of 12 has become a classic advanced undereraduate exoeriment. oarticularlv since The Electronic Spectrum of Iodine Revisited , . the expository paper of Stafford (2) and its extension by D'alterio et at. ( 3 ) . T h e readv availabilitv of minicomouters and programmable calculators means t h a t i t is possible for students to perform much more sophisticated analyses of experimentai data than has previously been the case. Staf- ford's analysis is quite restricted; with no more experimental work it is possible to extract very much more information on the spectroscopic constants of the iodine molecule and gain significant insight into a range of spectroscopic and quantum mechanical concepts. T h e data t h a t can be obtained include (a) the separat.ion between the minima of the potential curves, rr,, (1)) the rreqoencips and anharmonieities in each electronic state, w,, W d e , (c) the dissociatim energies in each electronic state, Dd,,, (dl the differences i n equilibrium bond lengths, ( e l maximumintensity transitions as well as information an the
vibrational wavehnctions. Spectroscopic Introduction The variation of potential energy of a diatomic molecule with internuclear distance is conveniently represented on a potential energy diagram. Figure 1 shows the variation in potential energy for the iodine molecule in its ground ( X ) electnmic state and its second (B) excited electronic state. This figure illustrates the parameters to he calculated and uses the standard spectroscopic notation (4). The iodine molecule gives rise t o well resolved vibronic hands between 500 nm and 620 nm. However, as can be seen from Figure 2, there is significant overlap between (u', 0), (u', 1 ) and ( u ' , 2) in the middle of this region. This makes the Hirge-Sponer plots (4, p. 438) of both Stafford and D'alterio Experimental Details The spectrometer used was a Unicam SP 1750 run a t 0.2 nm s-I with a band width of 0.2 nm, the spectrum being recorded on an AR 5 5 recorder. The spectrometer was calibrated as suggested by Stafford (2). The spectrum of gaseous iodine was run a t room temperature after placing several crystals in a 10-cm cell. I t was found unnecessary t o use higher tempera- tures provided that the sublimed iodine was removed from the windows prior to a run. Treatment of Experimental Data The first problem is to assign vihrational quantum numbers to the bands. The numhering given in Table 1 is based on that proposed by Steinfeld et al. ( 5 ) on the basis of intensity dis- tributions and proved hy Brown and James ( 6 ) from an analysis of the isotope effect.
In order to ensure that the assignments are consistent, i t is useful to prepare a Deslandres table (4, p. 40) as in Table 2. An inconsistency in the numbering will show up as a n in- consistency in the differences between rows or columns. This technique will not show up a consistent numbering error. Table 1. Numbering of Band Heads for Iodine d v" h i m d v" Alnm d ' Xlnm u.-2w.x. - .- > '' lNTERNUCLEAR DISTANCE Figure 1. Potential energy diagram for iodine. , : -~~ ,no $70 380 350 ,,! Figure 2. Overlapping of ( d . 0). ( d , 1) and ( d . 2) bands in iodine. Volume 57, Number2 February 1980 1 101 Directly Derived Parameters If 7" and T" represent the electronic term energies of the two states while G(u') and G(u") represent their vihrational energies then, ignoring rotational energy changes, the energy of a transition will he given by c = T. - T; + G ( d ) - G(u") (1) For the case studied here T i is Tero, because it refers to the ground electronic state while T , equals a,, the frequency of the hypothetical transition between the two minima of the
potential curves (Fig. 1). T h e vihrational term values can he written as GIU) = WJO + 'I2) - + %)2 + w a s ( " + %IS + . . . (2) where we is the frequency for infinitesimal amplitudes of vi- bration and m,~,, w,y,etc. are anharmonicity constants. If only the first two terms in this expression are taken, i.e. Glo) = we(" + %) - W.X,~U + %I2 (3) then the transition frequency is given by " = re + wp 10, + '14 - wpxp I", + 11d2 - w; i",, + '12) + w;x; ("I + 'I# (4) T h e usual procedure from this, point (2) is to use a Birge- Sponer p)): to ohtain w, and w&t,, then to use literature values for we, w,x, along with the observed a to determine a,. A much better way is to use the technique of mu!tiple linear regression (7) to determine a,,w,, were, we and w,x, directly from a, u' and u". Indirectly Derived Parameters Convergence Limit E * E* is the energy of the transition from u" = 0 to the top of the upper state potential well, i.e., the energy a t which the vihrational structure joins the continuum. I t is calculated as the energy of a transition ending on level u plus the sum of all the vihrational quanta ahove u t o the maximum urn.,. T h e energy of the vibrational quanta is given by G(u + 1) - G(u), so t h a t Retaining only quadratic terms in G(u) gives
G(u + 1) - G(u) = w; - 2wkxk(u + 1) ( 6 ) I t follows from this equation t h a t u,,, (i.e., t h a t u for which G(u + 1) - G(u) = 0) is given by u,,, = wpIiZwG1) - 1 17) Therefore EX = o, + 'lz[wk - Zwkxkiu + l)l(u,,, - u) (8) This equation is just the analytical form of the Birge-Sponer technique. Its advantage is that all the observed values of a, due to transitions from u" = 0 can be used to produce many independent values of E*, these values can then be used to ohtain some estimate of the error in E*. T h e graphical tech- nique is a one point method which gives no indication of the precision of the determined E*. This analytical technique assumes a linear Rirge-Sponer extrapolation, a valid assumption for the data ohtained in this experiment. Dissociation Energy D. The dissociation energy is the energy required to dissociate the molecule into atoms. (Dissociation into uositive and atom will dissociate into the upper state products. When separated the atoms have zero relative velocity. Increasing the energy ahove E* will give atoms having increasing relative velocity and kinetic energy. I t is this process which gives rise to the continuum observed in this exueriment a t short wave- lengths. T h e ground state dissociation energy measured from the
lowest vibrational level is given by where E(I*) is the difference in energy between a ground state iodine atom ( 2 P ~ l z ) and the first excited state of the iodine atom PPll2):It.has a value of 7589 cm-' (8). The ground state dissociation enerev measured from the bottom of its uotential T h e excited state dissociation energy measured from the hottom of its potential well is Force Constants k. When a bond is stretched there arises a restoring force which resists the stretching force. Within the simple harmonic oscillator approximation this restoring force is given by (4, p. 74) where r - r , is the extension and he is the force constant. The greater the force constant the more difficult it is to stretch the bond. T h e force constant is related to the curvature of the potential well and can he calculated from (4, p. 98) k, = [email protected]: (13) Morse Curve If the simple harmonic oscillator model did apply, the po- tential e n e r m curve would be a uarabola. i.e, its dissociation energy would be infinite. A more realistic potential energy curve is t h a t introduced by Morse (9) l i ( r - r e ) = ll.(p-RI'-'-) - (14) where 0 = nw,(2g~/D,h)'/~ (15) The importance of this potential function is that i t is possible to solve the Schrijdineer eauation for this ootential and ohtain ..
. the vihrational energy wavefunctions. These wavefunctions are finite and give eqn. 3 when appropriate identifications are made. If r - re is denoted by u then these wavefunctions R,iu) can be written ( 1 0 ) Rt(u)ru e-kz(k~)"2n-1L:,(k~) (16) where Table 2. Deslandres Table for IodineB d l / ' 0 1 2 19 17799 213 17586 212 17374 93 93 93 18 17706 213 17493 212 17281 94 94 94 17 17612 213 17399 212 17187 97 97 97 AII entries in cm-', corrected for vacuum (131. me difference between rowsequslr w; - 2w;xaiv+ 11 while the difference between columns equals w: - 2w;xi(v+ 11 where v is the lower of the two vibrational quantum numbers. 102 1 Journal of Chemical Education L k , ( k i ) = (kz)" - n ( h - n - l)(kr)"-'
+ % n ( n - l ) ( k - n - l ) ( k - n - 2 ) ( k ~ ) " - ~ . . . (17) where n is the vibrational quantum number. In this experi- ment we are interested in transitions from n = 0,1, and 2. The required prohahility density distrihutions are given by R i ( u ) a e - k ' ( h r ) k - ' (18) R : ( u ) a r-ki(ki)"-3[kz - (k - 2)la (19) R:(u)a eck2(k2)k-"[(kz)2 - 2(k - 3)kz + (k - 3)(k - 411' (20) However, for numerical evaluation these expressions are not very useful (they contain terms like ZOOzo0), so a scaled prohahility density distribution can he introduced, T i i ( u ) = ( R , , ( U ) I R , ( ~ ) ) ~ (21) Explicitly the scaled probability density distrihutions used here are , 2 = e - k ( a - l ) z k - l (22) R : ( ~ ) = e-klz-l)rk-3[kr - (k - 2)]2 (23) ~ l ( u ) = e-k'"-'~zh-5[(kz)Z - 2(k - 3)kz + (k - 3)(k - 4)12 (24) Figure 3 shows these functions superimposed on the Morse curve for the ground state of iodine. Also plotted are the squares of the wavefunctions for the harmonic oscillator. The harmonic oscillator probability density distrihutions are presented by (4, p. 78) +ga e-eu2 ( 2 5 )
+:a ou2 cnu2 (26) +arr (2au2 - 1 ) s e - ' ~ 2 (27) where a 4a2cpw,lh. I t is clear from Figure 3 t h a t as the vi- brational auantum number increases, the difference in bond h g t h Irtu'een thr maxima t ; ~ r the harmonic and anharmonic tr;cillators I w o m r ; morr prowuncrd. This is imlvxtant in determining the expected intensity distribution in the vihronic hands. Anharmonic Oscillator Maxima u:" The mnximn in thv ilnharm~,nicoscillat<,r distrihurionsrim t w calculntnl I'row the rollcrwing iormul:~r, where in t:nch case z is related t o u by the equation 1 u = - - I n * P (28) ur: z = (k - 1)Ik (29) u y : z = [(2k - 3) z t m I l 2 k (30) u p : (kz)" - (3k - 7 ) ( k # + (k - 3)(3k - 10)kr - (k - 3)(k - 4)(k - 5 ) = 0 (31) Figure 3. ~ a r i o n i c and anharmonic probability density distributions for io- dine. Although an analytical solution for eqn. (31) is obtainable, i t is easier to solve the equation numerically.
T h e classical turning points, u, for each of the three vibra- tional levels can he ohtained by solving the equation W;(U + II2) - W;X& + = D;(e-6"" - (32) Diflerence In Equilibrium Bond Lengths, re - r; As has been discussed by D'alterio e t al. (3), i t is possible to use the intensities of the (u', 0) bands to estimate the dif- ference between the equilibrium bond lengths in the two states. Because of the problem of overlapping bands men- tioned earlier, a simpler approach has been taken here. I t is considered sufficient merely to take the energy corresponding to the most intense transition, nnt. By the Franck-Condon principle (4, p. 1941, this t r ~ ~ i t i o n ~ w i l l originate and end a t the internuclear distance r ,, (not r e as claimed by D'alterio e t al.). I t is clear from Figure 1 t h a t ,J T h e plus sign is chosen if the hands are red degraded, i.e., rk > re, the negative sign if the hands are violet degraded, i.e., r , < r e . Transitions of Maximum Intensity Provided t h a t the vibrational quantum number is greater than about 10, the probability maximalie close to the potential curve (11). Assuming t h a t i t lies a t the same internuclear distance as the curve, and that the Franck-Condon principle holds, i t is possible to predict the most intense ( u ' , O), (u', 1) and (u', 2) transitions. They are ohtained by solving the fol- lowing equation for v' where u = u y + r ; - r ;
It should l,r nott!d that tl~iscdlculation r n h r n r ; - r. is used tu (t~lculatv the tnnxii~~um intensir., is nor related in a circulnr way t o the calculations in eqns. 33-35 (where the maximum intensity u' is used t o calculate r ; - r;) as i t brings in w, and wkx, which are independent of the parameters in those three equations. T h e calculations serve as a good check on the in- ternal consistency of the experimental data. If a plotter is available the relationship between the Morse curves and the vibrational wavefunctions can be shown as in Figure 4. Drawing a vertical transition from the maximum gives the value of U'(u), i t is instructive then t o draw in the appropriate upper state vibrational levels using eqn. (3). Figure 4. Derived potential curves for iodine. Volume 57, Number 2, February 1980 1 103 Table 3. Parameters for Iodine a Equation (4) Equation (42) vmax. intens. 27. 14. 9 26, 14. 10 All V B ~ Y B J in cm-' exceot where indicated. Rotational Constants 6. If the literature value for r, is assumed then i t is possible todetermine reand from these the rotational constants in both states as well as their variation with vibrational quantum number. The equilibrium rotational constants are determined
from R , = hl(8a2epr:) (37) T h e rotational constant in a given vibrational level is related to R. by the equation ( 4 , p. 106) & = R e - & + % ) (38) where LY, can h e determined from (4, p. 108) n, = 3 R . (a - 2R.lwJ (39) Equation (37) can he used t o verify t h a t the difference he- tween the hand head and hand origin can be ignored within the accuracy of these data. Equations (37)-(39) permit the calculation of the bond lengths ro, r l and rz in each of the three lowest vibrational levels. Choice of Equations Equation (39) is a modification of the one quoted hy Herzberg; the equation there involves the anharmonicity constant and therefore can have large error limits when de- termined from such low resolution d a t a as obtained in this experiment. T h e transformation uses the relationship w.x. = wa/4D, (4, p. 100). Similarly, the literature contains a number of different equations from those used here, the cri- terion of choice for this paper being those equations that contained precisely obtainable parameters. For example, al- ternative equations for some of the parameters are 6 = ( 8 a 2 ~ p ~ . ~ . l h ) 1 1 2 ( 9 ) (40) 1 h = -
X e (10) ( 4 1 ) Equations (40) and (41) involve t h e anharmonicity, for the Table 4. Summary of Relations multiole linear rearession multiple linear regression mullcple linear yression v,. = w./(~w.x.) - 1 E . = n, + %[w; - ~ W & Y + 1 ) I I v m ~ . - "1 Calculate for all o, originating from P = 0. Find the average and standard deviation. D ' - E . - - E(I.1 E(I') from literature D: = D; + %w. - '/q~Zx: ~ 4 = E' - c. t XW. ; !14~.a Do = D: - '12u. + % W ~ X ~ k = 4D&. 0 = a ~ d 2 c g / D ~ h ) " ~ a = [email protected] 1 z = ( k - 1 ) l k . U = - - I n = a ground state this is the least precisely determined parameter (see Table 3 ) . Eauation (41) does have the useful feature of showing t h a t as the anharmonicity increases the value of h decreases. Results T h e results of a typical careful run are collected together
in Table 3. T h e quoted errors are standard deviations. Comparison with the Literature I t is common nractice to comnare results with well tabulated literature values. Such a comparison must he approached cautiouslv for two reasons. T h e first reason is t h a t new ex- perimental data or reinterpretation of accepted data may lead to marked changes in the values. The second and more important reason is t h a t present lit- erature values are usually obtained using much more sophis- ticated term functions, e.g., eqn. (2) is usually expanded to higher powers in ( u + %), (for instance Le Roy (12) fitted the experimental data to an expansion up t o ( u + %)lo), thereby affecting the coefficients of the lower powers. This can have a marked effect, as can be seen in Table 3. The second column of figures was ohtained from the same experimental data using eqn. (42) in place of eqn. (4) n = a, + &(u' + '12) - w>;(u' + 1/2)2 + w>;(u' + MP - W:(U" + '12) + w.x;(ul' + 1/2)2 (42) I t is clear that the more sophisticated expression produces marked changes in the parameters and, in general, brings them closer to the accepted values. Because of these problems, the emphasis should he on precision (internal consistency, standard deviation) in eval- uating reports rather than accuracy (closeness to some tahu- lated accepted value). I t is one of the advantages of multiple linear regression that error estimates of the parameters can he ohtained easily, careful work produces small standard de-
104 / Journal of Chemical Education viations and should he treated as mare desirahle than work t h a t produces a mean value close to t h e accepted value h u t which has a large standard deviation. Table 4 summarizes t h e method of calculation of the pa- rameters i n a logical order. Conclusion This paper presents equations and techniques for calcu- lating and interpreting many of t h e spectroscopically im- portant parameters associated with the ground ('2:) and second excited (:'II:,) states of the iodine molecule. It also shows students that in physical chemistry the obtaining of experimental data in the lahoratory is often only a small part of the total time commitment compared to the detailed analysis of data. Acknowledgment The author wishes t o acknowledge his first contact with a rudimentary version of this experiment when he was a Senior T u t o r a t the University of Melbourne. Literature Cited !I1 M u l l i k e n R. S., J. C h e m I'hyr., 55.288(1971). 121 Stafhid F. E...I. CHEM. EDUC.. 39, ti26 119fi21. I:!) D'nlterii! H., Matts~rn H.. Harris. R.J. C H E M . EDIIC.SI.2S2 11974).
161 H e r r h e r ~ C.. "Mdrcular Spectra and Mdeculsr Structure I: Spectra of Diatnmir Mliind~s"Vsn N i ~ r t i a n d , I ' r i n m i m N.I. 2nd lid.. 1950. I?) S l & $ & i . I I.. %are R. N . . J m s % I. .. l.erk M . . a n d Klmmllarer W.. J. Then,. P I I ? ~ . 42. (81 l:syd,,n. A. c.. "l)irn,ria~ic,n Energies." Chapman and Hall, l.sndim.2nd lid.. Rex, 1 w . t . .. .,., . 191 MIIIIP. 1,. M.. P h j l K m . 31.17 (10291. l l 0 1 llunhnm. J. I. .. l'b% R n i . 14,438 119211. 1 1 1, Sch8n:l.. 1;'Quantum Mschnnic.,"MrCnr-Hill H w k O>..NewYork. 2nd Ed.. 1965. UP. 61-67, 1121 1.e Rliy. 1l..I..J. l'hrm i'hg*.. 52.26ti:i 119701. 1131 "Hsndhwlr i , f C h e m i s l r g md I'hyrirr." C11C I'reir. C l e u ~ l a n d , 55th Ed.. 1974, p. E YZ!. Volume 57, Number 2 February 1980 / 105 65 Electronic Spectrum of I2 Overview
In this experiment, the student will record and analyze the vibrational structure of the B – X electronic transition in molecular iodine to determine the dissociation energy of I2 in the B 3 Π state. The student will record the spectrum at several temperatures at low resolution to observe the gross effects of temperature on the overall structure of the spectrum and then record the spectrum at higher resolution for the analysis of the spectrum. Bands will be assigned and the dissociation energy calculated using a Bïrge- Sponer method. Theory The total energy of a molecule (under the Born-Oppenheimer approximation) can be expressed as a sum of electronic and vibrational energies. ETotal = Eelectronic + Evibrational
A simplified energy level diagram is shown below for two electronic states of the hypothetical diatomic molecule A2. Rotational energies are small enough that they are no resolved in this experiment and are thus ignored. The “term” value for a given quantum state is measured relative to the bottom of the potential energy well for the ground electronic state. By analyzing the 66 transitions between the two states, once can determine the energy differences labeled as Te’ (the electronic excitation term value of the excited electronic state), E* (the term value for the excited atomic state of A*), De” and De’ (the dissociation energy of the lower and upper electronic state respectively as measured from the potential well minima) and D0” and D0’ (the dissociation energy of the lower and upper electronic state respectively as measured from the v = 0 energy levels of the respective states).
The analysis in this experiment will focus on the energy levels of the upper state, so let’s write down an expression for these levels based on electronic and vibrational contributions to the total energy. For the purposes of this discussion, all energies will be expressed as term values which are really energies divided by the factor hc.) Tv = Te + Gv’ Taking the difference between successive levels gives the following result. Tv+1 – Tv = Gv+1 - Gv The difference Gv+1 - Gv is given the symbol ∆Gv+½. The vibrational term value can be expressed in terms of the vibrational constants ωe and ωexe and the vibrational quantum number v. Gv = ωe(v + ½) - ωexe(v + ½) 2 + ωeye(v + ½)
3 + … Neglecting any terms higher than ωexe, the total expression for ∆Gv+½ can be derived as follows: ∆Gv+½ = Gv+1 – Gv = ωe(v + 1 + ½) - ωexe(v + 1 + ½) 2 - ωe(v + ½) - ωexe(v + ½) 2 ∆Gv+½ = ωe – 2ωexe(v+1) According to theory, the dissociation limit occurs when the spacing between successive vibrational levels (∆Gv+½) is zero. Fitting ∆Gv+½ as a function of (v+1) should yield a straight line with slope -2ωexe and intercept ωe. The dissociation energy can be determined from the constants ωe and ωexe using the Birge-Spöner extrapolation. In this method, one plots
∆Gv+½ vs. (v+1). The area under the curve gives the dissociation energy (the sum of all of the vibrational spacings up to the dissociation limit – where ∆Gv+½ becomes zero.) The method typically over estimates the dissociation energy by approximately 15% (more for some molecules) due to neglecting higher-order terms in the vibrational term energy expression. The figure below shows an example of a Birge-Sponer plot. The deviation from the linear relationship occurs if high order terms such as ωeye or ωeze become important for a given molecule. 67 Finally, the dissociation energy of the lower state can be determined from Te, De’ and E* by noting that Te + De’ = De” + E*
Experimental Method The experimental data is taken on two different spectrometers. One is equipped with a temperature control and the other provides higher resolution. 1. Record the visible absorption spectrum of I2 at 30 o C, 40 o C and 50 o C using the Varian Bio 50 spectrophotometer. Use the highest resolution obtainable. 2. Record the same band system using the Perkin-Elmer Lambda 9 spectrophotometer. Seek the best possible signal to noise ratio and resolution combination on this instrument.
Analysis Organize your data into a table leaving room to report the wavenumber, vibrational assignment for each assignable transition. Also include a column for and ∆Gv+1/2 for the spacing between each pair of transitions. 68 Assign vibrational quantum numbers to the bands in the spectrum. Use the following assignments to get yourself started. These assignments are taken from J. Chem. Ed. 57, 101 (1980). v’ – v” λ (nm) 27-0 541.2 28-0 539.0 29-0 536.9 Birge-Spöner Method • Prepare a graph of ∆Gv+1/2 vs (v+1) and report the best-fit
line to your data. • From the fit, calculate values for ωe and ωexe. • Find vmax from this data. • Calculate the dissociation energy for the excited electronic state of I2. ∫ = += maxv 0v 2/1v0 dv∆GD • Also determine a value for De’ by calculating G(vmax). • From your fit of the data, calculate Te – the electronic excitation energy of the B state. • The value if E* is 7603 cm -1 (4). This is the energy of excitation for I* ( 2 P1/2) compared to I ( 2 P3/2). Use this information in conjunction with your above results to find a value for De” – the ground electronic state dissociation energy. Error Analysis
• Determine the uncertainties of the spectroscopic constants you report and compare all of your findings to values found in the literature. (A good source of literature values is the table of Constants of Diatomic Molecules found at http://webbook.nist.gov/.) Post Laboratory Questions 1. Explain the differences between the three spectra recorded on the Bio 50 instrument. 2. Derive the expression ∆Gv+½ = ωe – 2ωexe(v+1). 3. What would be the theoretical dissociation limit of a molecule that can be treated as a true harmonic oscillator? References 1. Experiments in Physical Chemistry, D. P. Shoemaker, C. W. Garland and J. W. Nibler, 7 th edition,
WCB/McGraw-Hill, New York, (2003) 2. Physical Chemistry: Methods, Techniques, Experiments, R. J. Sime, Saunders College Publishing, San Francisco, CA (1990) 3. Physical Chemistry, R. A. Alberty, R. J. Silbey and M. G. Bawendi, Physical Chemistry, 4 th ed., Chapter 16, John Wiley and Sons, Hoboken, NJ (2005) 4. Atomic Energy Levels, Vol. III (Molybdenum through Lanthanum and Hafnium through Actinium), Charlotte E. Moore, Circular of the National Bureau of Standards 467, U.S. Government Printing Office, Washington, DC (1958).

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Foglio1Pagina 14282015 125839 PMWavelength (nm) .docx

  • 1. Foglio1 Pagina 1 4/28/2015 12:58:39 PM Wavelength (nm) Absorbance 190 674 191 657 192 383 193 311 194 410 195 668 196 873 197 779 198 527 199 241 200 371 201 606 202 597 203 542 204 494 205 674 206 865 207 786 208 626 209 552 210 420 211 425 212 620 213 719 214 709
  • 2. 215 423 216 467 217 387 218 650 219 755 220 769 221 764 222 510 223 520 224 475 225 690 226 647 227 582 228 496 229 354 230 445 231 478 232 459 233 478 234 450 235 530 236 531 237 493 238 376 239 284 Foglio1 Pagina 2 240 322 241 283 242 362 243 352
  • 3. 244 331 245 427 246 475 247 524 248 471 249 386 250 489 251 452 252 525 253 454 254 488 255 499 256 344 257 372 258 350 259 353 260 364 261 352 262 362 263 362 264 348 265 370 266 410 267 393 268 333 269 381 270 369 271 441 272 477 273 512 274 526 275 547 276 524 277 454 278 418 279 401
  • 4. 280 395 281 514 282 560 283 560 284 666 285 807 286 1010 287 1134 288 1185 289 1257 290 1316 291 1302 292 1336 Foglio1 Pagina 3 293 1414 294 1589 295 1782 296 1948 297 2096 298 2165 299 2179 300 2150 301 2176 302 2118 303 2283 304 2362 305 2415 306 2370 307 2232 308 2307
  • 5. 309 2313 310 2446 311 2462 312 2454 313 2425 314 2336 315 2356 316 2301 317 2291 318 2301 319 2295 320 2306 321 2236 322 2201 323 2228 324 2223 325 2265 326 2286 327 2359 328 2446 329 2385 330 2326 331 2287 332 2250 333 2257 334 2260 335 2323 336 2352 337 2307 338 2242 339 2198 340 2190 341 2272 342 2316 343 2355 344 2310
  • 6. 345 2326 Foglio1 Pagina 4 346 2325 347 2279 348 2222 349 2163 350 2197 351 2155 352 2122 353 2076 354 2048 355 2006 356 1979 357 1946 358 1880 359 1812 360 1750 361 1703 362 1645 363 1579 364 1496 365 1429 366 1368 367 1314 368 1262 369 1201 370 1144 371 1090 372 1037 373 986
  • 7. 374 939 375 893 376 850 377 808 378 767 379 730 380 695 381 659 382 624 383 593 384 561 385 533 386 505 387 479 388 455 389 431 390 408 391 386 392 366 393 346 394 328 395 313 396 299 397 286 398 273 Foglio1 Pagina 5 399 260 400 248 401 237 402 225
  • 8. 403 215 404 205 405 197 406 190 407 184 408 177 409 171 410 165 411 162 412 158 413 155 414 150 415 147 416 146 417 146 418 146 419 145 420 145 421 146 422 146 423 148 424 149 425 151 426 153 427 155 428 158 429 161 430 165 431 168 432 173 433 179 434 185 435 190 436 195 437 202 438 209
  • 9. 439 217 440 225 441 234 442 242 443 250 444 257 445 267 446 277 447 287 448 298 449 306 450 316 451 328 Foglio1 Pagina 6 452 338 453 349 454 360 455 373 456 386 457 396 458 408 459 419 460 430 461 441 462 453 463 466 464 479 465 492 466 503 467 514
  • 10. 468 524 469 537 470 549 471 561 472 572 473 581 474 591 475 601 476 612 477 621 478 631 479 640 480 649 481 659 482 666 483 671 484 676 485 682 486 690 487 696 488 701 489 708 490 716 491 722 492 723 493 725 494 730 495 733 496 736 497 737 498 739 499 738 500 738 501 737 502 738 503 738
  • 11. 504 737 Foglio1 Pagina 7 505 733 506 731 507 729 508 724 509 720 510 716 511 711 512 705 513 699 514 695 515 690 516 684 517 677 518 671 519 664 520 656 521 649 522 642 523 634 524 626 525 616 526 609 527 599 528 587 529 576 530 567 531 562 532 553
  • 12. 533 543 534 534 535 523 536 513 537 503 538 494 539 484 540 473 541 464 542 456 543 446 544 435 545 425 546 416 547 407 548 399 549 390 550 381 551 370 552 360 553 352 554 344 555 338 556 328 557 319 Foglio1 Pagina 8 558 309 559 302 560 295 561 287
  • 13. 562 278 563 270 564 263 565 256 566 249 567 243 568 237 569 229 570 220 571 212 572 206 573 201 574 194 575 189 576 183 577 176 578 173 579 169 580 165 581 160 582 154 583 148 584 141 585 139 586 136 587 130 588 125 589 123 590 121 591 117 592 113 593 109 594 105 595 101 596 99 597 97
  • 14. 598 92 599 89 600 88 601 86 602 84 603 82 604 80 605 78 606 76 607 74 608 71 609 67 610 65 Foglio1 Pagina 9 611 66 612 67 613 65 614 62 615 59 616 57 617 56 618 57 619 57 620 54 621 52 622 50 623 51 624 50 625 47 626 44
  • 15. 627 43 628 45 629 44 630 42 631 42 632 42 633 43 634 43 635 41 636 40 637 41 638 40 639 40 640 38 641 38 642 37 643 37 644 37 645 39 646 39 647 38 648 37 649 34 650 32 651 34 652 34 653 34 654 31 655 30 656 30 657 31 658 31 659 31 660 30 661 31 662 30
  • 16. 663 29 Foglio1 Pagina 10 664 28 665 27 666 27 667 27 668 28 669 27 670 26 671 24 672 24 673 25 674 25 675 25 676 24 677 26 678 26 679 26 680 23 681 21 682 19 683 19 684 19 685 22 686 22 687 20 688 20 689 19 690 17 691 17
  • 17. 692 17 693 19 694 17 695 13 696 13 697 15 698 17 699 15 700 14 701 13 702 12 703 10 704 10 705 12 706 13 707 13 708 12 709 13 710 13 711 13 712 10 713 10 714 9 715 9 716 9 Foglio1 Pagina 11 717 10 718 11 719 9 720 7
  • 18. 721 7 722 6 723 8 724 9 725 10 726 9 727 8 728 7 729 6 730 8 731 8 732 3 733 3 734 3 735 5 736 5 737 8 738 9 739 6 740 5 741 5 742 5 743 4 744 3 745 2 746 2 747 1 748 0 749 -1 750 0 751 1 752 2 753 0 754 -1 755 -3 756 -4
  • 19. 757 -2 758 -1 759 -2 760 -2 761 0 762 1 763 0 764 -1 765 -2 766 -3 767 -3 768 -6 769 -4 Foglio1 Pagina 12 770 -3 771 -4 772 -6 773 -3 774 -5 775 -7 776 -6 777 -1 778 -1 779 0 780 -2 781 -5 782 -8 783 -8 784 -7 785 -7
  • 20. 786 -6 787 -5 788 -6 789 -8 790 -9 791 -8 792 -7 793 -5 794 -4 795 -5 796 -8 797 -7 798 -4 799 -4 800 -7 801 -12 802 -9 803 -8 804 -8 805 -8 806 -8 807 -9 808 -7 809 -3 810 -2 811 -7 812 -11 813 -11 814 -8 815 -7 816 -8 817 -14 818 -17 819 -19 820 -21 821 -19
  • 21. 822 -12 Foglio1 Pagina 13 823 -9 824 -11 825 -15 826 -14 827 -13 828 -12 829 -15 830 -16 831 -13 832 -10 833 -9 834 -7 835 -12 836 -18 837 -20 838 -12 839 -8 840 -11 Foglio1 Foglio1 Pagina 1 # Sample ID User name Date and Time 1 (nm) 1 (Abs) 2 (nm) 2 (Abs)
  • 22. 1 student 4/28/2015 12:58:39 PM Foglio1 In the Laboratory 336 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org The electronic absorption–emission–fluorescence spec- trum of I2 in the gas phase has long been a staple in physical chemistry laboratory instruction on spectroscopy (1–7). It is relatively easy to record the I2 absorption spectrum with reso- lution sufficient to measure many band heads in the B ← X electronic spectrum, and thereby to characterize the vibrational structure in the excited B state. By extrapolating to the point where the B-state vibrational intervals ∆Gυ� vanish (Birge– Sponer extrapolation), students can estimate the photodisso- ciation limit of the B state and thence the dissociation energy of the ground X state. Apart from the latter indirectly ob- tained quantity, the absorption spectrum provides much less information about the ground state, since normally bands can be assigned only up to ground-state vibrational levels υ� = 2 or 3, higher levels being insufficiently populated at typical cell temperatures due to small Boltzmann factors. The I2 B ↔ X fluorescence spectrum nicely comple- ments the absorption spectrum and is thus often included in the teaching instruction on this transition. The ideal fluo- rescence spectrum is produced by exciting a single (υ�, J�) level in the B state and consists of the P( J� + 1) and R( J� − 1) rotational lines (which, by convention, are labeled with the lower rotational level J �) going to a large range of vibrational levels in the X state (6, 7). Fluorescence thus offers not only
  • 23. information about the vibrational structure in the latter but also, with sufficient resolution, the rotational structure. Un- fortunately, the needs for suitable excitation sources and for a high-resolution spectrometer to record the spectrum make this experiment problematic in many teaching labs. The earliest descriptions employed 546.1-nm excitation from low-pressure Hg lamps, with photographic recording of the spectrum (1, 3); but such procedures can be tedious and time-demanding. The 514.5-nm line from an Ar-ion la- ser excites strong fluorescence; but unfortunately this source produces two main excited levels, (43, 12) and (43, 16), and the resulting fluorescence appears as narrow triplets, requir- ing high resolution for quantitative analysis. The widely avail- able He–Ne laser excites two well-characterized fluorescence series, from upper levels (6, 32) and (11, 127) (6, 7). The latter of these is particularly appealing, because the rotational doublet splitting (roughly proportional to J ) is large. How- ever, the absorption involves highly excited levels (υ� = 3 and 5, respectively), so with the readily available low-power He– Ne lasers, fluorescence intensities tend to be weak. Also, both upper levels may be excited together, depending on the op- erating characteristics of the laser (7). Such complications can make it difficult to complete the experiment “on schedule” in the teaching lab. For these reasons, I have generally treated this experi- ment as a “dry-lab”, done in conjunction with a demonstra- tion of laser-induced fluorescence excited by the Ar-ion laser and observed by students through a hand spectroscope. The data they then analyze are for the (6, 32) fluorescence, as re- corded at ∼1-cm resolution using a Raman spectrometer (6). In the latest edition of their laboratory text, Garland et al. (8) mention some of these laser sources and also suggest
  • 24. an appealing alternative—the doubled Nd�YAG laser, which is now readily available at ∼5 mW power in the form of the green laser pointer (GLP). When I used such a pointer re- cently to demonstrate I2 fluorescence, I observed a remark- able behavior, which bodes ill for the use of GLPs to excite fluorescence spectra for quantitative analysis. At the same time, this behavior does offer a “flashy” and compelling dem- onstration of some important properties of both laser-induced fluorescence (LIF) and lasers. It nicely complements and re- inforces the instructional value of a demonstration of LIF using the Ar-ion laser with I2. In following paragraphs, I de- scribe this behavior and explain how it arises from temporal variations in the output wavelength of the GLP. This expla- nation is corroborated with high-resolution spectra taken as a function of time for several different pointers. I also de- scribe the procedures used to photograph fluorescence spec- tra at low resolution, in color, using a digital camera in combination with a hand spectroscope. Some of these spec- tra are included in this article; however they are much better appreciated in color, to which end I have provided supple- mentary material for online viewing, in the form of a Power- Point document. The Supplemental MaterialW also includes four QuickTime movies of the behavior in question and of related phenomena involving the fluorescence and emission spectra of I2 in the gas phase. What Happens and Why? When the beam from a green laser pointer is directed through a cell containing I2 vapor at low pressure, the beam transiting the cell is alternately bright and vanishingly dim. This behavior was observed for four different GLP models (the only ones examined, details below), so it is clearly a prop- erty of the typical GLP. For none of these did the intensity of the laser itself seem to vary much, ruling out that obvious explanation. The correct interpretation is twofold: (1) the
  • 25. wavelength of the laser is varying with time and (2) at any time, the spectral purity of the laser is high compared with the inherent I2 linewidths (∼0.03 cm �1, dominated by Dop- pler broadening and molecular hyperfine structure; ref 9). Thus, as the wavelength of the laser changes with time, it alternately tunes itself into and out of resonance with lines in the absorption spectrum of I2. 1 Garland et al. (8) have included a figure (p 429) dis- playing the gain profile for a research-level doubled Nd�YAG laser, with prominent absorption lines identified. A much Laser-Induced Fluorescence in Gaseous I2 Excited with a Green Laser Pointer W Joel Tellinghuisen Department of Chemistry, Vanderbilt University, Nashville, TN 37235 Advanced Chemistry Classroom and Laboratory edited by Joseph J. BelBruno Dartmouth College Hanover, NH 03755 http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm
  • 26. l In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 337 more extensive characterization of the I2 absorption in this region has been given by Forkey et al. (10), who have listed all the detectable lines in absorption from υ� = 0, 2, and 3 in a ∼2-cm–1 interval.2 In fact, I have succeeded in photo- graphing spectra that are clearly attributable to these three υ� levels. Results are illustrated in Figure 1. The confirma- tion of excitation from the two excited levels (υ� = 2 and 3) is the observation of two and three anti-Stokes lines, respec- tively (anti-Stokes meaning lines falling at wavelengths shorter than that of the exciting line). Spectra of green laser pointers confirm that the output varies progressively rather than randomly with time. It seems likely that this behavior relates to thermal changes in com- ponents of the optical cavity or power supply as the laser warms up under power. As one indication of this, if the laser is allowed to operate for several minutes and is then turned off and on again quickly, it appears to cycle rapidly through its previous tuning cycle, producing two or three fluorescence “flashes” before it stabilizes again near its performance when turned off. In one particularly dramatic instance of such be- havior, through the spectroscope I observed in repeated fash- ion spectra like those illustrated in Figures 1B and 1C, with a cycle time of ∼2 s.3 The logic for high spectral purity is probabilistic in na- ture. The I2 B ← X absorption spectrum is so dense in this region that excitation at random wavelengths will produce
  • 27. strong fluorescence about half the time. For example, in early attempts to measure the continuous absorption between the lines of the B ← X system using atomic line sources, discrete absorption was evident for all but three of a dozen Ar atomic lines (11). In a later, more extensive study, almost all mea- surements had to be corrected for discrete absorption (9). When the present fluorescence “turns off ”, the beam can com- pletely disappear in a dark room; from observations of the strong fluorescence through neutral density filters, this means a drop in fluorescence intensity by a factor of at least 104. If the GLP emitted light in broad lines or in too many different modes, such “off ” behavior would be improbably rare. Green laser pointers operate on a doubled transition of Nd doped in a host crystal. The latter is often YVO4 (yttrium orthovanadate) instead of YAG (yttrium aluminum garnet, Y3Al5O12), because YVO4 is said to better couple optically with the 808-nm radiation from the laser diode pump (12). The IR laser output at 1064 nm is doubled in a crystal of KTP (potassium titanyl phosphate, KTiOPO4), which is contained within the laser optical cavity, where it is exposed to the high optical powers needed to achieve frequency doubling. The IR output is blocked with a filter, leaving only 532-nm radia- tion in the output beam.4 The lasers are said to operate in the single transverse optical mode TEM00. Longitudinal modes occur at frequency intervals of c �2d, where c is the speed of light and d is the length of the optical cavity (13). For the typical cavity lengths of ∼5 mm (12), this translates into a wavenumber spacing of ∼1 cm�1 in the IR fundamental, 2 cm�1 in the doubled visible output. Since the Nd gain profile is only several cm�1 wide, it is reasonable for the GLP to oper- ate on only one or two modes, as required probabilistically
  • 28. for the observed “off ” behavior of the fluorescence. Green laser pointers may operate either pulsed or CW (continuous wave). Pulsed operation can be demonstrated by sweeping the beam across a wall rapidly, giving a “dashed- line” display. Spectra of Green Laser Pointers Spectra were recorded at high dispersion (0.04 nm�mm) for three of the pointers (numbers 1, 2, and 4), revealing a surprising range of behaviors, with each displaying a spec- trum clearly distinct from those of the others. (Lasers are iden- tified in the Materials section.) Spectra were recorded with short exposures (0.3–1 s), slits almost closed (∼2 µm), and neutral density filters to further attenuate the light so as not to saturate the CCD array detector. For all three GLPs, some spectra were recorded in time sequence (< 1 min apart). Typi- cal results are illustrated in Figures 2 and 3. Figure 2 shows a behavior more complex than antici- pated. The strong modes are spaced ∼2.7 cm�1 apart, imply- ing a cavity length of 3.6 mm. However, there are other peaks, usually weak, that do not fit this pattern. The extra peaks do not appear to be due to different transverse modes; these Figure 1. Calibration and I2 fluorescence spectra for excitation with a green laser pointer: (A) reference spectra from Hg [bright green (546.1 nm) and yellow (577.0 and 579 nm)], Ne (red, from He– Ne laser at 632.8 nm), and from the green laser pointer (532 nm). Spectra (B–D) are I2 fluorescence spectra recorded at different times. (B) and (C) are dominated by fluorescence excited mainly from
  • 29. υ� = 0 and 2, respectively; (D) includes a detectable component ex- cited from υ� = 3 (note “blue” line at top). Figure 2. Representative spectra for GLP 4, as recorded for differ- ent “on” times and battery powers. These spectra were calibrated with Ne lines at 533.0778, 534.1094, and 534.3283 nm (stan- dard air), giving 532.05(1) nm or 18790.0(4) cm�1 for the zero point on the wavenumber scale. http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-000.jpg&w=215&h=120 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-001.jpg&w=234&h=131 In the Laboratory 338 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org would be expected to vary spatially in the laser spot, but no differences were observed in spectra obtained by directing different regions of the spot onto the spectrometer slit. Time sequence spectra for this GLP showed a smooth progression of changes, with modes drifting to lower wavenumber ac- companied by slow intensity changes. With fresh batteries, the longtime pattern was typically one strong mode and ei- ther one somewhat weaker mode (∼50% of main) or two much weaker modes (each ∼5% of the main mode) on ei- ther side.
  • 30. Figure 3 shows the much different output of the other two GLPs. GLP 1 displayed multiple peaks displaced con- siderably to the blue (spectrum C) when operated with weak batteries. On the other hand, with fresh batteries, only one of the many recorded spectra (D) showed more than one peak, and that an order of magnitude weaker than the main one. It is hard to interpret the peak spacing in spectra C and D in terms of longitudinal modes. Similarly, the mode pat- tern in the spectra of GLP 2 (A and B) is not simple; and in this case the pattern is similar for weak and fresh batteries and shows many peaks of significant intensity. Not surpris- ingly, the fluorescence beam excited by GLP 2 never weak- ened to the vanishing point in a dark room. It is noteworthy that GLP 1 operated CW while the other two were pulsed. To check on the possibility that the single-mode character was tied to this property, I recorded spectra for another CW GLP (not used in the fluorescence experiments) but found its spectrum more like that of GLP 4 in Figure 2. LIF in the Gas Phase: What Else Is Required? The green-laser-induced fluorescence dramatically illus- trates the prime condition necessary for LIF: the radiation must be absorbed by the fluorescing species. That this con- dition is necessary but not sufficient is nicely illustrated by directing the same laser beam through a cell containing a sig- nificant pressure of another gas in addition to the I2, as shown in Figure 4. I2 in its B electronic state is known to be de- stroyed by collisional predissociation (14), I + I + I2I2(B) + I2 (1)
  • 31. and I + I + Ar I2(B) + Ar (2) These processes represent radiationless decay of the B state, so the fluorescence is said to be quenched. Actually, the beam can be perceived weakly in a high-pressure cell like that shown in Figure 4, since the quenching is not 100% efficient. And interestingly, this weak beam seems less subject to the tem- poral variations of the exciting laser. This behavior, which has not yet been quantified, could be a consequence of pres- sure broadening of the I2 absorption lines, leading to a re- duced specificity in the excitation process, due to more extensive overlap of the absorption lines. So I2 visible fluorescence requires absorption into the B state at low pressures, in the absence of quenching gases. Is that sufficient? The answer is “No,” as is illustrated by Fig- ure 5, which shows several beams from an Ar-ion laser oper- ated multiline, directed through the same low-pressure I2 cell. Only the longest wavelength line, at 514.5 nm, excites fluo- rescence (spectrum in Figure 6). With a power meter, one can confirm that the other wavelengths are absorbed (9). However, all but the “green” line excite I2 above the B-state photodissociation limit, resulting in direct dissociation (hence no fluorescence), I + I*I2(X) + hν (3) where I represents (as before) the ground state (2P3/2) of the atom and I* the spin-orbit excited state (2P1/2). 5 Figure 6 includes the spectra produced by exciting from υ� = 0 at two different wavelengths, from which it is clear
  • 32. that the wavelength range of the fluorescence is determined firstly by the excitation wavelength of the source.6 This seems Figure 3. Spectra recorded for GLP 2 (A and B) and 1 (C–E). The wavenumber calibration is the same as in Figure 2. Figure 4. A GLP beam traverses (right to left) two cells containing I2 vapor at room temperature. The cell on the right contains only I2, while that on the left has ∼500 torr Ar in addition. The laser was positioned ∼7 m from the cell to give a less focused beam. http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-002.jpg&w=229&h=123 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-003.jpg&w=191&h=143 In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 339 at odds with comments made in a recent discussion of fluo- rescence and scattering in this Journal (15), in which the fluo- rescence wavelength was stated to be independent of exciting wavelength. The difference is that the present gas-phase fluo- rescence spectra are obtained under essentially collision-free conditions in the gas, while the fluorescence spectra discussed in ref 15 were observed in solution, where internal relaxation occurs in the excited electronic state prior to emission. In
  • 33. that sense the present fluorescence spectra are more like the resonance Raman spectra described in ref 15; in fact workers with a primary background in Raman spectroscopy have of- ten referred to such resonance fluorescence spectra in exactly that way, as resonance Raman spectra.7 The intensity patterns in Figures 1 and 6 reflect the vary- ing Franck–Condon factors (FCFs) for the transitions, since the intensities are proportional to the FCFs (6). Figure 7 shows these FCFs for υ� = 43 and 32, the latter being the lowest υ� level excited at 532 nm (10). Note especially the strong alternation in FCFs for the latter in the region υ� = 7–15—a behavior that is clearly evident in the spectra of Fig- ures 1B and 6B. Other Spectroscopic Processes in Gaseous I2 The phenomena discussed above all involve the B–X transition in I2, which occurs conveniently in the visible spec- tral region. However, I2 has a rich emission–fluorescence spec- trum at shorter wavelengths, involving a number of other electronic transitions, some of which can be illustrated easily as a part of a demonstration of LIF in the B–X system. Thus, for example, if a Tesla coil is discharged in contact with the two cells shown in Figure 4, the resulting optical emissions are radically different (see the Supplemental MaterialW): The low-pressure cell gives a weak, yellowish-white glow, which on spectral inspection can be seen to display many of the same B–X bands that are observed in absorption (16). On the other hand, the high-pressure cell now displays an in- tense blue emission. Both cells show strong emission in the UV, but again with radically different spectra. The low-pres- sure spectrum is dominated by an extensive series of diffuse bands known historically as the McLennan bands (17, 18), while the high-pressure cell shows several discrete band sys- tems (19). Similar (but weaker) spectra can be excited by a
  • 34. low-pressure mercury discharge lamp like that used to ob- tain the calibration spectra in Figures 1 and 6. The states responsible for the optical emissions in the UV are of ion-pair character and lie at much higher energies than the valence B state (20). Their behavior in the presence of inert gases (rare gases, N2) is in stark contrast with that of the B state. The low-lying ion-pair states are practically im- mune to quenching, permitting the excited molecules to re- lax into near-thermal υ and J population distributions prior to emission. The resulting fluorescence–emission spectra are relatively insensitive to excitation process, making this case more like that described for fluorescence in (15). This be- havior—immunity to collisional quenching—is also the hall- mark of the rare-gas halide (RgX) excimer lasers, which operate on similar charge-transfer electronic transitions. In- deed, the strongest transition in the high-pressure I2 emis- sion spectrum, known as D� → A�, was long ago made to lase in experiments like those which led to the development of the RgX lasers. Figure 5. Several beams from an Ar-ion laser operated multiline transit the low-P I2 cell. The beams are dispersed by a prism lo- cated ∼25 cm to the right of the cell, the window of which is viewed from the inside, looking through the cylindrical wall. Fluorescence is excited by only the beam at 514.5 nm. The spots on the en- trance window to the left of the fluorescence beam mark the points of entry for beams at 501.7 nm, 496.5 nm, 488.0 nm, and 476.5 nm, from right to left. The lower spots are from reflected beams, and the faint green and blue lines are “ghost” images. Figure 7. Franck–Condon factors for the lowest υ� level excited
  • 35. at 532 nm and for the υ� = 43 level excited at 514.5 nm. Figure 6. I2 fluorescence spectra excited by the Ar-ion laser at 514.5 nm (A) and, for comparison, spectrum (B) from Figure 1. Reference spectra are from Hg and (in A) from the laser, at wavelengths noted in Figure 3 (except missing the last). http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-004.png&w=215&h=138 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-005.jpg&w=191&h=143 http://pubs.acs.org/action/showImage?doi=10.1021/ed084p336& iName=master.img-006.jpg&w=232&h=188 In the Laboratory 340 Journal of Chemical Education • Vol. 84 No. 2 February 2007 • www.JCE.DivCHED.org Observing and Photographing the Fluorescence Spectra The reader may have noticed that the fluorescence beam in the low-pressure cell in Figure 4 appears as an elongated cone, with intensity clearly diminishing with distance from the entrance window. The laser light is absorbed so strongly
  • 36. that it is significantly attenuated by the time it exits the cell. For absorption near the centers of lines originating in υ� = 0, this attenuation is much greater than the ∼25% one might predict from the average molar absorptivity of ∼800 L mol�1 cm�1 (9), for a 10-cm cell at P = 0.3 torr (the approximate vapor pressure of I2 at room temperature). On the other hand, for observing the fluorescence spectrum, it is good to use a disperse laser beam (as in Figure 4) rather than a focused one, and then to view the fluorescence near the entrance window, where it is most intense. Lasers that operate single mode with high spectral purity can interact with only small velocity groups of molecules, so a focused beam can effectively de- plete the population of absorbers as it “burns” its way through the cell. The result is reduced absorption and fluorescence (and violation of Beer’s Law, which assumes constant con- centration of the absorbing species). In essence, this phenom- enon is the basis of one method of sub-Doppler spectroscopy, known as hole-burning. The gross attenuation of the beam seen in Figure 4 is also a dramatic illustration of the origins of nonlinear behavior in analytical fluorimetry: when the fluo- rescence intensity is observed some distance from the entrance window, it will first rise with increasing pressure of absorber, then decline as the absorption attenuates the exciting light before it reaches the observation region. Spectra like those shown in Figures 1 and 6 are readily observed “live” using a hand spectroscope; however, record- ing them required much trial-and-error work with a digital camera. This task provides a golden opportunity to get ac- quainted with the more subtle options in the camera’s menu. A tripod is necessary, since exposures of 1–8 s may be needed. It is helpful to use a bright source to achieve proper align- ment of the source, spectroscope, and camera. The fluores- cence excited by the Ar-ion laser can be bright enough for this purpose (depending on the laser power), but that pro-
  • 37. duced by the green laser pointer probably will not be. I used a mercury Pen-Ray lamp, the strong green and yellow lines from which could be seen easily on the camera’s LCD moni- tor at lowest magnification. I could then zoom in on these lines, adjusting the alignment to maintain good images. Manual focus is needed for this operation; about 60 cm was the optimal stated distance in my case, though the lens of the camera was actually only about 1 cm away from the eye- piece of the spectroscope. Manual settings for both the aper- ture and shutter speed are also required, and I found it necessary to operate at largest aperture (F2.8) to obtain good image quality. For the longer exposures it is also necessary to invoke the “noise reduction” option. Alternatively one may use the “night scene” setting, which automatically employs noise reduction. Finally, I had to tweak the white-balance to render the yellow Hg doublet as yellow; an enhancement of the red component sufficed. However, I was never able to record a full spectrum with the nuanced differences that are obvious to the eye, for example, in the several laser lines to shorter wavelength from the green line in the Ar-ion laser output. In part, this is likely a problem of color/intensity in- terdependence, which the eye can handle much better than the camera. Materials I observed the on–off behavior for all four of the laser pointers I tried: (1) from Sean and Stephan Corp. (Taiwan); (2) Model GP4 from Limate Corp. (Taiwan); (3) AltasNova (Mead, WA and Taiwan); and (4) Beta Electronics (Colum- bus, OH). All of these pointers are stated to be less than 5 mW, but some evidently push this limit harder than others, judging from differences in the observed fluorescence inten- sities. Pointer 1 appeared to be best overall. Pointers 1 and 3 appeared to operate CW, while pointers 2 and 4 were pulsed,
  • 38. with a much faster repetition rate for pointer 2. For pointer 1 the stated wavelength was 532 ± 10 nm, while that for pointer 2 was 500–550 nm! However, as shown above, all operate near the stated doubled Nd wavelength of 532 nm (12). The spectra illustrated in Figures 2 and 3 were obtained using a Jobin–Yvon 1.5-m spectrometer equipped with a 3600-groove�mm holographic grating and a CCD array (Photometrics CCD9000, 27-mm pixel width) as detector. The reciprocal dispersion in this region was obtained by re- cording several Ne lines (noted in Figure 2) from a discharge source. The spectroscope contains a prism as dispersing element and has an adjustable slit, which was set as narrow as pos- sible for recording the spectra. When focused, with narrow slits, this instrument easily resolves the Hg yellow lines, which are separated by 2.0 nm. The specific manufacturer’s infor- mation has been lost, but a very similar spectroscope is sold by Edmund Industrial Optics for less than $300. Cenco (Sargent–Welch) offers a prism model with adjustable slit and focus for less than $60, but the resolution is not stated. An Olympus Camedia model C-4000 (3× optical, ∼10× total zoom) was used to record the spectra. Procedures and settings have already been described, and similar procedures should work for other digital camera models. The mercury discharge source used for alignment and for reference–cali- bration was a Pen Ray lamp (Ultra-Violet Products, Inc.). The Ar-ion laser was a model Innova 300 from Coherent (15 W maximum power all lines). The Tesla coil was a model BD-20 from Electro-Technic Products. The cells were silica and were charged with I2 (and Ar, for the high-P cell) on a vacuum line and sealed off with a torch. Hazards
  • 39. It is never good to stare directly into any laser, includ- ing the Class IIIa devices that are approved for laser point- ers. However, numerous studies have shown that momentary exposure to such light is no sight hazard, and certainly not at the distances involved in purported cases of airplane “spot- ting” that made news not long ago (21). Ar-ion lasers, on the other hand, can operate at powers that represent great sight hazard, so any use of the 514.5 nm line from such a laser should be carefully planned so that observers are not exposed to either direct or reflected beam light. The Hg Pen- Ray lamp produces strong UV light at 253.7 nm, which can damage the cornea of the eye. Both glass and plastic safety glasses block this radiation, but side exposure should be avoided, also. Operation of the source at a distance of 1 m http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/ In the Laboratory www.JCE.DivCHED.org • Vol. 84 No. 2 February 2007 • Journal of Chemical Education 341 or more for only short times (1–2 min) should result in mini- mal exposure, with or without safety glasses. The Tesla coil output is very high voltage (up to 50 kV) but also very high frequency, so the spark is not dangerous (22). Instructors can “wow” students by boldly taking a ∼0.5- inch spark from the tip, though this is more comfortably done if the spark is directed to a coin held between thumb and forefinger.
  • 40. Both the Tesla coil and the Hg discharge lamp produce ozone at detectable levels (by smell), so should be left on only while needed. Summary A green laser pointer can be used in a “flashy” demon- stration of laser-induced fluorescence in the gas phase, by di- recting the beam of the laser through a cell containing I2 at its room temperature vapor pressure. The demonstration is a good one to provoke discussion, and the explanation of the on–off behavior provides valuable insight into the require- ments for LIF and the properties of lasers. On the other hand, if this source is to be used to record fluorescence spectra for quantitative analysis, it will be necessary to use an array-type detector and to be aware that the time-integrated spectrum may contain contributions from numerous (υ�, J�) levels. Under the assumption that it is changing cavity tem- perature that is responsible for the varying wavelength of the laser, it is conceivable that one can modify such a pointer to permit control of its temperature and hence its wavelength. I am looking into this possibility.8 WSupplemental Material Color versions of Figures 1, 4, 5, and 6 and four QuickTime movies illustrating the temporal behavior of the fluorescence and of the discharges obtained by bringing the low- and high-pressure cells into contact with an operating Tesla coil are available in this issue of JCE Online. Acknowledgment I thank Tim Hanusa for providing two of the green la- ser pointers and Mike Bowers and Laura Swafford for help
  • 41. setting up and operating the Ar-ion laser. Notes 1. This behavior is so astounding to observers that it readily elicits hypothetical explanations. In my experience on demonstrat- ing the phenomenon to a number of students and faculty, few of these are close to reality. 2. This wavelength region corresponds to a Franck–Condon “hole” in the absorption from υ� = 1, so no lines from this level are listed. 3. After about two minutes of operation, Pointer 1 produced a particularly intense spectrum that was dominated by excitation from υ� = 0. Then in the next minute it slowly drifted off this ex- citation (or excitations) and onto one involving predominantly υ� = 2. When the laser was clicked off and on again, it cycled back to the latter spectrum in about two seconds, displaying the former spectrum for about 1 second in the process. 4. Reference 12 includes many very instructive photographs and diagrams of green laser pointers and their components, in ad- dition to the diagram cited in the reference. 5. There is also a weaker continuous absorption in this same spectral region, designated 1Πu ← X, which yields two ground- state I atoms via photodissociation (9, 11). However, this transition never yields fluorescence and is in no way sensitive to the B-state
  • 42. photo- dissociation limit. 6. It is sheer coincidence that the 532-nm wavelength so nearly coincides with the third line in the fluorescence spectrum excited at 514.5 nm: the energy difference in the two wavelengths just happens to match closely the energy difference between υ� levels 0 and 3 in the X state. This is perhaps unfortunate, as it suggests a “symmetry” in nature that is not there. 7. The weak fluorescence that persists at high pressures shows less dependence on exciting wavelength, as the surviving B- state molecules have mostly relaxed into low υ� levels. This situation is thus closer to that described for fluorescence in ref 15. 8. Additional spectra recorded at lower dispersion show that GLP 1, which appears to operate single mode in Figure 3, actually operates on one or two additional modes spaced 2 nm (70 cm�1) apart most of the time. Such broad spacing is hard to explain from expectations for both the gain profile and the cavity length. Literature Cited 1. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 6th ed.; McGraw–Hill: New York, 1996; pp 425–434. 2. Stafford, F. E. J. Chem. Educ. 1962, 39, 626–629.
  • 43. 3. Steinfeld, J. I. J. Chem. Educ. 1965, 42, 85–87. 4. Hollenberg, J. L. J. Chem. Educ. 1970, 47, 2–8. 5. D’alterio, R.; Mattson, R.; Harris, R. J. Chem. Educ. 1974, 51, 282–284. 6. Tellinghuisen, J. J. Chem. Educ. 1981, 58, 438–441. 7. Muenter, J. S. J. Chem. Educ. 1996, 73, 576–580. 8. Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 7th ed.; McGraw–Hill: New York, 2003; pp 423–432. 9. Tellinghuisen, J. J. Chem. Phys. 1982, 76, 4736–4744. 10. Forkey, J. N.; Lempert, W. R.; Miles, R. B. Appl. Opt. 1997, 36, 6729–6738. 11. Tellinghuisen, J. J. Chem. Phys. 1973, 58, 2821–2834. 12. Goldwasser, S. M. Sam’s Laser FAQ. http://repairfaq.ece. drexel.edu/sam/laserssl.htm#sslafgl. See especially http://repairfaq. ece.drexel.edu/sam/dpss1.gif for a diagram of the optical cavity (both accessed Nov 2006). 13. Svelto, O. Principles of Lasers; Plenum: New York, 1976. 14. Masiello, T.; Vulpanovici, N.; Nibler, J. W. J. Chem. Educ. 2003, 80, 914–917. 15. Clarke, R. J.; Oprysa, A. J. Chem. Educ. 2004, 81, 705–707. 16. Singh, S. M.; Tellinghuisen, J. J. Mol. Spectrosc. 1973, 47, 409–419. 17. McLennan, J. C. Proc. Roy. Soc. 1913, A88, 289–291. 18. Tellinghuisen, J. Chem. Phys. Lett. 1974, 29, 359–363. 19. Guy, A. L.; Viswanathan, K. S.; Sur, A.; Tellinghuisen, J. Chem.
  • 44. Phys. Lett. 1980, 73, 582–588. 20. Mulliken, R. S. J. Chem. Phys. 1971, 55, 288-309. 21. The Register. http://www.theregister.co.uk/2005/01/06/ laser_man_letters/ (accessed Nov 2006). 22. McKeever, M. R.; Sur, A.; Hui, A. K.; Tellinghuisen, J. Rev. Sci. Instrum. 1979, 50, 1136–1140. http://www.jce.divched.org/ http://www.jce.divched.org/Journal/Issues/2007/ http://www.jce.divched.org/Journal/ http://www.jce.divched.org/Journal/Issues/2007/Feb/abs336.htm l http://www.jce.divched.org/Journal/Issues/1996/Jun/abs576.htm l http://repairfaq.ece.drexel.edu/sam/laserssl.htm#sslafgl http://repairfaq.ece.drexel.edu/sam/laserssl.htm#sslafgl http://repairfaq.ece.drexel.edu/sam/dpss1.gif http://repairfaq.ece.drexel.edu/sam/dpss1.gif http://www.jce.divched.org/Journal/Issues/2003/Aug/abs914.ht ml http://www.jce.divched.org/Journal/Issues/2004/May/abs705.ht ml http://www.theregister.co.uk/2005/01/06/laser_man_letters/ http://www.theregister.co.uk/2005/01/06/laser_man_letters/ Morse Oscillators, Birge-Sponer Extrapolation, and the Electronic Absorption Spectrum of l2 Leslie Lessinger Barnard College, New York. NY 10027
  • 45. The Absorption Spectrum of 12 The visible spectrum of gaseous I2 affords a most inter- esting and instructive experiment for advanced under- graduates. Analysis of the electronic absorption spectrum, measured at intermediate resolution (vibrational progres- sions resolved; rotational fine structure unresolved), was described in this Journal by Stafford (I), improving the ini- tial presentation of Davies (2). (Assignment of vibrational quantum numbers u' to the bands of the excited B elec- tronic state of 12 was corrected in two key papers, by Stein- feld et al. (3) and by Brown and James (41.) Improvements both in experimental techniques and in data analysis ap- plied to the 12 absorption spectrum continued to be pre- sented in this Journal (5-9). Instructions for the I2 experi- ment also appear i n several physical chemistry laboratory manuals (10-14). Steinfeld's excellent spectroscopy text- book (15) often reproduces parts of the actual spectrum of 12 as examples. Birge-Sponer extrapolation is one method often used to analyze the visible absorption spectrum of 12. All measured differences AG between adjacent vibrational energy levels u' + 1 and u' are plotted against vibrational quantum num- ber u'. The differences between all adjacent vibronic band head energies observed in the usual undergraduate labo- ratory experiment on 12 give a nicely linear Birge-Sponer plot. Thus, these vibrational energy levels (which do not include 20 to 30 levels a t very high values of u') of the ex- cited B electronic state can be accurately described as those of a n anharmonic oscillator with a single anharmo- nic term. This, in turn, implies that the potential well gov- erning the vibrations up to the maximum u' observed in these experiments is closely approximated by a Morse po- tential function.
  • 46. What is the Correct Procedure for BirgeSponer Extrapolat~on? Unfortunately, many sources, including some in the ref- erences, give incomplete, ambiguous, or self-contradictory descriptions of the Birge-Sponer extrapolation method. I N T E R N U C L E A R DISTANCE + Figure 1. Morse potential curve and vibrational energy levels. Equations for determining spectroscopic parameters pre- sented in different sources are not consistent with each other. Several erroneous presentations are common. This paper aims to give a complete, correct exposition of Birge- Sponer extrapolation for the special case in which the plot derived from the observed data is taken as exactly linear. If extrapolation beyond the observable differences is as- sumed to be strictly linear in both directions, and particu- larly toward the dissociation limit, then a Morse function would exactly describe the vibrational potential for the di- atomic oscillator. For real molecules this is not strictly cor- 388 Journal of Chemical Education rect, but the Morse potential is oRen a good appmxima- tion. Also. the analvsis of actual molecular s ~ e c t r a l data in terms of t ~ s exactiy soluble problem in quakum mechan- ics should be clear and unambirmous. .. A schematic diagram of a Morse potential, with its quan- tized vibrational enerw levels. and ~ictorial definitions of various terms used inyhe following isc cuss ion, is shown in
  • 47. Figure 1. Morse Oscillators In 1929, Morse (16) introduced a convenient two-param- eter analytical function to approximate the shape of the anharmonic potential energy curve for a diatomic molecu- lar oscillator: The Sehrodinger equation for a particle of reduced mass p i n this potential can be solved exactly. The energy levels are given by With the conventional spectroscopic units of wavenum- ber (cm-'1, these quantized levels are oRen written in the following form. There are no higher order terms. The energy levels of a Morse oscillator are given by a harmonic oscillator term plus a single anharmonic correction term, which is pre- cisely what is required for the linear Birge-Sponer extrap- olation procedure to be valid. D. is t h e depth of t h e vibrational potential well; P governs the curvature a t Re, and thusthe force constant k,. k. = wep2 ( 4 ) D. and p determine the fundamental vibration frequency, v. = cw., and the anharmonicity, v&, = ewe: Once we and we are found from analysis of the spec- trum, the well depth D. for the Morse oscillator can be de- termined exactly: The zero-point energy E(, = ,,, of the Morse oscillator is given by
  • 48. Subtracting E(, . o, from D. gives the bond dissociation energy DO: Do (0.- 0&J2 - = hc 4w& (9) A key feature of the vibrational energy levels in a Morse potential is that the number of bound states is finite; the integer vibrational quantum numbers u for the bound states have a maximum possible value u,, governed by the following inequality Moreover, in order to correspond to a finite, normalizable wave function, the highest quantized vibrational energy level must be less than the well depth; E(v,,) cannot be exactly equal to D.. The difference AED between the disso- ciation limit D. and the highest quantized level E(v,,) must obey a related inequality: Both inequalities result from the condition put on all phys- ically meaningful bound states: the vibrational wave func- tion must vanish as the internuelear distance R eoes to - infinity ( I 7). As a Morse oscillator approaches dissociation, the den- sity of states does not increase without limit, and the en- ergy level spacing does not approach zero, in contrast to the pattern in certain other bound systems, for example, the hydrogen atom. Finally, there is no necessary relation- ship between the values of the two parameters D. and P defining any Morse potential. Thus, w. and a&. can also have any arbitrary ratio greater than 1. BlrgeSponer Extrapolation Both we and w g e are often obtained from spectroscopic
  • 49. data by a graphical method introduced by Birge and Spo- ner (18). For any two adjacent vibrational energy levels of a Morse oscillator, AG(u) = G(u + 1 ) - G ( u ) = m e - 0 g e ( 2 u + 2 ) or equivalently, The spacings between adjacent vibrational enerw levels decrease as a-linear function of the quantum number v or, alternatively, of the variable v + 'h. The second differences are constant: All the Morse potential parameters can be found analyt- ically, of course, using equally well the line defined by ei- ther eq 12 or eq 13. However, a closer look a t the geometri- cal properties of the plot and its commonly presented interpretation shows why it is far better, particularly for pedagogical purposes, to make a Birge-Sponer extrapola- tion from a plot of AG vs. v + U rather than AG vs. u. A proper Birge-Sponer plot of AG vs. v + U, corresponding to the energy levels of Figure 1, is shown in Figure 2. The discrete values AG define a line (eq 13). From its slope, -2m&., the anharmonicity parameter w&. is found. This value is then used with the ordinate intercept of the line, w. - w&,, to determine the wavenumber w. corre- sponding to the fundamental vibration frequency v, in the h a m o n k oscillator approximation. All the of Volume 71 Number 5 May 1994 389 Figure 2. Linear BirgeSponer plot of A G v s . v + th.
  • 50. the Morse potential can now be determined. In particular, eqs 7 and 9 can be used to calculate D. and Do. Typical Sources of Error I n the Birg-Sponer plot (Fig. 2) the area of the triangle under the line between the ordinate intercept and the ab- scissa intercept is exactly equal to Ddhc (compare with eq 9): This is so because that area corresponds, as Figure 1 makes clear, to the following sum, by which the value ofDo can be expressed: The Contributions The last term in eq 15 is almost always either omitted, because a n unnecessary approximation is made, or erron- eously included within the sum, which is incorrectly writ- ten to run to u = urn.,. Clearly, Ddhc, the area of the large triangle in Figure 2, equals the sum of the areas of all the rectangular strips, each with height AG(u + h) and unit width, plus the area of the little shaded triangle. I will comment on each contribution in turn. If the plot were of AG vs. u rather than AG vs. u + h , the area under the line would not equal Ddhc. Then just a bit more than half the value of &GI, =ol, the area of the first and largest strip in the correct plot, would be left out, and DO would be seriously underestimated. The error in Ddhc would in general be For the B state of 12 this is = 65 em-'. If students are told to calculate the area of the triangle in the BirgeSponer plot to find Ddhc, the plot must be made correctly.
  • 51. The area of the little shaded triangle in Figure 2 is equal to (D, - E(u,))lhc. It corresponds exactly to the energy difference AED, shown in Figure 1, between the well depth D. and the highest quantized vibrational energy level E(u,,). Recall that AED bears no necessary relation to the pattern of quantized E values, or the corresponding AG values, for integral u. For the special case when D. acciden- tally equals exactly E(u, + 11, the numerical value of AEdhc is expressed by cyr. = AG(urn,). The highest bound state in this case, however, is still the one with u = u,,; there is no Morse oscillator wave function corresponding to E(u, + 1) = D.. Therefore, no rectangular strip is drawn for AEdhc in Figure 2, and i t is treated as a distinct sepa- rate term in eq 15. The highest vibrational quantum number urn, is the largest integer less than (we - w&.)l2w&, the value of the abscissa intercept. To what value does the abscissa inter- cept (which is generally nonintegral) itself correspond? Extrapolation in Incorrect Terms Birge-Sponer extrapolation is often incorrectly expli- cated in terms of the intercept a t AG = 0 and the supposed significance of the corresponding uib~=o,, with u treated as a continuous variable: when or equivalently, when The value U(AG= 0, does not correspond to urn, except ac- cidentally for the special case when
  • 52. m d h c = 0 ~ J 4 In general, uiAc=o, is a noninteger in the range Moreover, uiA~=o,, treating u as a continuous variable, does not correspond to UD, defined as the value of u a t the dissociation limit, because In short, the quantity uiAo=o, has no physical significance a t all. The correct value UD corresponding exactly to the disso- ciation limit, at which GI,=,, = DJhc, is This is also clearly shown by the proper BirgeSponer plot (Fig. 2) of AG vs. u + h , where uo exactly equals the value of the abscissa intercept. In this plot, each strip is bounded left and right by successive integer values of u. The abscissa intercept is simply the upper bound of a hy- pothetical last strip, withinteger lower bound u,, and an area AEdhc equal to the area of the shaded triangle. Gen- erally, U D is nonintegral, except for the special case when AEdhc = wo, its maximum ~ossible value. The disti%on between u = l, and u n can also be under- stood analvticallv. Eauations 12 and 13 fur AG are bv defi- - - nition expressions for finite differences Au = 1. 'This, the 390 Journal of Chemical Education condition AG = 0 does not correspond exactly to D d k , the maximum value of G. Then uD is defined as that value of u, treated as a continuous variable. for which G(u1 = D J k , the maximum in the parabolic function G(u). Thus, vD is
  • 53. correctlv found bv setting the derivative of G with respect to u equal to zero; and s o h g for u = UD: This gives for u~ the value in eq 19, precisely equal to the abscissa intercept on a Birge-Sponer plot of AG vs. u + U . Application of BirgeSponer Extrapolation to Real Data To apply the method to actual spectroscopic data, a least- squares line should be fit to a plot of the measured AG vs. u + 49. If "hot" bands are seen (so that AG data come from several vibronic progressions, originating in different vi- brational levels of the ground electronic state and going to vibrational levels of the same excited electronic state) then all the data should be used in the same Birge-Sponer plot to find the parameters of the upper state. For good results, it is important to calibrate the wavelength scale of the spectrophotometer. For I z , the band head positions (i.e., the data measured in the usual undergraduate experi- ment) are very good approximations to the positions of the band o r i ~ n s (6,151. ~tudeGts in the advanced laboratory course a t Barnard College apply linear Birge-Sponer extrapolation to band head data from the three overlapping vibrational progres- sions they see in the visible absorption spectrum arising from the B c X electronic transition of 12. Over the past 15 years typical student data bas yielded results for the B state in the ranges (cn-'1 shown in the table from mea- surements on the following transitions. McNaught warns against too facile comparison of stu- dent results with literature values (6). Older values may have been revised or reinterpreted. Newer data analyses use fitting methods more sophisticated than Birge-Sponer plots, so not all parameters are comparable. Student re- sults also depend on instrument calibration, resolution,
  • 54. and extent of data. We believe the most useful reference values for com~arison are those shown in the table. Accnrate high resolution spectroscopy shows that values of D, estimated by simple linear Birge-Sponer extrapola- tion are systematically incorrect. The discrepancy arises principally from the departure of successive term differ- ences from linearity. a t high u, approaching the dissocia- tion limit. The Morse potential does not adequately repre- sent the long-range attractive forces between the two atoms of a diatomic molecule at larrre separations. For the - - excited B state of I z , however, this inadequacy becomes ap- parent only a t very high values of u'. These are usually difficult or impossible to observe because these transitions have such low intensities. Correct Extrapolation a s v Approaches the Dissociation Limit More accurate methods for large u that take into account the actual long-range potentials are discussed in Steinfeld Typical Student Data Compared to Literature Values Student Data me 127-135 me% 0.94-1.05 Do/hC 4 1 7 1 4 2 2 D$hc 42364490 Lierature Values
  • 55. 0s 132.1 cm-' We& 1 .05 cm" (fit of band head data to a Morse potential (6)) Ddhc 4391 cm-' (estimate of actual well depth (3) (15). In an elegant application to 12, Le Roy and Bernstein (19) used a n extrapolation appropriate to a potential V ( R ) = D. - CR5 at large internuclear separation R. This is the correct potential between the J = l/z and J = 3 h atoms into which 12 in the B state dissociates. The fit to the 1 2 data a t very high values of u' is excellent. Thus, this ex- trapolation provides a n accurate determination of the en- ergy a t the dissociation limit, which lies just above u- = 8 7 ( ~ i n e a r BirgeSponer extrapolation U & ~ A G up to u' = 50 underestimates the dissociation limit of this state by 140 cm-' (201.1 An even more elaborate analysis, applying a power series in Rm, of the longe-range potential curves of the excited B electronic states in the series 12, Br,, and C12 was given by Le Roy (21). Literature Cited 1. Sfaffmd,F.E. J. Chem. Edue. 1 9 6 2 . 3 9 . 6 2 6 6 2 9 . 2. Danes, M . J . Chem. Educ. 1 9 5 1 , 2 8 , 4 1 4 4 7 7 . 3 . Steinfeld, J. I.; Zare, R.N.: Jones, L.;Lcek,M.;l(lemperer,W. J. Chem. Phys. 1866, 4 2 . 2 S 3 3 . 4 . Bmwn,R. L.; 3ames.T. C. J. Chem. Phys. 1 9 6 5 , 4 2 , 3 M 5 . 5 . D'altedo, R.; Matteon, R; Har6a.R. J . Chem. Edue.
  • 56. 1974,51,282-284. 6 . M a a u g h t , I. J. J. Chem Educ 1980.57.101-105. I . C a m a h t , H. M. J. Chem. Edue. 19&3,60,606607. 8 . Snadden, R.B. J . Chem.Educ. 1987, €4,919-921. 9 . Armanlous, M.: Shaja, M. J . Chem. Edue. 1%36,63,621- 628. 10. Bdm, A. G . I" Erpon"U"b in Physical chemistry, 2nd ed.: Wilson, J. M.; New- mmbe, R. J.; Densm, A. R.; Riekett, R. M. W , Eds.; P q a m n : O l f d , 1968; pp 303-306. 11. Salzbeq, H. W.; M m w , J. I.; Cohen, S. R.: Green, M. E. Phyrieol Chemisfm:A Modem Lobomfory Caum; Academic: New York, 1969, pp 2 3 6 2 4 9 : pp 443-445. 12. Hofacker, U. A. Chemiml Erperimtation; Free-: San Francisco, 1972; pp 4 2 51; p p 5 1 4 9 . 13. Findlqv'sPmcfimlPhysimlChomisfq,9thed.; Levitt, B. P., Ed.;hngnan: Londm. 1973; p 183. 14. White,J.M.PhysidChemialqLobomtoryE~p~iiitt:Pmtie~Hall:Engl ewood Cliffs, NJ, 1975: pp 3 8 1 3 8 8 . 15. Steinfe1d.J. I . MoleculesondRodlotion, 2nd ad.: M I T Cambddge, MA, 1985: C h a p ters 2 6 : e~peelally pp 128-134: pp 145-161. 16. M m e , P . MPhvs. Rou. 1 9 2 9 , 3 4 , 5 7 - M .
  • 57. 17. F M w , S. Pmetieol Quantum Mdmlulies; SpringerVerlag: New York, 1914: Vol. I. pp 1 8 2 1 8 6 . 18. Birge,R. T.; Sponer, H. Phys. Rau. 1928.28.259-283. 19. Ie Roy, R. J.; Bemstein, R. B. J. Md. Spoctmscopy 1911.37, 109-130. 20. Stelnfeld, J. I . ; Campbell, 3. D.: Weisa, N.A. J. Mol. Spetrosrnpy 1989,29,204-215. 21. Ie Ro5 R. J. Con. J. Phys. 1 9 7 4 , 5 2 , 2 4 6 2 5 6 . Volume 71 Number 5 May 1994 391 When experimental d a t a are obtained from expensive e t al. suspect a t lower values of the vihrational quantum equipment i t is desirable to extract as much information as number as i t is clear t h a t the (u', 0) hands for u' less than 14 possible from those data. I t is also desirable t o make the are swamped by the intensity of the (u', 1) and (u', 2) bands. analysis as self-contained as possible by minimizing the need Therefore, particular care must be taken in analyzing this to use literature values for quantities unobtainable by the region. Fortunately, the existence of these hot hands makes experimental design. i t possible, as will be shown below, to obtain much more in- The electronic spectrum of iodine has played a central role formation than is usually done. Ian J. McNaught University of Rhodesia P.O. Box MP 167
  • 58. Salisbury. Rhodesia in testing the cons&ency of quantum mechanics and observed spectroscopic fine structure. T h e results have been summa- rized in a masterly fashion by Mulliken ( I ). The analysis of the low resolution electronic spectrum of 12 has become a classic advanced undereraduate exoeriment. oarticularlv since The Electronic Spectrum of Iodine Revisited , . the expository paper of Stafford (2) and its extension by D'alterio et at. ( 3 ) . T h e readv availabilitv of minicomouters and programmable calculators means t h a t i t is possible for students to perform much more sophisticated analyses of experimentai data than has previously been the case. Staf- ford's analysis is quite restricted; with no more experimental work it is possible to extract very much more information on the spectroscopic constants of the iodine molecule and gain significant insight into a range of spectroscopic and quantum mechanical concepts. T h e data t h a t can be obtained include (a) the separat.ion between the minima of the potential curves, rr,, (1)) the rreqoencips and anharmonieities in each electronic state, w,, W d e , (c) the dissociatim energies in each electronic state, Dd,,, (dl the differences i n equilibrium bond lengths, ( e l maximumintensity transitions as well as information an the
  • 59. vibrational wavehnctions. Spectroscopic Introduction The variation of potential energy of a diatomic molecule with internuclear distance is conveniently represented on a potential energy diagram. Figure 1 shows the variation in potential energy for the iodine molecule in its ground ( X ) electnmic state and its second (B) excited electronic state. This figure illustrates the parameters to he calculated and uses the standard spectroscopic notation (4). The iodine molecule gives rise t o well resolved vibronic hands between 500 nm and 620 nm. However, as can be seen from Figure 2, there is significant overlap between (u', 0), (u', 1 ) and ( u ' , 2) in the middle of this region. This makes the Hirge-Sponer plots (4, p. 438) of both Stafford and D'alterio Experimental Details The spectrometer used was a Unicam SP 1750 run a t 0.2 nm s-I with a band width of 0.2 nm, the spectrum being recorded on an AR 5 5 recorder. The spectrometer was calibrated as suggested by Stafford (2). The spectrum of gaseous iodine was run a t room temperature after placing several crystals in a 10-cm cell. I t was found unnecessary t o use higher tempera- tures provided that the sublimed iodine was removed from the windows prior to a run. Treatment of Experimental Data The first problem is to assign vihrational quantum numbers to the bands. The numhering given in Table 1 is based on that proposed by Steinfeld et al. ( 5 ) on the basis of intensity dis- tributions and proved hy Brown and James ( 6 ) from an analysis of the isotope effect.
  • 60. In order to ensure that the assignments are consistent, i t is useful to prepare a Deslandres table (4, p. 40) as in Table 2. An inconsistency in the numbering will show up as a n in- consistency in the differences between rows or columns. This technique will not show up a consistent numbering error. Table 1. Numbering of Band Heads for Iodine d v" h i m d v" Alnm d ' Xlnm u.-2w.x. - .- > '' lNTERNUCLEAR DISTANCE Figure 1. Potential energy diagram for iodine. , : -~~ ,no $70 380 350 ,,! Figure 2. Overlapping of ( d . 0). ( d , 1) and ( d . 2) bands in iodine. Volume 57, Number2 February 1980 1 101 Directly Derived Parameters If 7" and T" represent the electronic term energies of the two states while G(u') and G(u") represent their vihrational energies then, ignoring rotational energy changes, the energy of a transition will he given by c = T. - T; + G ( d ) - G(u") (1) For the case studied here T i is Tero, because it refers to the ground electronic state while T , equals a,, the frequency of the hypothetical transition between the two minima of the
  • 61. potential curves (Fig. 1). T h e vihrational term values can he written as GIU) = WJO + 'I2) - + %)2 + w a s ( " + %IS + . . . (2) where we is the frequency for infinitesimal amplitudes of vi- bration and m,~,, w,y,etc. are anharmonicity constants. If only the first two terms in this expression are taken, i.e. Glo) = we(" + %) - W.X,~U + %I2 (3) then the transition frequency is given by " = re + wp 10, + '14 - wpxp I", + 11d2 - w; i",, + '12) + w;x; ("I + 'I# (4) T h e usual procedure from this, point (2) is to use a Birge- Sponer p)): to ohtain w, and w&t,, then to use literature values for we, w,x, along with the observed a to determine a,. A much better way is to use the technique of mu!tiple linear regression (7) to determine a,,w,, were, we and w,x, directly from a, u' and u". Indirectly Derived Parameters Convergence Limit E * E* is the energy of the transition from u" = 0 to the top of the upper state potential well, i.e., the energy a t which the vihrational structure joins the continuum. I t is calculated as the energy of a transition ending on level u plus the sum of all the vihrational quanta ahove u t o the maximum urn.,. T h e energy of the vibrational quanta is given by G(u + 1) - G(u), so t h a t Retaining only quadratic terms in G(u) gives
  • 62. G(u + 1) - G(u) = w; - 2wkxk(u + 1) ( 6 ) I t follows from this equation t h a t u,,, (i.e., t h a t u for which G(u + 1) - G(u) = 0) is given by u,,, = wpIiZwG1) - 1 17) Therefore EX = o, + 'lz[wk - Zwkxkiu + l)l(u,,, - u) (8) This equation is just the analytical form of the Birge-Sponer technique. Its advantage is that all the observed values of a, due to transitions from u" = 0 can be used to produce many independent values of E*, these values can then be used to ohtain some estimate of the error in E*. T h e graphical tech- nique is a one point method which gives no indication of the precision of the determined E*. This analytical technique assumes a linear Rirge-Sponer extrapolation, a valid assumption for the data ohtained in this experiment. Dissociation Energy D. The dissociation energy is the energy required to dissociate the molecule into atoms. (Dissociation into uositive and atom will dissociate into the upper state products. When separated the atoms have zero relative velocity. Increasing the energy ahove E* will give atoms having increasing relative velocity and kinetic energy. I t is this process which gives rise to the continuum observed in this exueriment a t short wave- lengths. T h e ground state dissociation energy measured from the
  • 63. lowest vibrational level is given by where E(I*) is the difference in energy between a ground state iodine atom ( 2 P ~ l z ) and the first excited state of the iodine atom PPll2):It.has a value of 7589 cm-' (8). The ground state dissociation enerev measured from the bottom of its uotential T h e excited state dissociation energy measured from the hottom of its potential well is Force Constants k. When a bond is stretched there arises a restoring force which resists the stretching force. Within the simple harmonic oscillator approximation this restoring force is given by (4, p. 74) where r - r , is the extension and he is the force constant. The greater the force constant the more difficult it is to stretch the bond. T h e force constant is related to the curvature of the potential well and can he calculated from (4, p. 98) k, = [email protected]: (13) Morse Curve If the simple harmonic oscillator model did apply, the po- tential e n e r m curve would be a uarabola. i.e, its dissociation energy would be infinite. A more realistic potential energy curve is t h a t introduced by Morse (9) l i ( r - r e ) = ll.(p-RI'-'-) - (14) where 0 = nw,(2g~/D,h)'/~ (15) The importance of this potential function is that i t is possible to solve the Schrijdineer eauation for this ootential and ohtain ..
  • 64. . the vihrational energy wavefunctions. These wavefunctions are finite and give eqn. 3 when appropriate identifications are made. If r - re is denoted by u then these wavefunctions R,iu) can be written ( 1 0 ) Rt(u)ru e-kz(k~)"2n-1L:,(k~) (16) where Table 2. Deslandres Table for IodineB d l / ' 0 1 2 19 17799 213 17586 212 17374 93 93 93 18 17706 213 17493 212 17281 94 94 94 17 17612 213 17399 212 17187 97 97 97 AII entries in cm-', corrected for vacuum (131. me difference between rowsequslr w; - 2w;xaiv+ 11 while the difference between columns equals w: - 2w;xi(v+ 11 where v is the lower of the two vibrational quantum numbers. 102 1 Journal of Chemical Education L k , ( k i ) = (kz)" - n ( h - n - l)(kr)"-'
  • 65. + % n ( n - l ) ( k - n - l ) ( k - n - 2 ) ( k ~ ) " - ~ . . . (17) where n is the vibrational quantum number. In this experi- ment we are interested in transitions from n = 0,1, and 2. The required prohahility density distrihutions are given by R i ( u ) a e - k ' ( h r ) k - ' (18) R : ( u ) a r-ki(ki)"-3[kz - (k - 2)la (19) R:(u)a eck2(k2)k-"[(kz)2 - 2(k - 3)kz + (k - 3)(k - 411' (20) However, for numerical evaluation these expressions are not very useful (they contain terms like ZOOzo0), so a scaled prohahility density distribution can he introduced, T i i ( u ) = ( R , , ( U ) I R , ( ~ ) ) ~ (21) Explicitly the scaled probability density distrihutions used here are , 2 = e - k ( a - l ) z k - l (22) R : ( ~ ) = e-klz-l)rk-3[kr - (k - 2)]2 (23) ~ l ( u ) = e-k'"-'~zh-5[(kz)Z - 2(k - 3)kz + (k - 3)(k - 4)12 (24) Figure 3 shows these functions superimposed on the Morse curve for the ground state of iodine. Also plotted are the squares of the wavefunctions for the harmonic oscillator. The harmonic oscillator probability density distrihutions are presented by (4, p. 78) +ga e-eu2 ( 2 5 )
  • 66. +:a ou2 cnu2 (26) +arr (2au2 - 1 ) s e - ' ~ 2 (27) where a 4a2cpw,lh. I t is clear from Figure 3 t h a t as the vi- brational auantum number increases, the difference in bond h g t h Irtu'een thr maxima t ; ~ r the harmonic and anharmonic tr;cillators I w o m r ; morr prowuncrd. This is imlvxtant in determining the expected intensity distribution in the vihronic hands. Anharmonic Oscillator Maxima u:" The mnximn in thv ilnharm~,nicoscillat<,r distrihurionsrim t w calculntnl I'row the rollcrwing iormul:~r, where in t:nch case z is related t o u by the equation 1 u = - - I n * P (28) ur: z = (k - 1)Ik (29) u y : z = [(2k - 3) z t m I l 2 k (30) u p : (kz)" - (3k - 7 ) ( k # + (k - 3)(3k - 10)kr - (k - 3)(k - 4)(k - 5 ) = 0 (31) Figure 3. ~ a r i o n i c and anharmonic probability density distributions for io- dine. Although an analytical solution for eqn. (31) is obtainable, i t is easier to solve the equation numerically.
  • 67. T h e classical turning points, u, for each of the three vibra- tional levels can he ohtained by solving the equation W;(U + II2) - W;X& + = D;(e-6"" - (32) Diflerence In Equilibrium Bond Lengths, re - r; As has been discussed by D'alterio e t al. (3), i t is possible to use the intensities of the (u', 0) bands to estimate the dif- ference between the equilibrium bond lengths in the two states. Because of the problem of overlapping bands men- tioned earlier, a simpler approach has been taken here. I t is considered sufficient merely to take the energy corresponding to the most intense transition, nnt. By the Franck-Condon principle (4, p. 1941, this t r ~ ~ i t i o n ~ w i l l originate and end a t the internuclear distance r ,, (not r e as claimed by D'alterio e t al.). I t is clear from Figure 1 t h a t ,J T h e plus sign is chosen if the hands are red degraded, i.e., rk > re, the negative sign if the hands are violet degraded, i.e., r , < r e . Transitions of Maximum Intensity Provided t h a t the vibrational quantum number is greater than about 10, the probability maximalie close to the potential curve (11). Assuming t h a t i t lies a t the same internuclear distance as the curve, and that the Franck-Condon principle holds, i t is possible to predict the most intense ( u ' , O), (u', 1) and (u', 2) transitions. They are ohtained by solving the fol- lowing equation for v' where u = u y + r ; - r ;
  • 68. It should l,r nott!d that tl~iscdlculation r n h r n r ; - r. is used tu (t~lculatv the tnnxii~~um intensir., is nor related in a circulnr way t o the calculations in eqns. 33-35 (where the maximum intensity u' is used t o calculate r ; - r;) as i t brings in w, and wkx, which are independent of the parameters in those three equations. T h e calculations serve as a good check on the in- ternal consistency of the experimental data. If a plotter is available the relationship between the Morse curves and the vibrational wavefunctions can be shown as in Figure 4. Drawing a vertical transition from the maximum gives the value of U'(u), i t is instructive then t o draw in the appropriate upper state vibrational levels using eqn. (3). Figure 4. Derived potential curves for iodine. Volume 57, Number 2, February 1980 1 103 Table 3. Parameters for Iodine a Equation (4) Equation (42) vmax. intens. 27. 14. 9 26, 14. 10 All V B ~ Y B J in cm-' exceot where indicated. Rotational Constants 6. If the literature value for r, is assumed then i t is possible todetermine reand from these the rotational constants in both states as well as their variation with vibrational quantum number. The equilibrium rotational constants are determined
  • 69. from R , = hl(8a2epr:) (37) T h e rotational constant in a given vibrational level is related to R. by the equation ( 4 , p. 106) & = R e - & + % ) (38) where LY, can h e determined from (4, p. 108) n, = 3 R . (a - 2R.lwJ (39) Equation (37) can he used t o verify t h a t the difference he- tween the hand head and hand origin can be ignored within the accuracy of these data. Equations (37)-(39) permit the calculation of the bond lengths ro, r l and rz in each of the three lowest vibrational levels. Choice of Equations Equation (39) is a modification of the one quoted hy Herzberg; the equation there involves the anharmonicity constant and therefore can have large error limits when de- termined from such low resolution d a t a as obtained in this experiment. T h e transformation uses the relationship w.x. = wa/4D, (4, p. 100). Similarly, the literature contains a number of different equations from those used here, the cri- terion of choice for this paper being those equations that contained precisely obtainable parameters. For example, al- ternative equations for some of the parameters are 6 = ( 8 a 2 ~ p ~ . ~ . l h ) 1 1 2 ( 9 ) (40) 1 h = -
  • 70. X e (10) ( 4 1 ) Equations (40) and (41) involve t h e anharmonicity, for the Table 4. Summary of Relations multiole linear rearession multiple linear regression mullcple linear yression v,. = w./(~w.x.) - 1 E . = n, + %[w; - ~ W & Y + 1 ) I I v m ~ . - "1 Calculate for all o, originating from P = 0. Find the average and standard deviation. D ' - E . - - E(I.1 E(I') from literature D: = D; + %w. - '/q~Zx: ~ 4 = E' - c. t XW. ; !14~.a Do = D: - '12u. + % W ~ X ~ k = 4D&. 0 = a ~ d 2 c g / D ~ h ) " ~ a = [email protected] 1 z = ( k - 1 ) l k . U = - - I n = a ground state this is the least precisely determined parameter (see Table 3 ) . Eauation (41) does have the useful feature of showing t h a t as the anharmonicity increases the value of h decreases. Results T h e results of a typical careful run are collected together
  • 71. in Table 3. T h e quoted errors are standard deviations. Comparison with the Literature I t is common nractice to comnare results with well tabulated literature values. Such a comparison must he approached cautiouslv for two reasons. T h e first reason is t h a t new ex- perimental data or reinterpretation of accepted data may lead to marked changes in the values. The second and more important reason is t h a t present lit- erature values are usually obtained using much more sophis- ticated term functions, e.g., eqn. (2) is usually expanded to higher powers in ( u + %), (for instance Le Roy (12) fitted the experimental data to an expansion up t o ( u + %)lo), thereby affecting the coefficients of the lower powers. This can have a marked effect, as can be seen in Table 3. The second column of figures was ohtained from the same experimental data using eqn. (42) in place of eqn. (4) n = a, + &(u' + '12) - w>;(u' + 1/2)2 + w>;(u' + MP - W:(U" + '12) + w.x;(ul' + 1/2)2 (42) I t is clear that the more sophisticated expression produces marked changes in the parameters and, in general, brings them closer to the accepted values. Because of these problems, the emphasis should he on precision (internal consistency, standard deviation) in eval- uating reports rather than accuracy (closeness to some tahu- lated accepted value). I t is one of the advantages of multiple linear regression that error estimates of the parameters can he ohtained easily, careful work produces small standard de-
  • 72. 104 / Journal of Chemical Education viations and should he treated as mare desirahle than work t h a t produces a mean value close to t h e accepted value h u t which has a large standard deviation. Table 4 summarizes t h e method of calculation of the pa- rameters i n a logical order. Conclusion This paper presents equations and techniques for calcu- lating and interpreting many of t h e spectroscopically im- portant parameters associated with the ground ('2:) and second excited (:'II:,) states of the iodine molecule. It also shows students that in physical chemistry the obtaining of experimental data in the lahoratory is often only a small part of the total time commitment compared to the detailed analysis of data. Acknowledgment The author wishes t o acknowledge his first contact with a rudimentary version of this experiment when he was a Senior T u t o r a t the University of Melbourne. Literature Cited !I1 M u l l i k e n R. S., J. C h e m I'hyr., 55.288(1971). 121 Stafhid F. E...I. CHEM. EDUC.. 39, ti26 119fi21. I:!) D'nlterii! H., Matts~rn H.. Harris. R.J. C H E M . EDIIC.SI.2S2 11974).
  • 73. 161 H e r r h e r ~ C.. "Mdrcular Spectra and Mdeculsr Structure I: Spectra of Diatnmir Mliind~s"Vsn N i ~ r t i a n d , I ' r i n m i m N.I. 2nd lid.. 1950. I?) S l & $ & i . I I.. %are R. N . . J m s % I. .. l.erk M . . a n d Klmmllarer W.. J. Then,. P I I ? ~ . 42. (81 l:syd,,n. A. c.. "l)irn,ria~ic,n Energies." Chapman and Hall, l.sndim.2nd lid.. Rex, 1 w . t . .. .,., . 191 MIIIIP. 1,. M.. P h j l K m . 31.17 (10291. l l 0 1 llunhnm. J. I. .. l'b% R n i . 14,438 119211. 1 1 1, Sch8n:l.. 1;'Quantum Mschnnic.,"MrCnr-Hill H w k O>..NewYork. 2nd Ed.. 1965. UP. 61-67, 1121 1.e Rliy. 1l..I..J. l'hrm i'hg*.. 52.26ti:i 119701. 1131 "Hsndhwlr i , f C h e m i s l r g md I'hyrirr." C11C I'reir. C l e u ~ l a n d , 55th Ed.. 1974, p. E YZ!. Volume 57, Number 2 February 1980 / 105 65 Electronic Spectrum of I2 Overview
  • 74. In this experiment, the student will record and analyze the vibrational structure of the B – X electronic transition in molecular iodine to determine the dissociation energy of I2 in the B 3 Π state. The student will record the spectrum at several temperatures at low resolution to observe the gross effects of temperature on the overall structure of the spectrum and then record the spectrum at higher resolution for the analysis of the spectrum. Bands will be assigned and the dissociation energy calculated using a Bïrge- Sponer method. Theory The total energy of a molecule (under the Born-Oppenheimer approximation) can be expressed as a sum of electronic and vibrational energies. ETotal = Eelectronic + Evibrational
  • 75. A simplified energy level diagram is shown below for two electronic states of the hypothetical diatomic molecule A2. Rotational energies are small enough that they are no resolved in this experiment and are thus ignored. The “term” value for a given quantum state is measured relative to the bottom of the potential energy well for the ground electronic state. By analyzing the 66 transitions between the two states, once can determine the energy differences labeled as Te’ (the electronic excitation term value of the excited electronic state), E* (the term value for the excited atomic state of A*), De” and De’ (the dissociation energy of the lower and upper electronic state respectively as measured from the potential well minima) and D0” and D0’ (the dissociation energy of the lower and upper electronic state respectively as measured from the v = 0 energy levels of the respective states).
  • 76. The analysis in this experiment will focus on the energy levels of the upper state, so let’s write down an expression for these levels based on electronic and vibrational contributions to the total energy. For the purposes of this discussion, all energies will be expressed as term values which are really energies divided by the factor hc.) Tv = Te + Gv’ Taking the difference between successive levels gives the following result. Tv+1 – Tv = Gv+1 - Gv The difference Gv+1 - Gv is given the symbol ∆Gv+½. The vibrational term value can be expressed in terms of the vibrational constants ωe and ωexe and the vibrational quantum number v. Gv = ωe(v + ½) - ωexe(v + ½) 2 + ωeye(v + ½)
  • 77. 3 + … Neglecting any terms higher than ωexe, the total expression for ∆Gv+½ can be derived as follows: ∆Gv+½ = Gv+1 – Gv = ωe(v + 1 + ½) - ωexe(v + 1 + ½) 2 - ωe(v + ½) - ωexe(v + ½) 2 ∆Gv+½ = ωe – 2ωexe(v+1) According to theory, the dissociation limit occurs when the spacing between successive vibrational levels (∆Gv+½) is zero. Fitting ∆Gv+½ as a function of (v+1) should yield a straight line with slope -2ωexe and intercept ωe. The dissociation energy can be determined from the constants ωe and ωexe using the Birge-Spöner extrapolation. In this method, one plots
  • 78. ∆Gv+½ vs. (v+1). The area under the curve gives the dissociation energy (the sum of all of the vibrational spacings up to the dissociation limit – where ∆Gv+½ becomes zero.) The method typically over estimates the dissociation energy by approximately 15% (more for some molecules) due to neglecting higher-order terms in the vibrational term energy expression. The figure below shows an example of a Birge-Sponer plot. The deviation from the linear relationship occurs if high order terms such as ωeye or ωeze become important for a given molecule. 67 Finally, the dissociation energy of the lower state can be determined from Te, De’ and E* by noting that Te + De’ = De” + E*
  • 79. Experimental Method The experimental data is taken on two different spectrometers. One is equipped with a temperature control and the other provides higher resolution. 1. Record the visible absorption spectrum of I2 at 30 o C, 40 o C and 50 o C using the Varian Bio 50 spectrophotometer. Use the highest resolution obtainable. 2. Record the same band system using the Perkin-Elmer Lambda 9 spectrophotometer. Seek the best possible signal to noise ratio and resolution combination on this instrument.
  • 80. Analysis Organize your data into a table leaving room to report the wavenumber, vibrational assignment for each assignable transition. Also include a column for and ∆Gv+1/2 for the spacing between each pair of transitions. 68 Assign vibrational quantum numbers to the bands in the spectrum. Use the following assignments to get yourself started. These assignments are taken from J. Chem. Ed. 57, 101 (1980). v’ – v” λ (nm) 27-0 541.2 28-0 539.0 29-0 536.9 Birge-Spöner Method • Prepare a graph of ∆Gv+1/2 vs (v+1) and report the best-fit
  • 81. line to your data. • From the fit, calculate values for ωe and ωexe. • Find vmax from this data. • Calculate the dissociation energy for the excited electronic state of I2. ∫ = += maxv 0v 2/1v0 dv∆GD • Also determine a value for De’ by calculating G(vmax). • From your fit of the data, calculate Te – the electronic excitation energy of the B state. • The value if E* is 7603 cm -1 (4). This is the energy of excitation for I* ( 2 P1/2) compared to I ( 2 P3/2). Use this information in conjunction with your above results to find a value for De” – the ground electronic state dissociation energy. Error Analysis
  • 82. • Determine the uncertainties of the spectroscopic constants you report and compare all of your findings to values found in the literature. (A good source of literature values is the table of Constants of Diatomic Molecules found at http://webbook.nist.gov/.) Post Laboratory Questions 1. Explain the differences between the three spectra recorded on the Bio 50 instrument. 2. Derive the expression ∆Gv+½ = ωe – 2ωexe(v+1). 3. What would be the theoretical dissociation limit of a molecule that can be treated as a true harmonic oscillator? References 1. Experiments in Physical Chemistry, D. P. Shoemaker, C. W. Garland and J. W. Nibler, 7 th edition,
  • 83. WCB/McGraw-Hill, New York, (2003) 2. Physical Chemistry: Methods, Techniques, Experiments, R. J. Sime, Saunders College Publishing, San Francisco, CA (1990) 3. Physical Chemistry, R. A. Alberty, R. J. Silbey and M. G. Bawendi, Physical Chemistry, 4 th ed., Chapter 16, John Wiley and Sons, Hoboken, NJ (2005) 4. Atomic Energy Levels, Vol. III (Molybdenum through Lanthanum and Hafnium through Actinium), Charlotte E. Moore, Circular of the National Bureau of Standards 467, U.S. Government Printing Office, Washington, DC (1958).