Raman observations of quantum interference in the $ν_1 / 2ν_2$ Fermi dyad region of carbon dioxide

0.34 MB·English

by  Craig W. McCluskey

Ramanobservations ofquantum interference inthe
ν1/2ν2 Fermidyad regionofcarbon dioxide
Craig W. McCluskey∗
Department of PhysicsC1600, TheUniversityof Texasat Austin, Austin, TX78712-0264, USA
David S. Stoker
6 Department of PhysicsC1600, TheUniversityof Texasat Austin, Austin, TX78712-0264, USA
0 and
0 TheCenterforNano-andMolecularScienceandTechnologyandtheTexasMaterialsInstitute,C2201
2 The University of Texas at Austin, Austin, TX 78712-1063, USA

n (Dated:February2,2008)
a
J Coherent anti-StokesRaman spectra (CARS)were obtained for CO2 in apositive column discharge. The
intensitiesoftheRamantransitionstotheν1/2ν2Fermidyaddecreasesignificantlywhenthedischargeisturned
4
onbecauseofequalizationofthelowerandupperstatepopulationsbyelectronimpactexcitation,inamanner
2 similartosaturation. Thestrengthofthe1285cm−1transitionisobservedtodecreasebyafactorof20greater
thanthe1388cm−1 lineduetoaquantuminterferencedecreasingthevibrationalrelaxationrateoftheupper
h] stateof the1285cm−1 transition. Thisinterferenceisverifiedbymeasurements ofthedecay ratesfromthe
p state.ExperimentsrulingoutStarkeffectsandpolarizationeffectsaredescribed.Supportingtheobservedrapid
- vibrationalexcitationbyelectrons,astronghotbandtransition2000←0220at1425.61cm−1 wasobserved.
m
TheseobservationsarecomparedwithrecentmeasurementsoftheRamanspectraofCO2heatedinaflame.
e
h
c I. INTRODUCTION allyindependentsimpleharmonicoscillatorsofmass1,to,
.
s
1 1
ic CO2 has been studied since the earliest days of both in- H = 2(η˙12+λ1η12)+ 2(η˙22+λ2η22)+···
s fraredandRaman spectroscopy. Rasetti wroteabouttheRa-
y + α η η η + β η η η η +··· .(1)
maneffectingases[1]andselectionrulesintheRamaneffect ijk i j k ijkl i j k l
h
p [2]in1929.Hehadnoticed,inparticular,thattheRamanlines Xijk Xijkl
[ observedinCO2werequitedifferentthanexpected,withtwo whereη arethenormalcoordinatesofthemotions.Thus,the
i
1 lines instead of a single line. Fermi [3] described the origin totalenergyisnolongerthesumofindependent(eventhough
of these strong Q-branch lines which now bear his name in
v anharmonic)oscillators.
1931. Discussions aboutquantuminterferencein transitions
2 In CO some of the coefficients α and β are equal
2 ijk ijkl
8 betweenthestatesofCO2 waiteduntilmuchlater. Zhuetal. to zero because the potential energy must be unchanged for
1 [4],forexample,wroteaboutquantuminterferenceintheex-
1 citation rates in 1991. Quantuminterferenceeffectsare also allsymmetryoperationsofthemolecule’spointgroup,D∞h.
Additionally, symmetry operations allow the antisymmetric
0 observableinrelaxationandarethesubjectofthiswork.
normal coordinates to occur in Eqn. 1 only in even powers.
6
0 Ofthetenuniquecubicterms,then,onlyα111,α122,andα133
/ arenon-zeroandthecubictermsofthepotentialenergyare,
s
ic II. THEORY α111η13+α122η1(η22a+η22b)+α133η1η32 . (2)
s
y
h A. AnharmonicityinCO2 α122 is the term that generates the anharmonic coupling be-
tweenν and2ν thatgivestheFermiresonance.
p 1 2
: TheQ-branchesinCO at1285cm−1 and1388cm−1 are NotethatinRamanspectroscopy,thesignalisproportional,
v 2 amongotherthings,tothederivativeofthepolarizabilitywith
i the components of the Fermi dyad, a feature due to an “ac-
X respect to position at the equilibrium distance. Because of
cidental” near degeneracy of one quanta of energy in sym-
this, thesymmetricstretchnormalmode(ν )producesaRa-
r metric stretch, ν , with two quanta of energy in bend, 2ν 1
a 1 2 man signal, while bend (ν ) and assymetric stretch (ν ) are
[3]. Becausethesetwofrequenciesaresoclosetoeachother 2 3
(δ = 7.87 cm−1 [5]) the states 1000 and 0200 strongly Ramaninactive.
perturbeachother.
(cid:12) (cid:11) (cid:12) (cid:11)
As Herzberg [6] shows, anhar(cid:12)monicities c(cid:12)hange the total
B. FermiSplittinginCO2
energyofa polyatomicmoleculefromthesumof3N mutu-
As Herzberg [6] also shows, in Fermi resonance the two
vibrationallevelsthathavenearlythesameenergyinzeroap-
∗Currentlywith: LosAlamosNationalLaboratory,N-2,AdvancedNuclear proximation(in CO2 ν1 and 2ν2) “repel” each other: one is
shiftedupandtheotherisshifteddownsothattheseparation
Technology, P.O. Box 1663, M/S B228, Los Alamos, NM 87545 USA;
[email protected] ofthetwolevelsismuchgreaterthanexpected.Inaddition,a
2
mixingoftheeigenfunctionsofthetwostatesoccurs.Thede- inEquations6(usingthemagnitudeofδ)gives,
viationoftheenergylevelsfromwhatisexpected,aswellas
themixingoftheeigenfunctions,becomegreaterasthesepa- a=0.734, b=0.679.
rationofthezeroapproximationenergiesbecomessmaller.
Inadditiontodependinginverselyontheseparationofthe Identifyingψ0 with 1000 and ψ0 with 0200 , expressing
n i
zero approximationenergies, the magnitude of the repulsion thecoefficientsassinesandcosines, andusingbracketnota-
(cid:12) (cid:11) (cid:12) (cid:11)
depends on the value of the corresponding matrix element tion,asdoZhuetal.[(cid:12)4],yields, (cid:12)
W oftheperturbationfunctionW:
ni
|ψi = cosθ 1000 −sinθ 0200
l
Wni = ψn0Wψi0∗dτ (3) |ψui = sinθ (cid:12)1000(cid:11)+cosθ(cid:12)0200(cid:11) , (7)
(cid:12) (cid:12)
Z
(cid:12) (cid:11) (cid:12) (cid:11)
whereWisessentiallygivenbytheanharmonic(cubic,quar- withθ =42.8°for12CO2(cid:12) (cid:12)
tic,etc.) termsofthepotentialenergydiscussedin§IIAand ThemagnitudeofδmustbeusedinEquations6becauseif
ψ0 andψ0 arethezeroapproximationeigenfunctionsofthe the minussign fromδ = −7.87cm−1 were used, the lower
n i
two levels. Note that since W has the full symmetry of the state, |ψ i, would have an excess 0200 instead of 1000 ,
l
molecule, as mentioned above, ψ0 and ψ0 must both have whichistheunperturbedstatewhichisactually“repelled”to-
n i
thesame symmetrytypein orderfortheintegralin(3)to be wardit. (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11)
non-zero. Thismeansthatonlyvibrationallevelsofthesame
symmetrycanperturbeachother. Thiscanbeseenintheen-
ergyleveldiagram,Fig.1, where1000and0200, bothbeing
C. NotationConventions
Σ+,areinvolvedinaFermiresonancewhile0220,being∆ ,
g g
isnot.
One of the problems in working with CO is the confu-
Iftheseparationofthezeroapproximationenergiesisfairly 2
sion engendered by the profusion of state naming and nota-
small,themagnitudeoftheshiftcanbecalculatedwithfirst-
tion conventions. Adel and Dennison [7] used the notation
orderperturbationtheoryfromtheseculardeterminant,
(v ,v ,v ,ℓ),whereℓisthevibrationalangularmomentum.
1 2 3
En0−E Wni Herzberg[6]usedthe notation(v1, v2ℓ, v3). AmatandPim-
= 0, bert [8] used notation similar to Herzberg’s but introduced
(cid:12)(cid:12) Win Ei0−E (cid:12)(cid:12) the idea of using 1 and 2 as ranking indices to denote the
(cid:12) (cid:12) higher and lower-energy peaks of a Fermi dyad. Schro¨tter
where, En0 andE(cid:12)(cid:12) i0 arethe unperturb(cid:12)(cid:12)edenergiesandWin = andBrandmu¨llerinFinsterho¨lzletal.[9]usedthenotationof
W∗ fromEqn.(3). Theperturbedenergiesarethen,
ni AmatandPimbert,butexplicitlylabeledstateswiththerank-
ingindex. They,forexample,labeledthe1388cm−1peakas
E =Eni± 12 4|Wni|2+δ2 , (4) (1000)I and the 1285 cm−1 peak as (1000)II. They also la-
where, p beledthe1410hotbandas(1110)I−(0110)andthe1265hot-
bandas(1110) −(0110),keepingforthehotbandssomeindi-
II
E = 1(E0+E0) cationofwhichlevelsareinvolved.RothmanandYoungused
ni 2 n i
notationsimilar tothatofSchro¨tterandBrandmu¨llerintheir
δ = (E0−E0) .
n i HITRAN (high resolution transmission) molecular absorp-
tion database intended for atmospheric physicists, but wrote
Theeigenfunctionsofthetworesultingstatesarelinearcom-
everything out on a single line suitable for a computerized
binations of the zero approximation eigenfunctions of the
database[10]. Intheirnotation,the1388cm−1 lineis10001
form,
andthe1285cm−1lineis10002.
ψ = aψ0−bψ0 Allthisconfusionissomewhatamelioratedbythe“Rosetta
n n i
ψ = bψ0+aψ0 , (5) Stone” of notation in Rothman and Young’s 1981 paper on
i n i
carbondioxide[11],reproducedbelowinTableI.
where,
1
4|W |2+δ2 +δ 2 TABLEI:ComparisonofnotationsforCO2 vibrationalenergylev-
a = ni , els.
p2 4|Wni|2+δ2 !
Herzberg[6] Amat[8] HITRAN[12]
1
4p|Wni|2+δ2 −δ 2 2000 (2000,0400)I 20001
b = . (6) 1200 (2000,0400) 20002
p2 4|Wni|2+δ2 ! 0400 (2000,0400)II 20003
III
PuttingHoward-LockpandStoicheff’svalues[5], 1220 (1220,0420) 12201
I
W1000−0200 = −51.232 cm−1 00444200 (12200,4040420)II 1024240021
δ = −7.87 cm−1 ,
3
Neglecting Fermi resonance Observed
3000
2000 S +g * 0440 G g0420D g * 11220200 DS g+g 22110022002200001212
* 0400 S + 20003
g
0001 S +u
2000 0330 F u * 1110 P u 11101
*
r 0310 P u 11102
e
b
m
1265.093 1409.476
u
n
e
v 1000 S +g 0220 D g 10001
Wa 0200 S +g 10002
1000
0110 P u 01101
1285.408 1388.132
0000 S +g 00001
0
FIG.1: EnergyLevelsofCO2. Energiesoflevelsmarkedwithanasteriskareuncertain. StrongestRamantransitionsaremarkedbyheavy
dottedlines.
As one progresses from Herzberg’s notation to the HI- III. EXPERIMENTALAPPARATUSANDCONDITIONS
TRAN notation, it becomes increasingly difficult to know
whichexactlevelsareinvolvedandthestatesinvolvedinthe Theexperimentalsetupusedforthiswork,describedprevi-
transitionbecomehidden. ouslyintheliterature[14],isshowninFig.2. Thespectrom-
eter consists of a neodymium-YAG laser (using the second
The vibrationalenergylevelsof CO are shown in Fig. 1.
2
In this figure, the strongest Raman transitionsof CO in the harmonicoutputatλ=532nm),atunabledyelaserpumped
2
bypartoftheoutputoftheYAGlaser,opticstooverlapboth
wavenumber range examined are marked by heavy dotted
lines. Herzberg’snotationisusedonthelefttoshowtheloca- spatiallyandtemporallytheremainderoftheYAGlaser’sout-
put(whichisrepresentedbytheboxlabeled“PumpLaser”in
tionsofwherethestateswouldbeiftheywerenotperturbed
byFermiresonance.Someoftheenergylevelsoftheselines, Fig.2)withtheoutputofthedyelaserinsidethesamplevol-
ume,andopticstodetectthenewlycreatedCARSbeamcom-
markedbyasterisks,weregiveninHerzberg[6]butwerenot
given in any of the other references examined. Herzberg’s ingfromthesamplecell.Aquartzblockinasqueezingmech-
anismisusedasavariablewave-plate(VWP)toadjustthepo-
(old)valuesoftheenergiesdidnotmakeanysenseconsider-
ingFermiresonanceandthe(newandmoreprecise)observed larizationoftheincidentYAGbeam.Fig.2alsoshowsoptics
and electronics for detection of Raman induced Kerr-effect
levels,whichareplottedontherightaccordingtoRothman’s
values[13]andmarkedwithHITRANnotation,sothevalues signals,whichwereusedpreviously[14].Forthisexperiment,
the“pick-off”prism(P2inFig.2),thedoublemonochroma-
oftheselevelsweremanuallyadjustedtobemorereasonable.
NotethatinFig.1,thenon-Fermiresonanceinfluencedlevels tor,andthephotomultipliertube(PMT)wereaddedtodetect
of 0220, 0330, and 0440 are notplotted on the rightside for thecoherentanti-StokesRamanspectroscopy(CARS)signal.
clarity.
4
Pump Probe 570 - 576 nm M 20000
Laser Laser
A CARS
532 nm GAP1 PMT
VWP L1 T MonDocohurbolmeeter nits) 15000
P1 u
494 - 499 nm b.
M ar
Glass Flat SaCmelplle L2 HNooltochg rFaiplhteirc nal ( 10000
g
P2 M RS Si
GAAP2 CA 5000
RIKES M
Differential
Detector Holographic 570 - 576 nm M
Notch Filter 0
1280 1320 1360 1400
FIG. 2: Spectrometer Optics Layout. A: aperture, GAP: Glan-air Wavenumber
polarizer,L:lens,M:mirror,P:prism,T:Galileantelescope,VWP:
FIG.4:Spectrumofcarbondioxidewithdischargeoff.
Variablewaveplate.
3500
The Pyrex glass linear discharge cell used for the major-
ity of the experimentsis shown in Fig. 3. It consists of two 3000
concentric glass tubes and has both a hollow cathode and a
hollowanode.Theinnerglasstubeis1′′O.D.attheendsand nits) 2500
u
b.
ar 2000
Stainless steel 5.1cm WO.aDte.r x j a6c1k ectm long Cooling water hose barbs Mounting/sealing gnal ( 1500
support block flanges Si
45.1 cm S
R
A 1000
C
500
40.6 cm
0
Wedged window Stainless wire connections
angled 10o Stainless hollow electrodes insulated with glass tubing 1280 1320 1360 1400
1.9 cm dia. x 7 cm long, 1.5 mm wall
Wavenumber (cm-1)
FIG.3:TheLinearDischargeCell.
FIG.5:Spectrumofcarbondioxidewithdischargeon.
necks down to an inner diameter of 0.375′′ over its central
16 inches. The smaller cross-section confines the glow dis- A. DropofBothFermiDyadPeaks
chargeandyieldsahighercurrentdensity.Theendsofthein-
nerglasstubeareopenformountingtostainlesssteelsupport
Thedropofbothpeakswhenthedischargeisturnedonis
blocksateachend.Theouterglasstubeformsawater-cooling
expected because the gas in the discharge is hotter and less
jacket which cools not only the region of the discharge, but
densethanthe gaswith no discharge. Itis also vibrationally
alsobeyondtheelectrodes.
heated, leading to the depletion of the groundstate. In their
When running, the cell provides a long positive column reportofspontaneousRamaninvestigationsofthepurerota-
glowdischargewhichextendsfourtofiveinchesonbothsides tional lines in CO Barrett et al. [15] stated the intensity of
2
of the focus of the YAG and dye laser beams so the region thelineswiththeirdischargeon(runat10mAandapressure
probedbythemostintenseportionsofthefocusedlaserbeams of 40 Torr) was about 1 the intensity with the dischargeoff.
5
iswithinthepositivecolumn.Atacurrentof100mA,thecur- Theycalculated the temperatureof their discharge(from the
rentdensityis1400A/m2(0.14A/cm2). strongestpurerotationalline)tobe730K.
Inthisexperiment,suchameasurementisnotpossible.The
temperaturecanbeestimated,however,bycalculatingtherel-
ativesignalstrengthofthe1285cm−1 and1388cm−1 peaks
IV. RESULTS to the 1265 cm−1 and 1410 cm−1 hotband peaks as a func-
tion of temperatureand thenmatchingthe ratio with whatis
As can be seen in Fig. 4 and 5, both peaks of the Fermi observed.
dyaddropwhenthedischargeisturnedon. Surprisingly,the Thepopulationofaspecificstateisgivenby,
1285 cm−1 line drops much more than the 1388 cm−1 line.
−Ei
This was true over broad ranges of both pressures and dis- gie kT
N =N , (8)
chargecurrents. i Q (T)
v
5
where,N isthetotalnumberofatoms,N ,g ,andE arethe off to 67% of the molecules with the discharge on and 25%
i i i
population,degeneracyfactor, and the energyof the ith state ofthemoleculesbeingshiftedfromthegroundstatetohigher
andQ (T)isthevibrationalpartitionfunction. states. Includingthedensityfactorwith thevibrationalheat-
v
TheCARSsignalisproportionaltothesquareofthediffer- ing, the 1285 cm−1 and 1388 cm−1 peaks should drop by a
encebetweentwostates,(∆N )2,sotheratiooftheintensity factorof1.9.
12
ofthegroundstatepeakstothatofthehotbandpeaksisgiven Since the 1285 cm−1 and 1388 cm−1 peaks decrease by
by, morethana factorof 19, muchmorethan Barrett’sfactorof
5,thereisanadditionalsourceofdecreasein(∆N )2.
12
2 −E2 2 One of the possibilities is that the ground state is being
N0−N2 = g0−g2e kT . (9) depopulated because the CO is being converted to other
(cid:18)N1−N3(cid:19) g1e−kET1 −g3e−EkT3 ! molecules. Barrett, for examp2le, consideredthat the drop in
signal might be due to the CO molecules being converted
Fig. 6 showsthe detailofthe baseline ofthe discharge-on 2
to CO +. He estimated the concentration of these ions in
spectrum. The large hump in the middle of the spectrum is 2
the discharge to be about 3 × 1011 ions/cm3, an insignifi-
backgroundnoise which remains when the cell is evacuated
cantconcentrationcomparedto his gasdensity of 1018/cm3.
andwhichmaybeaRamansignalfromtheoptics.
Mass spectrometric measurements of discharges have found
otherions,suchasO +,butthetotalconcentrationswerelow
300 2
[16,17,18,19,20]. Thiswouldalsoruleoutlargeconcentra-
tionsofdissociationproducts.
280
s)
nit
b. u 260 B. GreaterDropof1285cm−1Peak
ar
gnal ( In considering the greater drop of the 1285 cm−1 peak
S Si 240 (10002) than the 1388 cm−1 peak (10001), it must kept in
R
A mind that the magnitude of the CARS signal depends upon
C
220 the square of the difference in population of the upper and
lowerstates. Tohaveonepeakdropmorethantheother,then,
the ratio of the differences in populations must be affected.
200
Thissuggestsfourpossibilities:
1280 1320 1360 1400
Wavenumber (cm-1)
• The transition rate is altered or the degeneracy of the
FIG.6:BaselineofspectrumofFig.5showinghotbands. harmonicstatesbroken,
• Ramansaturationofupperlevels,
The ratio of the summed areas of the 1285 cm−1 and
1388cm−1groundstatepeaks(7393)tothesummedareasof • Thedischargesaturatesthefinalstateofthe1285cm−1
the1265cm−1and1410cm−1hotbandpeaks(172)is42.983.
transition,or,
(Notethatthebaselinewasmanuallyplacedforeachpeakto
compensateforthedispersivepartsofthelineshapes.) Solv- • Thedischargechangestherelaxationratesfromtheun-
ingEqn.9iterativelyforT givesadischarge-ontemperature perturbedstates.
of 373 K (quite a bit lower than Barrett’s 730 K). Note that
thisanalysisassumesthatthemechanismsexcitingamolecule Since both 10001 and 10002 begin in the ground state, it
from the ground state to one of the hotbands are in thermal is possible that the transition rate to 10002 is altered or the
equilibrium with the mechanisms de-exciting from the hot- near-degeneracyofthestates1000and0200ischanged.Such,
bandtothegroundstate. Aswillbeshownin§IVC,thisisan however,wouldrequireashiftofthewavenumberofthepeaks
invalidassumption. and none was observed (within a resolution of 0.2 cm−1)
Nevertheless,thisincreasedtemperaturewouldbeexpected when the discharge was turned on. No change in the ratio
to reduce the density and cause a drop of signal strength by ofintensitiesof10001and10002wasalsoobservedwhenthe
a factor of 1.60 but there is an additional contribution from electricfieldwasappliedwithoutadischarge.
depletion of the groundstate by vibrationalheating. To cal- If the molecule is illuminated by strong enough laser
culatethiscontribution,itisnecessarytouseEqns.8and9to beams, the transition could be saturated with equal popula-
calculatetheexactgroundstateandexcitedstatepopulations tionsintheupperandlowerstates. Totestthis,theYAGlaser
forthedischarge-offanddischarge-ontemperaturesof295K intensitywasvariedbyanorderofmagnitude.Nosignificant
and373K.SincetheexponentialsinQ rapidlybecomevery changeintheratioofintensitiesof10001and10002wasob-
v
smallonlya limitednumberoftermsareneeded. Thisanal- served.
ysis uses the states shown in Fig. 1, all of which are below The third possibility is not as easily addressed as the first
3000cm−1. The calculationshows the groundstate popula- two. Ifthedischargesaturatesthepopulationof10002,itsin-
tiondecreasingfrom92%ofthemoleculeswiththedischarge tensitywillbedecreasedrelativeto10001.Thiswouldrequire
6
anelectronexcitationrateof|ψiinEqn.7greaterthanthatof that the vibrationally inelastic excitation cross-section for e-
l
|ψ i. Zhuetal.[4],however,observedexactlytheopposite. CO scattering peaks around0.2 eV, quite close to the aver-
u 2
Theirexperimentalsetupwascompletelydifferentthanthe age electron energyin the dischargeof 0.1 eV mentioned in
oneusedinthisworkandtheyworkedatamuchlowerpres- theprevioussection.
sure(27.5milliTorr).
Theyproducedtranslationallyhotelectrons,e−*,bymulti- 0201
photonionizationofiodinewithanArFlaserat193nm: 1001
I2+n~ω(193nm)→I2++e−*. 4.3 m
These translationally hot electrons then collided with CO
2
moleculescausingrotationalandvibrationalexcitationofthe
10001and 10002states. By tuningtheir diodelaser, operat- y =sinq 1000 +cosq 0200
ingat(λ≈4.3µm),theycouldprobespecificrotationallines y u =cosq 1000 - sinq 0200
l 2
of the CO molecules by absorption in the strongly allowed
2
ν (anti-symmetricstretch)infraredbandusingthetransitions 1 0111
3
1000 → 1001 and 0200 → 0201. They concluded that the
upper state, |ψ i is significantly more populatedby electron 0000
u
scatteringthanthelowerstate,|ψi,byafactorof9.1to12.7,
l
dependingontheparticularJstateexamined. FIG.7:CO2excitationpaths.
Looking at the collisional excitation probability from the
groundstatetotheupperexcitedstate, Pgu, andtothelower The double quantum, direct excitation of 0200 from 0000
excitedstatePgl, is through the intermediate state of CO2−. CO2−, which
can have a lifetime as long as 90 µs in the gas phase [22],
Pgu ∝ sinθ 1000 V 0000 +cosθ 0200 V 0000 2 has a bond angle of 134.9° [23]), so the bend vibration is
Pgl ∝ (cid:12)(cid:12)cosθ(cid:10)1000(cid:12)(cid:12)V (cid:12)(cid:12)0000(cid:11)−sinθ(cid:10)0200(cid:12)(cid:12)V (cid:12)(cid:12)0000(cid:11)(cid:12)(cid:12)2 estnreornggylycuerxvceitsedofthCrOoug−hathree,dheocwayevoefr,CroOu2g−h.lyT3heeVpohteignhtiearl
2
whereV(cid:12)(cid:12)define(cid:10)sthei(cid:12)(cid:12)nter(cid:12)(cid:12)action(cid:11)potentia(cid:10)l. Ifo(cid:12)(cid:12)nea(cid:12)(cid:12)ssum(cid:11)es(cid:12)(cid:12)the than those of CO2 [24, 25], so this excitation channel was
electronexcitations 1000 V 0000 and 0200 V 0000 are opentoZhuetal.butclosedinthepresentexperiment.
nearly equal, Zhu’s results follow from these probabilities. Bothindirectpathsgothroughthesinglyexcitedbendstate
Theminussigninth(cid:10)eequ(cid:12)(cid:12)atio(cid:12)(cid:12)nfor(cid:11)|ψlith(cid:10)enca(cid:12)(cid:12)use(cid:12)(cid:12)sa“q(cid:11)uan- 0110 as an intermediate. Its excitation energy of 667 cm−1
tum interference” in its electron excitation rate, causing the (0.083 eV) is matched to the average electron energy in the
upperstate of|ψ itohavealowerpopulationthantheupper discharge. Thesecondhalfofthefirstindirectpath,0000 →
l
state of |ψ i. This is in contrast to the present observations 0110 → 1000, is a two quantum transition — it requires a
u
where|ψ i’sweakersignalimpliesa higherpopulationinits changeofv fromzeroto oneandasimultaneouschangeof
l 1
upperstate. Sinceonewouldexpectthatachangeinpopula- v from one to zero — whose probability is therefore low.
2
tionthataffectstheinfraredsignalinonewaywouldnotaffect Thesecondhalfofthesecondindirectpath,0000→0110→
theCARSsignalinanother,theremustbeadifferenceinthe 0200, is a single quantum transition and requires nearly the
experimentstocausedifferentresults. sameenergyasthefirsthalfoftheindirectpaths.Itthushasa
ThedifferenceisthatZhuetal.used“hot”electrons,asig- highprobabilityandistherouteforexcitationof0200inthis
nificantfractionofwhichhadenergiesaround3eV,whileour experiment.
experimentproducesmean electron energieson the order of ThedifferencebetweentheresultsseenbyZhuetal.’sex-
0.1eV.Theirhigherenergyelectronsopenedupanexcitation perimentandthepresentoneisthatZhuetal.haddirectex-
channelthatwasunavailabletothelowerenergyelectronsof citations from the ground state, 0000, to the final state, |ψ i
l
thisexperiment. or |ψ i, through two coherent channels, one exciting 1000
u
Countingasseparatetheunperturbedstatesmakingup|ψli andtheotherexciting0200. Becausethesetwochannelscan
and |ψ i, there are four paths of excitation from the ground occur coherently during a single electron excitation process,
u
stateto|ψiand|ψ i. AsshowninFig.7,therearetwodirect they saw interference between them and thus a lower exci-
l u
pathsmarkedwiththecircled“1”, tation probability for the production of |ψi. The present
l
experiment, using “cool” electrons, in contrast, has direct
0000 → 1000
excitation 0000 → 1000, but indirect, two-step excitation
0000 → 0200, 0000 → 0110 → 0200. Since these two processesare inco-
herentthereisnointerferenceandtheexcitationprobabilities
andtwoindirectpathsmarkedwiththecircled“2”,
of|ψ iand|ψ iaresimilar,leadingtotheconclusionthatthe
l u
0000 → 0110 → 1000 discharge could not have preferentially saturated one of the
observedstates.
0000 → 0110 → 0200.
Thefourthandremainingpossibilityforexplainingthisre-
Thesinglequantum,directexcitationof1000from0000oc- sultisthatthedischargechangestherelaxationratesfromthe
cursmostreadilyinthedischarge.Mazevetetal.[21]showed unperturbedstates. Heretheminussignof|ψiinEqn.7does
l
7
explain what was seen. If turning on the discharge causes 2500.00
thetransitionratesfortheunperturbedstates,1000and0200,
to becomemoreequal, thequantuminterferenceengendered 2000.00
g 2.4
by the minus sign in Eqn. 7 will cause |ψli to have a much din 4.0 Torr Torr
tshmaanll|eψr rie.l(aIxtactaionnreraadteilytobethseeelnowtheartltehveerlse,la0x1a1ti0onanradte0s0c0a0n, er Rea 1500.00 1.6
be diffeurentbetweendischargeon and dischargeoff because gitiz Torr 1.2 Torr
when the discharge is off, relaxation occurs by spontaneous ge Di 1000.00 0.8 Torr
emissionandinelasticcollisionswithmoleculesandthewalls ar
h
C
(collision induced emission). When the discharge is turned 500.00
on, relaxationadditionallyoccursbyinelastic andsuperelas-
tic collisionswith free electronsand ion andelectron charge
0.00
(E~ field)inducedemission.) 0 2 4 6 8 10
The significant drop of the 1285 cm−1 peak can thus be Time (seconds)
explainedbythedischargecausingchangesin therelaxation
ratesfromtheunperturbedstatesandsaturatingthestate|ψi. FIG.9:Riseof1388cm−1signalofCO2withturn-offofdischarge.
l
Notethatthemajorityofobserveddropofthe1388cm−1line
couldalsobeduetothestate|ψ ibeingpartiallysaturatedby
u
2000
thedischarge.
To test this hypothesis, the change of 1285 cm−1 and
1388cm−1 CARSsignalsasafunctionoftimeafterthedis-
ctoha4rgTeorwr.aTshtuersneeddaotaffawreasgmraepahseudreindFfoigr.p8reasnsudr9es.between0.8 counts) 1500 tt 21 == 12.37.05 ±± 05..142 s escec
y (
2500.00 nsit 1000
e
4.0 Torr 0 - Int 1.6 Torr
ng 2000.00 2.4 Torr 200 500
di
a
e
er R 1500.00 1.6 Torr
z
giti 0
arge Di 1000.00 1.2 Torr 0 5 10 Time (s) 15 20 25
h
C
500.00 FIG.10: Riseof1285cm−1 signalofCO2 at1.6Torrwithdouble
0.8 Torr exponentialfit.
0.00
0 5 10 15 20 25 30
Time (seconds) Roche et al. [26], for example, measured relaxation of the
ν /2ν Fermi dyad with Raman-infrared double resonance
1 2
FIG.8:Riseof1285cm−1signalofCO2withturn-offofdischarge. using a CO laser to continuously monitor populations via
2
the 9.4 or 10.4 µm CO laser transitions much in the same
2
way as Zhu et al.used their diode laser. Instead of excit-
For a pressure of 1.6 Torr, for example, the 1388 cm−1 ing CO molecules with hot electrons or H(D) atoms, they
2
signal increased as an exponentialwith a time constantτ ≈ selectively excited the molecules with a doubled-YAG/dye
2.0±0.3s. Theincreaseinsignalisduetoboththedecrease laser setup similar to the one used in the present work —
ofpopulationin|ψuiandtherepopulationofthegroundstate thoughtheirsachievednarrow-banddyeoperationwithanar-
from other levels by collisions. This vibrational relaxation gonlaserpumpedCW dyeoscillator. Theymeasuredthevi-
timeforrepopulatingthegroundstatedecreaseswithincreas- brationalrelaxationrateversuspressurefromthe1388cm−1
ingpressureasexpected. peak.Sincethe1388cm−1peakisverynarrowtherotational
Under the same conditions, the growth of the 1285 cm−1 linesblendatlowpressures(0.006amagat,5Torr). Because
signal due to |ψli exhibits a double-exponential behavior of this, they excited all of the rotational lines, which lead
whose shorter time constant is consistent with the repopula- to achieving rotational equilibrium more rapidly and mak-
tion ofthe groundstate. Thesedata are re-plottedin Fig. 10 ingthevibrationalrelaxationmoreefficient. Theywereable
with the curvefit. The results of this analysis are consistent to resolve the 1285 cm−1 Q-branch and used that to mea-
with the model of quantum interference altering the vibra- sure the rotational relaxation rate versus pressure from that
tionalrelaxationratefrom|ψlicomparedwiththatfrom|ψui. state. Unfortunately, they do not give the vibrational relax-
Others have measured relaxation rates from CO states ation rate from the 1285 cm−1 peak and do not give any
2
but their results are not applicable to this experiment. results for CO in a discharge. Dang et al. [27] measured
2
8
therelaxationtimesofCO vibrationallevelsinadischarge. 300
2
Theydisturbedthedischarge-onequilibriumwithaCO laser
2
pulseandthenwatchedasthatequilibriumwasre-established 280
with thedischargeremainingon. Inthis experiment,in con- s)
trast,thedischarge-onrelaxationratesarecomparedwiththe unit
discharge-offrelaxationratesanda changeintherateswhen arb. 260
thedischargeisturnedoniswhatcausesthe1285cm−1peak al (
todropmorethanthe1388cm−1peak. Sign 240
S
R
A
C
C. VisibilityofSaturationof|ψliinSpectra 220
If,aspositedintheprevioussection,the1285cm−1signal 200
dropsmore than the 1388cm−1 signal because its lower re- 1280 1320 1360 1400
laxationratecausesasaturationoftheupperlevelofthetran- Wavenumber (cm-1)
sition,thatsaturatedupperlevelshouldthenbethelowerlevel
ofhotbandtransitionsthatappearwhenthedischargeisturned FIG. 11: Baseline of CO2 spectrum showing 1299.61 cm−1 line
on. The possible transitions out of both the 1285 cm−1 and whendischargeisturnedon.
1388cm−1stateswithenergiessimilartothoseofthepresent
investigation are shown in Table II. Of the ten candidates,
cross-sectionforthetransition.Itisalsopossiblethatthethat
TABLE II: Hotband transitions out of 1285 cm−1 and 1388 cm−1 350
states.
300
10001 LowerState 10002 units) 250
UpperState (1388.18cm−1) (1285.41cm−1) arb.
20003 (2548.37cm−1) 1160.18 1262.96 gnal ( 200
12202 (2585.02cm−1) 1196.84 1299.61 RS Si 150
20002 (2671.14cm−1) 1282.96 1385.73 A
C
12201 (2760.72cm−1) 1372.54 1475.32 100
20001 (2797.14cm−1) 1408.95 1511.73
50
1360 1370 1380 1390 1400 1410 1420 1430
six are within the 1245 cm−1 to 1430 cm−1 range exam- Wavenumber (cm-1)
inedexperimentally. Ofthesesix,thethreefrom1285cm−1
should be strongerthan the three from 1388 cm−1. In addi- FIG. 12: Baseline of CO2 spectrum showing 1385.73 cm−1 line
tion,twoofthetransitions(12201←10001at1372.54cm−1 (solid arrow) and additional lines at 1366, 1424, and 1426 cm−1
and12202 ←10002at1299.61cm−1) involvethechangeof whenthedischargeisturnedon(hollowarrows).
onlythenumberofquantaofthebendvibration,ν . Asmen-
2
tioned in §IIA, the ν vibration is Raman inactive and so, the cross-products between a line and its neighbors and be-
2
absentanyconcomitantchangeinthesymmetricstretch(ν ) tweenalineandthebackgroundinherentinthe|χ(3)|2probed
1
quantumnumberorFermimixing,shouldnotbevisibleatall. byCARSdestructivelyinterfereandreducethestrengthofthe
Thebaselinesofmanydischarge-onspectrawereexamined line.
forevidenceoflinesatthesewavenumbers. Ofthesixcandi- Fig. 13 shows a comparison of the no discharge and
dates,1299.61cm−1,1385.73cm−1,and1408.95cm−1sug- 100 mA discharge spectra of 11101 ← 01101 hotband at
gestthepossibilitiesoflines. 1409.476cm−1. Note that the discharge on peak is broader
Thespectrumexaminedpreviouslyforcomputationofthe and is less symmetric than the discharge off peak, being
temperatureofthedischarge(Fig.6)isreproducedinFig.11 shaded to the lower energyside, where the 20001 ← 10001
withan arrowpointingatthepossiblelineat1299.61cm−1, hotband would be. In contrast to the amplitude of the
heavilyinfluencedbynoise. Itsweaknessisnotsurprisingin 1385cm−1 linejustdiscussed, herethereis noquestionthat
viewoftheRamaninactivityofthetransition. thecross-productsinthe|χ(3)|2 changetheshapeandplace-
Fig.12showstheregionofthe1388cm−1 Q-branchwith ment of the lines. This effect could be avoided by using
thesolidarrowpointingatthepossiblelineat1385.73cm−1. RIKES instead of CARS. Since this line arises from the up-
The weakness of this line is surprising given the degree of per state of the 1388 cm−1 Q-branch, |ψui, its weakness is
saturation expected for the upper state of the 1285cm−1 Q- expected.
branch,|ψi,thoughitcouldbeexplainedbyasmallRaman Interestingly, however, it was noticed that when the dis-
l
9
1500 220 thermalequilibriumwereachieved.
Discharge off CARS Signal (arb. units) 1500000 D10is0c mhaArg dei socfhfarge 111112024680000000 Discharge on CARS Signal (arb. units) apstettshaihtleoimskeolcInelnbwtoq.nayfnunIrpcaonaoeopptntlmnaoFrehtertcuniiraevggtmianrci.Bhsoott1ed.ionunr4etlsiott,cplmezyrrtrmtehepffhaseeoaaessrrncieuentwvndriteeencieovrssenrytsiksrsdtiihrinsebiteoeninunetwlthtflesgiiecoatnaohtmn1if,mlv2oaeteeh8wnusxe5dcpa[ph2coesar8tixpmnelm]ycudsoaisliwtaln1rueaetd3rhdireo8tatihsnont8stepaowfrocetunmemfchretnaalu−rwiltaanp1hcd,oepeaislusesisspnlrtsde[rbeh2ilcesbeeo9texuvrwi]nas---;
0 80 QuantuminterferenceispresentinothertransitionsofCO2
1406 1408 1410 1412 1414 as well. Table II shows that a line at 1511 cm−1 due to the
Wavenumber (cm-1) transition 20001 ← 10002 should be observed. Since this
transitionisduetoanincreaseoftheRaman-activesymmet-
FIG. 13: Baseline of CO2 spectra showing comparison of
1409.476cm−1 hotbandwithdischargeoff(solidline)andon(dot- ric stretch quantum number, one would expect it to be quite
visible.Experimentally,however,thislineisnotseen,evenin
tedline).
spectraat 40Torr. In thiscase, the level20001iscomposed
ofthestates2000+1200+0400andthelevel10002iscom-
chargewason,therewererepeatable,clearlyresolvablelines posedofthestates1000−0200. Hereexcitationbythepath
at 1366, 1424, and 1426 cm−1. These lines are marked in 1200 ←0200(withtheminussign)interfereswithexcitation
Fig. 12 with hollow arrows. From Rothman’s data [13], the bythepath2000 ← 1000andthetotalexcitationprobability
1366cm−1 line isduetothetransition10011 ← 00011,the vanishes.Itisalsointerestingtonotethatthe1409cm−1hot-
1424cm−1 lineto21101 ← 11101,andthe1426cm−1 line band, perhapsseen only as a broadeningin Fig. 13, is more
to 12201 ← 02201, where 02201 is the two quanta bend distinctasaseparatepeakinFig.14.
vibrational state with two quanta of vibrational angular mo-
mentumthatisnotinvolvedin Fermiresonancewith 10001.
Thoughhotbandtransitionsdirectlyfromthesaturated10001 250
and 10002 states are not observed, the strong hotband from
02201 only 50 cm−1 away suggests large populations of all
threelevels. Thisvalidatesthesaturationmodel. nits) 200
u
NotethatinFig.12,theamplitudeofthelineat1424cm−1 b.
(gwrehaitcehr tohraigninthaatetsoffrothme tlhinee0a2t2104110stactme−at11(3w3h5i.c1h5ocrmig−in1a)teiss a)(ar1/2 150 Experiment
fromthe01101stateat667.3799cm−1),whereasaBoltzman w dat 100
distribution at 373 K would require the opposite. This indi- S ra
catesthatthegastemperaturecalculatedin§IVAfromthera- R
A
tioofthepopulationsisincorrect,thatthevibrationalenergy (C 50 Theory
of the moleculesis not in thermal equilibriumwith the tem-
peratureofthegasbutismoreontheorderofthetemperature
0
oftheelectrons,whichforenergiesof0.1–0.2eVisroughly 1380 1390 1400 1410 1420 1430 1440 1450 1460
1000–2000K. Thisisnotsurprisingbecausethemolecules Wavenumber (cm-1)
are excitedto upperlevelsby electronimpactwhile theyare
de-excited back to the ground state by collisions with other FIG.14:HighpressureCO2spectrashowinghigher-orderhotbands.
molecules.Infact,withseparateprocesses,onewouldbesur- Notethat the 1388 cm−1 peak in theexperimental (upper) traceis
saturated;thestrongsignalsat40Torroverloadtheelectronics.
prisediftheexcitationandde-excitationrateswereequaland
[1] F.Rasetti,Nature123,205(1929). VanNostrandCompany,Inc.,1945).
[2] F.Rasetti,Nature123,757 (1929). [7] A.AdelandD.M.Dennison,PhysicalReview43,716 (1933).
[3] E.Fermi,Zeitschriftfu¨rPhysik71,250 (1931). [8] G. Amat and M. Pimbert, Journal of Molecular Spectroscopy
[4] L.Zhu,S.A.Hewitt,andG.W.Flynn,TheJournalofChemical 16,278 (1965).
Physics94,4088 (1991). [9] H.Finsterho¨lzl,H.W.Klo¨ckner,K.Srinivasan,H.W.Schro¨tter,
[5] H.E.Howard-Lock andB. P.Stoicheff, Journal of Molecular andJ.Brandmu¨ller,IndianJournalofPure&AppliedPhysics
Spectroscopy37,321 (1971). 16,370 (1978).
[6] G.Herzberg,MolecularSpectraandMolecularStructure,vol. [10] L. Rothman, ed., The HITRAN Database (Harvard-
II. Infrared and Raman Spectra of Polyatomic Molecules (D. Smithsonian Center for Astrophysics, World Wide Web,
10
http://www.hitran.com,2002). ElectricalEngineers126,356 (1979).
[11] L. S. Rothman and L. D. G. Young, Journal of Quantitative [21] S.Mazevet,M.A.Morrison,L.A.Morgan,andR.K.Nesbet,
SpectroscopyandRadiativeTransfer25,505 (1981). PhysicalReviewA64,040701(2001).
[12] R.A.McClatchey,W.S.Benedict,S.A.Clough,D.E.Burch, [22] R.N.Compton,P.W.Reinhardt,andC.D.Cooper,TheJournal
R. F. Calfee, K. Fox, L. S. Rothman, and J. S. Garing, Tech. ofChemicalPhysics63,3821 (1975).
Rep., Air Force Geophysics Laboratory, Hanscom AFB, MA [23] A.R.RossiandK.D.Jordan,TheJournalofChemicalPhysics
01731(1973). 70,4422 (1979).
[13] L. S. Rothman, R. L. Hawkins, R. B. Wattson, and R. R. [24] D.SpenceandG.J.Schulz, TheJournalofChemicalPhysics
Gamache, Journal ofQuantitativeSpectroscopy andRadiative 60,216 (1974).
Transfer48,537 (1992). [25] C. R.Claydon, G.A.Segal, and H.S.Taylor, TheJournal of
[14] P.S.Bhatia,J.P.Holder,andJ.W.Keto,JournaloftheOptical ChemicalPhysics52,3387 (1970).
SocietyofAmericaB14,263 (1997). [26] C.Roche,G.Millot,R.Chaux,andR.Saint-Loup,TheJournal
[15] J. J. Barrett and A. Weber, Journal of the Optical Society of ofChemicalPhysics101,2863 (1994).
America60,70 (1970). [27] C.Dang,J.Reid,andB.K.Garside,AppliedPhysicsB31,163
[16] J.M.AustinandA.L.S.Smith,JournalofPhysicsD:Applied (1983).
Physics5,468 (1972). [28] R. J. Blint and J. H. Bechtel, Journal of Quantitative Spec-
[17] M. Saporoschenko, Physical Review A (General Physics) 8, troscopyandRadiativeTransfer23,89 (1980).
1044 (1973). [29] R. L. Farrow, P. E. Schrader, T. A. Reichardt, and R. J. Gal-
[18] J. A. J. III and R. Ramaiah, Physical Review A (General lagher,in2000GlassIndustryProjectReview,OakRidge,Ten-
Physics)36,774 (1987). nessee (CombustionResearchFacility,SandiaNational Labo-
[19] S.V.K.KumarandV.S.Venkatasubramanian,TheJournalof ratories,Livermore,CA94550,2000).
ChemicalPhysics79,6423 (1983).
[20] J. A. Rees and S. R. Alger, Proceedings of the Institution of