MULTIPLE- · chemistry and for laser-induced isotope sep-aration. However, in the early 1970s another rather surprising feature was discovered in-volving the interaction of infrared

The multiple-photonMULTIPLE-effect, in w

a single polyatomicmolecule

absorbs manyinfrared photons,

is proving to bea complex and

mysteriousstillphenomenon

in laserphotochemistry.

EXCITATIONby John L. Lyman, Harold W. Galbraith, and Jay R. Ackerhalt

T here is no doubt that lasers haverevolutionized photochemistry.Initially, the main attraction ofl a s e r s w a s t h e h i g h l y

monochromatic nature of the light; manyresearchers hoped that this feature mightserve as the basis for bond-selective photo-chemistry and for laser-induced isotope sep-aration. However, in the early 1970s anotherrather surprising feature was discovered in-volving the interaction of infrared laser lightand polyatomic molecules.

This phenomenon, multiple-photon excita-tion, is the absorption of many infraredphotons of the same frequency by a singlemolecule. Observation of this phenomenonwas only possible with the high light in-tensities typical of lasers. It was a surprisingeffect because multiple-photon excitation didnot fit the established theoretical pictures ofhow molecules absorb radiation. While con-siderable experimental and theoretical workhas now been directed toward understandingthis phenomenon, much remains to be ex-plained.

One can think of multiple-photon excita-tion or absorption as high-intensity spec-troscopy. As such, it is qualitatively different

66

from normal low-intensity spectroscopy. Atlow intensities relatively small numbers ofphotons delicately probe individual energytransitions. Typically, the molecules occupya known equilibrium distribution of energystates, and, for a given absorption frequency,only a small fraction of these are moved to asingle excited state. On the other hand,multiple-photon excitation necessarily affectslarge fractions of the molecules and drivesthem through many energy states. In fact,since the absorbed energy becomes vibra-tional energy, the resulting high degree ofvibrational excitation drastically alters thechemical nature of the molecule, even to thepoint of dissociation. The dissociative reac-tion led to fulfillment of the early hope forlaser isotope separation.

As a result of these differences, the in-terpretation of data for multiple-photon ex-citation is much more difficult than fornormal absorption spectra; a given absorp-tion feature represents the sum effect ofmany energy transitions in a collection ofmolecules driven rapidly into a nonequi-librium distribution. Dissociation data, aswell as absorption data, are necessary tocharacterize that distribution. Also, the theo-

retical picture dealing with the effect isnecessarily both statistical and dynamical innature; it is concerned with the rates atwhich molecules are driven through the meshof energy states.

This article will describe the currentunderstanding of multiple-photon excitationand outline both the experimental and theo-retical development of this fascinating re-search area. While the article emphasizesLos Alamos work, we note that scientists inmany laboratories have contributed substan-tially to the understanding of the effect.Multiple-photon excitation is an importantphenomenon in laser isotope separation (see“Separating Isotopes With Lasers” in thisissue) and is attractive for applications likepurification of chemicals, synthesis of newspecies, and study of chemical reactions. Butwe have limited our discussion to thephenomenon and its historical development.

The Old Picture

The understanding of the interaction be-

tween polyatomic molecules and intense in-frared radiation is evolving to give a verydifferent picture from what conventional wis-

LOS ALAMOS SCIENCE

dom portrayed ten years ago. In the late

sixties only the most naive thought that onecould use infrared lasers to induce dissocia-tion (photolysis) of molecules. Those whoknew something of infrared spectroscopyand molecular physics used the followinglogic to discount the possibility of infraredphotolysis.

The photon energy of infrared light issubstantially less than the energy required toinduce most chemical reactions. In fact,photolysis of molecules that are stable atroom temperature requires the energyequivalent of 20 to 50 carbon-dioxide (C02)laser photons.

A molecule absorbs infrared radiation byinteraction of its vibrating electric dipolewith the oscillating electric field of the radi-ation. The radiant energy becomes vibra-tional energy in the molecule. Because mo-lecular vibrational energy is quantized, thisabsorption is best treated as a match be-tween the energy of the photon being ab-sorbed and the gap between the energy levelsinvolved in the transition.

An ideal harmonic oscillator. that is, anoscillator with two masses and a restoringforce that remains proportional to the sepa-ration between those masses, has equallyspaced energy levels. However, all realdiatomic molecules deviate from this idealbecause the restoring force typically dropstoward zero as the bond stretches furtherand fur ther toward breakage . Thesemolecules are vibrationally enharmonic andthe spacing between adjacent vibrationalenergy levels decreases with excitation.

One can think of the vibrational energylevels as rungs in an in tera tomic-potential-energy ladder (Fig. 1). The rungsget closer as one moves up the ladder. At thetop, where the levels are so close as to be, ineffect, continuous, the bond breaks and the

diatomic molecule dissociates.Now if the frequency of the photon being

absorbed matches the lowest energy gap(between n = O and n = 1 in Fig. 1), it willnot match the next gap, which is smaller.

68

Bond Length

Fig. 1. The potential-energy well of a diatomic molecule. The horizontal linesrepresent the quantized vibrational energy levels of the molecule with the ground statelabeled by n = 0. Near the ground state the potential-energy well is approximatelyharmonic and the energy levels approximately equally spaced. As the molecular bondstretches, the potential-energy curve becomes less steep and the levels merge to a

The mismatch becomes progressively largerwith increasing vibrational excitation andprecludes the possibility that one diatomicmolecule will absorb many photons.

Molecular vibrations of polyatomicmolecules are somewhat more complex.However. one can resolve a particular vibra-tional motion into a superposition of severalvibrations called normal modes. To a firstapproximation, each normal mode vibratesindependently and maintains the character-istics of a vibrating, enharmonic diatomicm o l e c u l e . T h e r e f o r e , v i b r a t i o n a lanharmonicity restricts absorption ofphotons by polyatomic molecules in basi-cally the same way it does for diatomicmolecules.

If this were not enough to prevent absorp-tion of large amounts of infrared energy,another severe restriction exists. Allmolecules in a gas sample do not respondidentically to infrared radiation because ofdifferences in rotational and translationalenergy of the molecules.

In the case of rotational energy, themolecules of a gas sample are distributedamong many quantized rotational states.Each rotational state may contribute dif-ferently to the total change in internal energyduring an absorption, which means thatmolecules in different rotational states re-quire different infrared frequencies for effi-cient absorption. The result is the rotationalstructure discussed in this issue in “TheModern Revolution in Infrared Spec-

troscopy.”Likewise. the translational energies (or

velocities) of the molecules cover a widethermal distribution. Each velocity compo-nent along the path of the light propagationwill produce a different Doppler shift in thefrequency of the infrared light as viewed bythe molecule. Because the optimum frequen-cy for absorption varies with the rotationaland translational energy state. the fraction ofmolecules that can absorb even the firstphoton for vibrational excitation is greatlyreduced.

LOS ALAMOS SCIENCE

MULTIPLE-PHOTON EXCITATION

Fig. 2. Homogeneous and effective homogeneous linewidth.The low-intensity spectrum of an absorption band representsthe distribution of molecules among various molecular energystates, each state with absorbing transitions at differentfrequencies. For the absorption on the left, the laser photonsinteract only with that fraction of the molecules whose stateshave transition frequencies falling approximately within thelaser bandwidth. This fraction of molecules are said to beabsorbing homogeneously because they all interact in the same

way with the photons. For this absorption the homogeneouslinewidth matches the laser bandwidth. However, certainmechanisms, such as collisions that change energy states ormultiphoton absorption, can cause molecules outside the laserbandwidth to absorb photons in a manner identical to thosewithin the homogeneous linewidth. The effective homogeneouslinewidth (right) has a larger frequency range that includesthis larger fraction of absorbing molecules.

Molecules that all interact in a like manner another mechanism broadening thewith the photons are said to absorbhomogeneously, Molecules in different statesthat interact differently with the photons aresaid to absorb inhomogeneously. Certainmechanisms. such as collisions, can alter therotational and translational energy states.Thus, during a laser pulse molecules from

one homogeneous collection that is not ab-sorbing strongly can be transferred by thesemechanisms into another homogeneous col-lection that is absorbing. This increases thefraction of molecules affected in a like man-ner by the photons . The ef fec t ivehomogeneous linewidth is a measure, interms of frequency, of the extent of suchmechanisms (Fig. 2). However, calculationssuggest that many collisions during a pulsefrom an infrared laser are necessary for evena small fraction of the molecules to absorbthe photons needed for dissociation. Thosesame collisions would tend to distribute theabsorbed energy randomly among manymolecules in the sample. The net effect ofthis particular broadening mechanism, then,would be similar to heating the sample bymore conventional methods.

Photolysis might be accomplished by mul-tiphoton absorption, which is a single, reso-nant interaction between several photons andtwo widely separated states. This effect is

homogeneous absorption linewidth; it re-quires only that the sum of the photonenergies match the gap between the initialand final states, thus bypassing unmatchedintermediate states. However, the thir-ty-photon process that would be necessaryfor dissociation would have a vanishingly

small probability of occurrence.(Catastrophic dielectric breakdown could

lead to dissociation, but this sudden ioniza-tion can only be produced in a molecular gaswith extremely high-intensity infrared radi-ation and is quite different from multi-ple-photon dissociation.)

The line of reasoning outlined above led tothe conclusion that a single polyatomicmolecule could not, in the absence of col-lisions. absorb sufficient energy from aninfrared laser pulse to dissociate. The argu-ments appeared to have no serious deficien-cies; they were based, for the most part, onsound principles of infrared spectroscopy.The conclusion, however, was wrong.

The Historical Development of theMultiple-Photon Effect

In the late sixties people in several labora-tories began to induce chemical reactionswith the newly developed C02 laser. The

experiments tended to fall into two catego-ries. In the first category, absorbed laserenergy induced reactions that involved thecollision between two molecules. The re-searchers concluded that a single absorbedphoton enhanced the reaction rate. In the

second category, the pressure of the samplewas high enough to produce rapid collisional

scrambling of the absorbed energy. Bothtypes of experiments gave results that tit wellinto the old picture we portrayed above.

With the advent of high-intensity, pulsedinfrared sources, such as a pulsed version ofthe CO2 laser, that picture began to showdefects.

In 1971 researchers for the National Re-search Council of Canada showed that whenthey irradiated low-pressure SiF4 gas withintense CO2 laser pulses, the molecules dis-sociated to give electronically excited SiFfragments. The laser intensity that producedthese reactions was close to, but below, thethreshold for dielectric breakdown. At theselow pressures, the absorption took placeunder nearly collisionless conditions. Thisexperiment showed that a single. isolatedmolecule (and its fragments) could absorbwell over one hundred photons during a laserpulse.

During the next year experiments at LosAlamos demonstrated that increased pres-

LOS ALAMOS SCIENCE 69

sure, and so increased collisions, impeded,r a t h e r t h a n e n h a n c e d , t h e r a t e o flaser-induced dissociation of N2F4 to NF2. Adissociation reaction that depended on sim-ple conversion of laser energy to thermalenergy in the gas would have shown theopposite pressure effect. However, these ex-periments were still not an unequivocal dem-onstration of laser-induced dissociation inthe complete absence of collisions.

In the period from 1972 to 1974 severalexperiments gave strong evidence thatC O2-laser pulses of modest fluence(time-integrated intensity per unit area) couldinduce chemical reactions producing atomicfragments. At Los Alamos CO2-laser pulseswere used to initiate explosive reactions inmixtures of H2 with either the absorber N2F4

or SF6. These experiments suggested thatduring the laser pulse the radiant energyfragmented the absorber molecule (N2F 4 orSF6) to produce fluorine atoms. These atomsthen reacted rapidly with the H2 to generateenergy and more reactive species. This ac-celerating chain reaction resulted in ex-plosion of the gas mixture. An importantfinding was the fact that the laser energyabsorbed was significantly less than wouldbe required in an equivalent thermally in-duced reaction.

Experiments at the Institute of Spec-troscopy in the Soviet Union demonstratedthat the laser-induced reaction of BC13 in thepresence of oxygen produced electronically.

excited BO molecules. These reactions wereisotonically selective. When the CO2-laserfrequency was near the vibrational frequencyof a particular isotopic form of BC13, the BOfragment favored that boron isotope. Aboutthe same time researchers at the NationalBureau of Standards demonstrated thelaser-induced, isotonically selective reactionof BC13-H2S mixtures. The isotopic selectivi-ty in the latter two experiments was particu-larly strong evidence of a nonthermal, in-frared photolytic reaction.

But the experiments that demonstratedunequivocally the phenomenon of multi-

70

pie-photon excitation were performed by theInstitute of Spectroscopy in late 1974 andLos Alamos in early 1975. These experi-ments demonstrated direct photolysis of SF6

with extremely high isotopic selectivity underconditions where molecular collisions couldnot have played a major role. Similar demon-strations with other species followed rapidly,as well as a broad range of explanations forthe phenomenon.

The Experimental Characterizationof Multiple-Photon Excitation

Multiple-photon excitation experimentsexhibit many features that are independent ofthe particular molecular species. These fea-tures, which must be included in any suc-cessful theory of the process, will be present-ed below. We will start by giving a quali-tative picture of the absorption and dissocia-tion processes and then show how genericproperties of the molecules, the laser pulse,a n d t h e g a s sample influence thephenomenon.

Q U A L I T A T I V E F E A T U R E S . T y p i c a llaser-induced photolysis experiments ex-amine only the end products of the dis-sociative reaction. How then can multi-ple-photon excitation be distinguished fromother possible dissociation mechanisms? Forexample, the dominant process might belaser ionization, rather than vibrational ex-citation, of the absorbing molecules. In laserionization the observed products would begenerated by ion-molecule reactions ratherthan single-molecule, or unimolecular, disso-ciation. But, since experiments have shownthat laser pulses produce no ions underconditions of extensive chemical reaction,researchers now feel that the general mecha-nism most consistent with available data isone in which the polyatomic molecule ab-sorbs the infrared laser energy in the form ofvibrational energy. Then, if the moleculeabsorbs sufficient energy, the molecule maydissociate into fragments.

What type of molecules undergo multi-ple-photon excitation? Many of the earlysuccessful experiments were with highlysymmetric molecules like SF6 and SiFq. Ishigh symmetry an essential feature? Doesmolecular size play a role? From Table I,which gives a partial list of molecules thatdissociate by multiple-photon excitation, wesee that it is a general phenomenon. Molecu-lar symmetry is not a restriction. Molecularsize is only a partial restriction: diatomicmolecules do not dissociate; some triatomicmolecules like OCS and 03 dissociate whenirradiated with very intense infrared radi-ation; and very large molecules likeuranyl-bis-hexafluoracetylacetonate tetra-hydrofuran dissociate very easily.

One of the impressive features of multi-ple-photon excitation is high isotopic selec-tivity. We define isotopic selectivity as theratio of dissociation probabilities of twoisotopic forms of a molecule. With ap-propriate laser frequencies and gas pressuresone can induce isotonically selective reac-tions in most polyatomic molecules.

Isotopic selectively depends strongly onthe isotopic shift of absorption features in theinfrared spectrum. Basically, the shift is dueto a change in the vibrational frequency ofthe molecule when one of the vibratingatoms has a different isotopic mass. As aresult, isotopic selectivities tend to be largefor light isotopes, where the relative changein the mass is large, and small for heavyisotopes, where the relative change is small.For example, isotopic selectivities for heavymetals like molybdenum (in MoF6), osmium(in 0s04), and uranium [in U(CH30)6] fall inthe range between 1.0 and 1.1. Selectivitiesfor light isotopes like hydrogen (in CF3H)may be as high as 20,000. For thewell-studied molecule SF6, the selectivity for33S relative to 32S is about eight, and for 34Srelative to 32S, it is about fifty. These valuesare typical for intermediate atomic weightisotopes.

Isotopic selectivity observed in a widerange of molecules gave clear evidence that a

LOS ALAMOS SCIENCE


TABLE I

SPECIES DISSOCIATED BY MULTIPLE-PHOTON EXCITATION

OCS0 3

NH3

H 2C 0BC13

CCI4

CF2HCICF2C12

CF31Cr02C12

HCOOH0 s 04

CHF3

CH30HCH3CNCH3NCC2H3CIC2H3FCF2CH2

C2H4

N2F4

SF6

SF5CICH3NH2

CH 3N 02

SeF6

single molecule can absorb many infraredphotons with no help from collisional proc-esses and that the resulting vibrationalenergy is sufficient to induce a unimolecularchemical reaction.

These results suggested that the reactionsmight be bond selective. That is, if moleculesabsorbed laser energy under collisionlessconditions, perhaps the vibrational energyremained in a single normal mode andcleaved the chemical bond that experiencedthe highest vibrational amplitude. If this wereso, then one could select the point of reactionwithin the molecule by matching the laserfrequency to the frequency of the ap-propriate normal mode.

Arguments for bond-selective dissociationby multiple-photon excitation rely heavily onmany features of the old picture for theabsorption of infrared radiation. Because theold picture can’t explain multiple-photonexcitation in the first place, one should behighly suspicious of the possibility of such areaction. Many people claimed to have dem-onstrated bond-selective reactions, or at leastsome degree of vibrational-energy local-ization within a polyatomic molecule. How-ever. in all cases that we are aware of,alternate explanations are more plausible.

Researchers at the Berkeley campus of theUniversity of California performed experi-ments tha t not only sugges ted tha tbond-selective reactions were unlikely, butalso demonstrated the collisionless nature ofthe multiple-photon excitation process.These experiments involved dissociating themolecules in a molecular beam with

MoF6

UF6

CF3CHC12

CF3CH3

C2H5FC2H5NCC2H5OHSF5NF2

CF3COCF 3

S2F10

C6F5HU(OCH 3)6

CH2FCH2CI

C O2-laser pulses. A molecular beam is alow-density stream of molecules passingthrough a vacuum. These conditions vir-tually eliminate collisions between moleculesduring the time the beam is in the irradiationregion. When the molecules dissociate, recoilof the separating fragments frequentlycauses the reaction products to leave thepath of the beam. This altered trajectorypermits detection and identification of thefragments,

Reactions of over fifteen species contain-ing from four to eight atoms per moleculewere studied. Multiple reaction pathwayswere available for many of these species.However, all species reacted by theenergetically most favorable pathway. Thisresult is consistent with statistical redistribu-tion of the available vibrational energy priorto dissociation. The velocities of the separat-ing molecular fragments were also consistentwith a statistical energy distribution.

DENSITY OF VIBRATIONAL STATES. Wenoted that heavier molecules dissociate moreeasily than lighter ones. People generallyattribute this fact to the increasing density ofvibrational states with increasing mass of themolecule. The density of states is simply thenumber of available vibrational states perunit energy interval of the molecule and isone of the profound differences between adiatomic molecule and a larger polyatomicmolecule. (“The Modern Revolution in In-frared Spectroscopy” discusses the origin ofthis high density of vibrational states forpolyatomic molecules.)

BenzeneCyclopropanePropyleneHexafluorocyclobutenePerfluorocyclobutaneEthylvinyletherEthyl acetatesec- Butyl acetateTetramethyldioxetaneUranyl-bis-hexafluoroacetyl-

acetonate.tetrahydrofuran

From spectroscopic constants one canobtain an adequate estimate of the density ofvibrational states. Figure 3 shows how thisdensity depends on vibrational energy forseveral representative molecules. We see ahuge difference in the state density between asmall molecule like OCS and a larger onelike S2F10, even at modest energies.

A high state density gives an immensestatistical advantage to absorption by largemolecules. For example, at an energy nearthe dissociation threshold of S2F IO (20,000cm -1), the densities of vibrational states forS 2F 10 and OCS differ by a factor of 1O22.This means that if molecules of both specieswere in equilibrium with some environment(such as a radiation field or a thermal gas),the probability that the S2FIO molecule is in avibrational state near 20,000 cm-l could beas much as 1022 times the same probabilityfor the OCS molecule. Of course, absorptionof laser radiation is certainly not anequilibrium process. Moreover. as the energyor temperature of the environment increasesand both molecules become distributedthroughout a large range of energy levels, theprobability ratio will decrease below the 1022

factor. Nevertheless, the huge statistical ad-vantage remains for larger molecules.

Recent dissociation experiments withpolyatomic molecules have demonstrated theeffect of the density of vibrational states onmultiple-photon excitation. To facilitate

comparison of these experiments we define

dissociate 1% of the molecules in the laserbeam, and N( 10,500), the density of vibra-


tional states for each molecule at 10,500cm–’, which is about half the energy neededto dissociate a typical molecule.

Figure 4 shows the relationship betweenthese two quantities in many different experi-ments with a wide range of experimentalconditions. We see the trend of decreasing

N( 10,5OO) (higher density of vibrationalstates).

There are, of course, many other variablesin these experiments. These include: absorp-

tion cross section, chemical bond strength,frequency relative to the infrared absorptionband, gas pressure, laser pulse shape, andlaser spectral width. To eliminate some ofthis diversity we report the results of a set ofexperiments performed under more con-trolled conditions (Table II). These results allcame from the same laboratory (Los Ala-mos), and the laser pulse shape and gaspressure were nearly the same for all experi-ments. The species all have the structureSF5X allowing the laser frequency to be setnear the center of a strong infrared absorp-tion band of a sulfur-fluorine stretching vi-bration for each species. The main parameterthat still varies is the strength of the weakestchemical bond: from 50 kilocalories permole for SF5NF2 to 93 kilocalories per molefor SF6. However, the most dramatic effect

with increasing density of states.The strength of the chemical bond that

breaks when the molecule dissociates plays a

ple, SiF 4 and OsO 4 have similar vibra-tional-state densities. But SiF4 has a bondstrength of 142 kilocalories per mole and a

OsO d has a strength of only 73 kilocalories

centimeter.The correlation with bond strength is not

as strong as the correlation with certain of’the other factors. In fact, it is not difficult tofind a pair of molecules where the morestrongly bound molecule dissociates more

72

Energy of Vibration (cm-1)

Fig. 3. Density of vibrational states for various molecules as a function of vibrationalenergy. The density of vibrational states, that is, the number of available states perunit energy interval, increases for all molecules as vibrational excitation increases.However, the large molecules have considerably larger densities than the smallmolecules at all but the lowest energies. [Here, frequency in wavenumbers is used asan energy unit: 1 reciprocal centimer (cm–1) = 1.24 X 10 -4 electron volt (e V). Alsonote the logarithmic scale for density of vibrational states. ]

Fig. 4. The dissociation efficiency for various molecules as a function of vibrational

dissociation. The overall trend is easier dissociation for larger molecules with highervibrational state densities (here measured for each molecule at an energy of 10500cm –1).

LOS ALAMOS SCIENCE


TABLE IICOMPARISON OF FLUENCE NECESSARY TO DISSOCIATE 1% OF MOLECULES IN BEAM

Species N(105OO) (l/cm-1) Bond Strength (kcal/mole)

SF 6 8.5 X 108 93SF@ 2.1 x 109 61SF5NF2 2.6 X 1012 50S2F10 3.1 x 1017 58

Time

Fig. 5. The intensity-fluence relationship. Laser intensity (power per unit area) is givenas a function of time by the height of these idealized laser-pulse curves. The fluence(energy per unit area) is the time-integrated laser intensity, or the area under eachcurve. (a) If the laser pulse is attenuated (by placing a partial absorber in the beam orby refocusing to a larger area) both the intensity and the fluence are reduced. (b) If thelaser pulse is attenuated but, at the same time, the pulse length is increasedproportionately, the fluence can be held constant while the intensity is reduced.

easily. Again, from Table II, the density ofvibrational states for S2F 10 is larger than forS F5N F2, and this fact appears to be moreimportant for ease of dissociation than theweaker bond strength of SF5NF2.

FLUENCE EFFECTS. One of the more easilyvaried parameters in a multiple-photon ex-citation experiment is the laser fluence. Thefluence can be varied by placing attenuatorsin the beam or by focusing the beam dif-ferently in the sample (Fig. 5a). As observedby many researchers, variations in fluenceproduce profound changes in the dissocia-

LOS ALAMOS SCIENCE

tion probability and the absorption crosssection.

dissociation probability of a large molecule(SF5NF2). Figure 7 shows the correspondingcurves for a somewhat smaller molecule(C2H4). Both the differences and similaritiesbetween these figures are instructive.

At the threshold for dissociation, the

a factor of about 103. However, the absorp-

differs by about this factor, only inversely, so

2.80.60.050.018

the cross section is proportional to the

Thus both molecules need to absorb approx-imately the same energy to reach thethreshold for dissociation. Furthermore, theobserved value of ( Eab)TH corresponds toabout half the energy needed to dissociateeither of the molecules. If only a smallfraction of the molecules were absorbing theenergy, the dissociation reaction would havestarted at a much lower fluence. These factssuggest that the molecules behave similarlyat high levels of excitation and that theabsorbed energy must be distributed amongmany molecules.

For the large molecule (Fig. 6), the crosssection remains almost constant from lowfluence up to the threshold for dissociation(for both long and short pulses). The drop incross section at this point is probably due,for the most part, to dissociation of themolecules during the laser pulse. In otherwords, the probability for absorption of aphoton by the molecule remains constantfrom the low-fluence limit where themolecule’s initial state of excitation is low tonear the threshold for dissociation where themolecule is highly excited. This behaviorcontrasts dramatically with the old-pictureprediction that the absorption cross sectionwould drop rapidly with excitation due to agrowing enharmonic mismatch betweenphoton energy and the gaps in the vibra-tional energy ladder.

A high absorption cross section over abroad range of fluence suggests that all ofthe molecules are absorbing laser radiationregardless of their translational or rotationalenergy states. In terms of linewidth, thismeans the effective homogeneous linewidth

73

is on the order of the frequency width of theentire low-intensity absorption band: abroadening mechanism is apparently atwork. The high cross section at high fluencesuggests than any enharmonic frequencyshift with increasing vibrational excitation isless than this effective homogeneous line-width.

We reach opposite conclusions with thesmaller molecule (Fig. 7). The low absorp-tion cross section suggests that at lowfluence only a small fraction of the moleculesare able to absorb laser radiation. The im-mediate drop in cross section with increasingfluence suggests that this small fraction iseasily depleted. Thus. the homogeneous line-

width for the absorption is much less than

the frequency width of the low-intensityabsorption band and there is no significantbroadening mechanism at low fluence. Thelow, but finite, cross section at high fluencesuggests that the enharmonic frequency shiftsubstantially reduces, but does not eliminate,the probability that an excited moleculeabsorbs more photons.

Other types of experiments confirm theseconclusions. Increasing the pressure (col-lision rate) increases the absorption crosssection for small molecules at high fluencewhereas pressure has little effect on the crosssection for large molecules. As pointed outearlier, collisions change the energy state ofthe molecules, bringing more of them intostates where absorption is favorable. Similarchanges in large molecules are insignificantbecause the already larger homogeneouslinewidth makes absorption somewhat inde-pendent of energy state.

In summarizing the above findings forsmall molecules, we see an intriguing dif-ference between the extremes of low and highfluence. At low fluence only a small fractionof the small molecules absorb photons. Thisconclusion follows from the large effectpressure has on the effective homogeneouslinewidth and the early drop in cross sectionwith increasing fluence. At high fluence amuch larger fraction of the small moleculesabsorb photons. This follows from the factsthat, on the average, the molecules absorb

74

Fig. 6. Absorption cross section (dashed curve) and dissociation probability (solidcurve) for SF5NF2. For this large molecule the absorption cross section remains fairly

proportional to the average energy absorbed per molecule at threshold conditions; thisaverage is about 1O-19 joule and represents about half the energy needed for amolecule to dissociate. Data are from John L. Lyman, Wayne C. Danen, Alan C.Nilsson, and Andrew V. Nowak, Journal of Chemical Physics, 71,12061210 (1979).

dissociation probability (solidcurve) for C2H4. The absorption cross section for this smaller molecule decreases withincreasing fluence long before the threshold fluence for dissociation is reached.Despite large differences in threshold fluence and cross section between this moleculeand SF5NF2 (Fig. 5), the energy absorbed per molecule at the dissociation threshold isapproximately the same. This suggests that both molecules behave similarly at highlevels of excitation. The resemblance in the rise of dissociation probability with fluence

for both molecules further strengthens this suggestion. Data are from O. N. Avatkov,V. N. Bagratashvili, I. N. Knyazev, Yu. R. Kolomiiskii, V. S. Letokhov, V. V. Lobko,and E. A. Ryabov, Soviet Journal of Quantum Electronics 7, 412-417 (1977).

LOS ALAMOS SCIENCE


4x10–’7

930 940 950

Frequency (cm –1)

Fig. 8. Fluence dependence of the v3 absorption band of SFb at 300 kelvin. Thelow-fluence spectrum was obtained at a fluence typical of conventional spectroscopy,whereas the two high-fluence spectra were obtained with a pulsed CO2 laser. The mostobvious effect of high fluence is a significant drop in the absorption cross section.There is also an indication of a shift to lower frequencies. The high-fluence data are

from Wei-shin Tsay, Clyde Riley, and David O. Ham, Journal of Chemical Physics70, 3558-3560 (1979); the low-fluence data are from A. V. Nowak and J. L. Lyman,Journal of Quantitative Spectroscopy and Radiative Transfer 15,945 (1975).

Low Fluence I

930 940 950

Frequency (cm–1 )

Fig. 9. Fluence dependence of the v3 absorption band of SF6 at 500 kelvin. Theincrease in temperature @rem 300 kelvin in Fig. 7) eliminates the structure in thelow-fluence spectrum. Also, the drop with increasing fluence in the absorption crosssection is less precipitous, making obvious the shift to lower frequencies. The data arefrom the same references cited in Fig. 8.

LOS ALAMOS SCIENCE

half the energy needed for dissociationbefore reaching the threshold, and that there-after, the reaction probability increases asrapidly as for large molecules.

INTENSITY EFFECTS. When the fluence isvaried with attenuators or by focusing thebeam differently, the intensity (laser powerper unit area) also changes (Fig. 5a). Toseparate the effects of intensity and fluence,researchers usually vary the intensity whileholding the fluence constant. This is a dif-ficult task because it requires changing thetemporal shape of the laser pulse whilekeeping the total pulse energy constant (Fig.5b). For instance, rapid electro-optical shut-ters can change the pulse length at thesample. but, at the same time, one musteither refocus the beam or change the at-tenuation to keep the total energy per unitarea constant.

The general observation from experimentsof this type is that, unlike the effects offluence, only in special situations does in-tensity play a major role. One of thesesituations is for high gas pressures whenfrequent collisions redistribute the absorbedenergy. Thus, the critical effect is the numberof these collisions that take place during thelaser pulse, rather than the interaction be-tween the laser field and an isolatedmolecule. As intensity is varied by changingpulse length, the total number of collisionsduring the pulse will change. altering theultimate energy distribution.

FREQUENCY DEPENDENCE. Polyatomicmolecules absorb low-fluence infrared radi-ation in frequency bands that correspond tothe frequencies of normal-mode vibrations(or some combination of those frequencies).We would expect multiple-photon excitationto have a similar frequency response, butwith some modification because of the factthat high-fluence laser pulses severely per-turb the absorbing sample. Early experi-ments verified this approximate relationshipbetween low- and high-fluence spectra.

Figures 8, 9, and 10 compare low- andhigh-fluence spectra in the region of an

75

absorption band for SF6 at two temperaturesand for S2F 10 at one temperature. In all threeexamples we see a similarity between thespectra at high and low fluences, but with ashift to lower frequencies at high fluence.This is most certainly related to theenharmonic frequency shift. High-fluenceabsorption involves excitation up severalrungs of the enharmonic ladder with theresonant frequency quickly becoming mis-matched. A somewhat lower frequency thatmatches an intermediate rather than thelowest energy gap is optimum because itaverages the mismatch over more levels,Moreover, as suggested earlier, multiphotonabsorption necessarily occurs at lower fre-quencies. As one would expect from ourprevious discussion, we also see a decrease inabsorption cross section at high fluence, andthat decrease is most pronounced for thelighter molecules (SF6) at the lower tem-

perature (300 kelvin).The effect of temperature on high-fluence

spectra is most interesting. For SF6 t h eabsorption cross section drops quickly withincreasing fluence at 300 kelvin but drops

much less quickly at 500 kelvin. This couldbe related to the degree of initial vibrationalexcitation in the molecule because most SF6

molecules are not excited at 300 kelvinwhereas most are excited at 500 kelvin. Forlarger molecules, such as S2F 10, even at 300kelvin there is little loss of cross section withincreasing fluence. Again, this could be re-lated to vibrational excitation because thesemolecules have pliant bending modes withlower energy levels and even at 300 kelvinvirtually all molecules have some degree ofvibrational excitation. In terms of linewidth,it appears that increased vibrational excita-

tion increases the effective homogeneouslinewidth so that a larger fraction of the

molecules are able to absorb photons. Thistrend with vibrational excitation parallels thepreviously observed trends with density ofvibrational states and molecular size. Thus,vibrational excitation may be one, butperhaps not the only, reason for the

76

Chemical Physics 69, 1858-1864 (1978).

state-density trend.

EXPERIMENTAL CONCLUSIONS. Clearly,conclusions drawn from the data for multi-ple-photon excitation experiments do not fitthe old picture of the absorption of photons.

Most significantly, at high fluence the laser

frequencies need not match the absorptionfeatures exactly. We suggest four phe-nomena that contribute to this observation.

1. Absorption in the wings of a spectralline is possible because the cross sectiondoes not go to zero, but decreases with thesquare of the distance from line center.

2. The laser field itself can broaden theabsorption lines by a process called Rabibroadening (this will be discussed in moredetail later).

3 . Mul iphoton absorpt ion , whichbypasses one or more intermediate states,occurs at frequencies displaced slightlyfrom the single-photon spectrum.

4. Most of the larger molecules havesome degree of vibrational excitation prior

to laser irradiation (because of the higherdensity of vibrational states), while thesmaller molecules do not, and this excita-

tion may play a role similar to collisions inallowing for homogeneous absorption.

The first effect means that there will alwaysbe a small amount of off-resonant energyabsorbed and that this amount will increasewith fluence. The next two effects increasethe effective homogeneous linewith with in-creasing fluence, and the last effect accountsfor differences between large and smallmolecules at low fluence.

Collisionless Multiple-Photon Excita-tion Theory

A theory of multiple-photon excitationshould be consistent with the experimentalobservations that we have outlined above.The theory should explain the effect ofvibrational-state density, It should give theproper fluence and frequency dependence. Itshould give an increasing homogeneous line-width with the degree of vibrational excita-tion. And it should also suggest additionalexperiments whereby one could furthercheck the theory.

Early attempts at theories ranged fromstrict compliance with the old picture tocomplete rejection of that picture. Enoughexperimental evidence has accumulated now

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to force recognition of two points: multi-ple-photon excitation is a general, not apeculiar, limited, phenomenon; and theo-retical explanation will require an expansionand modification of the old picture in termsof a variety of effects such as absorption linebroadening, multiphoton absorption, high-er-order interactions among normal-modestates, and intramolecular energy flow be-tween states.

THEORETICAL APPROXIMATIONS. Gen-erally, one bases a theory of light absorptionon a quantum-mechanical description of theabsorbing medium and either a classical orquantum-electrodynamic description of thelight. In the case of multiple-photon excita-tion, the question is not whether fundamentalconcepts such as the Schrodinger equationor Maxwell’s equations are correct, but rath-er, what degree of approximation is neces-sary in the application of the basic theoriesto this complex problem. In our opinion, themajor problem with the old picture was thatit used approximations appropriate for spec-troscopy problems but not for the multi-ple-photon problem.

Before we begin to consider the moredetailed physical process of multiple-photonabsorption in polyatomic molecules, let usreview those properties of a molecule thatallow it to absorb electromagnetic radiationand point out in this review the variousapproximations that are needed. First, amolecule consists of some number of atomsbound together by interatomic elec-tromagnetic forces at fixed separation dis-tances and at fixed relative orientations.These positions are obviously not absolutelyrigid, but are only relative equilibrium loca-tions for each nucleus.

As a result of the very large mass dif-ference between the electrons and the nuclei,it is valid to treat the electronic and nuclearmotions as independent. This approximation,due to Born and Oppenheimer, assumes thatthe electrons produce a potential in whichthe nuclei vibrate. For the SF6 molecule, the

approximation reduces the complexity of thedescription of vibrational motion from 77particles (70 electrons and 7 nuclei) to 7particles.

Because the negative charge of the elec-trons and the positive charge of the nucleiare spatially separated, it is possible toestablish or change an electric dipole mo-ment in the molecule by displacement of anycharged particle from its equilibrium posi-tion. Typically, it is photons in the infraredwhose frequencies match those of the chang-ing dipole moments generated by nucleardisplacements (vibrations). The ensuing in-teraction leads to the absorption of infraredphotons.

Displacements of the nuclei from theirequilibrium positions can be described byharmonic motions that are referred to as thenormal modes of the molecule. The nor-mal-mode approximation allows one to de-scribe the motion as many uncoupled mo-tions. This further reduces the 7-particleproblem to a set of l-particle problems.Furthermore, in conventional infrared spec-troscopy one treats interactions between nor-mal modes as minor perturbations on theinitial uncoupled motions. This set of approx-imations gives excellent solutions to in-frared-spectroscopy problems when themolecules are in or near the ground vibra-tional state. This approach is discussed atseveral levels of perturbation in “The Mod-ern Revolution in Infrared Spectroscopy.”

In any octahedral XY6 molecule there arefifteen normal modes, but, because sym-metry leads to mode degeneracies, only sixare at different frequencies. Of these only thetriply degenerate modes v3 and v4 generate achanging electric dipole moment and there-fore absorb infrared radiation. These normalmodes behave initially like harmonic os-cillators, but as energy is put into thesemotions their enharmonic nature becomesmore pronounced until dissociation isreached.

In addition to the vibrational motion justdiscussed, a molecule also undergoes rota-

tional and translational motion. Becausetranslational motion does not affect the in-ternal molecular structure, but leads only tosmall Doppler broadening of the absorptionfeatures, we will neglect it in further dis-cussions. The combined motion of vibrationand rotation leads to a greater wealth ofpossible transitions so that a single vibra-tional absorption feature actually consists ofa broad absorption band containingthousands of ind iv idual ro ta t iona l -vibrational transitions. This rotationalbroadening is not unlike Doppler broadeningin nature, but the effects in frequency dis-persion are far greater and must be includedhere.

So far our discussion has dealt mainlywith theory used for conventional spec-troscopy, in which only the lowest vibra-t i o n a l e x c i t a t i o n s a r e induced bylow-intensity light. Do the same assumptionsapply when vibrational excitation is highenough to cause dissociation of themolecule? The most well-established theoryfor the unimolecular dissociation ofpolyatomic molecules is the so-calledRRKM unimolecular reaction-rate theory.Without describing the RRKM theory indetail, we can say that it retains the firstapproximation of separability of electronicand nuclear motion. However, instead ofassuming a very weak coupling among nor-mal modes of vibration, the theory assumesthat the interaction among the vibrationalstates at high vibrational energies is strongenough to continuously maintain a statisticaldistribution of population among thosestates. The normal-mode approximation isused in the RRKM theory only to countvibrational states at energies near thatneeded to dissociate the molecule. This theo-ry explains a large body of data on uni-molecular reaction rates for dissociation ofmolecules with high levels of vibrationalenergy.

A successful theory of multiple-photonexcitation would probably contain elementsof both of the previous theories. The initial


MixingWith

Resonant OtherMode Vibrational

Ladder Modes

absorption of intense infrared radiation pro-duces low vibrational excitation where thenormal-mode approximation of the old theo-ry applies. But at excitation energies near thedissociation limit, the molecule is in a statewhere the approximations of the RRKMtheory apply. Also, a theory for multi-

ple-photon excitation can no longer treat thelaser radiation simply as a probe because itseverely perturbs the absorbing sample.

We will further restrict our theoreticaldescription to the absorption of infrared laserphotons by isolated molecules. The introduc-tion of collisions into the theory precludesany first principle description of the problem

and. in fact, masks the more interestingphenomena we wish to study here.

A NEW PICTURE. we begin a theoreticalexplanation of collisionless multiple-photonexcitation by sketching a new picture withthe three major parts shown in Fig. 11: theregion of discrete, low-energy levels de-scribed by conventional spectroscopic theo-ry: the region of high-energy levels and highdensity of states described by RRKM theo-ry; and a region called the molecularquasicontinuum that connects these two ex-tremes. The discrete, low-energy region ofFig. 11 shows resonant absorption at thefrequency of the laser, vL, in the fundamentalof an infrared-active mode, such as v, forS F3. A s e x c i t a t i o n a d v a n c e s , t h e

anharmonicity necessarily causes a gradualmismatch between the laser frequency andthe resonance frequency for excitation to thenext higher state.

However, at about the energy level wherethe mismatch becomes significant, the multi-ple-photon excitation process merges into thesecond part of our theoretical picture. In thismiddle region the excited states of the reso-nant mode mix with other “background”states (vibrational states at the same energywith different normal mode character.) Brief-ly. this mixing occurs because the normalmode vibrations are not truly independent,but rather are loosely coupled to each other.

78

RRKMUnimolecularReaction-RateTheory

Molecule

Dissociates

Start ofQuasicontinuum

Fig. 11. Theoretical framework of multiple-photon excitation.

Most importantly, this mixing processcauses shifts and broadening of the excitedstates of the molecule. With enough mixing afine mesh of states, the molecular quasicon-tinuum. is formed that compensates for theanharmonicities and allows further absorp-tion of laser photons. The mixing allowsaccess to the full density of vibrational stateswith its huge statistical advantage forabsorption, Laser excitation through the

quasicontimuum is resonant, but of a naturedifferent from the simple normal mode ex-citation that occurs at the low molecularlevels: the quasicontinuum states obey quitedifferent dynamics owing to their “strongadmixture” character.

Finally, the third part of our theoreticalpicture in Fig. 11 is reached when enoughenergy for dissociation has been absorbed. Inthis region, a statistically random dissocia-

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9 4 4 948

Laser Frequency (cm –1 )

Fig. 12. Experimental and theoretical high-fluence spectra in the region of the v,absorption of SF6. The theoretical spectrum was computed for coherent transitionswithin the three-level system of Ov3, Iv3, and 2V3. Since all molecules were assumed tobe initially in the ground state, the theoretical spectrum does not include the hot-band

spectrum is based on data from S. S. Alimpiev, N. V. Karlov, S. M. Kikiforov, A. M.Prokhorov, B. G. Sartakov, E. M. Kohkhlov, and A. L. Shtarkov, Optics Communica-tions 31, 309-312 (1979).

tion occurs that can be described by theRRKM unimolecular reaction-rate theory.

Let us now proceed more slowly throughthe stages of excitation within the frameworkof a more detailed model constructed byworkers at Los Alamos and applied to SF6,S 2F 10, and UF6. All of these molecules havebeen dissociated with infrared laser photonsand, therefore, must proceed through thesestages of excitation. A molecule that cannotbe dissociated may simply be reaching abottleneck at the first stage of excitationbecause the quasicontinuum may not occurlow enough in the molecule’s vibrationalladder to overcome the anharmonicity.

THE DISCRETE, LOW-ENERGY REGION.

We have obtained very-high-resolution spec-tra, for both SF6 and UF 6, of the v3 fun-damental band (transitions from the ground

the second overtone of v3 (transitions from

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the ground state to the third excited state: O

basis, a great wealth of information has beenobtained about the multiple-photon ladder inthe discrete. low-energy region for thesemolecules.

Two important conclusions follow fromthis work. First, vibrational splittings thatrelate to the octahedral symmetry of the SF6

molecule are a dominant feature of thedegenerate energy levels of the v3 overtonesand may provide levels that compensate foranharmonicity in the discrete, low-energyregion. These octahedral splittings are due tointeractions between the degenerate compo-nents within the v3 vibrational ladder. Assuch, octahedral splittings are not the sameas the mixings that generate the molecularquasicontinuum, namely, the mixings be-

tween levels with different normal-modecharacter. Octahedral splittings were antici-pated in early work at Los Alamos, but there

was no empirical confirmation until infraredspectra of overtones had been obtained.These splittings are discussed in “The Mod-ern Revolution in Infrared Spectroscopy. ”

The second conclusion drawn from thespectra is the fact that the quasicontinuumand coupling to other vibrational modes playno role below or even at the 3 v, level. This isa surprising result because the background

states per cm -1 at 3V 3. The data clearlyindicate a very weak coupling strength be-tween the 3V3 overtone levels and this multi-tude of background states, although the datado not preclude mixing within the back-ground states. The data also show us thatsimple state counting is not enough for adetermination of the energy level that marksthe beginning of the quasicontinuum. In fact,a determination of the level and the mecha-nism for the start of the quasicontinuum withrespect to the background state mixing is oneof the major outstanding theoretical prob-lems.

These conclusions about the start of thequasicontinuum apply only to molecules thatare initially in the ground vibrational state.Excitation in lower vibrational states of othermodes prior to laser excitation may substan-tially alter the quasicontinuum character ofthe absorber by providing an interactionpathway between the v3 mode and states thatare not directly coupled to v3-mode states.Because temperature alters the population ofthe lower vibrational states, absorption fea-tures whose transitions originate in these

states are usually referred to as hot bands.At O kelvin all of the molecules are in theground state.

The analysis of multiple-photon absorp-tion up the v3 ladder begins with the solutionof the time-dependent Schrodinger equationfor multiple-level systems. The Hamiltonianused is

Here Hm describes the energy levels of the

79

teraction between the laser photons and themolecule. The molecular Hamiltonian, Hm,has been developed to second order of

perturbation for octahedral molecules, andmany of the molecular parameters have beenevaluated using the spectra of v3 and 3V 3

(“The Modern Revolution in Infrared Spec-troscopy”). Typically, Hm contains terms ofboth scalar and tensor nature that account,respectively, for energy-level shifts andenergy-level splittings, including the vibra-tional octahedral splittings.

The wave function for the total Hamil-tonian, H, is expanded in the basis of thefirst-order molecular energy states so that alloperators, except those associated with theoctahedral splitting and the photon-moleculeinteraction, are already diagonal. Ignoringcollisions, each rotational state in the groundvibrational state can be chosen to be inde-pendent; that is, the absorption for differentrotational states is inhomogeneous. Excita-tion modeling is then done individually foreach rotational ground state and the result-ing populations are averaged with the ap-propriate Boltzmann distribution factors atthe end of the calculation.

If the number of quanta in the pumpedmode is small, simple solutions to theSchrodinger equation can be found by eigen-vector techniques using the rotational-stateangular momentum as the Hamiltonian ma-trix index. The multiple-photon spectrumcomputed for the lowest levels of the v3 modeof SF6 is compared in Fig. 12 with ex-perimental data obtained at the LebedevInstitute in the Soviet Union. Considering

cm -1 is not included in our calculation, wefind a very good agreement with the meas-ured multiple-photon spectrum. By includingthe coherent dynamics of the time-dependentSchrodinger equation for the three-level sys-tem, the calculated resonances are in betterproportion to the experimental spectrumthan if one had simply used the popu-lation-counting technique typical of conven-

80

920 940 960

Frequency (cm–1 )

Fig. 13. Range of possible absorption frequencies for the v3 mode of SF6. The white

rotational structure. The black area represents the increase in the range of absorptionfrequencies that results from the octahedral splitting of v3 overtone levels. This latterrange increases with vibrational excitation in such a way that it compensates for muchof the enharmonic shift up to 10v3.

tional spectroscopic analysis. This resultshows that the anharmonicities of the pump-ed mode lead naturally to the observed shiftto lower frequencies that is characteristic ofhigh-fluence, multiple-photon absorption.

At this fluence (0.09 joule per squarecentimeter) the average number of photons

some molecules are not excited, this averageimplies levels of excitation possibly as highas 4v3 or 5V3 for the absorbing molecules.However, above 2V3 the octahedral splittingoperator effectively couples all rotationalstates, making both the detailed structureunobservable and this method of solutionintractable. New methods to cope with this

complication are currently being developed.The computed spectrum in Fig. 12, however,reflects the structure of only one and twoquanta absorption.

Before leaving the discrete low-energyregion of multiple-photon excitation, we needto discuss two important issues: 1) the levelNo at which resonant normal-mode couplingoccurs. that is, the energy level for the startof the quasicontinuum, and 2) the role of“coherence” in the absorption process.

First, consider Fig. 13, which illustrateshow the range of possible absorption fre-quencies in the v3 mode of SF6 changes as

vibrational excitation increases. For exam-ple, n = 1 on the vertical axis represents the

n = 1 from about 942 cm–] to 951 cm -1

represents the width of the v3 absorptionband resulting from the various rota-tional-energy transitions. At higher excita-tion energies this rotational width is shownas the white area split into its traditional low-and high-frequency branches. The black areaexpanding upward between the rota-tional-energy branches represents the addi-tional frequency range for absorption thatresult from the octahedral splitting of v 3

overtone levels. We note that Fig. 13 is anidealization: many holes are certainly pres-ent within the given range where no transi-tions exist. However, the figure can be usedto determine for a given frequency how highin the vibrational energy ladder it may bepossible to resonantly excite the v3 mode.

As a result of the growing enharmonicmismatch, the center of the frequency rangeshifts to lower frequencies. However, theincrease in frequency range due to the oc-tahedral splitting compensates for theenharmonic shift for many levels of excita-tion. About eight photons resonant with thecenter of the v3 fundamental band at 948c m–[ can be absorbed before the next excita-

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tion is outside the range of possible absorp-tion frequencies. If the laser is tuned to lowerfrequencies, say around 944 cm-’, about tenphotons can be resonantly absorbed up thev g ladder of SF6. This calculated increaseexplains, in part, the observed shift ofhigh-fluence absorption spectra to lower fre-

quencies.Conversely, for the excitation to proceed

much above 10v3 we must have couplingwith other normal modes, that is, the start ofthe quasicontinuum must be below 10v3 orSF6 would never dissociate. By consideringan explicit expansion of the molecular vibra-tional potential in normal coordinates andstrong tensor splitting of background states,we have predicted the start of the quasiconti-

most recent work leads to a new counting ofall coupled states and shows that the averagedensity of interacting states becomes rough-tly constant as the molecule is excited to highlevels,

The above arguments allow roughlyone-quarter of the excitation dynamics to bedescribed by the time-dependent Schrodingerequation in the discrete levels. This descrip-tion requires the excitation to be coherent,reflecting the fact that the molecule developsa transition dipole moment in phase with thelaser. In particular, populations in near-resonant or resonant laser-coupled energylevels are seen to cycle between these levels,that is, to “Rabi oscillate.” The absorptionand ensuing stimulated emission of energydepends on the product of the magnitudes ofthe electric field and the transition dipolemoment,

When the difference between the laserfrequency and the transition resonance fre-quency is large, that is, for off-resonantstates, the populations oscillate rapidly atthis detuning frequency, but with very littlenet transfer of population between states.However, because the Rabi frequency de-pends on the magnitude of the electric field,increasing the intensity decreases the effectof the detuning; that is, at higher laser

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powers an off-resonant state behaves in aresonant manner with population flow be-tween states. This phenomenon is the Rabibroadening mentioned earlier.

An inhomogeneously broadened absorp-tion feature consists of many independenttransitions that are separately coherent. Forexample, the time-dependent Schrodingerequation correctly gives the dynamics ofeach transition in a typical, rotationallybroadened absorption band. However, theexcitation that is observed experimentally (asin Fig. 12) is made up of a sum over all thesetransitions. Because each transition has aslightly different detuning frequency de-termined by its location in the band, thecoherent Rabi oscillation period for eachtransition is also slightly different. Thesedifferences give rise to a dephasing, or loss ofcoherence, in the dynamics observed for theentire band. In addition, each rotationaltransition has only a small fraction of thepopulation available to it (determined by theBoltzmann thermal distribution of rotationalenergy states). Therefore, the overall excita-tion may not execute simple coherent Rabioscillation.

The inhomogeneous character of the totalabsorption process helps explain the ob-served weak dependence of multiple-photonexcitation on laser intensity. Solutions of theSchrodinger equation show strong intensitydependence, especially when multiple-photonresonances are important, However, solu-tions that are summed over rotational-energystates are a much weaker function of theintensity than the individual terms. Thus, theobserved weak intensity dependence in theabsorption shows that the excitation doesnot have the character expressed by a singleSchrodinger equation.

Of course, if the laser power is sufficientlygreat, the middle, or quasicontinuum, regionof the molecule also contributes to the excita-tion. Most models of multiple-photon excita-tion that incorporate the molecular quasicon-tinuum ignore coherent effects and use in-stead a set of population-rate equations for

the dynamics. These rate equations are sin-gle-step in nature and do not exhibit Rabioscillation. They automatically provide forfluence-dependent absorption. The rate equa-tions are derived on the basis of a collision-less, unimolecular damping mechanismwhereby energy flows out of the resonantmode into other coupled vibrational back-ground states. However, the fact that thephenomenon can be described as a sum ofmany different coherent processes showsthat the data do not require a rate-equationdescription.

To test these hypotheses experiments areneeded that measure the lifetimes of excitedstates, thereby providing information aboutdamping mechanisms by which energy flowsout of one state into others. The Heisenberguncertainty principle relates the lifetime of astate to frequency linewidths. Thus, measure-ments are needed of the linewidths of variousprocesses, such as absorption from excitedstates, double resonance absorption, andfluorescence.

EXCITATION THROUGH THE QUASICON-TINUUM. The model for multiple-photonexcitation developed at Los Alamos in 1978is illustrated in Fig. 14. To be definite, wewill use as an example the absorption ofC 02-laser photons by SF 6. The discretelevels of the v3 resonant-mode ladder aretreated as described above. However, atsome level of excitation, N., a couplingoccurs from the v3 overtones (the leftmostcolumn in Fig. 14) to the background sub-density of states (the second column) having( N0–1) v 3 quanta, but still at the totalenergy N 0v3. (Figure 14 is drawn, for sim-

plicity and definiteness, with N. = 3.) Thisbackground subdensity is, in turn, coupled tothe subdensity having (N0–2) v3 quanta(third column), and so forth. The model istherefore distinguished by the selection rule

for the vibrational excitation of v3.The loss of a single v3 quantum to another

vibrational mode causes the molecule to

81

drop back one level in its v3 content. In otherwords. this construction is convenient sincethe transition dipole moment is carried byt h e r e s o n a n t v3 m o d e . F u r t h e r , t h e

because v3 is the highest energy mode in SF6

and a constant energy change of two ormore v, quanta would require coupling tomany other modes, making such couplingshigher order and much less probable.

At a sufficiently large subdensity of states,

Fermi’s Golden Rule can be used to describethe rate at which the population flows from agiven state in the v3 ladder to the coupledbackground states with one less v3 quantum.The Golden Rule rate is proportional to thedensity of the coupled background states andto the enharmonic coupling strength aver-aged over the states with one less v3 quan-tum. The problem of low-level excitation inthe v3 ladder coupled with leakage into thequasicontinuum is independent of the dy-namics of further excitation or energy trans-fer within the quasicontinuum. This firstproblem can thus be solved exactly, givingthe population at different locations in thequasicontinuum as a function of time.

Heller and Rice have shown that. formolecules in the quasicontinuum. if the signs

of the level coupling are random. then se-quential population flow is observed to thesubdensities having (N0 – 2), (N0 – 3), ..., v3

quanta. This leakage leads to phase interrup-tion and hence loss of coherence. The dy-namics can then be determined by popu-lation-rate equations provided that the Gold-en Rule rates exceed the laser upward pump-ing rates. Our calculations are madeself-consistently as follows. We assume thepopulation-rate equations are valid and thenobtain the coupling constant and, thus, theGolden Rule rates by fitting the data forabsorption versus fluence. We then check tosee if these rates do, in fact, exceed the laserpumping rates for the range of intensitiesconsidered. Our tit to such absorption data(circles) is shown in Fig. 15. We have alsocomputed the fraction of molecules dis-sociated using standard RRKM theory

82

Fig. 14. Transfer of vibrational energy from the resonant, absorbing mode to othervibrational modes. Each column represents a vibrational ladder with the large stepsapproximately equal to one v3 quantum of energy. The left column is the ladder for thepure v, mode. The next column represents the situation in which one quantum of v3

energy has leaked into other vibrational modes. This loss of v3 energy causes themolecule to drop back one level in its v, content. Thus a new “ground state”(combination modes with no v3 content) is fixed at 1v3 of total energy. Each furthercolumn depicts the same situation with an additional quantum of v3 energy transferredto other modes and new “ground states ’’for the v3 mode at ever increasing levels oftotal vibrational excitation. Multiple states are drawn at each level to depict thedensity of modes coupled between ladders. The size of the arrow’s and arrowheadsbetween ladders indicates the relative strength of coupling in both directions. Thelowest level for which coupling is significant (here depicted for convenience at the 3V3

level) is defined as the start of the quasicontinuum.

(black region). The lower edge of the regionwas defined by using RRKM theory justduring the laser pulse, and the upper edgewas defined by continuing the calculationafter the pulse for all population above the33v 3 level. Molecular-beam data (triangles)for SF6 at 140 kelvin, a temperature at whichalmost all of the molecules are initially in theground vibrational state, fit the theoreticalcurve well. The other data (squares) areincluded to show a typical range for dissocia-tion data, but are not expected to correspondas well to the calculated curves because ofless favorable experimental conditions (aclosed cell rather than a molecular beamexperiment; a temperature of 300 kelvin and,therefore, higher initial vibrational excita-tion).

Figure 16 shows the calculated populationdistributions versus the level of excitation (interms of CO2 photons) for various laserenergies. Below a laser fluence of 0.5 jouleper square centimeter we see a populationbottleneck due to the transition from thediscrete, low-energy region to the quasicon-tinuum. Below the bottleneck we see that afraction of the molecules are unaffected bythe laser photons and remain in groundstates because of inhomogeneous absorption.We also note that the population distribu-tions are quite broad. The tail of each curveabove the 33VJ level represents the molecules

with enough energy to dissociate.There is no way to measure these popu-

lation distributions separately: however. wecan compare our results with those predicted

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2

Laser Fluence (J/cm2)

Fig. 15. Theoretical fit for SF6 of absorption data and the fraction of moleculesdissociated as a function of fluence. Absorption data (circles) for SF6 [T. F. Deutsch,Optics Letters 1, 25 (1977)] has been used to evaluate the single coupling constant inthe Los Alamos quasicontinuum model for multiple-photon excitation. The fraction ofdissociated SF6 molecules can be calculated when the RRKM unimolecular dissocia-tion theory is combined with the Los A lames quasicontinuum theory. The curvedefining the lower edge of the black region is for the dissociation that occurs duringthe laser pulse only; the curve defining the upper edge is the result of continuing therate calculations after the laser pulse for molecules excited above the 33v3 level. Thetriangles are molecular-beam data for dissociation of SF6 at 140 kelvin [F. Brunnerand D. Proch, Journal of Chemical Physics 68, 4978 (1978)], which agress well withthe maximum dissociation curve. The squares are other data [J. G. Black, P.Kolodner, M. J. Shultz, E. Yablonovitch, and N. Bloembergen, Physical Review A 19,704-716 (1979)] taken under conditions less favorable for comparison with theory.

Level of Excitation (number of v3 quanta)

Fig. 16. The distribution of molecules with respect to level of excitation. Thedistribution curves calculated from the Los Alamos quasicontinuum model becomebroader with increasing fluence so that more molecules are in the high-energy tail ofthe dissociation region above 33v3. The fraction of molecules below the 3V3 bottleneckrepresent those molecules that are unaffected by the laser photons because, forinstance, they are in rotational-energy states that do not absorb at that frequency.

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by a thermal quasicontinuum model. Thismodel is based on the idea that laser excita-tion leads to a thermal population distribu-tion, that is, a Boltzmann distribution amongall vibrational modes. A key assumption forthe model is constant absorption cross sec-tion in the quasicontinuum. This assumptionis equivalent to strong coupling between all

modes as opposed to our assumption of the

weak coupling in the quasicontinuum.An assumption of constant cross section

for all levels from the ground state up leadsto a Boltzmann thermal distribution with no3V 3 bottleneck and a narrow distributionwith negligible dissociation (see curve withlong dashes in Fig. 17). Surprisingly, how-ever, if the thermal model is solved with a v3

energy ladder, that is, discrete energy levelsare used for the energy levels below N0, thepopulation distribution (short dashes) isalmost identical with that calculated from theLos Alamos model (solid line). While thethermal model with the v3 ladder does notresult in a large inhomogeneous fraction ofmolecules below 3V 3, the fraction ofmolecules in the high-energy tail is almostthe same.

In summary. the modeling efforts to date

on the SF6-C0 2 laser multiple-photon prob-lem have demonstrated a possible mecha-nism for this phenomenon. The observedeffects of fluence and intensity and the shiftto lower frequencies for high-intensityabsorption are mostly accounted for by thehot bands and the discrete. low-energy re-gion of the model. The trends in absorptioncross section due to temperature and molecu-lar size are accounted for by the vibrationalstate density of the quasicontinuum. Also.the resonant-mode vibrational ladder hasbeen identified as a critical element of themodel in that, without it, the calculateddistribution curves are too narrow for signifi-cant d association to occur.

But there are difficulties. Calculations inthe resonant-mode ladder above the firstovertone are formidable, and the data appearto be unable to distinguish quasicontinuum

83

models that originate from opposinghypotheses. It is our belief that newquasicontinuum models must be based upon(if not explicitly derived from) accuratepotential-energy surfaces for the molecule. Anew model based upon a potential-energyexpansion already indicates that theso-called weak coupling model with the

number of coupled states. The overcountingresults from the fact that the selection ruleapplies only to changes in v3, thus limitingthe vibrational coupling of v3 with othermodes to low order, but not limiting in asimilar manner the couplings between allother modes in the quasicontinuum.

There are at least two other importantaspects that need eventually to be in-corporated into the model for multi-ple-photon excitation. The first is the effectof initial vibrational excitation. This effectmay explain important differences in absorp-tion behavior for different molecules andtempera tures by provid ing a l te rna tepathways for multiple-photon excitation. Thesecond is the effect of collisions, an extreme-ly difficult modeling problem. Nevertheless,many photochemical or laser isotope separa-

tion schemes require high molecular concen-trations to insure the generation of signif-icant amounts of reaction product. An ade-

Fig. 17. Distribution curves for thermal and quasicontinuum models. The thermalmodel (long dashes) results in a Boltzmann thermal distribution with no 3v3

bottleneck and a narrow overall distribution. The addition of a v3 energy ladder to thethermal model broadens the distribution (short dashes). Surprisingly, the high-energytail nearly matches the tail predicted by the Los Alamos quasicontinuum model (solidcurve).

quate model for such conditions must neces-sarily account for collisions.

Concurrently with theoretical advances,experimental techniques need to be de-veloped that reveal more detail about thedistribution of vibrational energy, bothamong the collection of molecules and withingiven molecules. Presently, dissociation ex-periments measure the total number ofmolecules above the dissociation energy, thatis, the area under the tip of the high-energytail of the population distribution curve.Likewise, absorption experiments determineonly the average number of photons ab-sorbed per molecule, that is, the mean of the

distribution curve, Experiments need to bedesigned that map the detailed shape of thecurve and so distinguish between alternatemodels. Also, once photons are absorbed,how is the energy distributed among thevibrational modes of the molecule? Are therepathways of rapid energy flow between cer-tain states, but restricted flow between oth-ers?

Much has been learned about the surpris-ing phenomenon of multiple-photon excita-tion. But there are experimental and theo-retical challenges yet to be overcome—manyof which undoubtedly harbor further sur-

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Further Reading

N. Bloembergen and E. Yablonovitch, “Infrared-laser-induce unimolecularreactions,” Physics Today 31, No. 5, 23-30(1 978).

Ahmed H. Zewail, “Laser selective chemistry—is it possible?,” PhysicsToday 33, No. 11, 27-33 (1980).

V. S. Letokhov, “Laser-induced chemical processes,” Physics Today 33, No.11,34-41 (1980).

J. L. Lyman, G. P. Quigley, and O. P. Judd, “Single-infrared-frequencystudies of multiple-photon excitation and dissociation of polyatomicmolecule s,” in Multiple-Photon Excitation and Dissociation of PolyatomicMolecules, C. D. Cantrell, Ed. (Springer-Verlag, Berlin, to be published).

N. R. Isenor and M. C. Richardson, “Dissociation and breakdown ofmolecular gases by pulsed CO2 laser radiation,” Applied Physics Letters 18,224226 (1971).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (JohnWiley & Sons, New York, 1975).

LOS ALAMOS SCIENCE


Harold W. Galbraith earned his Bachelor of Science degree in 1967 atPennsylvania State University and a Ph.D. in physics from the University ofPennsylvania in 1971. He came to Los Alamos on a postdoctoral appointmentin 1973 to work with James D. Louck on group theoretical approaches to thefew-nucleon problem. He became interested in molecular spectroscopy in 1974and began working on problems of interest to the Applied PhotochemistryDivision. His research interests moved to quantum dynamics in 1976 with thediscovery of isotopic selectivity in multiple-infrared-photon dissociation ofpolyatomic molecules. (Photo by Henry F. Ortega)

Jay R. Ackerhalt earned his Bachelor of Science (1969) from Hobart Collegeand his Master of Arts (1972) and his Ph.D. (1 974) in physics, specializing inquantum optics, from the University of Rochester. Before joining theLaboratory in 1977, he held postdoctoral positions at the Institute forTheoretical Physics of the University of Warsaw, Poland, at the Departmentof Physics and Astronomy of the University of Rochester, and at the PhysicsDepartment of Johns Hopkins University. At Los Alamos he has worked on atheory of multiple-photon excitation in polyatomic molecules, primarily SF6

and UFb, and has shared with Harold Galbraith the Laboratory’s Distin-guished Service Award for this research and its application to the uraniumenrichment program. His present interests include self-focusing effects inpolyatomic molecules. (Photo by Henry F. Ortega)

John L. Lyman received his Bachelor of Science and his Ph.D. in chemistryfrom Brigham Young University in 1968 and 1973, respectively. He performedthe research for the Ph.D. degree at Los Alamos under Reed J. Jensen. Thatresearch included some of the first experimental studies of infrared-laser-induced reactions of polyatomic molecules. He also worked on HFchemical lasers. After joining the Laboratory in 1973, Lyman, along withStephen Rockwood, Reed Jensen, C. Paul Robinson, and Jack Aldridge, wasactive in the early research on multiple-photon excitation. These scientists holdthe patent for isotope separation by that process. Lyman has authored overthirty-five papers, including several review articles, on this and relatedphenomena. He is currently the Assistant Group Leader of the LaserChemistry Group in the Applied Photochemistry Division. Most of his time isspent working on problems related to the uranium enrichment project. Theseproblems include infrared self-focusing effects in UFb gas and the chemicalreactions of UFb dissociation fragments. (Photo by Henry F. Ortega)

AUTHORS

I

R. V. Ambartsumyan, Yu. A. Gorokhov, V. S. Letokhov, and G. N.Makarov, “Separation of sulfur isotopes with enrichment coefficient >103through action of CO2 laser radiation on SF3 molecules,” JETP Letters 21,171 (1975).

F. Brunner and D. Proch, “The selective dissociation of SF6 in an intense irfield: A molecular beam study on the influence of laser wavelength andenergy,” Journal of Chemical Physics 68,4936-4940 (1978).

S. S. Alimpiev. N. V. Karlov. S. M. Nikiforov. A. M. Prokhorov. B. G.John L. Lyman, Reed J. Jensen, John Rink, C. Paul Robinson, and StephenD. Rockwood, “Isotopic enrichment of SF6 in S34 by multiple absorption ofCO, laser radiation,” Applied Physics Letters 27,87-89 (1975).

J. R. Ackerhalt and J. H. Eberly, “Coherence versus incoherence in stepwiselaser excitation of atoms and molecules,” Physical Review A 14, 1705-1710(1976).

N. Bloembergen and E. Yablonovitch, “Collisionless Multiphoton Dissocia-tion of SFb: A Statistical Thermodynamic Process,” in Laser Spectroscopy111, J. L. Hall and J. L. Carlsten, Eds. (Springer-Verlag, New York, 1977).

T. F. Deutsch, “Optoacoustic measurements of energy absorption in C02

TEA-laser-excited SF6 at 293 and 145 K,” Optics Letters 1,25-27 (1977).

LOS ALAMOS SCIENCE

Sartakov, ‘E. M. Khokhlov, and A. L. Shtarkov, “Spectral Characteristics ofthe SFb Molecules Excitation by a Strong IR Laser Field at ContinuouslyTuned Radiation Frequency,” Optics Communications 31,309-312 (1979).

Jerry G. Black, Paul Kolodner, M. J. Shultz, Eli Yablonovitch, and N.Bloembergen, “Collisionless multiphoton energy deposition and dissociationof SFb,” Physical Review A 19, 704-716 (1979).

O. P. Judd, “A quantitative comparison of multiple-photon absorption inpolyatomic molecules,” Journal of Chemical Physics 71,4515-4530 (1979).

Harold W. Galbraith and Jay R. Ackerhalt, “Vibrational Excitation inPolyatomic Molecules, “ in Laser-Induced Chemical Processes, J. I. Stein-feld, Ed. (Plenum Press, New York, 198 1), pp. 1-44.

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Documents

MULTIPLE- · chemistry and for laser-induced isotope sep-aration. However, in the early 1970s another rather surprising feature was discovered in-volving the interaction of infrared