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The importance of spectroscopy for infrared multiphoton excitation
Prog.Quam. Electr. 1981, Vol. 7, pp. 117-151
0079-6727/81/020117-35517.50/0 Copyright © 1981 Pergamon Press Ltd.
Printed in Great Britain. All rights reserved
THE IMPORTANCE OF SPECTROSCOPY FOR INFRARED MULTIPHOTON EXCITATION W. Fuss and K. L. KOMPA Projektgruppe f'tir Laserforschung der Max-Planek-Crv~llsehaft zur F6rderung der Wissensehaften e.V., D-8046 Garehing b. Miinchen
CONTENTS Introduction Anharmonicity Rotational Compensation and Anharmonic Splitting Nearly Degenerate Modes The Width of the Resonances, Intensity and Fluence Dependence Collisionless Vibrational Relaxation Rotational Hole Burning, Direct Two-Photon Absorption, Two-laser Experiments, and Temperature Effects 8. Collision Effects 9. Conclusion References
1. 2. 3. 4. 5. 6. 7.
117 121 123 128 131 135 137 139 142 144
It is substantiated by examples that the infrared spectra of molecules in high vibrational states are similar in width to those of the ground states. Therefore in order to explain collisionless infrared multiphoton excitation, the existence of resonance has to be checked, not only for the first three steps, but for all of them. That is, their (low resolution) spectra should be studied. This review summarizes the spectroscopic mechanisms contributing to multiphoton excitation, which have been suggested to date, including several kinds of rotational compensation and of vibrational level splitting, which cooperate to overcome the anharmonic shift. The spectral quasicontinuum, generated by intensity borrowing, must neither be very broad nor dense, and collisionless vibrational relaxation is only important at very high energies. Knowledge of relatively few spectroscopic details helps to understand many details and many differences in multiphoton excitation. 1. I N T R O D U C T I O N
Since the first demonstration of the collisionless infrared multiphoton excitation *~76~in 1973, the whole field has experienced a boom. Several reviews are already available, ~l-s} and in more than 100 molecules-from 3 to 62 atoms in size--unimolecular dissociation, isomerization (CHCI=CHCI, cyclo-C4F6, cyclo-C3H6, RNC~ electronic excitation (CRO2C12, OsO4, SiF4 and several dissociation fragments) or electron detachment (C6HsCH2) have been triggered in this way (Table 1). Part of the stimulation certainly came from the possibility that isotopes can be separated in this w a y ; (6'193-196) its feasability in a laboratory scale has been demonstrated, tt 93,x94~Also the fact that many radicals have become conveniently accessible in this way (see the compilation in ~s,,xs9~) plays a role. But more important is probably the fact that collisionless multiphoton excitation is an unexpected new phenomenon. Although not more than one photon should be resonantly absorbed, many molecules were found to absorb 30-50 photons directly from the field of a pulsed high power infrared laser (mostly the CO2 laser). Many theories and models ~24~-31s~have been developed to explain this phenomenon. These agree only partly with one another. The problem is basically spectroscopic in nature. It concerns the question what does the spectra of excited molecules look like; are they shifted, split or broadened compared to the ground state spectra? In spite of these obvious questions most theories and models have taken details of molecular spectroscopy into account only for the first three to five steps. Part * The term "multiphoton excitation" is only intended to count the photons. When we want to emphasize the contribution of real intermediate states, we use the term "stepwise multiphoton excitation", in contrast to "direct multiphoton excitation" which indicates virtual intermediate states. 117
TABLE 1. Molecules which have been multiphoton excited. The references quoted refer to experimental work related to multiphoton excitation, but not necessarily demonstrating it. A similar list is contained in;(S) a few molecules and references have been added; but molecules like SO2 (*16,243) or NH3 (244'2443) whose dissociation seems to be collision induced have not been included. Triatomics
8-atomics
OCS(9-1o,) 0 3 (9) NO2a)(1t.~2) SO2a). 1 , )
CCIF2-CH3 (313,206) CCIF2_CF3 (31a,361-363) CHCI2-CF31~:;'~::: CDCI2-CF 3 , CH3_CF3(2os. zo9) CF3:CF3 (21°) CH2D-CH2CI(211) CH2Br-CH2 F(209) CH2CI-CH2 F(212) CH3CH2 F(1033) CH3_CHF2( 1o3~,213--214.376) CH3COOH (s4") CH3_CCI3 (313.206) C2H3C1(21,206.206a) CzHsF (2°9) CF2-S (215) I I S-CF 2 HCIC = CHBCI2 (37s)
4-atomics
ECI3(I 3-19b,34,88) H2COt HDCO~D2CO (2°-25,s4,3~) CI2CS (374) F2CO (245) HN3, DN3 (26) 5-atomics
CCl4(27-29) CC13F(3O.31) CBr2F2(3~ 34) CC12F2(31-41,ss) CHCI2F(3 l a) CHCIFz(31~,33,42-44) CDF3(364--365) CH2F2(31a,45) CBrF3(31,46-493) CF3Br+(356) CC1F3(31.46.226a,366,367) CF3I(31,46.4~,49 62,190d,e,193,377) CF3I+(356,357) CF30-(359) HCOOH(63, 6'*) CrO2Cl2(62, 65-65d) OsO4 (65b-75a) SiF4(34,3a,76-ao) CDF3(81. 82) CH 2 = CO (83-34~) 6-atomics
CH3OH(22. aS-as) CH3CN (as-91) CHaNC (92) CCIF = CF2 (88'93 93c) CHCI = CF2 (31a'94) CHCI = CHCI (62'94-1°°) CHCI = CCl (31,) CH2 = CC12 ~(~4) CH 2 = CHCI (94.101-102) CH 2 = CHF(103-104) CH 2 = CF2 (103,1033) CH 2 = CH2(84,88,105-11 t) HC = C - CHO a)(112-112b) N2F4.(313.I 13.114) N2H4 (t~s) SF5 +(116,117)
9-atomics
C2HsCN (216) C2HsNC (92) C2HsOH(853,216,217) CHaOCH3(218) CF3CH2OH (853,b) C3F~-(219.22o) C3H61221) cyclo.C3H6(221-223) SFsNF2 (224) CH3CICH~ -(3sO) l O-atomics
C2HsNH2 (216) CF 3COCF 3(62,225-229) cycio_C4F6 (230.230.) 11- and 12-atomics C3FTI (62) n_C3H 7OH(217) trans-CH3CH = C H C H 3 "°°) CH3COCOCH3O~ 36s.373) CH2-CH2(369) I I CH2-CO H2C = CHCH2COOH (379) cyclo.C4Fa(231) C6HF5 (232) $2F10 (178) more than 12 atoms
C][..'[3NO 2(88.89,122-123)
H O O C _ C H 2 _ C O _ C O O H (379) C6HsCN +(3sl) C2HsO-CH_--CH2(233-233) CHaOCH2CH----CHz (23s) C H s C O - O - C O C H 3 (s4,s4")
CH3ONO (121) CH3OHF-(3ss) CH 2 = CHCN(9S,12 '),125) SF5C1(126.127) SF6(28.31 a.38.47.80.111.128-196.355) M o F (197) SeF (~93) UF6~199-205)
C6HsCH ~ ( 2 4 2 ) CH3COO-sec-C4H~ (21~) (C2Hs)2OHO(C2H~)~(219.2zo)(CH3)2C_C(CH3)2(23~ 239) I L o-o U(OCH3)6 (24°) UO2(hfacac)2" T H F °')(241-241b) (UO2 (hfacac)2)2(241~)
7-atomics
CH3NH20 ! sA ls-120)
(a) after electronic excitation (b) hfacac = CF3CCHCCF 3, T H F = (CH2)40
o
o
c'~
r,~r,
~
(44,235,236)
The importance of spectroscopy for infrared multiphoton excitation
119
of the reason is apparent from Table 1: This list of molecules suggests that a universal excitation mechanism exists, while spectroscopy is different for every molecule all examples show the same basic behaviour. The most popular of the global models is the quasicontinuum model (see the reviews, (1-3) especially (3)). It assumes that in the region of high density of states no selection rule can be strict enough to suppress all the transitions to these states; therefore the molecule will always b e in resonance, even if the principal absorption has been offset by anharmonicity. It has to be noted, however, that none of the general models explains the excitation of every molecule of the table. For example the density of states of the triatomics and of HN 3, C2H 4, H2CO and other molecules is too small to form a quasicontinuum. Furthermore molecules seem to exist which cannot be dissociated by multiphoton excitation (Table 2), although they are structurally quite similar to others which have been dissociated. It is appropriate at this point to consider the following two arguments. (1) Many details of the multiphoton process can already be understood with relatively little spectral information, e.g. at low resolution, and this report tries to give examples for that. Such information may not be difficult to get, since low resolution spectra are not very different for systems of low and high densities of states. As an example consider Fig. 1, which TABLE 2. Unsuccessful multiphoton dissociation experiments
molecule CH3I CD3I COF 2 NF3
Vmmlec(335)
Vla.,er
882.4cm - t 951 1049 965 1032 985
(e) 944.2cm - t (at) 944.2 (e) (a~) 951.2 (al) 1031.5 (al + e) 983.3
ElM© r
(lit focus
Pdi~
20J 20
220J/cm 2 220
<3.10 -3 <3.10 -3
130 110 110
<8.10 -3 < 8.10- 3 <8.10 -3
4 3.5 3.5
The CO2 laser oF x33) (50 ns spike with 30 % of the energy, 3/ts tail, beam cross section 5 x 5era 2 for 944.2cm -1 and 3 x 4era 2 for other wavelengths) was focused ( f = 20 cm) into a cell of 23 cm a volume. Conversions were < 10- 5/pulse in all cases. The energy density ~ focus and dissociation probability Poi,s in the focus were calculated using the focal dimensions 3 × 3 × 10ram 3 (at 944.2cm - t ) and 1.5 × 2 × 10ram s (elsewhere). For COF 2 and NFa 1 mbar of H2 was added as scavenger. o/
cm2
'
'
k
r
'
'
'1
I
L
o/
1049
10-20
10-22 CF3 CO CF 3
OCS
10-21 J , i ~ 1 0 "23 900 gho 1000cn'rI 1000 IOr--JO 11{X)cm-1 [nfrsLt~dspectraofasmal]molecule(carbo~ylsuLfide)andofamo]ecule(hexafluoroacetone~
F~G. 1. which is in the quasicontinuum of states at 313 K, the temperature of measurement, Spectral slit width was 1 c m - 1. Higher resolution reveals rotational structure and a slightly higher cross section for COS.
120
W. F u s s and K. L. KOMPA
compares the infrared spectra of a small and a medium sized molecule. The latter, hexafluoroacetone, has an average vibrational energy of 1500 cm- 1 at 310 K. The transitions shown from 1500cm -1 to 2500cm -~ connect regions of 700 and 2.104 states per cm -1. Nevertheless the band is not broader than the one shown in OCS, where the density of states is < 1/cm- 1. Obviously (approximate) selection rules suppress the many possible transitions outside the band. It is interesting to note that the absorption cross section of hexafluoroacetone around 971 cm- 1 drops by not more than a factor 3 when the molecule is excited up to 10 quanta/molecule (see below and (225)). Every model based On background absorption in the wings which according to Fig. 1 is very weak will have difficulties to explain this fact. (2) Table 1 conceals striking differences which are not predicted by global models. For example, Fig. 2 shows the absorption cross sections (=absorbed energy per molecule/incident energy per cm 2) plotted versus the mean excitation energy. Obviously this cross section drops by orders of magnitudes for some molecules, whereas in others it remains nearly constant up to almost the dissociation energy or even above. The lower states cannot be responsible alone for this difference. Figure 3 shows the dissociation yield as a function of wavelength for three molecules. Its optimum wavelength is shifted from that of the linear
~. r'//i
I
I
i 10-18
10-19
I 0 -is
~ S F s
944.2 crn-'
|
10"20 I0-t
I0 -2
r-'~/
d/cruZ ,o.~,/
I I
I0
I
I
II
I0"I?!~
~
$2 Fro
io-te
crn"l
=,-,,,,
10733 cm NN
°""R
\
%.
10-19
~o~
I
I0 "I
I
1
1
10quonta/rnolec
FIG. 2. Multiphoton absorption cross section ~r v e n u s average excitation energy q (quanta/molecule). Since only a fraction of the molecules take part in the excitation, q is only a lower b o u n d of the actual energy of the probed molecules. These plots have been derived from measurements of q as a function of the incident energy @ (photorm/cm 2) via ¢r := q/~l). The differential cross sections are smaller by a factor 1 to 2 in the cases shown, b u t are less accurate. Data for SF 6 from (284) which agree with the compilation in (s) for H2C = CHCI from, (1o1.~,,~os) S2Flo,(l~s) CF3COCF3,(22s) CF3I, (53,347) C2F4S2 .(2151
The importanceof spectroscopyfor infraredmultiphotonexcitation
SF 6
/
/--
9 0
950
BCI3
/
121
/""',.
c m -1
%
/
.,t
900
\' /\", 950 cmq
' SiFz,
I
p
~/{~x\
/
IO00
x
l~ '
1050 cm "1
. . . . . r o,t o t (tas) (see alsot h e c om pilatlon FIG.3. Frequencydependenceof dissociationprobaoihty o,, " in (5)k BCl3(19a) and SiF,t (79) comparedto their small signalspectra (e).
absorption maximum, sometimes to the blue (SiF,), in other cases to the red (SF6, BCI3). The shapes and positions of these dissociation spectra (which are crucial for isotope selectivity) are usually thought to be determined by the spectroscopy of the first few steps. Note, however, that isotope selectivity has been found by irradiation of a band of CHaNO 2, in which there is no spectral isotope shift.(t22) Obviously in higher steps another mode contributes whose frequency does depend on the particular isotopic content. A substantial difference has also been found for the internal energy of the fragment NH 2 depending on whether it was produced by dissociation of N2H , or CH3NH2. (t is) This fragment energy indicates how far the parent molecule has been excited beyond the first dissociation threshold. Obviously in N2H ¢ the optical excitation finds an exit channel closely above the dissociation energy, whereas CH3NH 2 allows absorption continuing to higher energies although the number of atoms in the molecules differs by only one hydrogen atom. All these examples demonstrate that one should study the spectroscopy of excited molecules in more detail. This report is meant as an encouragement to do so, even if the necessary spectroscopic information is not easily accessible. 2. ANHARMONICITY Collisionless infrared multiphoton absorption would be no problena for a harmonic oscillator. Because of its ladder of equidistant energy levels it would not get out of resonance after absorption of any number of photons. But normal molecules are anharmonic. The vibrational energy levels of a one-dimensional anharmonic oscillator (e.g. a diatomic molecule) can be represented in good approximation by E v / h c = re1 v + x v ( v -
1)
(1)
(Vet fundamental frequency, x anharmonicity, both in crn-t). The mismatch between v-th level and the energy of v photons therefore increases quadratically with v. Power broadening is only proportional to v/~ + 1 :
(gg) = (
,0,Ek/v + 1
(2)
Therefore it will be able to compensate the anharmonic shift in a few steps at most, even if certain modifications of eq. (2) for degenerate modes are taken into account.(249'2s°) Under
122
W. Fuss and K. L. KOMPA
multiphoton excitation conditions a power broadening of ( f i o l E ) / h c = 0.1 to 1 cm -~ is typical. Anharmonicities are generally negative. For diatomic molecules their magnitude is a few per cent of Voi, i.e. more than 10cm-1. Close to the dissociation limit, the level distance approaches zero. Therefore it is believed that diatomic molecules cannot be excited up to dissociation. Similar behavior is expected for vibrational coordinates of polyatomics which directly lead to dissociation. Therefore it will not be possible to excite a single bond in a molecule selectively until it breaks. (For a possible exception see below under rotational compensation). Since the vibrations ofpolyatomics are often delocalized over several bonds, the amplitudes oftheindividual atoms are smaller, resulting also in smaller anharmonicities than in diatomics. Furthermore the anharmonicity of one mode contains contributions from several displacement coordinates, which can mutually compensate. Positive contributions often arise from the curvilinear nature of deformation coordinates. 13~7) The exceptionally small anharmonicity x44 = + 0 . 1 c m -1 of the v4 mode of SF 6 at 615 c m - ~¢31s>may be an example for such a compensation. Even smaller anharmonicities are entirely conceivable. The tetrahydrofurane adduct of uranyl hexafluoroacetylacetonate¢241) seems to have such a harmonic vibration: If it is excited by a CO 2 laser, its infrared absorption band is neither shifted nor broadened. 13~m The absorption cross section of a harmonic oscillator does not depend on excitation energy. For several molecules (Fig. 2) linear absorption has in fact been found; in these cases, however, the interpretation of section 4 (contribution of different modes) is more probable. The vibrational energy levels of a polyatomic molecule can usually be written Ev/hc
= E i viv i + Z i x , v i ( v i -
1) + E i , j > i x o v i v j
(la)
This is an extension ofeq. (1). v~are the observed fundamental frequencies, and x , and x o are called diagonal and cross anharmonicities, respectively. The most frequent deviations from eq. (1 a) arise from anharmonic resonances like Fermi resonance. These interactions mix states of one mode with' combination states lying close-by. The resulting energy correction is often also bilinear or quadratic in v~. In these cases, the correction to (1 a) can be contracted with one of the anharmonicities. Anharmonic resonances are always important in the quasicontinuum of states (Section 5). They cause displacements from eq. (la) which often compensate. So eq. (la) can remain valid although the mode parentage of the states change. The quadratic dependence of the detuning on the quantum number will drive the molecule out of resonance after very few steps. How in a rotating polyatomic molecule this mismatch can be overcome is shown in the following sections. Often this mismatch is discussed in terms of a transition whose frequency shifts linearly with the step number; this would be correct if the laser would be tuned at each step to follow this shift (v 1 = driven mode): (Ev,+ lvj - E~,~j)/hc = (vl + Z j , l x l . a v j )
+ 2xxlvl
(3)
A linear relationship with a quantum number is also valid for the detuning of the laser to an nphoton resonance with the v = n state of a single oscillator: E,/nhc = v + x(v -
1)
(4)
When an unknown number of modes has been excited in the previous steps, in the n-th step a modification of eq. (3) is sometimes employed: (E,,~ + 1~ -
Eo,v~)/hc = v 1 + 2~vl
(5)
with an unknown type of average anharmonicity ~, whose value is close to a thermal average. ~3 But Section 4 shows that the spectral shift is not even monotonic in n, if more than one mode contribute. Another modification of eq. (3) results for a model molecule whose anharmonic shift only depends on the energy, irrespective of its distribution over the modes; i.e. for a molecule for which 2 x l ~ = av~ xij = avj,j ~ 1
(6)
The importanceof spectroscopyfor infrared multiphotonexcitation
123
At the energy Evlv2... the resonant mode will then be detuned by (Ev, + iv2". - Ev,~2)/hc -
vl = aE~,~2.../hc
(7)
from v~. The oscillator strength of vl can be distributed by intensity borrowing over a certain width of close lying combination transitions (see s and 2s4). In this case the molecule continues to absorb by means of these transitions, and eq. (7) gives the detuning from the absorption maximum. It must again be pointed out, however, that in real molecules with less uniform anharmonicities the anharmonic shift is not a simple monotonic function of the step number (Section 4). 3. ROTATIONAL COMPENSATION AND ANHARMONIC SPLITTING At room temperature, many rotational states of the vibrational ground state are populated. The excitation process has to be studied separately for each rotational state. To the vibrational energy of eq. (2) a rotational term has to be added. This term reads for symmetrical tops Erot/h¢
=
BJ(J + 1) + (A - B ) K 2 ~ 2 A ~ I K
(8a)
and for spherical tops Er°t/sc = BJ(J + 1) + B~[R(R + 1) - J(J + 1) - l(l + 1)]
(8b)
A and B are the rotational constants for rotation around the molecular axis and an axis vertical to it, ~ is the Coriolis coupling parameter, 1 is the quantum number of vibrational angular momentum and R is the rotation of the molecular frame (R = J - 1). Relatively strict selection rules apply to the rotational quantum numbers J and K: For transitions parallel to the molecule axis AK = 0, AJ = _+1,0,AI = 0
(9a)
A K = + I, AJ = _+I,O, AI = _+I.
(9b)
for transitions vertical to it
Of special interest is the case where in a sequence of vibrational levels J regularly increases or decreases from Jo in the state v = 0 to Jo +_ v in the state v. Insertion o f J = Jo +_ v, K = Ko (parallel transitions) into eq. (6) or eq. (7) yields terms linear and quadratic in v, which, when added to the vibrational energies eq. (2), may be contracted with the fundamental frequency and the anharmonicity: Y/.eff -~-
Y i ~---
2B(Jo + 1 ___ 1)
(10) (11)
Xii,eff = Xii -[- B
Similarly for vertical transitions the sequence K = Ko +_ v, l = v can be described by the effective frequencies and anharmonicities v~.cff=
{
v/+2(A(1-~) v~+2(A(1 ~)
B)K o forJ=Jo B)Ko+_2B(Jo+I_I)
f x , + A ( 1 - ~) - B Xii'eff= Xii + A(1 ~)
forJ=Jo+_V
forJ=Jo for J = Jo +- v
(12)
(13)
These relations can easily be refined to take the v dependence of A and B into account. Equations (10) and (12) are analogous to the frequency shifts of the P and R branches of fundamentals. Equations (11) and (13) show that the vibrational anharmonicity has to be corrected by rotational terms. A compensation arises when these terms are positive. A and B are always positive, and - 1 ~< ~ ~< 1. This compensation is independent of the initial rotational state (Jo, Ko); all excitation frequencies within the rotational contour of the fundamental will see
124
W. Fuss and K. L. KOMPA
the same anharmonicity. A and B vary over a wide range. For example for = 4 cm- 1. For SF 6 the correction can nearly be neglected; for C2H4, however, possibly "~77,©ff > 0. It is also remarkable that this type of compensation--if it is perfect--makes dissociative exaltation of diatomics possible, too. A combination of the subcases of ext. (13) has been considered in ~1o7,) for ethylene, which is a nearly symmetric top. For asymmetric tops, analogous closed form relations do not exist. Another type of rotational compensation has long been knownJ 1'2~ Consider the three consecutive vibrational absorption steps with change of rotational quantum number Jo ~ Jo - 1 ~ Jo - 1 ~ Jo in a spherical top like SF 6 for example. These transitions will occur at the approximate frequencies v o - 2 B J , v o + 2 x , v o + 4 x + 2 B J o respectively. All three frequencies are the same if SF 6 Bcff = B(1 - () = 0.028 c m - 1 and for ethylene A - B
(14)
BJ o = - x
For appropriate Jo, ¢q. (14) can always be fulfilled. On the other hand, only few of the thermally populated rotational levels can take part in such a three-photon PQR resonance. The width and spectral position of it are equal to the width and position of the Q-branch of the v = 1 ~ 2 transition. However, these positions do not coindde with the most prominent peaks found in the multiphoton absorption of SF6 ¢176.177) and dissociation of C2HsC1 (2°6"), both measured by continuously tunable lasers. These peaks rather coincide with the Q-branches of direct two- and three-photon resonances, according to spectroscopic evidence. (According to ¢qs. (3) and (4), a direct two-photon resonance, e.g. occurs at re1 - x, whereas the PQR resonance is expected at v01 - 2x for the simplest case.) It can be concluded that PQR resonances are only of limited importance. A third type of rotational compensation has been suggested by Platonenko. t27°~ This type exists if the vibrational levels consist of several sublevels. The first three steps are again a three-photon P Q R resonance, as described above. Thereafter a stimulated P or Q branch emission takes place from the populated v = 3 to a sublevel of v = 2, which was not populated directly by absorption. The whole sequence may then start again from this sublevel. This mechanism is considered in more detail below. In several molecules (SF6, UF6, SiF4, OsO4, CC14, BC13 and others) exhibiting multiphoton absorption, the laser excites a degenerate vibration. If these modes were harmonic, they would cause a degeneracy gv of the level v~32°'~ gv =
(v+d1) d- 1
(14)
(d = mode degeneracy). Above v = 1 this formula yields degeneracies higher than required by symmetry. So anharmonicities will split them. In a first approximation the level energies can be regarded as a special case of ¢q. (2): EJhc
= vv + E x,~v~(v~ -
l) + ½ E x~pv~vp
a
(15)
~-¢,#
where ~ and B run over a, b . . . (the components of the degenerate mode), v = ~v~, and x~ = xp~ .... x ~ = xBv. . . . The wave functions are assumed in this approximation to be products of single coordinate functions (symmetrizcd with respect to permutations of ~/~ .... ): Iv) = Slv~)lvB)...
(16)
In an octahedral molecule like SF 6 the a, b, c components arc the cartesian components of the vibration. The splitting of the higher levels means that it makes a difference in energy if the quanta are tither concentrated in one component or distributed over more of them. A different approximation 32°~ transforms from the coordinates a, b, (c) to polar coordinates r, ¢p, (~). The wave functions consist of a radial part multiplied by a cylindrical or spherical harmonic: [v~ = [v,>ll, m r ~ (17)
The importance of spectroscopy for infrared multiphoton excitation
125
The energies are Ev/hC = vv + x v ( v -
1) + g(/2 _ v)
(18)
for the two-dimensional oscillator (d = 2) and EJhc
= vv +
xv(v - 1) + g[l(l + 1) - 2v]
(19)
for d = 3. ! is the quantum number of internal vibrational angular momentum. It can take on the values 1 = v, v - 2 .... 1 or 0. If either x.a >i - x~(>/0)
(20a)
g I> - x (>10)
(20b)
or
the splittingof the multiplets will be larger than theirmean anharmonic shift,i.e.they will extend on both sides of the harmonic position vv. In this case any number of quanta can be absorbed via strongly allowed near resonant transitions.Evidence has been found that eq. (20a) may apply to SiF 4.(266)But even iftbesplittingissmaller than the mean shift,itforms-in cooperation with rotational compcnsation--a very powerful means to support multiphoton excitation.Before explaining this,we should refine the approximations of eq. (15) and eq. (19). In these approximations certain levels stillhave a higher degeneracy than required by symmetry. This degeneracy isremoved by terms in the hamiltonian which are nondiagonal in the representation ofeq. (16) or eq. (17).They are nonzero for statesdifferingby transferof two quanta from one to another mode (V is a parameter): (v~v~ - 2 . . . IH[v~ - 2 v p . . . )
= yx/v~(v~ - 1)vp(v~ - 1)
(21)
and the corresponding matrix elements with the representation ofeq. (17) can be derived by angular momentum theory. Although nondiagonal elements may be very small, they should not be neglected, as they couple states of equal or nearly equal energy. They do not affect the center of gravity of the multiplets,but the total width of the splitting is increased a little bit. In order to check for resonances, a diagram plotting directly the energies is not sufficiently accurate.it is more useful to plot instead of the total energy E, (of levels close to n hvll,,r) only the anharmonic deviation E , - n E t versus the step number n (E 1 = hc v01). This is shown for the v3 states of SF 6 in Fig. 4. The levels are enclosed by the two parabolas predicted by the diagonal approximations eq. (15) and eq. (19), although nondiagonal terms are included in the Fig. Multiples oflaser quantum energies n ht)laser lie on straight lines in such a diagram. For example, the central straight line denoted by 0Q in Fig. 4 represents the CO2 laser P201ine. This lineyields resonances and near resonances for several steps while crossing the band of levels between the parabolas. The Fig. also illustrates the quadratical increase of detuning after the last resonant step. Up to now in this section only nonrotating molecules were considered. If rotation is separable from vibration, a simple way to take rotation into account is to go into a moleculefixed coordinate system. An oscillator in an initial rotational state Jo sees an amplitudemodulated laser field, appearing as the superposition of the laser frequency itself (which even may.be missing for certain orientations) with two sidebands at a distance +2B*J. For a spherical top, e.g., B* ffi B(1 - ~) is the effectiverotational constant. A consequence ofthis fact is the usual observation of P, Q, and R transitions. The two lines called Op and °R in Fig. 4 represent such sidebands of the 944.2 cm- t laser frequency for J0 m 55 for SFe. Unfortunately in this molecule second order Coriolis interaction is very important, i.e. the effective rotational constant strongly depends on the vibrational state ° 5 ° and rotation is not separable from vibration. Although Fig. 4 neglects this effect, it serves to illustrate the powerful possibilities of cooperation between rotational compensation and vibgational multiplet splitting. The initial rotatiomtl state selected in the exampl e~has a Pbranch transition resonant in the first step;
126
W. Fuss and K. L. KOMPA
-~O
OR
0O
0p
..-.
\
"
r
t /
\
/ /
~
"---~"
~
/
/ /
/
/// // L
.... - 150
, .... - 100
,
i
/
/
/ JP //
l
i
L
]
l
i ~
50
~ 50
E n - nE/cm
-lp
/
" .~, ""x~x,
"I
FIG. 4. Energies E, minus the corresponding harmonic energies n 'E 1 o f S F 6 vibrational levels close to n'Eco2, multiples of the CO2 laser quantum energy. For Eco2/hc = 944.2crn-1, the latter lie approximately on the straight line denoted by 0Q. The anharmonic constants of (354)have been used. The straight lines are given by E, - nE 1 = n'(Eco2 - El) - (E~t - E~°t) where the second bracket contains the difference of rotational energies before and after excitation by a pure sequence ofARP,aRQor ARRtransitions, respectively,neglecting second order Coriolis coupling. These lines connect (virtual or real) vibrational states which could be reached by such pure sequences; actual transitions are parallel to them (fat line). Broken lines indicate AR # 0 transitions. The P and R lines actually deviate from straight lines t o lower energyfor increasing energy due to the change ofJ with n, an effectwhich is not noticeable in this scale for SF 6. Decrease of Btff with n (as it is common for many molecules, whereas SF6 behaves more complicated(as4)bends the lines into the opposite direction; this effect has been neglected in the Fig. thereafter follows a n o t h e r stepin the P branch, one stepin the Q b r a n c h (parallel to the line OQ), two R branch transitions (parallel to °R), and so on (fat line). Such a sequence of transitions is similar to the second type o f rotational compensation, i.e. a P Q R sequence for a onedimensional oscillator. The first type of rotational compensation, which takes a c c o u n t of the change o f J with the step n u m b e r n, also has implications for Fig. 4. It leads to a slight deviation (of magnitude B * n ( n 1) o f the 0p and °R lines from straight lines. The third type of rotational compensation, suggesting alternation of absorption and emission steps, is visible in the upper part of the excitation chain (strong solid line in Fig. 4). By this mechanism the molecule is very efficiently excited until it reaches the upper p a r a b o l a and the line OR. Crossing the line Op m a k e s difficulties: The optical field tends to couple only to the internal angular m o m e n t u m i of a degenerate mode, whereas it leaves the rotation R = J - I of the molecular skeleton unaffected; in other words, because of the selection rule AR = 0 . A s i n t h e g r o u n d s t a t e l = 0 , a l l t h e p o p u l a t e d s t a t e s h a v e R = J . T h e m a x i m u m / v a l u e in each multiplet is equal to n. The range of J values which can be populated by the laser therefore spans the range R - n to R + n. An emission step from thelevel n = 8, J = R + 8 in a P b r a n c h (J ~ J + 1, vertical transition in Fig. 4 ) , i s n o t possible since a level n = 7, J = R + 9 does not exist. O n l y the levels within the triangle limited by the lines op a n d °R can be reached by AR = 0 transitions. The selection rule AR = 0 is not exact, since in a vibrational potential of lower than spherical s y m m e t r y I and therefore R are not g o o d q u a n t u m numbers. AR # 0 transitions are p r o b a b l y i m p o r t a n t for transitions b e t w e e n higher states. K n y a z e v e t aL (271) therefore suggested to take them into account. T w o m o r e P, Q, and R branches are possible, the frequencies of which are given by (z71) -
v = vo + 2B(1 - ~)J(AJ + AR(/(1 - ~))
(22)
These transitions are parallel to the b r o k e n lines in Fig. 4, d e ¢ o t e d by ~Rp, ~RQ, XRR. Such transitions are p r o b a b l y i m p o r t a n t for transitions to states above v3 ~, 2 in S F 6 . T h e y permit the construction of a lot m o r e resonant excitation chains. F u r t h e r m o r e these chains are not
The importance of spectroscopyfor infrared multiphoton excitation
127
limited to any triangle like those with A R = 0. So in principle one single but anharmonically split mode can be excited up to dissociation. It is however very probable that the detuning happens at a certain step to be large enough ( > I crn- i) to stop the absorption. Then the molecule has to continue in another channel (see next section). But suppose the light source provides, e.g. by powder broadening, sufficientbandwidth to overcome these detunings and cvcn to cover most part of the anharmonicaUy split spectrum. (The latter is probably not broader than the thermal spectrum at the same energy.(332)Then the single degenerate mode will stay in resonance with the field.That isthe molecule willapproximately behave like alinear absorber, in spite of its anharmonicity and in contrast to narrow band excitation. W h e n the molecular absorption in the firstfew steps isbroad enough, e.g.by virtue of thermal preheating, the cross section of this linear absorber will be identical to its small signal absorption coefficient.Both predictions are verifiedby recent absorption measurements of SF6 employing picosecond pulses of up to 10 GW/crn 2 power density.(3ss)Since pumping ofalinear absorber generates Boltzmann type population distribution,(2s4'296) the success of a thermal model (s'3°3'3°4) for short pulse excitation is not surprising, although the use of thermal absorption coefficients is not appropriate. (349) The suggested alternating absorption and emissions at high intensitiesshould accelerate the molecular rotation to a much higher degree than proposed by.°I 6) However this expectation was not confirmed by a calculation of the excitation of a rotating classical anharmonic oscillator. It has been tacitly assumed up to now, that all the transitions in the excitation chain are allowed. The zig-zag line in Fig. 4 has even been drawn with a width (indicating power broadening) based on transition moments of the form #,,_ 1,,, = #oxv"n
(23)
Actually the transition moments worked out by (32L322) for a rotating spherical top roughly agree with eq. (23). Only the A~s levels require special consideration. For these levels R = J. As long as AR = 0 is postulated, these levels are not optically connected to the (A~) ground state unless the resultant change in J is zero. For example the A~g level of SF 6 at E 2 - 2E1 = - 7 crn- 1 cannot be pumped by a P- and an ensuing Q-branch transition which would be in resonance with the CO 2 laser P24 line. If however after the P branch transitions to the v3 = 1-state a collision causes a rotational relaxation restoring R = J, this bottleneck is removed. This peculiarity complies with the observation (16s) that the absorption cross section in this wavelength region is much more pressure sensitive than for shorter wavelengths. See also Section 8. Up to this point it has been shown that for certain excitation frequencies and for certain initial rotational states, the anharmonic splitting in cooperation with rotational compensation provides excitation channels up to fairly high energies. In other cases, such resonant chains arc not possible and one has to postulate contributions by other mechanisms, such as absorption by other modes (Section 4), direct multiphoton excitation (Section 7), absorption far in the wings of a band (Section 5), and intramolecular vibrational relaxation (Section 6); although the role of the latter two contributions has probably been overestimated in the past (Sections 5 and 6). Another case in which selection rules require special considerations is COS which has been dissociated by the CO2 laser. (9) The CO 2 laser is resonant with the first overtone of the doubly degenerate bending vibration v2. Figure 5 shows the higher states of interest. The series of levels with l = v2 - 2 only has a small anharmonicity, whereas the ! = 0 series (v2 even) is more anharmonic because of a Fermi interaction with the states (0, v2 - 2,1). The only levels which are reasonably in resonance with the employed CO2 laser frequencies arc the l = v2 - 2 states (v2 even). However, these states can only be populated by Al = 2 transitions, which are forbidden by the Al = 0, + 1 selection rule. Direct multiphoton transitions to these states are not possible either, since the selection rules for them are the same as for the stepwise multiphoton transition. Only in a very high order of approximation, these nearly coincident transitions can acquire intensity.(323) Their cross section will be much smaller than the 10- 20 cm ~ found for the v2 = 0 --* 2 transition (Fig. 1 ). But then the molecule will have difficulties to absorb sufficient photons at 60 J/era-2 ( = 3 x 1021 photons/era 2) to
128
W. Fuss a n d K. L. KOMPA I
I
I
n
OCS O
n 5 -00--
i -I00
X~ ~ N ,
P
'~0(~n_2)01 /~" I ~
02n60 •o~
, 00~'~./ -50
I
• experimental o extrapolated
F n - n . 1047.1 c m "1
02nI'0 0 2nZ0 o~ ~o~
i SO
I I O O c m -1
FIG. 5. Vibrational energies E, modulo E 1 of COS vibrational states close to n • Eco~. The lines P, Q and R arc similar to Fig. 4; excitation is at 1 0 4 3 . 2 c m - ' . Spectroscopic data from. '323}
generate the 60% dissociation observed. Maybe direct multiphoton absorption to an unknown higher state plays a role. This would be a bottleneck which may be overcome only by high power densities. Below these powers the absorbed energy should temporarily leveloff. This has in fact bern found. (l°a) Consistent with a highly nonlinear absorption is also the very steep rise of the dissociation probability from 0 to close to I in a narrow range of laser fluence. (a~On the other hand, recent experiments with picosecond CO2 laser pulses failed to dissociate COS. (360) 4. N E A R L Y
DEGENERATE
MODES
The existence of multipiets, connected by strongly allowed transitions, is not confined to degenerate normal coordinates. Nearly degenerate modes generate a similar level structure. The principal difference is that already the first excited level is split, e.g. because of lack of symmetry. Thus ethyl chloride, C2H5C1, has five infrared active CH stretch vibrations between 2900 and 3010cm-', (2tJ which may be correlated with the partially degenerate vibrations a,~, e,, eg and a2, of C2H 6. The multiphoton excitation of C2H5C1 around 3000 cm- ~has been assigned to detailed sublevels of such multiplet bands, which in turn were studied by overtone spectroscopy. (2') Due to the cross anharmonicities, and due to the differences of the fundamental frequencies the energetic spread of the multiplets increase with the excitation energy, similarly as for degenerate modes. Differences of the diagonal anharmonicities also contribute to this spread. This contribution may be positive or negative. If the highest frequency modes have the largest (negative) anharmonicities, it may thus happen that the width of the multiplet bands first increase, but decrease again for higher excitation energies. Such a case is the CH stretch vibration of benzene. Their spectral width is 90 c m - t in the region of the fundamental (~3000cm-1); it increases up to the 4th overtone (~14000cm -1) to 109cm-', but decreases again to 90 cm- ' in the 6th overtone ( ~ 16550 cm- ').(324) On the other hand, the width of these overtone bands has been widely interpreted (324-33~} as a lifetime broadening due to a supposed internal relaxation (see Section 6). Decreasing band spread is of course unfavorable for multiphoton excitation, and no successful excitation of such a molecule is known to date. In this context the l o c a l {317) mode representation should be mentioned. It describes excited states as a linear combination of states ofanharmonic local oscillators (e.g. the six CH stretch oscillators of benzene). Normal mode splitting is introduced by offdiagonal terms in the Hamiltonian. If this splitting is small compared to anharmonic corrections (as for the benzene overtones), the local mode representation is more useful than the normal mode representation. In the latter picture, anharmonicity often yields large offdiagonal terms in the Hamiltonian. The states described by both bases are, of course, identieal. Table 3 compiles more examples of molecules which have boon dissociated by multiphoton excitation and which have closely spaced vibrations. If infrared active combination vibrations and overtones are included, as is done in the table, it seems to be wide-spread thata new mode can come into resonance just at the time when the mode first excited shifts out of resonance. The whole idea shall be described in more detail for S F 6 .
The importance of spectroscopy for infrared multiphoton excitation
129
TABLE 3. Infrared bands close to multiphoton excitation frequencies
Molecule
excited at (era-a)
0 3
1037.4 (9)
CF3Br
1040-1082.3 (46-*91
CFaI
1028.1-1082.3 ~46'49-621
CF2CI 2
920.8-931.0 ° t ~ l ) 1081.1-1080.0
CrO2Cl 2
969-984 (651
CH3CN
933.0, 1046.9 (es-911
N2F 4
944.2, 975.9 (31.'113'1141
CH aN-H 2
1031.5-1070.5 (HS"t18-12°)
IR bands (era-t) and relative intensities 1042.1 o351 1103.2 1084 (1)°33'3361 1095 (0.1) 1119.6 (0.1) 1028.1 (0.1)°35,337) 1075.2 (1) 1081.0 (0.1) 882 (0.3)O3s) 992 (1)
1102 (1) 990 single band(631 10OO 920 (11(336) 1041 (1) 933 to 103113391 (7 strong bands) 931 (0.t) (33e) 945 (0.1) 1044 (1)
114o (o.3) SFsCI
904.9, 914.9 (t26,t271
SF 6
930-1056 (51
909 (34°1 947 1029 948 (11(333) 992 (0.01)
114o (o.ool) C2HsCi
973.3 (206) 2850-3050 (211
C2F 6
1087.9 (21°)
CFsCOCF 3
944.2-983.3 ~223-2291
SFsNF 2
908.5-957.8 (22.)
S2FI o
920.8_944.2 (t ~s)
974 (2 modes) (1)121'3351 1081 (0.116) 2880 to 3013 (5 strong bands) 1025 (0.005)O361 1063 (0.OO5) 1117 (1) 971 (1)(336) 1027 (0.05) 1070 (0.05) 885 (1) (2241 910 (1) 950 (1)
and several less intense bands 909 938 960 970
For simplification assume that all anharmonic shifts are only energy dependent, no matter over which modes this energy is distributed. Mathematically this means for the v 3 constants of S F 6 x ai oc vi
(24)
Such a model has been suggested for SF6 (3321 and only slight deviations from it were found in. (3331 Deviations up to a factor of 2 from eq. (24) will not change the principle of the following. Equation (24) implies that when in the v 3 v 3 multiplet of states the v a quanta are successively replaced by v 2 + v6 quanta (which have an energy higher by only 45 crn- 1), the new multiplets (v 3 - v2)v3 + v2 (v2 + %) are generated. They are equidistant from each other (distance 45 era-t); each o f them is displaout from its corresponding harmonic position by equal amounts. Such bands,are shown in Fig. 6. Obviously at the time (or even before) the molecule'reaches the limit of one band of levels, it can cross over to ,the next one; that is a v 2 + v6 transition has moved i n t o resonanee, when the v a transitions have shifted out of resonance. Within the newband, v 3 transitiom are again possible (Fig. 6), until these again become detuned. Obviously increase of v 2 and v6 switches the v 3 absorption back from long
130
W. Fuss and K. L. KOMPA f
°
[
i
,
,
......
. . . . . .
f
I
i
,
f
J
]
i
,
J
l
I
f
l
,
,
I
l
,
\ \."e.,, \ .,,-."....'./, \ \ "z. \ \>\"
\.I
N ........ \ -I00
-SO E n
- n E I/era
(3
r
l
....
L -150
'
50
-I
FIG. 6. Vibrational levels of the v3 ladder and a few ladders containing v2 + v~ quanta. Fat line connects populated vibrational states. Broken line arrows indicate coincident absorptions and coincident emissions described in, (~sT) which were interpreted as evidence for the shown excitation path.(~s~)
to short wavelengths. Such a nonmonotonous shift of the "resonant mode" could not have been anticipated by consideration of spectra alone, but demonstrates the importance of level schemes, as was already pointed out in Section 2. By an excitation path like in Fig. 6 it was possible to explain (187) the conspicuous short wavelength emission and the constant position of the v2 + v6 absorption observed (~sm when SF 6 was pumped by 944 cm-1 radiation. The idea was based on the equidistance of pumped rotational-vibrational levels and on the equidistance of bands like in Fig. 6. Thus certain (rotational-vibrational) v2 + v6 transitions superimpose at a constant distance from the pump frequency. In this way they have a good chance to be observed within a region seriously congested by absorptions and emissions. On the basis of the observed spectra it was estimated in, (ls7) that this mechanism--alternate absorption of a few v3 quanta and one v2 + v6 q u a n t u m - - h o l d s to at least the twentieth step. To reach levels of this or even higher energy, Platonenko's rotational compensation (Section 3) certainly contributes; that is, intermediate emission steps (Fig. 6) convert the energy difference of vibrational sublevels into rotational energy. While Fig. 6 is drawn for 944.2 crn-1 ~xcitation, similar Figs. will account for every frequency to the red side of the Q branch (948 crn- ~). But for shorter wavelengths the difficulty is that the v3-band will be detuned before the v2 + v6 transitions come into resonance. However, Fig. 6 is incomplete as certain sublevels ofv3v3 + v2 + v6 have been omitted which are only weakly optically coupled to the v3v3 levelsJ 1ST) They may be responsible for shorter wavelength excitation. An indication of such a level has been found in (~sT) by overtone spectroscopy. After several steps at short wavelength, the next combination vibration v4 + v5 (Table 3) may also come into resonance. This would explain the v4 emission observed in "92) and (334). The emissions in (~9/) however, are probably already strongly affected by collisional redistribution. Another interpretation" s7.334) pointed to the levels v3v3 + 2v4 + 2v6, which are slightly higher than (v3 + 2)v3. But these levels can only be reached by XilAvi[ = 5 transitions which will be very weak. It has to be noted that the excitation mechanism suggested in this section only employs the two strongest transitions in the 950cm- t region. Assuming that every step is sufficiently well in resonance (see next section), the rate determining step will be the v3v3 -~ v3v3 + v2 + v6 transitions. Their first member (v3 = 0) is 100 times weaker than the v3 band. Increased intensity borrowing by Fermi resonance for higher v3 tends to level offthis difference to some extent. Imagine a molecule where the corresponding pair of transitions has comparable oscillator strength from the beginning, then the absorption cross section would not drop, but remain at its small signal value. Such a molecule, though anharmonlc, would behave in absorption like a harmonic oscillator up to an energy where also the second mode shifts out of
The importance of spectroscopy for infrared multiphoton excitation
131
resonance. Figure 2 shows, along with the more typical case ofSF6, that for several molecules the multiphoton absorption probability is not much smaller than the spectroscopic absorption cross section. In fact for CF3I, SFsC1, SFsNF2, CrO2C12 and other molecules (Table 3) adjacent transitions have similar strengths. To explain the ease of excitation of such molecules, the quasicontinuum (see next section) has been invoked. (1~s) But every broadening mechanism has to reduce the cross section to below its small-signal value. It may be of interest to discuss in this context the possibility of mode selectivity. Obviously in the suggested mechanism the v3 and the v2 + v6 mode contribute; six degrees of freedom (part of them are degenerate) have been used to draw Fig. 6. On the other hand, the degeneracy of states actually populated only varies between 1 and 3; that is, it is nearly independent of the excitation energy: g. ~c E °
(25)
This is similar to a classical s-dimensional oscillator g, oc E~,- 1
(26)
with s = 1. (Quantum mechanics replaces eq. (26) by a factorial function, (32°) a detail which is unimportant here.) A higher s would apply if either the excitation path branches again and again, or if more and more background states become optically connected by intensity borrowing (next section) to the "primary states" of Fig. 6. A small s has been postulated for SF6, (4'2s3'2s4) hexafluoroacetone (225) and tetrafluoro-l.3-dithietane (21s) on the basis of an analysis of absorption and dissociation data by means of a rate equations model. Such an analysis predicts a population of the energy levels, which is a product of a statistical weight (as eq. (26) for example) times a population per state. (284) The latter can be deduced from the dependence of absorbed on incident energy. A clearcut ease is a linear absorber like C2F4S2;(215) in this ease, incoherent optical absorption always produces a Boltzmann distribution of population, (284.296)independent of complications like rotational hole burning in the initial state. In contrast to the absorbed energy, the dissociation depends strongly on the weight function g,. Large s implies a function of dissociation probability on laser energy which is steeper than found experimentally for molecules like SF6 ('t'283'284) C F 3 C O C F 3 (225), C2F4S2(2 ! 5), as well as for C F 3I, S2F 10, SFsNF2 and SFsC1. On the other hand the difference is not very large. By a similar analysis (based however only on a single set ofdatalike dissociation or quantum yield as a function of incident or absorbed energy) other authors(285,288,297,304,305a) therefore found consistency with experiment when they assumed the maximum possible s. This assumption will certainly be correct for levels close to the dissociation energy, where the energy can freely flow between degenerate states (Section 6). So there will be no chemical consequence (mode selective excitation or not) if in the preceeding steps all or not all the degenerate states are populated. The principal effect is the energetic width of the population distribution: For large s it is very narrow, whereas for small s it is broad. (2s4) 5. THE WIDTH OF THE RESONANCES, INTENSITY AND FLUENCE DEPENDENCE
Up to this point the multiphoton excitation was discussed in terms of sharp states and transitions. It is now necessary to discuss power broadening at this point. For SFe at 50MW/(~rl 2 this broadening is already (~(v~-l)-.v~) ~/hc ~ ~ 3 - 1 c r n -1, whereas the average distance of v3 transitions inferred from Fig. 4 is about 3 cm- 1. But if only power broadening compensates for detuning, the whole process would predominantly depend on power density. This is in contrast to what has been observed for SF6: Nearly identical dissociation yields and absorbed energies have been measured for this molecule when CO2 laser pulses of very different pulse lengths and shapes but equal energy fluence (time integrated intensity) were employed.(!53-156)The most popular model of multiphoton excitation therefore asstunes in the higher steps a dense Cquasicontinuous') spectrum, giving rise to fluence dependence, whereas isolated resonances and coherent excitation are assumed for the lower steps.
132
W. Fuss and K. L. KOMPA
The coherent steps can be quantitatively described by adding to the Sch6dinger equation of the molecule
- ih~-~~
=/./~o1~
(27)
the (dipole) interaction with the time dependent field/~(t)
-ih~o
= (/./tool + k£(t))(p
(28)
When the rotating-wave approximation (which is not equally applicable for resonance and nonresonance t287~) is adopted, a time independent part can be separated: ei~oi = ¢
(I'l~o°l+ ~oEo)~oj --*
t
(29)
where Eo is the field amplitude and fl~j are the transition moments between the molecular states i andj. Detailed derivations are given in textbooks. °42~ Solution ofeq. (29) reduces to the problem of finding the eigenfunctions q~ (the so-called dressed states) and eigenvalues e~ (the so-called quasi-energies) of the matrix in eq. (29). If the molecular energies/./mot and transition moments #o are available, one can easily calculate the time evolution of a system of 100 levels, for example. (This number is not so large in view of the large number of rotational-vibrational sublevels.) The method (and another one based on the nearly equivalent Bloch equations) has repeatedly been applied to multiphoton excitation, with various assumptions on the molecular parameters, t246-262.2a°'2sl~ It is important to note that reliable predictions are also possible and have been made on direct two- or more-photon excitation t2*6-2s6~ and on coherent propagation effects, cat s~ If the calculations are done in terms of density matrices, relaxation effects can also be included, t3°°-3°2~ Recently another quantum mechanical method has been suggested which, although based on perturbation theory, should be capable of any desired accuracy, t269) For systems of isolated levels these calculations yield oscillations of populations, a result which is typical for a coherent excitation. The oscillation frequencies depend on detunings and transition moments. If the laser excites many closely spaced sublevels or many levels with random dipole moment, their populations will oscillate at different frequencies. When added together, the oscillations will disappear in the sums, and these coarse-grained populations will only smoothly change. Such behavior is typical of incoherent excitation. Similarly, if the laser excites molecules in several different (thermally populated) initial states and when the measurement only concerns their average (like the absorbed energy or the dissociation yield), the excitation will also appear incoherent. From one of these conditions or a combination of them, rate equations can be derived t2s~'29t-293~ and have been used to analyze multiphoton excitation, t*'2sa~2as'2aa-29°'294-3°°~ If only step-by-step transitions are considered, they write ~2s,~ dN~ d~-
I~ N~-
g~ N,+ 1 +Ie~_ 1 N~_ 1 g~+l
-~ N~
(30)
which can easily be solved for the populations N~ of the levels of energy ehvma~e~, if the absorption cross-sections tr~ and the degeneracies g~ ( = number of states in level e taking part) are known. Division ofeq. (30) by the laser intensity I (t) and substitution ofd~ for I dt shows that eq. (30) does not depend on pulse shape or length, but only on the energy per c n ' l 2 (fluence) tb defined by ~ =
f2
I(t)dt
(31)
The solutions of two special cases t2s4~ are worth noting: (1) if ~ = %e + const. (as for a harmonic oscillator), N, e g, exp( - e/~. q~)
(32)
The importance of spectroscopy~forinfrared multiphoton excitation
133
which is a (pointwise) Bottzmann, distribution. (Remember that n i~ integer.) The absorbed energy is (~) oc Oo~
(33)
2) Ifo~ = ~ro, a Cranssian distribution results N~ • g~exp( - ~2/200 ~ )
(34)
The absorbed energy is
In both cases, the dissociation probability P is an Arrhenius-like function of P ~ exp(-@o/@)
(36)
(@o is a parameter), if g, does not much vary; it rises more steeply with • if g~ increases strongly with ~. The latter will happen if all the modes of a molecule contribute to the absorption. Terms describing dissociation.rates and collisional or collisionless deactivation rates may be added to the fight hand side ofeq. (30): (4,285,~ss-290,294--298) The transformation from a time to afluence derivative isthen not possible any longer, and a certain intensity dependence is generated ~2s9,29°~ although the dissociation yield again becomes fluence dependent when for very high excitation a steady state is reached by the competition between pumping and dissociation. ~zsg) It has been shown on this basis, that pumping rates can be derived by measuring how quickly P (~)converges to 132s9~Another intensity dependence is introduced by the assumption that g~ (in particular go, the number of rotational states removed from the vibrational ground state)depends on the power broadened bandwidth ( ~ / ] ) . However, the more logical way to treat such effects is to combine the coherent description for the lower levels with the rate equations for the higher ones. ~2s°'2sl~ Such a calculation also reveals a strong intensity dependence ~es°~whicharises if the laser frequency hits at least one transition only far in the wing of its power broadened width ("intermediate level bottleneck"). Passing such a bottleneck has been called "leakage" intcCthe quasiconfinuum. ~'62~To overcome this, a certain minimum intensity is required. This is in contrast to the 11/2 dependence expected at low ! for the number of rotational states removed from the ground state (the "rotational bottleneck "~343~ and to the approximate I - 1 dependence of dissociation yield expected at high intensity on account of the finite rates of dissociation and internal energy redistribution. By varying the p~se length the question has been examined for several molecules to which extent the reaction probabilities depend on energy fluence or on intensity. The most extreme examples are the molecular ions C3F ~- and [(CeH5)20]2 H+, 100% of which dissociate within seconds when irradiated by a continuous-wave CO 2 laser at intensities as low as a few watts/era232 ~9,e20~If the excitation would reallybe collisionless, the detuning of every step in the ladder had to be smaller than the estimated power broadening of 10 -5 cm-~ for every molecule. However, the promoting effect of the background gas was shown in322°~ Furthermore the ions have been generated in high vibrational states by the electron impact ion source. The dissociation of SF 6 has been investigated using various pulse lengths cl53,15,t~and shapes, ~1s s, t 56~ as mentioned above. The energy absorptions have also been measured by these pulses. ~163'r6~) :Far from the dissociation threshold the absorbed energy is much larger for the higher intensities. But the difference is still of a magnitude expected for rotational hole burning (Section 7). But even at average energies of 35 quanta/molecule the absorption keeps increasing with intensity. ~3s s) This has been ascribed to the presence ofnarrow spectral features at these high levels of energy~3s s) (see also Section 3). A larger difference has been,found for dissociation of CF3I close to the threshold335) An intermediate |evel bottleneck, seems to play a role. Other examples of such a bottleneck are CHCIF2 ~ ~'3"~and OsO~.~73)In other molecules where strong intensity effectshave been found up to now (C2 H~CI,~ ~0 ~~ H z C O t22'2 s)) the rotational structure of the first absorption step is so wide that the rotational h o l e b u r n i n g will dominate in dissociation and absorption
134
W. FUSS and K. L. KOMPA
measurements. No example of an inverseintensity dependence ofabsorption or dissociation is known. Although it is not established for many molecules, it is generally believed that multiphoton excitation depends predominantly on energy fluence and that intensity effects are more or less small corrections to this limiting case. It has been the merit of the quasicontinuum model to predict this fluence dependence. An early version ~344~of this model suggested that the first few steps occurred via rotational compensation, whereas in the later steps the molecules have a large number of densely spaced Cquasicontinuous") absorption lines which are distributed over such a large spectral range that there is always resonance. The idea ofa quasicontinuous broad spectrum was inferred from the very large number of vibrational combination states at high energies, i.e. the number N of combinations of vibrational quantum numbers by which a given energy E can be realized :~345a~ N(E)=
~[ (v i + d i - l ) ! vi~-_--l~ "
~,
(37)
~,vihvi=E i = 1
(d~ = mode degeneracy, s = E d~) Approximately N oc (E + a) ~- 1, where a is a fitting parameter, t34s~ The broad spectral quasicontinuum eliminates the need to study spectra of highly excited molecules. It has been recognized, ca) however, that selection rules, even approximate ones, tend to concentrate the oscillator strength within a spectral range of a few cm-1 width only. Thus the v3 band halfwidth of S F 6 heated to 1780 K (= 15000 cm-1 average vibrational energy)is 42 cm- 1 o n l y , (34"3) whereas the calculated width of the rotational envelope at 1780 K is 24 cm. That is, the vibrational contribution to the width is small. Similar conclusions result from the often narrow IR spectra oflarge molecules at room temperature (see Section 1 and Fig. 1) and from the sharp ( ~ 5 cm-1) structures observed in double resonance experiments ~ls6J where the s p e c t r u m of S F 6 pumped by a CO z laser was measured. The effectiveness of nonrigorous selection rules has also long been known in electronic absorption or emission spectroscopy, where transitions to very high vibrational states ( ~ 30000 cm- 1) are neither broadened nor congested by other modes, e.g. P F a, or the acetylenes,taz°b~ However, the small halfwidth does not exclude the existence of far reaching wings of the absorption bands of highly excited molecules, tzs4~ These wings as well as the (possibly quasicontinuous) substructure of the bands can best be described in terms of intensity borrowing: Start from a (not necessarily harmonic) zero-order basis in which oscillator strength is localized in a few transitions only. Consider an infrared inactive transition ~,o -~ ¢~ which is close (with energy distance AE) to an active transition ¢o '-* ¢°. The anharmonic potential terms in the Hamiltonian will in general admix a little bit of ¢° to ¢,~, the admixture coefficient co~ being c.~ oc H . J A E
(38)
in perturbation theory. The transition probability for ~'o ~ ( ~ + c°i~'°) is then #2 oc c°2
(39)
The AE denominator ofeq. (38) is responsible for the clustering of transition probability in a narrow spectral range. The interaction matrix element can be extremely small. As a rule of t h u m b ~32°a) it drops exponentially with increasing total difference Av Av = ~
Iva - v~l
(40)
modes
of the quantum numbers v° and vi of the zero-order harmonic states ~. and ~t. As a consequence, the optical excitation tends to use only a small subspace of the vibrational degrees of freedom. It has, however, been pointed out ~6°'75") that states differing by a large Av can also be mixed by a sequence of intermediate steps coupled in lowest order. In this case the simple relation (38) has to be replaced by an eigenvalue problem. In this way more transitions can acquire intensity than predicted by the consideration above, although they will still cluster
The importance of spectroscopyfor infrared multiphoton excitation
135
in narrow energy ranges. Further investigations are necessary to clarify whether the density of spectral lines is equal to the density of states, or whether they differ by orders of magnitude. Thus it is conceivable that thenumber of spectral lines per cm -1 is much smaller than the number of states per cm- I which is e,g. around 109 cm- t for SF6 at 20000 era- 1. It should be pointed out,in this context that a few lines within the power broadened bandwidth ofthe laser (often ,~ 1 cm ~-1) are sufficient to make the excitation incoherent (fluence dependent). (2sa) This number may be even smaller if a random distribution of the transition moment is taken into account.(291.292) In the next section evidence is presented that not the full number of states, available at each energy, is populated. This will also be evidence for a sparse spectral line density. 6. COLLISIONLESS VIBRATIONAL RELAXATION In a molecular beam experiment it has been found that the multiphoton dissociation of SF 6 does not release any kinetic energy into the fragments. (t49' 150)This fact primarily proves that there is no potential barrier in the dissociation channel. Most importantly, however, it proves that before dissociation the energy can freely flow around among the vibrational degrees of freedom :(I so) I~ instead, the energy would be localized, a nonnegligible fraction of the energy of the last absorbed quantum should appear as kinetic energy. Vibrational energy redistribution on a time scale of the order of a vibrational period is the basic assumption of modem unimolecular reaction theory (RRKM theory). °45) Subsequently many multiphoton dissociation experiments have been found to be consistent with this theory. (6) Such collisionless vibrational redistribution can occur under two conditions. Firstly, both the energy and the anharmonidty have to be large enough. A quantitative criterion does not exist. It is helpful, however, to investigate the question by classical calculations of trajeetories on an anharmonic potential sLLrface.(273-2~5) These trajectories become (locally) unstable (i.e. change the mode parentage) in certain high energy regions of the potential. (2~5) Furthermore an energy threshold seems to exist above which the trajectories very rapidly lose their memories for the initial conditions and above which they seem to become ergodic. (273) On the basis of such classical calculations it has been suggested as a rule of thumb that (apart from a few narrow ranges of configuration space) vibrational redistribution can generally occur above about two thirds of the dissociation energy (34s) although it may be slow even above dissociation threshold in less common cases like van der Waals molecules. However, the possibility of coUisionless vibrational energy flow is seriously restricted by the second ~condition, which is based on energy conservation. It allows redistribution only between states of (within the uncertainty principle) exactly equal energy. (24s) This uncertainty width is (in the absence of collisions) equal to the reciprocal of the radiative lifetime of a few tens of milliseconds usually, that is ~ 10-9 cn]-1. (Note that this natural width is 6 orders of magnitude smaller than when electronic excitation is involved.) Only a density of states much larger than 109/era -I will provide su~cient acceptor states for redistribution. (24s) This number is reached for SF 6 above 20000cm -1 ( = 6 0 ~ of the dissociation energy), but not reached below dissociation threshold, e.g, neither for ethylene nor vinyl chloride, not to speak of the triatomics. So for molecules up to the size of SF 6 collisionless energy redistribution cannot occur during most of the excitation steps, although it does occur (shortly below and) above the dissociation energy. This statement only refers to narrow bandwidth excitation. Quantum mechanically, temporal evolution results from superposition of states of different energy, and the narrowest bandwidth of states to be superimposed is the natural width. Larger widths, and therefore faster temporal evolutions, are generated by external means like power broadening or the transform limited width of short pulses. Generally only those states exchange population with each other which are directly optically excited. That is, a probe 0ike the pump laser itself) which monitors the population of the whole set of states together will not see any temporal evolution after the end of pumping, whereas a narrow-band source can see it. In agreement with this expeetation, "collisiouless relaxation" of IR absorption of pumped SF 6
136
W. Fuss and K. L. KOMPA
has been seen (~s4~ when monitored by a cw CO 2 laser (< 1 MHz width) but has not been found when probed by a TEA CO 2 laser (~1 GHz width).(~st) In the case of transform limited pulses, the temporal evolution (of sublevels) will be the faster, the shorter ( = larger bandwidth) the pump pulse; a probe pulse of equal length, however, will not see any temporal evolution apart from the one associated with absorption or emission. Such a case applies to an experiment~s7"~ in which the transmission of SF 6 pumped by a picosecond CO 2 laser pulse was probed by a delayed pulse of the same kind; the transmission returned to its initial value already during the pump pulse. This temporal evolution must be caused by excitation and subsequent deexcitation similar to adiabatic fast passage, (187") but not by relaxation among sublevels which was suggested by (lS7a) as the preferred alternative. The same reason will have caused the restoration of absorption of a (narrow-band) cw laser probe after pumping SF 6 by nanosecond pulses. (lss'xs9~ Since the energy can only flow between states which are directly populated by the excitation laser, the number of contributing states is again governed by the density of spectral lines, which can be very much smaller than the density of states (Section 5). Furthermore the number of lines excited via power broadening should not be overestimated: When the spectral intensity is distributed over more and more lines, the individual transition moments (and the power broadened widths associated with them) decrease. It should also be noted that the term power broadening only concerns the spectral response to the pump; the states are not broadened but remain sharp as before (apart from two sidebands of each state due to modulation splitting), a fact which can be demonstrated by a weak tunable probe. Nevertheless many published models assume collisionless vibrational redistribution already in low lying levels.3'299-3°5a) Reduced density matrix techniques(3°°-3°2~ or rate equations (299'3°5~) are used to describe the process quantitatively. Usually it is not explicitly stated why coltisionless relaxation is postulated and why it should support the excitation. Some models seem to invoke the broadening by spectral redistribution of oscillator strength. Others apparently assume that the anharmonic shift only depends on the quantum number of the "driven mode" (v3 in SF6), which is lowered by relaxation. As it was pointed out in,¢3) the probability distribution of this quantum number would be exponential for a microcanonlcal ensemble (i.e. for equal population of states of a given energy), if the number of degrees of freedom would be large enough. Therefore it was suggested(3) that the absorption spectrum of each level with the energy E should be (nearly) identical to the spectrum of a canonical ("thermal") ensemble of molecules with the average energy E. This assumption was quantitatively analyzed by means of rate equations for excitation of S F 6 .(34.9) Gross disagreement with absorption measurements below 10 quanta/molecule was found; especially meaningful was the comparison C3~9~with picosecond absorption, because in this case all rotational states are excited and no ambiguity is left about the width of a rotational hole. In the same reference (349) the mierocanonical assumption was also checked: Thermal spectra of SF 6 were considered as a superposition of spectra ~r~ of the energy levels E. Deconvolution from the Boltzmann distribution yielded the individual ~rE which were then inserted into the rate equations (Section 5). Again gross deviations from measurements of collisionless multiphoton absorption were found. Presence of collisions improved the agreement, as expected. A further evidence against microcanonical distributions are the spectra of highly excited SF6 .~186) The narrow structures found therein do not comply with the broad bands postulated because of the distribution ofanharmonic shifts. By probing the U V absorption of OsO 4 it was found, that the (UV active) iJ1 mode was excited only after the laser pulse which pumped the (IR active) v3 mode; (TS) unfortunately, the absorbed energy was not given. As explained in Section 4 and in (334) (by the v2 + v6 mode of SF6), emission of a mode different from the pumped one,¢192'334) does not prove equlpartion. Stimulated emission as observed in the v,t region of SF 6 when v3 was pumpe d{33o) even proves the existence of narrow spectral features: if everything would be broad, stimulated emission would be superimposed and quenched by v4 absorption of more strongly populated lower states. It should also be noted that, if the molecule had to wait for a relaxation process in an early
The importance of spectroscopyfor infrared multiphoton excitation
137
step, the absorption should be least effective for the shortest pulses (of equal energy). The absorbed energy q should approximately be inversely proportional to intensity I,
qoc1-1 Thereby even the increased hole burning by power broadening (oc 11/2)should be overcome. But an increase of absorbed energy and dissociation yield with I has always been observed. ~s) 7. ROTATIONAL HOLE BURNING, DIRECT TWO-PHOTON ABSORPTION, TWO-LASER EXPERIMENTS, AND TEMPERATURE EFFECTS At room temperature many rotational states are populated. Rotation splits each vibrational transition into three lines or less, but not more. Any additional (rotational) splitting, e.g. by centrifugal distortion and Coriolis effects or by the hyperfine structure, is due to molecules in different initial states. Nearly each absorption line corresponds to a different molecule, and the rotational structure is therefore basically inhomogeneous. That is, excitation by a (e.g. power broadened) bandwidth smaller than the width of one rotational branch will burn a hole into this branch. The fraction of molecules removed from the ground state is approximately equal to the ratio of these two widths. If the first step is within the (narrow) Q branch, this fraction is high, but may be reduced in the next step which may employ an R branch, for example. Repeatedly attempts have been made to determine quantitatively the extent of rotational hole burning, tS'53"Ss'66~ It is known that the dissociation probability of several compounds (e.g. SF6, {133}CF3I,<53}$2F10,~1~s) CF3COCF3~22s)) reaches = 80 ~o already at a modest flux, whereas on the basis of the ratio of 21~E/hc (power broadening at the corresponding intensity) to the rotational width only less than half this value would have been predicted. Other evidence at lower flux~s3.Ss'66) though less direct, also indicates that more molecules are excited than expected. An almost 100 ~o depletion of rotational states was concluded from three other experiments. {11,xs6.190) The first one of them found an absorption independent of pulse length which was varied from the collisionless to collision dominated regime) 111) Collisions should have filled up the hole and increased the absorption. But as pointed out by,{5) the smaller bandwidth of the longer pulses counteracts this increase and may compensate it. An additional compensating effect is due to the change of the spectrum in the higher steps, as explained in the next section. Other experiments {1s6. t 9o) found an unexpected strong bleaching of absorption of rotational states of SF 6 when the molecule was pumped by various other frequencies. However, according to a later interpretation <~sT~this bleaching was only feigned by emissions from higher populated levels to unpopulated sublevels. A few collisions seem to suffice to equalize these sublevel populations, thus quenching the emissions, whereas the rotational quantum number changes by a few units only ( ~ 1 per collision). Thus the rotational absorption hole survives nearly unchanged, {1s7) and the "spectra of pumped SF 6 after a few collisions"{~s6) present a visualization of this hole. To summarize, it remains to be explained why about a factor of two more rotational states respond to the laser radiation than expected on the basis of power broadening. Most probably the cause lies in direct two-photon transitions. ~ls6,26s) Their spectrum is shifted by virtue of anharmonicity to longer wavelengths and contains rotational branches with AJ = __.2, ___1 and 0 (and perhaps also some with AR ~ 0~271)).In general, it partially overlaps the one-photon spectrum. Thus additional rotational states can be excited. The importance of direct multiphoton transitions has been discussed and confirmed in terms of the Schr6dinger equation.{2,~-2s 6}The most direct experimental evidence is the observation that the excitation probability of molecules like boron trichloride <19a>or hexafluoroacetone ~22s)has its maximum well outside the range of the linear (one-photon) spectrum. Furthermore, two- and threephoton resonances have been found in the form of sharp spectral peaks around 3000 cm- 1 in the dissociation probability of C2HsC1.~2°6") At room temperature only the small fraction
J~>QE 7 : 2 - E
138
w. Fuss and K. L. KOMPA
(Qvib is the vibrational partition function) of the molecules is in the vibrational ground state. It decreases with s, the number of modes, especially if they have low frequencies. The other molecules are in low lying excited states. The transitions starting from those states ("hot bands") are slightly shifted due to anharmonicity, but are otherwise similar to the fundamental band. It is appropriate to return again here to Fig. 1 which compares the room temperature absorption spectra of a small and a large molecule. The larger one has an average vibrational energy of 1500 crn- 1. At this energy the density of states is approximately 700/cm-1. Therefore molecules of this size have been said to "be in the quasicontinuum already at room temperature. ''ta'l ~8) As was pointed out before, however, the spectra in Fig. 1 do not look very different with regard to their absorption width. No high resolution I R spectrum of a highly energized molecule is known. But probably each rotational state will have several vibrational transitions, since close lying states in the upper level will mix and thereby exchange spectral intensity. Thus the homogeneity of the spectrum is increased and hole burning effects are reduced. Complete homogeneity of room temperature S2F 10 has been invoked in a 78) because of the absence of any pressure effect on absorption up to 5 quanta/molecule. On the other hand at higher energies, the absorption cross section was increased by addition of buffer gas. This effect was assigned to V T relaxation, tS~ However, a simpler explanation based on the change from linear to nonlinear absorption is presented in the next section. Multiphoton absorption via hot bands can be described by a level scheme similar to Fig. 4. Preexcitation places the molecules into a level above the lowest end of a parabolic funnel. Ifa molecule in the uppermost sublevel ofa multiplet is excited at long wavelength (as sketched in Fig. 7), it remains for a large number of steps within the limits of the two parabolas, i.e. in near resonance with the most intense band of transitions. Approximately linear absorption is expected in this case. In fact when preexcitation was achieved for SF 6, the absorption cross section at longer wavelengths (measured opto-acoustically) was nearly independent of the probe laser energy between 0.1 and 1J/cm2. tla2) This cross section is found to be ~ 10-18 era-2 (via calibration by single laser absorption measurements of ~182) and t186~). This magnitude is equal to the one found in a similar experiment tla6~ where the probe laser had only 0.5 mJ/crn 2. Thus the linear range is rather large. This observation has important implications for isotope separation in cases where the spectral isotope effect is small: A first
n
9 6 3
En-nE1
0
FIG. 7. Effectof preexcitationfor moleculeswith bands of levels(due to anharmonicsplittingor to nearlydegeneratemodes).(a) Whenpreexited(thermallyor by laser) to the highenergyborderline,a very long wavelength(straightline) remains for a long time within this band of levels (within the resonantmode);(b) Whensimultaneouslyexcitedby two frequencies,the moleculestaysin resonance for a prolonged time by alternate long-wavelengthabsorption and short-wavelengthemission.
The importance of spectroscopyfor infrared rnultiphoton excitation
139
weak laser excites one isotope to a level which has a longer wavelength absorption, C4~'8°~far beyond the band origins of the other isotopes. A second laser drives the molecules at this long wavelength up to dissociation. This methOd has successfully been applied to OsO4. ~6s'71,72'74~ Since the absorption probability is (and remains) high, the second laser must not be intense either; thus, single frequency excitation of a second isotope (which requires higher powers) is avoided and selectivity is better preserved than in single frequency methods and methods with a second, intense nonresonant laser. ~3-t'xs°) This method not only reduces the power required, but also dramatically increases the quantum yield. The latter effect is also due to the high absorption probability, which has been shown to be of crucial importance for economic utilization of photons. ~x9a) The absorption may be further improved by irradiating the sample by both frequencies simultaneously. ~1s 1) As shown in Fig. 7, the molecule can then climb up in a zig-zag path, employing long wavelength for absorption and short wavelength for emission. Unfortunately, simultaneous two-frequencies irradiation reduces the isotopic selectivity; given one of the frequencies Vl, the second frequency v2 will promote the excitation, be it at long wavelength (absorption continues via v2, Fig. 7a) or at short wavelength (v2 stimulates emission so that vx can be absorbed again, Fig. 7b). So the frequency response is also broadened to the blue side, as found in. ~181) Broadening of course reduces the selectivity. This enhanced dissociation on the blue side by simultaneous twofrequency excitation has not been explainable by other theories. Let us return to thermal preexcitation. Again the long-wavelength absorption will be very much enhanced and its linearity will be improved. Both effects have been found when SF 6 was heated up to 600°C. ~6~) No enhancement should appear at short wavelengths according to Fig. 7 and a slight decrease has even been observed in. (161) This is probably due to thermal population of rotational and vibrational states whose absorptions scatter over a broader spectral range (preferably at longer wavelengths) than at low temperature. At 944.2 cm- x the absorption was nearly temperature independent. ~16~)The dissociation yield at this frequency, however, increased strongly below about 200K. ~16°) For quantitative calculations of hot band effects on absorption compare. ~255) 8. COLLISION EFFECTS Lyman et al. tS) have given an extensive compilation of collision effects in multiphoton excitation. The following principal effects were discussed: (1) Collision induced pressure broadening, which will preferentially have an influence on the coherent steps of the excitation. This effect has yet to be seen experimentally. (2) Rotational hole-filling increasing both absorption and dissociation, which has often been observed. The effect is strongest when the hole is narrow compared to the rotational structure. (3) V-R, T relaxation withdrawing vibrational energy from the molecule, thereby decreasing dissociation, but increasing absorption. This effect has been observed in molecules with small rotational constants, where the hole extends over a large part of the spectrum. (4) Intramolecular vibrational energy redistribution altering degeneracies, the consequences of which are less distinct. In (1) to (4) buffer gases as well as the pumped gases are effective, which is in contrast to the following. (5) Vibrational energy exchange between molecules of the same kind either enhancing or diminishing the absorption, depending on how it modifiesl the vibrational energy distribution produced by the laser. Processes (3)-(5) have another consequence which has apparently not received due attention. They change the absorption spectrum of the individual molecules. This shall be illustrated'by considering two examples. Consider the first few Steps of pmnping of a degenerate mode whose higher states are split by anharmonicity into multiplets, Thelaser only populates certain sublevels of each multiplet (Fig. 4). Collisions will, redistribute this population into all sublevels. With the rule of thumb
140
W. Fuss and K. L. KOMPA
that t r a n s i t i o n s from one e n d of the level b a n d to the other e n d are less p r o b a b l e t h a n others (derived from the a p p r o x i m a t e selection rule AI ----_ 1, see Section 3 a n d (t s7)), the spectrum is predicted to shift to longer wavelengths. This collisional shift will be the larger t h e higher the excitation, since the a n h a r m o n i c splitting increases with energy. It m a y overcome a n y spectral b r o a d e n i n g which is c o m m o n l y looked u p o n as the m a j o r effect of collisions. I n Fig. 8 a b s o r p t i o n cross sections of SF6 at two different pressures are compared. At frequencies lower t h a n 944 c m - 1 the cross section increases with pressure, b u t at higher frequencies there is n o
930
lO-te
9t, O
950 cm "t
i l l l l l i l l l l l i l
61cma O.SsO,T "0~ ~0~.0,.,~ ~ 0
/
0.3 Jcm "2
""o
o" 1J cm"z
e ~
I0-,9
X
: i
t
I
930
t
t
i
i
I
9/.0
~
I
I
i
\.
I t 950 cm "1
FIG. 8. Absorption cross sections cr (defined in Fig. 2) of SF6 for various laser enerEies.(347) Measurements were made with the multimode TEA CO 2 laser (70ns spike, 3p.s tail) and the waveguidecell (10ram diameter, 1 m length) of.(t ss) The latter was used in double-~r~_~;this makes a very homogeneous distribution of intenmty even for absorption as high as 50~o, so that the evaluation~s)is very simple. SF~ densities n were used such that ~. n. 200cm = 0.2-0.7. The second part of the Fig. shows the effectof density on the spectrum. Other absorption data are compiledin.(s~
The
importance of spectroscopy
for infrared m u l t i p h o t o n excitation
141
change. Such an effectis generated by raising the absorption everywhere by means of rotational hole-filling and by simultaneous shift due to sublevel repopulation. The pressure independence of SF6 absorption at 944.2cm-1 has been a matter of some debate (see the review 5). The explanation presented here does not have to assume an excessivelybroad hole in the rotational contour. At the same frequency, dissociation is strongly diminished by addition of buffer gases. This has been ascribed to V - T relaxation. ¢159~But both V - T relaxation and rotational hole filling increase the absorption, in contrast to observation. In, ~349)the change of the spectrum of SF6 by collisions has been quantitatively taken into account. As a second example consider a molecule which is a linear absorber owing to the following peculiar mechanism: Suppose the molecule has two or more modes of similar oscillator strength, the second one of which comes just into resonance with the pump frequency when the first one moves out. In such a way the molecule can stay in resonance over a considerable number of steps, as explained in Section 4. The molecule will behave as alinear absorber if for all steps e the net absorption is constant:
~
-
a 7 = ~o
(42)
Here a7
= ~ - I "g~-
l/g~
(43)
The observed cross section is then constant, too: O'obs =
~,N,(o: bs - -
O~era) = a o
(44)
where N, is the relative population of the level 8 and Y,,N~ = 1. Condition (42) would also be met by a harmonic oscillator, for which (45) 0"~cm ~
0 o "
8/d
(46)
(d = mode degeneracy). Consider collisional redistribution between (rotational or vibrational) levels of a spacing small compared to the laser quantum. This relaxation will fill up the (rotational) hole in the initial wbrational state, but will shift the molecule out of resonance in the later steps. The latter effect can occur although the spectra can have a width of several crn- 1, since the spectra of vibrational sublevels differ considerably. Rotational and vibrational sublevel redistribution will proceed with collisional efficiency of almost unity, that is to say at a nearly equal rate. Denote by r o the vibrational-rotational sublevel resonant in an individual step of an excitation route like in Fig. 4. r o will be a function of the step number 8. Nonresonant levels will be called r'. Then eq. (30) can be extended to read dN~'°=IA(a,_L,og~_,.,oAN~-I"° I dt g.- 1,ro 1
~, ( k , o ~ r ' N ~ , o - k , , ~ , o N v , , )
(47)
r" • ro
The buffer gas pressure is contained in the rate constants k. The question to ask then is whatwill be their influence on the absorbed number of quanta (e). First sum eq. (47) over r. The relaxation terms will cancel, as expected, as e is not changed by them: dN~ = IA(tr._, ,.g.-t ,o AN~- t"*l dt ' ' g~- 1.,. I
(48)
(N~ := E,N~ and ~r~, = a~, ° 6,, 0 were used). Multiplying by e and summing over it, one obtains by rearrangement
d<8>. d N._t, * dt "= ~ Z , eN, = I E,e~_ 1,,og,_I.,0A g,- I,,.
(49)
This is equivalent to eq. (44) ifone defines :=
(50)
142
W. Fuss and K. L. KOMPA
Thus d
(51)
In general aobs will depend on relaxation rates via N~,o, which have to be calculated from eq. (47). For a linear absorber, however, eq. (42) may be used to get trob.~ = ao,° X~N~, °
(52)
This sum does not depend on relaxation, as may be shown by summation ofeq. (47) over e. (The independence of the k's of e has to be used thereby.) It is equal to the fractional population N~o of the (rotational) state before excitation. Therefore aob~ = ao, oN,o
(53)
The physical interpretation referring to the question why the absorbed energy of a linear absorber is insensitive to sublevel relaxation, is that as many molecules are brought into resonance with the first excitation step as are removed from resonance with the higher steps. For a nonlinear oscillator, however, (for which the net absorption drops with e) hole filling in the initial level must not counteract the "antihole" broadening in the higher levels, so foreign gas usually improves the absorption. Only at high intensities where all the rotational states are excited, the buffer gas effect on absorption will again diminish. It has been observed (17s) that addition of buffer gas to $2F10 has no effect on aobs at low laser energies (where S2F~o behaves like a linear absorber), but increases the absorption at higher energies (where the absorption is less than linear). On the other hand a buffer gas improves the absorption ofSF 6 (nonlinear absorber) at low energies, but has no effect at high energies "sl) (where most of the rotational states are excited). This difference is difficult to understand without the sublevel relaxation mentioned above. What will be the effect of this relaxation on the dissociation probability? Before relaxation, the population distribution of a linear absorber will be: ~2s4'296)
N~,oOCN, og,, ° exp( - e/eo)
(54)
(e) o]. I Another fraction 1--N~o will where eo is related to the mean energy per oscillator eo ~ sN, remain unexcited. Sublevel relaxation increases the excited fraction N,o and possibly also the number of oscillators, but keeps the absorbed energy (e) constant. The corresponding drop of the tail of the exponential eq. (54)--which is responsible for dissociation--will in general not be overcome by the increase of N,o and g~o. Thus sublevel relaxation can reduce the dissociation yield without affecting the absorbed average energy. It must be suspected that a similar phenomenon will occur also in nonlinear absorbers. 9. CONCLUSION It has been pointed out in the introduction that the absorption spectra of highly excited molecules are not wider than the corresponding ground state spectra. Recently this fact has been demonstrated for molecular ions of energies of only 1000 era- 1 below dissociation :(356) CFaBr + and CF3 I+ were generated in an ion source which apparently produced part of them in highly excited states. A continuous-wave CO 2 laser then dissociated them in a singlephoton absorption step, which was inferred from the linear dependence of dissociation probability on laser intensity.(356) The frequency dependence was only 10 cm- 1 wide and the wings were very weak, comparable to infrared spectra of similar molecules; e.g. CFaI has a halfwidth of 20 cm- 1. Whereas evidence of this kind shows that vibrational oscillator strength cluster in a range not much wider than a typical rotational structure, it is even conceivable that under the envelopes even narrower vibrational structures are concealed. Such an assumption allows to explain elegantly why SF6 absorbs ps pulses much more effectively than ns pulses :0 s s) If the gaps (ofa few cm- 1) in the anharmonically split va transitions are not levelled off, a narrow-band radiation (like a ns pulse) will sooner or later experience a poor resonance
The importance of spectroscopy for infrared multiphoton excitation
143
whereas a broad-band radiation (a ps pulse) will not feelit very much; so the latterwill be absorbed nearly likeby alinear absorber, much more effectivelythan a ns pulse.If~however, the gaps would be smoothed out above a certain level,the differentialabsorption cross-section above this energy level would be nearly constant and high in both cases in contrast to the findings.(See also Section 3.).The same experiment also clearlydemonstrates that collisionless relaxation isnot important for excitationofSF~: Ifitwould be necessary (i.e.rate determining) in a certain step, excitation by shorter pulses should be much less effective(Section 6). The opposite was found, however. The similarityof spectra implies that there isno reason to restrictthe study of spectroscopy to the firstthree steps: First of all,the simplicity of spectra even at high energies (at low resolution) is encouraging. Second, the spectra will be of comparable importance in the first few and later steps. Then, however, it has again to be asked how molecules can continue to absorb photon after photon and how they sometimes even ,behave like linear absorbers in spite of their anharmonic shift.W e have shown in the preceding sections,that by considering not only the firsttwo but all the contributing mechanisms of Table 4, a wide variety of phenomena can be understood at leastin principle.These variationscome up not only because spectroscopy is differentin every molecule, but also because every molecule (and even every rotational or vibrational state and every laser wavelength) selectsits own set of contributing mechanisms.
TABLE 4. Mechanisms contributing to multiphoton excitation Mechanism
Important
Function
Section
(1) Spectral intensity borrowing of background states (quasicontinuum)
in large molecules and at high energies
causes energy instead of intensity dependence; compensates small detunings, less well also large detunings by absorption wings
5
(2) Collisionless vibrational relaxation
in very large molecules and close broadens spectrum (due to to dissoc, energy distrib, of anharmonicities and to short lifetime), causes RRKM type dissociation
6
(3) Raising of rotational quantum number J
where rotational constant B is large; for all J
compensates part of anharmonicity; x, ff = x + B
3
(4) P, Q and R type transitions
for rotational states for which J = anharm, detuning/2B
provides three resonances for each vibrational transition
3
(5) Weakly forbidden rotational transition (e.g. AR # 0 in spherical tops or AK d: 0 in parallel transitions of symmetric tops)
in higher states where spherical or cylindrical symmetry is a bad approximation
provides three more resonances for each vibrational transition
3
(6) Multiplet level bands of nearly degenerate modes or anharmonically split degenerate
for very many molecules
reduces anharmoni¢ shift of some sublevels; cooperation with (4) and (5) very effective
3
where multiplet bands are available
converts vibrational into rotational energy
3
where available
can restore resonance with the original mode
4
modes (7)Intermediatestimulated emissionin a (e.g.)P type transition (8)Absorptionvia (anharmonicallyshifted)shorter wavelengthmode (9)Power broadening (10) Direct two- and threephoton transitions
where broadening by (1) and (2) compensates small detunings, is unimportant causes a certain intensity dependence
2, 5
especially if enhanced by near resonance
7
increased number of rotational states can take part; causes sharp resonances and a certain intensity dependence
144
W. Fuss and K. L. K O M P A
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0079-6727/81/020117-35517.50/0 Copyright © 1981 Pergamon Press Ltd.
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THE IMPORTANCE OF SPECTROSCOPY FOR INFRARED MULTIPHOTON EXCITATION W. Fuss and K. L. KOMPA Projektgruppe f'tir Laserforschung der Max-Planek-Crv~llsehaft zur F6rderung der Wissensehaften e.V., D-8046 Garehing b. Miinchen
CONTENTS Introduction Anharmonicity Rotational Compensation and Anharmonic Splitting Nearly Degenerate Modes The Width of the Resonances, Intensity and Fluence Dependence Collisionless Vibrational Relaxation Rotational Hole Burning, Direct Two-Photon Absorption, Two-laser Experiments, and Temperature Effects 8. Collision Effects 9. Conclusion References
1. 2. 3. 4. 5. 6. 7.
117 121 123 128 131 135 137 139 142 144
It is substantiated by examples that the infrared spectra of molecules in high vibrational states are similar in width to those of the ground states. Therefore in order to explain collisionless infrared multiphoton excitation, the existence of resonance has to be checked, not only for the first three steps, but for all of them. That is, their (low resolution) spectra should be studied. This review summarizes the spectroscopic mechanisms contributing to multiphoton excitation, which have been suggested to date, including several kinds of rotational compensation and of vibrational level splitting, which cooperate to overcome the anharmonic shift. The spectral quasicontinuum, generated by intensity borrowing, must neither be very broad nor dense, and collisionless vibrational relaxation is only important at very high energies. Knowledge of relatively few spectroscopic details helps to understand many details and many differences in multiphoton excitation. 1. I N T R O D U C T I O N
Since the first demonstration of the collisionless infrared multiphoton excitation *~76~in 1973, the whole field has experienced a boom. Several reviews are already available, ~l-s} and in more than 100 molecules-from 3 to 62 atoms in size--unimolecular dissociation, isomerization (CHCI=CHCI, cyclo-C4F6, cyclo-C3H6, RNC~ electronic excitation (CRO2C12, OsO4, SiF4 and several dissociation fragments) or electron detachment (C6HsCH2) have been triggered in this way (Table 1). Part of the stimulation certainly came from the possibility that isotopes can be separated in this w a y ; (6'193-196) its feasability in a laboratory scale has been demonstrated, tt 93,x94~Also the fact that many radicals have become conveniently accessible in this way (see the compilation in ~s,,xs9~) plays a role. But more important is probably the fact that collisionless multiphoton excitation is an unexpected new phenomenon. Although not more than one photon should be resonantly absorbed, many molecules were found to absorb 30-50 photons directly from the field of a pulsed high power infrared laser (mostly the CO2 laser). Many theories and models ~24~-31s~have been developed to explain this phenomenon. These agree only partly with one another. The problem is basically spectroscopic in nature. It concerns the question what does the spectra of excited molecules look like; are they shifted, split or broadened compared to the ground state spectra? In spite of these obvious questions most theories and models have taken details of molecular spectroscopy into account only for the first three to five steps. Part * The term "multiphoton excitation" is only intended to count the photons. When we want to emphasize the contribution of real intermediate states, we use the term "stepwise multiphoton excitation", in contrast to "direct multiphoton excitation" which indicates virtual intermediate states. 117
TABLE 1. Molecules which have been multiphoton excited. The references quoted refer to experimental work related to multiphoton excitation, but not necessarily demonstrating it. A similar list is contained in;(S) a few molecules and references have been added; but molecules like SO2 (*16,243) or NH3 (244'2443) whose dissociation seems to be collision induced have not been included. Triatomics
8-atomics
OCS(9-1o,) 0 3 (9) NO2a)(1t.~2) SO2a). 1 , )
CCIF2-CH3 (313,206) CCIF2_CF3 (31a,361-363) CHCI2-CF31~:;'~::: CDCI2-CF 3 , CH3_CF3(2os. zo9) CF3:CF3 (21°) CH2D-CH2CI(211) CH2Br-CH2 F(209) CH2CI-CH2 F(212) CH3CH2 F(1033) CH3_CHF2( 1o3~,213--214.376) CH3COOH (s4") CH3_CCI3 (313.206) C2H3C1(21,206.206a) CzHsF (2°9) CF2-S (215) I I S-CF 2 HCIC = CHBCI2 (37s)
4-atomics
ECI3(I 3-19b,34,88) H2COt HDCO~D2CO (2°-25,s4,3~) CI2CS (374) F2CO (245) HN3, DN3 (26) 5-atomics
CCl4(27-29) CC13F(3O.31) CBr2F2(3~ 34) CC12F2(31-41,ss) CHCI2F(3 l a) CHCIFz(31~,33,42-44) CDF3(364--365) CH2F2(31a,45) CBrF3(31,46-493) CF3Br+(356) CC1F3(31.46.226a,366,367) CF3I(31,46.4~,49 62,190d,e,193,377) CF3I+(356,357) CF30-(359) HCOOH(63, 6'*) CrO2Cl2(62, 65-65d) OsO4 (65b-75a) SiF4(34,3a,76-ao) CDF3(81. 82) CH 2 = CO (83-34~) 6-atomics
CH3OH(22. aS-as) CH3CN (as-91) CHaNC (92) CCIF = CF2 (88'93 93c) CHCI = CF2 (31a'94) CHCI = CHCI (62'94-1°°) CHCI = CCl (31,) CH2 = CC12 ~(~4) CH 2 = CHCI (94.101-102) CH 2 = CHF(103-104) CH 2 = CF2 (103,1033) CH 2 = CH2(84,88,105-11 t) HC = C - CHO a)(112-112b) N2F4.(313.I 13.114) N2H4 (t~s) SF5 +(116,117)
9-atomics
C2HsCN (216) C2HsNC (92) C2HsOH(853,216,217) CHaOCH3(218) CF3CH2OH (853,b) C3F~-(219.22o) C3H61221) cyclo.C3H6(221-223) SFsNF2 (224) CH3CICH~ -(3sO) l O-atomics
C2HsNH2 (216) CF 3COCF 3(62,225-229) cycio_C4F6 (230.230.) 11- and 12-atomics C3FTI (62) n_C3H 7OH(217) trans-CH3CH = C H C H 3 "°°) CH3COCOCH3O~ 36s.373) CH2-CH2(369) I I CH2-CO H2C = CHCH2COOH (379) cyclo.C4Fa(231) C6HF5 (232) $2F10 (178) more than 12 atoms
C][..'[3NO 2(88.89,122-123)
H O O C _ C H 2 _ C O _ C O O H (379) C6HsCN +(3sl) C2HsO-CH_--CH2(233-233) CHaOCH2CH----CHz (23s) C H s C O - O - C O C H 3 (s4,s4")
CH3ONO (121) CH3OHF-(3ss) CH 2 = CHCN(9S,12 '),125) SF5C1(126.127) SF6(28.31 a.38.47.80.111.128-196.355) M o F (197) SeF (~93) UF6~199-205)
C6HsCH ~ ( 2 4 2 ) CH3COO-sec-C4H~ (21~) (C2Hs)2OHO(C2H~)~(219.2zo)(CH3)2C_C(CH3)2(23~ 239) I L o-o U(OCH3)6 (24°) UO2(hfacac)2" T H F °')(241-241b) (UO2 (hfacac)2)2(241~)
7-atomics
CH3NH20 ! sA ls-120)
(a) after electronic excitation (b) hfacac = CF3CCHCCF 3, T H F = (CH2)40
o
o
c'~
r,~r,
~
(44,235,236)
The importance of spectroscopy for infrared multiphoton excitation
119
of the reason is apparent from Table 1: This list of molecules suggests that a universal excitation mechanism exists, while spectroscopy is different for every molecule all examples show the same basic behaviour. The most popular of the global models is the quasicontinuum model (see the reviews, (1-3) especially (3)). It assumes that in the region of high density of states no selection rule can be strict enough to suppress all the transitions to these states; therefore the molecule will always b e in resonance, even if the principal absorption has been offset by anharmonicity. It has to be noted, however, that none of the general models explains the excitation of every molecule of the table. For example the density of states of the triatomics and of HN 3, C2H 4, H2CO and other molecules is too small to form a quasicontinuum. Furthermore molecules seem to exist which cannot be dissociated by multiphoton excitation (Table 2), although they are structurally quite similar to others which have been dissociated. It is appropriate at this point to consider the following two arguments. (1) Many details of the multiphoton process can already be understood with relatively little spectral information, e.g. at low resolution, and this report tries to give examples for that. Such information may not be difficult to get, since low resolution spectra are not very different for systems of low and high densities of states. As an example consider Fig. 1, which TABLE 2. Unsuccessful multiphoton dissociation experiments
molecule CH3I CD3I COF 2 NF3
Vmmlec(335)
Vla.,er
882.4cm - t 951 1049 965 1032 985
(e) 944.2cm - t (at) 944.2 (e) (a~) 951.2 (al) 1031.5 (al + e) 983.3
ElM© r
(lit focus
Pdi~
20J 20
220J/cm 2 220
<3.10 -3 <3.10 -3
130 110 110
<8.10 -3 < 8.10- 3 <8.10 -3
4 3.5 3.5
The CO2 laser oF x33) (50 ns spike with 30 % of the energy, 3/ts tail, beam cross section 5 x 5era 2 for 944.2cm -1 and 3 x 4era 2 for other wavelengths) was focused ( f = 20 cm) into a cell of 23 cm a volume. Conversions were < 10- 5/pulse in all cases. The energy density ~ focus and dissociation probability Poi,s in the focus were calculated using the focal dimensions 3 × 3 × 10ram 3 (at 944.2cm - t ) and 1.5 × 2 × 10ram s (elsewhere). For COF 2 and NFa 1 mbar of H2 was added as scavenger. o/
cm2
'
'
k
r
'
'
'1
I
L
o/
1049
10-20
10-22 CF3 CO CF 3
OCS
10-21 J , i ~ 1 0 "23 900 gho 1000cn'rI 1000 IOr--JO 11{X)cm-1 [nfrsLt~dspectraofasmal]molecule(carbo~ylsuLfide)andofamo]ecule(hexafluoroacetone~
F~G. 1. which is in the quasicontinuum of states at 313 K, the temperature of measurement, Spectral slit width was 1 c m - 1. Higher resolution reveals rotational structure and a slightly higher cross section for COS.
120
W. F u s s and K. L. KOMPA
compares the infrared spectra of a small and a medium sized molecule. The latter, hexafluoroacetone, has an average vibrational energy of 1500 cm- 1 at 310 K. The transitions shown from 1500cm -1 to 2500cm -~ connect regions of 700 and 2.104 states per cm -1. Nevertheless the band is not broader than the one shown in OCS, where the density of states is < 1/cm- 1. Obviously (approximate) selection rules suppress the many possible transitions outside the band. It is interesting to note that the absorption cross section of hexafluoroacetone around 971 cm- 1 drops by not more than a factor 3 when the molecule is excited up to 10 quanta/molecule (see below and (225)). Every model based On background absorption in the wings which according to Fig. 1 is very weak will have difficulties to explain this fact. (2) Table 1 conceals striking differences which are not predicted by global models. For example, Fig. 2 shows the absorption cross sections (=absorbed energy per molecule/incident energy per cm 2) plotted versus the mean excitation energy. Obviously this cross section drops by orders of magnitudes for some molecules, whereas in others it remains nearly constant up to almost the dissociation energy or even above. The lower states cannot be responsible alone for this difference. Figure 3 shows the dissociation yield as a function of wavelength for three molecules. Its optimum wavelength is shifted from that of the linear
~. r'//i
I
I
i 10-18
10-19
I 0 -is
~ S F s
944.2 crn-'
|
10"20 I0-t
I0 -2
r-'~/
d/cruZ ,o.~,/
I I
I0
I
I
II
I0"I?!~
~
$2 Fro
io-te
crn"l
=,-,,,,
10733 cm NN
°""R
\
%.
10-19
~o~
I
I0 "I
I
1
1
10quonta/rnolec
FIG. 2. Multiphoton absorption cross section ~r v e n u s average excitation energy q (quanta/molecule). Since only a fraction of the molecules take part in the excitation, q is only a lower b o u n d of the actual energy of the probed molecules. These plots have been derived from measurements of q as a function of the incident energy @ (photorm/cm 2) via ¢r := q/~l). The differential cross sections are smaller by a factor 1 to 2 in the cases shown, b u t are less accurate. Data for SF 6 from (284) which agree with the compilation in (s) for H2C = CHCI from, (1o1.~,,~os) S2Flo,(l~s) CF3COCF3,(22s) CF3I, (53,347) C2F4S2 .(2151
The importanceof spectroscopyfor infraredmultiphotonexcitation
SF 6
/
/--
9 0
950
BCI3
/
121
/""',.
c m -1
%
/
.,t
900
\' /\", 950 cmq
' SiFz,
I
p
~/{~x\
/
IO00
x
l~ '
1050 cm "1
. . . . . r o,t o t (tas) (see alsot h e c om pilatlon FIG.3. Frequencydependenceof dissociationprobaoihty o,, " in (5)k BCl3(19a) and SiF,t (79) comparedto their small signalspectra (e).
absorption maximum, sometimes to the blue (SiF,), in other cases to the red (SF6, BCI3). The shapes and positions of these dissociation spectra (which are crucial for isotope selectivity) are usually thought to be determined by the spectroscopy of the first few steps. Note, however, that isotope selectivity has been found by irradiation of a band of CHaNO 2, in which there is no spectral isotope shift.(t22) Obviously in higher steps another mode contributes whose frequency does depend on the particular isotopic content. A substantial difference has also been found for the internal energy of the fragment NH 2 depending on whether it was produced by dissociation of N2H , or CH3NH2. (t is) This fragment energy indicates how far the parent molecule has been excited beyond the first dissociation threshold. Obviously in N2H ¢ the optical excitation finds an exit channel closely above the dissociation energy, whereas CH3NH 2 allows absorption continuing to higher energies although the number of atoms in the molecules differs by only one hydrogen atom. All these examples demonstrate that one should study the spectroscopy of excited molecules in more detail. This report is meant as an encouragement to do so, even if the necessary spectroscopic information is not easily accessible. 2. ANHARMONICITY Collisionless infrared multiphoton absorption would be no problena for a harmonic oscillator. Because of its ladder of equidistant energy levels it would not get out of resonance after absorption of any number of photons. But normal molecules are anharmonic. The vibrational energy levels of a one-dimensional anharmonic oscillator (e.g. a diatomic molecule) can be represented in good approximation by E v / h c = re1 v + x v ( v -
1)
(1)
(Vet fundamental frequency, x anharmonicity, both in crn-t). The mismatch between v-th level and the energy of v photons therefore increases quadratically with v. Power broadening is only proportional to v/~ + 1 :
(gg) = (
,0,Ek/v + 1
(2)
Therefore it will be able to compensate the anharmonic shift in a few steps at most, even if certain modifications of eq. (2) for degenerate modes are taken into account.(249'2s°) Under
122
W. Fuss and K. L. KOMPA
multiphoton excitation conditions a power broadening of ( f i o l E ) / h c = 0.1 to 1 cm -~ is typical. Anharmonicities are generally negative. For diatomic molecules their magnitude is a few per cent of Voi, i.e. more than 10cm-1. Close to the dissociation limit, the level distance approaches zero. Therefore it is believed that diatomic molecules cannot be excited up to dissociation. Similar behavior is expected for vibrational coordinates of polyatomics which directly lead to dissociation. Therefore it will not be possible to excite a single bond in a molecule selectively until it breaks. (For a possible exception see below under rotational compensation). Since the vibrations ofpolyatomics are often delocalized over several bonds, the amplitudes oftheindividual atoms are smaller, resulting also in smaller anharmonicities than in diatomics. Furthermore the anharmonicity of one mode contains contributions from several displacement coordinates, which can mutually compensate. Positive contributions often arise from the curvilinear nature of deformation coordinates. 13~7) The exceptionally small anharmonicity x44 = + 0 . 1 c m -1 of the v4 mode of SF 6 at 615 c m - ~¢31s>may be an example for such a compensation. Even smaller anharmonicities are entirely conceivable. The tetrahydrofurane adduct of uranyl hexafluoroacetylacetonate¢241) seems to have such a harmonic vibration: If it is excited by a CO 2 laser, its infrared absorption band is neither shifted nor broadened. 13~m The absorption cross section of a harmonic oscillator does not depend on excitation energy. For several molecules (Fig. 2) linear absorption has in fact been found; in these cases, however, the interpretation of section 4 (contribution of different modes) is more probable. The vibrational energy levels of a polyatomic molecule can usually be written Ev/hc
= E i viv i + Z i x , v i ( v i -
1) + E i , j > i x o v i v j
(la)
This is an extension ofeq. (1). v~are the observed fundamental frequencies, and x , and x o are called diagonal and cross anharmonicities, respectively. The most frequent deviations from eq. (1 a) arise from anharmonic resonances like Fermi resonance. These interactions mix states of one mode with' combination states lying close-by. The resulting energy correction is often also bilinear or quadratic in v~. In these cases, the correction to (1 a) can be contracted with one of the anharmonicities. Anharmonic resonances are always important in the quasicontinuum of states (Section 5). They cause displacements from eq. (la) which often compensate. So eq. (la) can remain valid although the mode parentage of the states change. The quadratic dependence of the detuning on the quantum number will drive the molecule out of resonance after very few steps. How in a rotating polyatomic molecule this mismatch can be overcome is shown in the following sections. Often this mismatch is discussed in terms of a transition whose frequency shifts linearly with the step number; this would be correct if the laser would be tuned at each step to follow this shift (v 1 = driven mode): (Ev,+ lvj - E~,~j)/hc = (vl + Z j , l x l . a v j )
+ 2xxlvl
(3)
A linear relationship with a quantum number is also valid for the detuning of the laser to an nphoton resonance with the v = n state of a single oscillator: E,/nhc = v + x(v -
1)
(4)
When an unknown number of modes has been excited in the previous steps, in the n-th step a modification of eq. (3) is sometimes employed: (E,,~ + 1~ -
Eo,v~)/hc = v 1 + 2~vl
(5)
with an unknown type of average anharmonicity ~, whose value is close to a thermal average. ~3 But Section 4 shows that the spectral shift is not even monotonic in n, if more than one mode contribute. Another modification of eq. (3) results for a model molecule whose anharmonic shift only depends on the energy, irrespective of its distribution over the modes; i.e. for a molecule for which 2 x l ~ = av~ xij = avj,j ~ 1
(6)
The importanceof spectroscopyfor infrared multiphotonexcitation
123
At the energy Evlv2... the resonant mode will then be detuned by (Ev, + iv2". - Ev,~2)/hc -
vl = aE~,~2.../hc
(7)
from v~. The oscillator strength of vl can be distributed by intensity borrowing over a certain width of close lying combination transitions (see s and 2s4). In this case the molecule continues to absorb by means of these transitions, and eq. (7) gives the detuning from the absorption maximum. It must again be pointed out, however, that in real molecules with less uniform anharmonicities the anharmonic shift is not a simple monotonic function of the step number (Section 4). 3. ROTATIONAL COMPENSATION AND ANHARMONIC SPLITTING At room temperature, many rotational states of the vibrational ground state are populated. The excitation process has to be studied separately for each rotational state. To the vibrational energy of eq. (2) a rotational term has to be added. This term reads for symmetrical tops Erot/h¢
=
BJ(J + 1) + (A - B ) K 2 ~ 2 A ~ I K
(8a)
and for spherical tops Er°t/sc = BJ(J + 1) + B~[R(R + 1) - J(J + 1) - l(l + 1)]
(8b)
A and B are the rotational constants for rotation around the molecular axis and an axis vertical to it, ~ is the Coriolis coupling parameter, 1 is the quantum number of vibrational angular momentum and R is the rotation of the molecular frame (R = J - 1). Relatively strict selection rules apply to the rotational quantum numbers J and K: For transitions parallel to the molecule axis AK = 0, AJ = _+1,0,AI = 0
(9a)
A K = + I, AJ = _+I,O, AI = _+I.
(9b)
for transitions vertical to it
Of special interest is the case where in a sequence of vibrational levels J regularly increases or decreases from Jo in the state v = 0 to Jo +_ v in the state v. Insertion o f J = Jo +_ v, K = Ko (parallel transitions) into eq. (6) or eq. (7) yields terms linear and quadratic in v, which, when added to the vibrational energies eq. (2), may be contracted with the fundamental frequency and the anharmonicity: Y/.eff -~-
Y i ~---
2B(Jo + 1 ___ 1)
(10) (11)
Xii,eff = Xii -[- B
Similarly for vertical transitions the sequence K = Ko +_ v, l = v can be described by the effective frequencies and anharmonicities v~.cff=
{
v/+2(A(1-~) v~+2(A(1 ~)
B)K o forJ=Jo B)Ko+_2B(Jo+I_I)
f x , + A ( 1 - ~) - B Xii'eff= Xii + A(1 ~)
forJ=Jo+_V
forJ=Jo for J = Jo +- v
(12)
(13)
These relations can easily be refined to take the v dependence of A and B into account. Equations (10) and (12) are analogous to the frequency shifts of the P and R branches of fundamentals. Equations (11) and (13) show that the vibrational anharmonicity has to be corrected by rotational terms. A compensation arises when these terms are positive. A and B are always positive, and - 1 ~< ~ ~< 1. This compensation is independent of the initial rotational state (Jo, Ko); all excitation frequencies within the rotational contour of the fundamental will see
124
W. Fuss and K. L. KOMPA
the same anharmonicity. A and B vary over a wide range. For example for = 4 cm- 1. For SF 6 the correction can nearly be neglected; for C2H4, however, possibly "~77,©ff > 0. It is also remarkable that this type of compensation--if it is perfect--makes dissociative exaltation of diatomics possible, too. A combination of the subcases of ext. (13) has been considered in ~1o7,) for ethylene, which is a nearly symmetric top. For asymmetric tops, analogous closed form relations do not exist. Another type of rotational compensation has long been knownJ 1'2~ Consider the three consecutive vibrational absorption steps with change of rotational quantum number Jo ~ Jo - 1 ~ Jo - 1 ~ Jo in a spherical top like SF 6 for example. These transitions will occur at the approximate frequencies v o - 2 B J , v o + 2 x , v o + 4 x + 2 B J o respectively. All three frequencies are the same if SF 6 Bcff = B(1 - () = 0.028 c m - 1 and for ethylene A - B
(14)
BJ o = - x
For appropriate Jo, ¢q. (14) can always be fulfilled. On the other hand, only few of the thermally populated rotational levels can take part in such a three-photon PQR resonance. The width and spectral position of it are equal to the width and position of the Q-branch of the v = 1 ~ 2 transition. However, these positions do not coindde with the most prominent peaks found in the multiphoton absorption of SF6 ¢176.177) and dissociation of C2HsC1 (2°6"), both measured by continuously tunable lasers. These peaks rather coincide with the Q-branches of direct two- and three-photon resonances, according to spectroscopic evidence. (According to ¢qs. (3) and (4), a direct two-photon resonance, e.g. occurs at re1 - x, whereas the PQR resonance is expected at v01 - 2x for the simplest case.) It can be concluded that PQR resonances are only of limited importance. A third type of rotational compensation has been suggested by Platonenko. t27°~ This type exists if the vibrational levels consist of several sublevels. The first three steps are again a three-photon P Q R resonance, as described above. Thereafter a stimulated P or Q branch emission takes place from the populated v = 3 to a sublevel of v = 2, which was not populated directly by absorption. The whole sequence may then start again from this sublevel. This mechanism is considered in more detail below. In several molecules (SF6, UF6, SiF4, OsO4, CC14, BC13 and others) exhibiting multiphoton absorption, the laser excites a degenerate vibration. If these modes were harmonic, they would cause a degeneracy gv of the level v~32°'~ gv =
(v+d1) d- 1
(14)
(d = mode degeneracy). Above v = 1 this formula yields degeneracies higher than required by symmetry. So anharmonicities will split them. In a first approximation the level energies can be regarded as a special case of ¢q. (2): EJhc
= vv + E x,~v~(v~ -
l) + ½ E x~pv~vp
a
(15)
~-¢,#
where ~ and B run over a, b . . . (the components of the degenerate mode), v = ~v~, and x~ = xp~ .... x ~ = xBv. . . . The wave functions are assumed in this approximation to be products of single coordinate functions (symmetrizcd with respect to permutations of ~/~ .... ): Iv) = Slv~)lvB)...
(16)
In an octahedral molecule like SF 6 the a, b, c components arc the cartesian components of the vibration. The splitting of the higher levels means that it makes a difference in energy if the quanta are tither concentrated in one component or distributed over more of them. A different approximation 32°~ transforms from the coordinates a, b, (c) to polar coordinates r, ¢p, (~). The wave functions consist of a radial part multiplied by a cylindrical or spherical harmonic: [v~ = [v,>ll, m r ~ (17)
The importance of spectroscopy for infrared multiphoton excitation
125
The energies are Ev/hC = vv + x v ( v -
1) + g(/2 _ v)
(18)
for the two-dimensional oscillator (d = 2) and EJhc
= vv +
xv(v - 1) + g[l(l + 1) - 2v]
(19)
for d = 3. ! is the quantum number of internal vibrational angular momentum. It can take on the values 1 = v, v - 2 .... 1 or 0. If either x.a >i - x~(>/0)
(20a)
g I> - x (>10)
(20b)
or
the splittingof the multiplets will be larger than theirmean anharmonic shift,i.e.they will extend on both sides of the harmonic position vv. In this case any number of quanta can be absorbed via strongly allowed near resonant transitions.Evidence has been found that eq. (20a) may apply to SiF 4.(266)But even iftbesplittingissmaller than the mean shift,itforms-in cooperation with rotational compcnsation--a very powerful means to support multiphoton excitation.Before explaining this,we should refine the approximations of eq. (15) and eq. (19). In these approximations certain levels stillhave a higher degeneracy than required by symmetry. This degeneracy isremoved by terms in the hamiltonian which are nondiagonal in the representation ofeq. (16) or eq. (17).They are nonzero for statesdifferingby transferof two quanta from one to another mode (V is a parameter): (v~v~ - 2 . . . IH[v~ - 2 v p . . . )
= yx/v~(v~ - 1)vp(v~ - 1)
(21)
and the corresponding matrix elements with the representation ofeq. (17) can be derived by angular momentum theory. Although nondiagonal elements may be very small, they should not be neglected, as they couple states of equal or nearly equal energy. They do not affect the center of gravity of the multiplets,but the total width of the splitting is increased a little bit. In order to check for resonances, a diagram plotting directly the energies is not sufficiently accurate.it is more useful to plot instead of the total energy E, (of levels close to n hvll,,r) only the anharmonic deviation E , - n E t versus the step number n (E 1 = hc v01). This is shown for the v3 states of SF 6 in Fig. 4. The levels are enclosed by the two parabolas predicted by the diagonal approximations eq. (15) and eq. (19), although nondiagonal terms are included in the Fig. Multiples oflaser quantum energies n ht)laser lie on straight lines in such a diagram. For example, the central straight line denoted by 0Q in Fig. 4 represents the CO2 laser P201ine. This lineyields resonances and near resonances for several steps while crossing the band of levels between the parabolas. The Fig. also illustrates the quadratical increase of detuning after the last resonant step. Up to now in this section only nonrotating molecules were considered. If rotation is separable from vibration, a simple way to take rotation into account is to go into a moleculefixed coordinate system. An oscillator in an initial rotational state Jo sees an amplitudemodulated laser field, appearing as the superposition of the laser frequency itself (which even may.be missing for certain orientations) with two sidebands at a distance +2B*J. For a spherical top, e.g., B* ffi B(1 - ~) is the effectiverotational constant. A consequence ofthis fact is the usual observation of P, Q, and R transitions. The two lines called Op and °R in Fig. 4 represent such sidebands of the 944.2 cm- t laser frequency for J0 m 55 for SFe. Unfortunately in this molecule second order Coriolis interaction is very important, i.e. the effective rotational constant strongly depends on the vibrational state ° 5 ° and rotation is not separable from vibration. Although Fig. 4 neglects this effect, it serves to illustrate the powerful possibilities of cooperation between rotational compensation and vibgational multiplet splitting. The initial rotatiomtl state selected in the exampl e~has a Pbranch transition resonant in the first step;
126
W. Fuss and K. L. KOMPA
-~O
OR
0O
0p
..-.
\
"
r
t /
\
/ /
~
"---~"
~
/
/ /
/
/// // L
.... - 150
, .... - 100
,
i
/
/
/ JP //
l
i
L
]
l
i ~
50
~ 50
E n - nE/cm
-lp
/
" .~, ""x~x,
"I
FIG. 4. Energies E, minus the corresponding harmonic energies n 'E 1 o f S F 6 vibrational levels close to n'Eco2, multiples of the CO2 laser quantum energy. For Eco2/hc = 944.2crn-1, the latter lie approximately on the straight line denoted by 0Q. The anharmonic constants of (354)have been used. The straight lines are given by E, - nE 1 = n'(Eco2 - El) - (E~t - E~°t) where the second bracket contains the difference of rotational energies before and after excitation by a pure sequence ofARP,aRQor ARRtransitions, respectively,neglecting second order Coriolis coupling. These lines connect (virtual or real) vibrational states which could be reached by such pure sequences; actual transitions are parallel to them (fat line). Broken lines indicate AR # 0 transitions. The P and R lines actually deviate from straight lines t o lower energyfor increasing energy due to the change ofJ with n, an effectwhich is not noticeable in this scale for SF 6. Decrease of Btff with n (as it is common for many molecules, whereas SF6 behaves more complicated(as4)bends the lines into the opposite direction; this effect has been neglected in the Fig. thereafter follows a n o t h e r stepin the P branch, one stepin the Q b r a n c h (parallel to the line OQ), two R branch transitions (parallel to °R), and so on (fat line). Such a sequence of transitions is similar to the second type o f rotational compensation, i.e. a P Q R sequence for a onedimensional oscillator. The first type of rotational compensation, which takes a c c o u n t of the change o f J with the step n u m b e r n, also has implications for Fig. 4. It leads to a slight deviation (of magnitude B * n ( n 1) o f the 0p and °R lines from straight lines. The third type of rotational compensation, suggesting alternation of absorption and emission steps, is visible in the upper part of the excitation chain (strong solid line in Fig. 4). By this mechanism the molecule is very efficiently excited until it reaches the upper p a r a b o l a and the line OR. Crossing the line Op m a k e s difficulties: The optical field tends to couple only to the internal angular m o m e n t u m i of a degenerate mode, whereas it leaves the rotation R = J - I of the molecular skeleton unaffected; in other words, because of the selection rule AR = 0 . A s i n t h e g r o u n d s t a t e l = 0 , a l l t h e p o p u l a t e d s t a t e s h a v e R = J . T h e m a x i m u m / v a l u e in each multiplet is equal to n. The range of J values which can be populated by the laser therefore spans the range R - n to R + n. An emission step from thelevel n = 8, J = R + 8 in a P b r a n c h (J ~ J + 1, vertical transition in Fig. 4 ) , i s n o t possible since a level n = 7, J = R + 9 does not exist. O n l y the levels within the triangle limited by the lines op a n d °R can be reached by AR = 0 transitions. The selection rule AR = 0 is not exact, since in a vibrational potential of lower than spherical s y m m e t r y I and therefore R are not g o o d q u a n t u m numbers. AR # 0 transitions are p r o b a b l y i m p o r t a n t for transitions b e t w e e n higher states. K n y a z e v e t aL (271) therefore suggested to take them into account. T w o m o r e P, Q, and R branches are possible, the frequencies of which are given by (z71) -
v = vo + 2B(1 - ~)J(AJ + AR(/(1 - ~))
(22)
These transitions are parallel to the b r o k e n lines in Fig. 4, d e ¢ o t e d by ~Rp, ~RQ, XRR. Such transitions are p r o b a b l y i m p o r t a n t for transitions to states above v3 ~, 2 in S F 6 . T h e y permit the construction of a lot m o r e resonant excitation chains. F u r t h e r m o r e these chains are not
The importance of spectroscopyfor infrared multiphoton excitation
127
limited to any triangle like those with A R = 0. So in principle one single but anharmonically split mode can be excited up to dissociation. It is however very probable that the detuning happens at a certain step to be large enough ( > I crn- i) to stop the absorption. Then the molecule has to continue in another channel (see next section). But suppose the light source provides, e.g. by powder broadening, sufficientbandwidth to overcome these detunings and cvcn to cover most part of the anharmonicaUy split spectrum. (The latter is probably not broader than the thermal spectrum at the same energy.(332)Then the single degenerate mode will stay in resonance with the field.That isthe molecule willapproximately behave like alinear absorber, in spite of its anharmonicity and in contrast to narrow band excitation. W h e n the molecular absorption in the firstfew steps isbroad enough, e.g.by virtue of thermal preheating, the cross section of this linear absorber will be identical to its small signal absorption coefficient.Both predictions are verifiedby recent absorption measurements of SF6 employing picosecond pulses of up to 10 GW/crn 2 power density.(3ss)Since pumping ofalinear absorber generates Boltzmann type population distribution,(2s4'296) the success of a thermal model (s'3°3'3°4) for short pulse excitation is not surprising, although the use of thermal absorption coefficients is not appropriate. (349) The suggested alternating absorption and emissions at high intensitiesshould accelerate the molecular rotation to a much higher degree than proposed by.°I 6) However this expectation was not confirmed by a calculation of the excitation of a rotating classical anharmonic oscillator. It has been tacitly assumed up to now, that all the transitions in the excitation chain are allowed. The zig-zag line in Fig. 4 has even been drawn with a width (indicating power broadening) based on transition moments of the form #,,_ 1,,, = #oxv"n
(23)
Actually the transition moments worked out by (32L322) for a rotating spherical top roughly agree with eq. (23). Only the A~s levels require special consideration. For these levels R = J. As long as AR = 0 is postulated, these levels are not optically connected to the (A~) ground state unless the resultant change in J is zero. For example the A~g level of SF 6 at E 2 - 2E1 = - 7 crn- 1 cannot be pumped by a P- and an ensuing Q-branch transition which would be in resonance with the CO 2 laser P24 line. If however after the P branch transitions to the v3 = 1-state a collision causes a rotational relaxation restoring R = J, this bottleneck is removed. This peculiarity complies with the observation (16s) that the absorption cross section in this wavelength region is much more pressure sensitive than for shorter wavelengths. See also Section 8. Up to this point it has been shown that for certain excitation frequencies and for certain initial rotational states, the anharmonic splitting in cooperation with rotational compensation provides excitation channels up to fairly high energies. In other cases, such resonant chains arc not possible and one has to postulate contributions by other mechanisms, such as absorption by other modes (Section 4), direct multiphoton excitation (Section 7), absorption far in the wings of a band (Section 5), and intramolecular vibrational relaxation (Section 6); although the role of the latter two contributions has probably been overestimated in the past (Sections 5 and 6). Another case in which selection rules require special considerations is COS which has been dissociated by the CO2 laser. (9) The CO 2 laser is resonant with the first overtone of the doubly degenerate bending vibration v2. Figure 5 shows the higher states of interest. The series of levels with l = v2 - 2 only has a small anharmonicity, whereas the ! = 0 series (v2 even) is more anharmonic because of a Fermi interaction with the states (0, v2 - 2,1). The only levels which are reasonably in resonance with the employed CO2 laser frequencies arc the l = v2 - 2 states (v2 even). However, these states can only be populated by Al = 2 transitions, which are forbidden by the Al = 0, + 1 selection rule. Direct multiphoton transitions to these states are not possible either, since the selection rules for them are the same as for the stepwise multiphoton transition. Only in a very high order of approximation, these nearly coincident transitions can acquire intensity.(323) Their cross section will be much smaller than the 10- 20 cm ~ found for the v2 = 0 --* 2 transition (Fig. 1 ). But then the molecule will have difficulties to absorb sufficient photons at 60 J/era-2 ( = 3 x 1021 photons/era 2) to
128
W. Fuss a n d K. L. KOMPA I
I
I
n
OCS O
n 5 -00--
i -I00
X~ ~ N ,
P
'~0(~n_2)01 /~" I ~
02n60 •o~
, 00~'~./ -50
I
• experimental o extrapolated
F n - n . 1047.1 c m "1
02nI'0 0 2nZ0 o~ ~o~
i SO
I I O O c m -1
FIG. 5. Vibrational energies E, modulo E 1 of COS vibrational states close to n • Eco~. The lines P, Q and R arc similar to Fig. 4; excitation is at 1 0 4 3 . 2 c m - ' . Spectroscopic data from. '323}
generate the 60% dissociation observed. Maybe direct multiphoton absorption to an unknown higher state plays a role. This would be a bottleneck which may be overcome only by high power densities. Below these powers the absorbed energy should temporarily leveloff. This has in fact bern found. (l°a) Consistent with a highly nonlinear absorption is also the very steep rise of the dissociation probability from 0 to close to I in a narrow range of laser fluence. (a~On the other hand, recent experiments with picosecond CO2 laser pulses failed to dissociate COS. (360) 4. N E A R L Y
DEGENERATE
MODES
The existence of multipiets, connected by strongly allowed transitions, is not confined to degenerate normal coordinates. Nearly degenerate modes generate a similar level structure. The principal difference is that already the first excited level is split, e.g. because of lack of symmetry. Thus ethyl chloride, C2H5C1, has five infrared active CH stretch vibrations between 2900 and 3010cm-', (2tJ which may be correlated with the partially degenerate vibrations a,~, e,, eg and a2, of C2H 6. The multiphoton excitation of C2H5C1 around 3000 cm- ~has been assigned to detailed sublevels of such multiplet bands, which in turn were studied by overtone spectroscopy. (2') Due to the cross anharmonicities, and due to the differences of the fundamental frequencies the energetic spread of the multiplets increase with the excitation energy, similarly as for degenerate modes. Differences of the diagonal anharmonicities also contribute to this spread. This contribution may be positive or negative. If the highest frequency modes have the largest (negative) anharmonicities, it may thus happen that the width of the multiplet bands first increase, but decrease again for higher excitation energies. Such a case is the CH stretch vibration of benzene. Their spectral width is 90 c m - t in the region of the fundamental (~3000cm-1); it increases up to the 4th overtone (~14000cm -1) to 109cm-', but decreases again to 90 cm- ' in the 6th overtone ( ~ 16550 cm- ').(324) On the other hand, the width of these overtone bands has been widely interpreted (324-33~} as a lifetime broadening due to a supposed internal relaxation (see Section 6). Decreasing band spread is of course unfavorable for multiphoton excitation, and no successful excitation of such a molecule is known to date. In this context the l o c a l {317) mode representation should be mentioned. It describes excited states as a linear combination of states ofanharmonic local oscillators (e.g. the six CH stretch oscillators of benzene). Normal mode splitting is introduced by offdiagonal terms in the Hamiltonian. If this splitting is small compared to anharmonic corrections (as for the benzene overtones), the local mode representation is more useful than the normal mode representation. In the latter picture, anharmonicity often yields large offdiagonal terms in the Hamiltonian. The states described by both bases are, of course, identieal. Table 3 compiles more examples of molecules which have boon dissociated by multiphoton excitation and which have closely spaced vibrations. If infrared active combination vibrations and overtones are included, as is done in the table, it seems to be wide-spread thata new mode can come into resonance just at the time when the mode first excited shifts out of resonance. The whole idea shall be described in more detail for S F 6 .
The importance of spectroscopy for infrared multiphoton excitation
129
TABLE 3. Infrared bands close to multiphoton excitation frequencies
Molecule
excited at (era-a)
0 3
1037.4 (9)
CF3Br
1040-1082.3 (46-*91
CFaI
1028.1-1082.3 ~46'49-621
CF2CI 2
920.8-931.0 ° t ~ l ) 1081.1-1080.0
CrO2Cl 2
969-984 (651
CH3CN
933.0, 1046.9 (es-911
N2F 4
944.2, 975.9 (31.'113'1141
CH aN-H 2
1031.5-1070.5 (HS"t18-12°)
IR bands (era-t) and relative intensities 1042.1 o351 1103.2 1084 (1)°33'3361 1095 (0.1) 1119.6 (0.1) 1028.1 (0.1)°35,337) 1075.2 (1) 1081.0 (0.1) 882 (0.3)O3s) 992 (1)
1102 (1) 990 single band(631 10OO 920 (11(336) 1041 (1) 933 to 103113391 (7 strong bands) 931 (0.t) (33e) 945 (0.1) 1044 (1)
114o (o.3) SFsCI
904.9, 914.9 (t26,t271
SF 6
930-1056 (51
909 (34°1 947 1029 948 (11(333) 992 (0.01)
114o (o.ool) C2HsCi
973.3 (206) 2850-3050 (211
C2F 6
1087.9 (21°)
CFsCOCF 3
944.2-983.3 ~223-2291
SFsNF 2
908.5-957.8 (22.)
S2FI o
920.8_944.2 (t ~s)
974 (2 modes) (1)121'3351 1081 (0.116) 2880 to 3013 (5 strong bands) 1025 (0.005)O361 1063 (0.OO5) 1117 (1) 971 (1)(336) 1027 (0.05) 1070 (0.05) 885 (1) (2241 910 (1) 950 (1)
and several less intense bands 909 938 960 970
For simplification assume that all anharmonic shifts are only energy dependent, no matter over which modes this energy is distributed. Mathematically this means for the v 3 constants of S F 6 x ai oc vi
(24)
Such a model has been suggested for SF6 (3321 and only slight deviations from it were found in. (3331 Deviations up to a factor of 2 from eq. (24) will not change the principle of the following. Equation (24) implies that when in the v 3 v 3 multiplet of states the v a quanta are successively replaced by v 2 + v6 quanta (which have an energy higher by only 45 crn- 1), the new multiplets (v 3 - v2)v3 + v2 (v2 + %) are generated. They are equidistant from each other (distance 45 era-t); each o f them is displaout from its corresponding harmonic position by equal amounts. Such bands,are shown in Fig. 6. Obviously at the time (or even before) the molecule'reaches the limit of one band of levels, it can cross over to ,the next one; that is a v 2 + v6 transition has moved i n t o resonanee, when the v a transitions have shifted out of resonance. Within the newband, v 3 transitiom are again possible (Fig. 6), until these again become detuned. Obviously increase of v 2 and v6 switches the v 3 absorption back from long
130
W. Fuss and K. L. KOMPA f
°
[
i
,
,
......
. . . . . .
f
I
i
,
f
J
]
i
,
J
l
I
f
l
,
,
I
l
,
\ \."e.,, \ .,,-."....'./, \ \ "z. \ \>\"
\.I
N ........ \ -I00
-SO E n
- n E I/era
(3
r
l
....
L -150
'
50
-I
FIG. 6. Vibrational levels of the v3 ladder and a few ladders containing v2 + v~ quanta. Fat line connects populated vibrational states. Broken line arrows indicate coincident absorptions and coincident emissions described in, (~sT) which were interpreted as evidence for the shown excitation path.(~s~)
to short wavelengths. Such a nonmonotonous shift of the "resonant mode" could not have been anticipated by consideration of spectra alone, but demonstrates the importance of level schemes, as was already pointed out in Section 2. By an excitation path like in Fig. 6 it was possible to explain (187) the conspicuous short wavelength emission and the constant position of the v2 + v6 absorption observed (~sm when SF 6 was pumped by 944 cm-1 radiation. The idea was based on the equidistance of pumped rotational-vibrational levels and on the equidistance of bands like in Fig. 6. Thus certain (rotational-vibrational) v2 + v6 transitions superimpose at a constant distance from the pump frequency. In this way they have a good chance to be observed within a region seriously congested by absorptions and emissions. On the basis of the observed spectra it was estimated in, (ls7) that this mechanism--alternate absorption of a few v3 quanta and one v2 + v6 q u a n t u m - - h o l d s to at least the twentieth step. To reach levels of this or even higher energy, Platonenko's rotational compensation (Section 3) certainly contributes; that is, intermediate emission steps (Fig. 6) convert the energy difference of vibrational sublevels into rotational energy. While Fig. 6 is drawn for 944.2 crn-1 ~xcitation, similar Figs. will account for every frequency to the red side of the Q branch (948 crn- ~). But for shorter wavelengths the difficulty is that the v3-band will be detuned before the v2 + v6 transitions come into resonance. However, Fig. 6 is incomplete as certain sublevels ofv3v3 + v2 + v6 have been omitted which are only weakly optically coupled to the v3v3 levelsJ 1ST) They may be responsible for shorter wavelength excitation. An indication of such a level has been found in (~sT) by overtone spectroscopy. After several steps at short wavelength, the next combination vibration v4 + v5 (Table 3) may also come into resonance. This would explain the v4 emission observed in "92) and (334). The emissions in (~9/) however, are probably already strongly affected by collisional redistribution. Another interpretation" s7.334) pointed to the levels v3v3 + 2v4 + 2v6, which are slightly higher than (v3 + 2)v3. But these levels can only be reached by XilAvi[ = 5 transitions which will be very weak. It has to be noted that the excitation mechanism suggested in this section only employs the two strongest transitions in the 950cm- t region. Assuming that every step is sufficiently well in resonance (see next section), the rate determining step will be the v3v3 -~ v3v3 + v2 + v6 transitions. Their first member (v3 = 0) is 100 times weaker than the v3 band. Increased intensity borrowing by Fermi resonance for higher v3 tends to level offthis difference to some extent. Imagine a molecule where the corresponding pair of transitions has comparable oscillator strength from the beginning, then the absorption cross section would not drop, but remain at its small signal value. Such a molecule, though anharmonlc, would behave in absorption like a harmonic oscillator up to an energy where also the second mode shifts out of
The importance of spectroscopy for infrared multiphoton excitation
131
resonance. Figure 2 shows, along with the more typical case ofSF6, that for several molecules the multiphoton absorption probability is not much smaller than the spectroscopic absorption cross section. In fact for CF3I, SFsC1, SFsNF2, CrO2C12 and other molecules (Table 3) adjacent transitions have similar strengths. To explain the ease of excitation of such molecules, the quasicontinuum (see next section) has been invoked. (1~s) But every broadening mechanism has to reduce the cross section to below its small-signal value. It may be of interest to discuss in this context the possibility of mode selectivity. Obviously in the suggested mechanism the v3 and the v2 + v6 mode contribute; six degrees of freedom (part of them are degenerate) have been used to draw Fig. 6. On the other hand, the degeneracy of states actually populated only varies between 1 and 3; that is, it is nearly independent of the excitation energy: g. ~c E °
(25)
This is similar to a classical s-dimensional oscillator g, oc E~,- 1
(26)
with s = 1. (Quantum mechanics replaces eq. (26) by a factorial function, (32°) a detail which is unimportant here.) A higher s would apply if either the excitation path branches again and again, or if more and more background states become optically connected by intensity borrowing (next section) to the "primary states" of Fig. 6. A small s has been postulated for SF6, (4'2s3'2s4) hexafluoroacetone (225) and tetrafluoro-l.3-dithietane (21s) on the basis of an analysis of absorption and dissociation data by means of a rate equations model. Such an analysis predicts a population of the energy levels, which is a product of a statistical weight (as eq. (26) for example) times a population per state. (284) The latter can be deduced from the dependence of absorbed on incident energy. A clearcut ease is a linear absorber like C2F4S2;(215) in this ease, incoherent optical absorption always produces a Boltzmann distribution of population, (284.296)independent of complications like rotational hole burning in the initial state. In contrast to the absorbed energy, the dissociation depends strongly on the weight function g,. Large s implies a function of dissociation probability on laser energy which is steeper than found experimentally for molecules like SF6 ('t'283'284) C F 3 C O C F 3 (225), C2F4S2(2 ! 5), as well as for C F 3I, S2F 10, SFsNF2 and SFsC1. On the other hand the difference is not very large. By a similar analysis (based however only on a single set ofdatalike dissociation or quantum yield as a function of incident or absorbed energy) other authors(285,288,297,304,305a) therefore found consistency with experiment when they assumed the maximum possible s. This assumption will certainly be correct for levels close to the dissociation energy, where the energy can freely flow between degenerate states (Section 6). So there will be no chemical consequence (mode selective excitation or not) if in the preceeding steps all or not all the degenerate states are populated. The principal effect is the energetic width of the population distribution: For large s it is very narrow, whereas for small s it is broad. (2s4) 5. THE WIDTH OF THE RESONANCES, INTENSITY AND FLUENCE DEPENDENCE
Up to this point the multiphoton excitation was discussed in terms of sharp states and transitions. It is now necessary to discuss power broadening at this point. For SFe at 50MW/(~rl 2 this broadening is already (~(v~-l)-.v~) ~/hc ~ ~ 3 - 1 c r n -1, whereas the average distance of v3 transitions inferred from Fig. 4 is about 3 cm- 1. But if only power broadening compensates for detuning, the whole process would predominantly depend on power density. This is in contrast to what has been observed for SF6: Nearly identical dissociation yields and absorbed energies have been measured for this molecule when CO2 laser pulses of very different pulse lengths and shapes but equal energy fluence (time integrated intensity) were employed.(!53-156)The most popular model of multiphoton excitation therefore asstunes in the higher steps a dense Cquasicontinuous') spectrum, giving rise to fluence dependence, whereas isolated resonances and coherent excitation are assumed for the lower steps.
132
W. Fuss and K. L. KOMPA
The coherent steps can be quantitatively described by adding to the Sch6dinger equation of the molecule
- ih~-~~
=/./~o1~
(27)
the (dipole) interaction with the time dependent field/~(t)
-ih~o
= (/./tool + k£(t))(p
(28)
When the rotating-wave approximation (which is not equally applicable for resonance and nonresonance t287~) is adopted, a time independent part can be separated: ei~oi = ¢
(I'l~o°l+ ~oEo)~oj --*
t
(29)
where Eo is the field amplitude and fl~j are the transition moments between the molecular states i andj. Detailed derivations are given in textbooks. °42~ Solution ofeq. (29) reduces to the problem of finding the eigenfunctions q~ (the so-called dressed states) and eigenvalues e~ (the so-called quasi-energies) of the matrix in eq. (29). If the molecular energies/./mot and transition moments #o are available, one can easily calculate the time evolution of a system of 100 levels, for example. (This number is not so large in view of the large number of rotational-vibrational sublevels.) The method (and another one based on the nearly equivalent Bloch equations) has repeatedly been applied to multiphoton excitation, with various assumptions on the molecular parameters, t246-262.2a°'2sl~ It is important to note that reliable predictions are also possible and have been made on direct two- or more-photon excitation t2*6-2s6~ and on coherent propagation effects, cat s~ If the calculations are done in terms of density matrices, relaxation effects can also be included, t3°°-3°2~ Recently another quantum mechanical method has been suggested which, although based on perturbation theory, should be capable of any desired accuracy, t269) For systems of isolated levels these calculations yield oscillations of populations, a result which is typical for a coherent excitation. The oscillation frequencies depend on detunings and transition moments. If the laser excites many closely spaced sublevels or many levels with random dipole moment, their populations will oscillate at different frequencies. When added together, the oscillations will disappear in the sums, and these coarse-grained populations will only smoothly change. Such behavior is typical of incoherent excitation. Similarly, if the laser excites molecules in several different (thermally populated) initial states and when the measurement only concerns their average (like the absorbed energy or the dissociation yield), the excitation will also appear incoherent. From one of these conditions or a combination of them, rate equations can be derived t2s~'29t-293~ and have been used to analyze multiphoton excitation, t*'2sa~2as'2aa-29°'294-3°°~ If only step-by-step transitions are considered, they write ~2s,~ dN~ d~-
I~ N~-
g~ N,+ 1 +Ie~_ 1 N~_ 1 g~+l
-~ N~
(30)
which can easily be solved for the populations N~ of the levels of energy ehvma~e~, if the absorption cross-sections tr~ and the degeneracies g~ ( = number of states in level e taking part) are known. Division ofeq. (30) by the laser intensity I (t) and substitution ofd~ for I dt shows that eq. (30) does not depend on pulse shape or length, but only on the energy per c n ' l 2 (fluence) tb defined by ~ =
f2
I(t)dt
(31)
The solutions of two special cases t2s4~ are worth noting: (1) if ~ = %e + const. (as for a harmonic oscillator), N, e g, exp( - e/~. q~)
(32)
The importance of spectroscopy~forinfrared multiphoton excitation
133
which is a (pointwise) Bottzmann, distribution. (Remember that n i~ integer.) The absorbed energy is (~) oc Oo~
(33)
2) Ifo~ = ~ro, a Cranssian distribution results N~ • g~exp( - ~2/200 ~ )
(34)
The absorbed energy is
In both cases, the dissociation probability P is an Arrhenius-like function of P ~ exp(-@o/@)
(36)
(@o is a parameter), if g, does not much vary; it rises more steeply with • if g~ increases strongly with ~. The latter will happen if all the modes of a molecule contribute to the absorption. Terms describing dissociation.rates and collisional or collisionless deactivation rates may be added to the fight hand side ofeq. (30): (4,285,~ss-290,294--298) The transformation from a time to afluence derivative isthen not possible any longer, and a certain intensity dependence is generated ~2s9,29°~ although the dissociation yield again becomes fluence dependent when for very high excitation a steady state is reached by the competition between pumping and dissociation. ~zsg) It has been shown on this basis, that pumping rates can be derived by measuring how quickly P (~)converges to 132s9~Another intensity dependence is introduced by the assumption that g~ (in particular go, the number of rotational states removed from the vibrational ground state)depends on the power broadened bandwidth ( ~ / ] ) . However, the more logical way to treat such effects is to combine the coherent description for the lower levels with the rate equations for the higher ones. ~2s°'2sl~ Such a calculation also reveals a strong intensity dependence ~es°~whicharises if the laser frequency hits at least one transition only far in the wing of its power broadened width ("intermediate level bottleneck"). Passing such a bottleneck has been called "leakage" intcCthe quasiconfinuum. ~'62~To overcome this, a certain minimum intensity is required. This is in contrast to the 11/2 dependence expected at low ! for the number of rotational states removed from the ground state (the "rotational bottleneck "~343~ and to the approximate I - 1 dependence of dissociation yield expected at high intensity on account of the finite rates of dissociation and internal energy redistribution. By varying the p~se length the question has been examined for several molecules to which extent the reaction probabilities depend on energy fluence or on intensity. The most extreme examples are the molecular ions C3F ~- and [(CeH5)20]2 H+, 100% of which dissociate within seconds when irradiated by a continuous-wave CO 2 laser at intensities as low as a few watts/era232 ~9,e20~If the excitation would reallybe collisionless, the detuning of every step in the ladder had to be smaller than the estimated power broadening of 10 -5 cm-~ for every molecule. However, the promoting effect of the background gas was shown in322°~ Furthermore the ions have been generated in high vibrational states by the electron impact ion source. The dissociation of SF 6 has been investigated using various pulse lengths cl53,15,t~and shapes, ~1s s, t 56~ as mentioned above. The energy absorptions have also been measured by these pulses. ~163'r6~) :Far from the dissociation threshold the absorbed energy is much larger for the higher intensities. But the difference is still of a magnitude expected for rotational hole burning (Section 7). But even at average energies of 35 quanta/molecule the absorption keeps increasing with intensity. ~3s s) This has been ascribed to the presence ofnarrow spectral features at these high levels of energy~3s s) (see also Section 3). A larger difference has been,found for dissociation of CF3I close to the threshold335) An intermediate |evel bottleneck, seems to play a role. Other examples of such a bottleneck are CHCIF2 ~ ~'3"~and OsO~.~73)In other molecules where strong intensity effectshave been found up to now (C2 H~CI,~ ~0 ~~ H z C O t22'2 s)) the rotational structure of the first absorption step is so wide that the rotational h o l e b u r n i n g will dominate in dissociation and absorption
134
W. FUSS and K. L. KOMPA
measurements. No example of an inverseintensity dependence ofabsorption or dissociation is known. Although it is not established for many molecules, it is generally believed that multiphoton excitation depends predominantly on energy fluence and that intensity effects are more or less small corrections to this limiting case. It has been the merit of the quasicontinuum model to predict this fluence dependence. An early version ~344~of this model suggested that the first few steps occurred via rotational compensation, whereas in the later steps the molecules have a large number of densely spaced Cquasicontinuous") absorption lines which are distributed over such a large spectral range that there is always resonance. The idea ofa quasicontinuous broad spectrum was inferred from the very large number of vibrational combination states at high energies, i.e. the number N of combinations of vibrational quantum numbers by which a given energy E can be realized :~345a~ N(E)=
~[ (v i + d i - l ) ! vi~-_--l~ "
~,
(37)
~,vihvi=E i = 1
(d~ = mode degeneracy, s = E d~) Approximately N oc (E + a) ~- 1, where a is a fitting parameter, t34s~ The broad spectral quasicontinuum eliminates the need to study spectra of highly excited molecules. It has been recognized, ca) however, that selection rules, even approximate ones, tend to concentrate the oscillator strength within a spectral range of a few cm-1 width only. Thus the v3 band halfwidth of S F 6 heated to 1780 K (= 15000 cm-1 average vibrational energy)is 42 cm- 1 o n l y , (34"3) whereas the calculated width of the rotational envelope at 1780 K is 24 cm. That is, the vibrational contribution to the width is small. Similar conclusions result from the often narrow IR spectra oflarge molecules at room temperature (see Section 1 and Fig. 1) and from the sharp ( ~ 5 cm-1) structures observed in double resonance experiments ~ls6J where the s p e c t r u m of S F 6 pumped by a CO z laser was measured. The effectiveness of nonrigorous selection rules has also long been known in electronic absorption or emission spectroscopy, where transitions to very high vibrational states ( ~ 30000 cm- 1) are neither broadened nor congested by other modes, e.g. P F a, or the acetylenes,taz°b~ However, the small halfwidth does not exclude the existence of far reaching wings of the absorption bands of highly excited molecules, tzs4~ These wings as well as the (possibly quasicontinuous) substructure of the bands can best be described in terms of intensity borrowing: Start from a (not necessarily harmonic) zero-order basis in which oscillator strength is localized in a few transitions only. Consider an infrared inactive transition ~,o -~ ¢~ which is close (with energy distance AE) to an active transition ¢o '-* ¢°. The anharmonic potential terms in the Hamiltonian will in general admix a little bit of ¢° to ¢,~, the admixture coefficient co~ being c.~ oc H . J A E
(38)
in perturbation theory. The transition probability for ~'o ~ ( ~ + c°i~'°) is then #2 oc c°2
(39)
The AE denominator ofeq. (38) is responsible for the clustering of transition probability in a narrow spectral range. The interaction matrix element can be extremely small. As a rule of t h u m b ~32°a) it drops exponentially with increasing total difference Av Av = ~
Iva - v~l
(40)
modes
of the quantum numbers v° and vi of the zero-order harmonic states ~. and ~t. As a consequence, the optical excitation tends to use only a small subspace of the vibrational degrees of freedom. It has, however, been pointed out ~6°'75") that states differing by a large Av can also be mixed by a sequence of intermediate steps coupled in lowest order. In this case the simple relation (38) has to be replaced by an eigenvalue problem. In this way more transitions can acquire intensity than predicted by the consideration above, although they will still cluster
The importance of spectroscopyfor infrared multiphoton excitation
135
in narrow energy ranges. Further investigations are necessary to clarify whether the density of spectral lines is equal to the density of states, or whether they differ by orders of magnitude. Thus it is conceivable that thenumber of spectral lines per cm -1 is much smaller than the number of states per cm- I which is e,g. around 109 cm- t for SF6 at 20000 era- 1. It should be pointed out,in this context that a few lines within the power broadened bandwidth ofthe laser (often ,~ 1 cm ~-1) are sufficient to make the excitation incoherent (fluence dependent). (2sa) This number may be even smaller if a random distribution of the transition moment is taken into account.(291.292) In the next section evidence is presented that not the full number of states, available at each energy, is populated. This will also be evidence for a sparse spectral line density. 6. COLLISIONLESS VIBRATIONAL RELAXATION In a molecular beam experiment it has been found that the multiphoton dissociation of SF 6 does not release any kinetic energy into the fragments. (t49' 150)This fact primarily proves that there is no potential barrier in the dissociation channel. Most importantly, however, it proves that before dissociation the energy can freely flow around among the vibrational degrees of freedom :(I so) I~ instead, the energy would be localized, a nonnegligible fraction of the energy of the last absorbed quantum should appear as kinetic energy. Vibrational energy redistribution on a time scale of the order of a vibrational period is the basic assumption of modem unimolecular reaction theory (RRKM theory). °45) Subsequently many multiphoton dissociation experiments have been found to be consistent with this theory. (6) Such collisionless vibrational redistribution can occur under two conditions. Firstly, both the energy and the anharmonidty have to be large enough. A quantitative criterion does not exist. It is helpful, however, to investigate the question by classical calculations of trajeetories on an anharmonic potential sLLrface.(273-2~5) These trajectories become (locally) unstable (i.e. change the mode parentage) in certain high energy regions of the potential. (2~5) Furthermore an energy threshold seems to exist above which the trajectories very rapidly lose their memories for the initial conditions and above which they seem to become ergodic. (273) On the basis of such classical calculations it has been suggested as a rule of thumb that (apart from a few narrow ranges of configuration space) vibrational redistribution can generally occur above about two thirds of the dissociation energy (34s) although it may be slow even above dissociation threshold in less common cases like van der Waals molecules. However, the possibility of coUisionless vibrational energy flow is seriously restricted by the second ~condition, which is based on energy conservation. It allows redistribution only between states of (within the uncertainty principle) exactly equal energy. (24s) This uncertainty width is (in the absence of collisions) equal to the reciprocal of the radiative lifetime of a few tens of milliseconds usually, that is ~ 10-9 cn]-1. (Note that this natural width is 6 orders of magnitude smaller than when electronic excitation is involved.) Only a density of states much larger than 109/era -I will provide su~cient acceptor states for redistribution. (24s) This number is reached for SF 6 above 20000cm -1 ( = 6 0 ~ of the dissociation energy), but not reached below dissociation threshold, e.g, neither for ethylene nor vinyl chloride, not to speak of the triatomics. So for molecules up to the size of SF 6 collisionless energy redistribution cannot occur during most of the excitation steps, although it does occur (shortly below and) above the dissociation energy. This statement only refers to narrow bandwidth excitation. Quantum mechanically, temporal evolution results from superposition of states of different energy, and the narrowest bandwidth of states to be superimposed is the natural width. Larger widths, and therefore faster temporal evolutions, are generated by external means like power broadening or the transform limited width of short pulses. Generally only those states exchange population with each other which are directly optically excited. That is, a probe 0ike the pump laser itself) which monitors the population of the whole set of states together will not see any temporal evolution after the end of pumping, whereas a narrow-band source can see it. In agreement with this expeetation, "collisiouless relaxation" of IR absorption of pumped SF 6
136
W. Fuss and K. L. KOMPA
has been seen (~s4~ when monitored by a cw CO 2 laser (< 1 MHz width) but has not been found when probed by a TEA CO 2 laser (~1 GHz width).(~st) In the case of transform limited pulses, the temporal evolution (of sublevels) will be the faster, the shorter ( = larger bandwidth) the pump pulse; a probe pulse of equal length, however, will not see any temporal evolution apart from the one associated with absorption or emission. Such a case applies to an experiment~s7"~ in which the transmission of SF 6 pumped by a picosecond CO 2 laser pulse was probed by a delayed pulse of the same kind; the transmission returned to its initial value already during the pump pulse. This temporal evolution must be caused by excitation and subsequent deexcitation similar to adiabatic fast passage, (187") but not by relaxation among sublevels which was suggested by (lS7a) as the preferred alternative. The same reason will have caused the restoration of absorption of a (narrow-band) cw laser probe after pumping SF 6 by nanosecond pulses. (lss'xs9~ Since the energy can only flow between states which are directly populated by the excitation laser, the number of contributing states is again governed by the density of spectral lines, which can be very much smaller than the density of states (Section 5). Furthermore the number of lines excited via power broadening should not be overestimated: When the spectral intensity is distributed over more and more lines, the individual transition moments (and the power broadened widths associated with them) decrease. It should also be noted that the term power broadening only concerns the spectral response to the pump; the states are not broadened but remain sharp as before (apart from two sidebands of each state due to modulation splitting), a fact which can be demonstrated by a weak tunable probe. Nevertheless many published models assume collisionless vibrational redistribution already in low lying levels.3'299-3°5a) Reduced density matrix techniques(3°°-3°2~ or rate equations (299'3°5~) are used to describe the process quantitatively. Usually it is not explicitly stated why coltisionless relaxation is postulated and why it should support the excitation. Some models seem to invoke the broadening by spectral redistribution of oscillator strength. Others apparently assume that the anharmonic shift only depends on the quantum number of the "driven mode" (v3 in SF6), which is lowered by relaxation. As it was pointed out in,¢3) the probability distribution of this quantum number would be exponential for a microcanonlcal ensemble (i.e. for equal population of states of a given energy), if the number of degrees of freedom would be large enough. Therefore it was suggested(3) that the absorption spectrum of each level with the energy E should be (nearly) identical to the spectrum of a canonical ("thermal") ensemble of molecules with the average energy E. This assumption was quantitatively analyzed by means of rate equations for excitation of S F 6 .(34.9) Gross disagreement with absorption measurements below 10 quanta/molecule was found; especially meaningful was the comparison C3~9~with picosecond absorption, because in this case all rotational states are excited and no ambiguity is left about the width of a rotational hole. In the same reference (349) the mierocanonical assumption was also checked: Thermal spectra of SF 6 were considered as a superposition of spectra ~r~ of the energy levels E. Deconvolution from the Boltzmann distribution yielded the individual ~rE which were then inserted into the rate equations (Section 5). Again gross deviations from measurements of collisionless multiphoton absorption were found. Presence of collisions improved the agreement, as expected. A further evidence against microcanonical distributions are the spectra of highly excited SF6 .~186) The narrow structures found therein do not comply with the broad bands postulated because of the distribution ofanharmonic shifts. By probing the U V absorption of OsO 4 it was found, that the (UV active) iJ1 mode was excited only after the laser pulse which pumped the (IR active) v3 mode; (TS) unfortunately, the absorbed energy was not given. As explained in Section 4 and in (334) (by the v2 + v6 mode of SF6), emission of a mode different from the pumped one,¢192'334) does not prove equlpartion. Stimulated emission as observed in the v,t region of SF 6 when v3 was pumpe d{33o) even proves the existence of narrow spectral features: if everything would be broad, stimulated emission would be superimposed and quenched by v4 absorption of more strongly populated lower states. It should also be noted that, if the molecule had to wait for a relaxation process in an early
The importance of spectroscopyfor infrared multiphoton excitation
137
step, the absorption should be least effective for the shortest pulses (of equal energy). The absorbed energy q should approximately be inversely proportional to intensity I,
qoc1-1 Thereby even the increased hole burning by power broadening (oc 11/2)should be overcome. But an increase of absorbed energy and dissociation yield with I has always been observed. ~s) 7. ROTATIONAL HOLE BURNING, DIRECT TWO-PHOTON ABSORPTION, TWO-LASER EXPERIMENTS, AND TEMPERATURE EFFECTS At room temperature many rotational states are populated. Rotation splits each vibrational transition into three lines or less, but not more. Any additional (rotational) splitting, e.g. by centrifugal distortion and Coriolis effects or by the hyperfine structure, is due to molecules in different initial states. Nearly each absorption line corresponds to a different molecule, and the rotational structure is therefore basically inhomogeneous. That is, excitation by a (e.g. power broadened) bandwidth smaller than the width of one rotational branch will burn a hole into this branch. The fraction of molecules removed from the ground state is approximately equal to the ratio of these two widths. If the first step is within the (narrow) Q branch, this fraction is high, but may be reduced in the next step which may employ an R branch, for example. Repeatedly attempts have been made to determine quantitatively the extent of rotational hole burning, tS'53"Ss'66~ It is known that the dissociation probability of several compounds (e.g. SF6, {133}CF3I,<53}$2F10,~1~s) CF3COCF3~22s)) reaches = 80 ~o already at a modest flux, whereas on the basis of the ratio of 21~E/hc (power broadening at the corresponding intensity) to the rotational width only less than half this value would have been predicted. Other evidence at lower flux~s3.Ss'66) though less direct, also indicates that more molecules are excited than expected. An almost 100 ~o depletion of rotational states was concluded from three other experiments. {11,xs6.190) The first one of them found an absorption independent of pulse length which was varied from the collisionless to collision dominated regime) 111) Collisions should have filled up the hole and increased the absorption. But as pointed out by,{5) the smaller bandwidth of the longer pulses counteracts this increase and may compensate it. An additional compensating effect is due to the change of the spectrum in the higher steps, as explained in the next section. Other experiments {1s6. t 9o) found an unexpected strong bleaching of absorption of rotational states of SF 6 when the molecule was pumped by various other frequencies. However, according to a later interpretation <~sT~this bleaching was only feigned by emissions from higher populated levels to unpopulated sublevels. A few collisions seem to suffice to equalize these sublevel populations, thus quenching the emissions, whereas the rotational quantum number changes by a few units only ( ~ 1 per collision). Thus the rotational absorption hole survives nearly unchanged, {1s7) and the "spectra of pumped SF 6 after a few collisions"{~s6) present a visualization of this hole. To summarize, it remains to be explained why about a factor of two more rotational states respond to the laser radiation than expected on the basis of power broadening. Most probably the cause lies in direct two-photon transitions. ~ls6,26s) Their spectrum is shifted by virtue of anharmonicity to longer wavelengths and contains rotational branches with AJ = __.2, ___1 and 0 (and perhaps also some with AR ~ 0~271)).In general, it partially overlaps the one-photon spectrum. Thus additional rotational states can be excited. The importance of direct multiphoton transitions has been discussed and confirmed in terms of the Schr6dinger equation.{2,~-2s 6}The most direct experimental evidence is the observation that the excitation probability of molecules like boron trichloride <19a>or hexafluoroacetone ~22s)has its maximum well outside the range of the linear (one-photon) spectrum. Furthermore, two- and threephoton resonances have been found in the form of sharp spectral peaks around 3000 cm- 1 in the dissociation probability of C2HsC1.~2°6") At room temperature only the small fraction
J~>QE 7 : 2 - E
138
w. Fuss and K. L. KOMPA
(Qvib is the vibrational partition function) of the molecules is in the vibrational ground state. It decreases with s, the number of modes, especially if they have low frequencies. The other molecules are in low lying excited states. The transitions starting from those states ("hot bands") are slightly shifted due to anharmonicity, but are otherwise similar to the fundamental band. It is appropriate to return again here to Fig. 1 which compares the room temperature absorption spectra of a small and a large molecule. The larger one has an average vibrational energy of 1500 crn- 1. At this energy the density of states is approximately 700/cm-1. Therefore molecules of this size have been said to "be in the quasicontinuum already at room temperature. ''ta'l ~8) As was pointed out before, however, the spectra in Fig. 1 do not look very different with regard to their absorption width. No high resolution I R spectrum of a highly energized molecule is known. But probably each rotational state will have several vibrational transitions, since close lying states in the upper level will mix and thereby exchange spectral intensity. Thus the homogeneity of the spectrum is increased and hole burning effects are reduced. Complete homogeneity of room temperature S2F 10 has been invoked in a 78) because of the absence of any pressure effect on absorption up to 5 quanta/molecule. On the other hand at higher energies, the absorption cross section was increased by addition of buffer gas. This effect was assigned to V T relaxation, tS~ However, a simpler explanation based on the change from linear to nonlinear absorption is presented in the next section. Multiphoton absorption via hot bands can be described by a level scheme similar to Fig. 4. Preexcitation places the molecules into a level above the lowest end of a parabolic funnel. Ifa molecule in the uppermost sublevel ofa multiplet is excited at long wavelength (as sketched in Fig. 7), it remains for a large number of steps within the limits of the two parabolas, i.e. in near resonance with the most intense band of transitions. Approximately linear absorption is expected in this case. In fact when preexcitation was achieved for SF 6, the absorption cross section at longer wavelengths (measured opto-acoustically) was nearly independent of the probe laser energy between 0.1 and 1J/cm2. tla2) This cross section is found to be ~ 10-18 era-2 (via calibration by single laser absorption measurements of ~182) and t186~). This magnitude is equal to the one found in a similar experiment tla6~ where the probe laser had only 0.5 mJ/crn 2. Thus the linear range is rather large. This observation has important implications for isotope separation in cases where the spectral isotope effect is small: A first
n
9 6 3
En-nE1
0
FIG. 7. Effectof preexcitationfor moleculeswith bands of levels(due to anharmonicsplittingor to nearlydegeneratemodes).(a) Whenpreexited(thermallyor by laser) to the highenergyborderline,a very long wavelength(straightline) remains for a long time within this band of levels (within the resonantmode);(b) Whensimultaneouslyexcitedby two frequencies,the moleculestaysin resonance for a prolonged time by alternate long-wavelengthabsorption and short-wavelengthemission.
The importance of spectroscopyfor infrared rnultiphoton excitation
139
weak laser excites one isotope to a level which has a longer wavelength absorption, C4~'8°~far beyond the band origins of the other isotopes. A second laser drives the molecules at this long wavelength up to dissociation. This methOd has successfully been applied to OsO4. ~6s'71,72'74~ Since the absorption probability is (and remains) high, the second laser must not be intense either; thus, single frequency excitation of a second isotope (which requires higher powers) is avoided and selectivity is better preserved than in single frequency methods and methods with a second, intense nonresonant laser. ~3-t'xs°) This method not only reduces the power required, but also dramatically increases the quantum yield. The latter effect is also due to the high absorption probability, which has been shown to be of crucial importance for economic utilization of photons. ~x9a) The absorption may be further improved by irradiating the sample by both frequencies simultaneously. ~1s 1) As shown in Fig. 7, the molecule can then climb up in a zig-zag path, employing long wavelength for absorption and short wavelength for emission. Unfortunately, simultaneous two-frequencies irradiation reduces the isotopic selectivity; given one of the frequencies Vl, the second frequency v2 will promote the excitation, be it at long wavelength (absorption continues via v2, Fig. 7a) or at short wavelength (v2 stimulates emission so that vx can be absorbed again, Fig. 7b). So the frequency response is also broadened to the blue side, as found in. ~181) Broadening of course reduces the selectivity. This enhanced dissociation on the blue side by simultaneous twofrequency excitation has not been explainable by other theories. Let us return to thermal preexcitation. Again the long-wavelength absorption will be very much enhanced and its linearity will be improved. Both effects have been found when SF 6 was heated up to 600°C. ~6~) No enhancement should appear at short wavelengths according to Fig. 7 and a slight decrease has even been observed in. (161) This is probably due to thermal population of rotational and vibrational states whose absorptions scatter over a broader spectral range (preferably at longer wavelengths) than at low temperature. At 944.2 cm- x the absorption was nearly temperature independent. ~16~)The dissociation yield at this frequency, however, increased strongly below about 200K. ~16°) For quantitative calculations of hot band effects on absorption compare. ~255) 8. COLLISION EFFECTS Lyman et al. tS) have given an extensive compilation of collision effects in multiphoton excitation. The following principal effects were discussed: (1) Collision induced pressure broadening, which will preferentially have an influence on the coherent steps of the excitation. This effect has yet to be seen experimentally. (2) Rotational hole-filling increasing both absorption and dissociation, which has often been observed. The effect is strongest when the hole is narrow compared to the rotational structure. (3) V-R, T relaxation withdrawing vibrational energy from the molecule, thereby decreasing dissociation, but increasing absorption. This effect has been observed in molecules with small rotational constants, where the hole extends over a large part of the spectrum. (4) Intramolecular vibrational energy redistribution altering degeneracies, the consequences of which are less distinct. In (1) to (4) buffer gases as well as the pumped gases are effective, which is in contrast to the following. (5) Vibrational energy exchange between molecules of the same kind either enhancing or diminishing the absorption, depending on how it modifiesl the vibrational energy distribution produced by the laser. Processes (3)-(5) have another consequence which has apparently not received due attention. They change the absorption spectrum of the individual molecules. This shall be illustrated'by considering two examples. Consider the first few Steps of pmnping of a degenerate mode whose higher states are split by anharmonicity into multiplets, Thelaser only populates certain sublevels of each multiplet (Fig. 4). Collisions will, redistribute this population into all sublevels. With the rule of thumb
140
W. Fuss and K. L. KOMPA
that t r a n s i t i o n s from one e n d of the level b a n d to the other e n d are less p r o b a b l e t h a n others (derived from the a p p r o x i m a t e selection rule AI ----_ 1, see Section 3 a n d (t s7)), the spectrum is predicted to shift to longer wavelengths. This collisional shift will be the larger t h e higher the excitation, since the a n h a r m o n i c splitting increases with energy. It m a y overcome a n y spectral b r o a d e n i n g which is c o m m o n l y looked u p o n as the m a j o r effect of collisions. I n Fig. 8 a b s o r p t i o n cross sections of SF6 at two different pressures are compared. At frequencies lower t h a n 944 c m - 1 the cross section increases with pressure, b u t at higher frequencies there is n o
930
lO-te
9t, O
950 cm "t
i l l l l l i l l l l l i l
61cma O.SsO,T "0~ ~0~.0,.,~ ~ 0
/
0.3 Jcm "2
""o
o" 1J cm"z
e ~
I0-,9
X
: i
t
I
930
t
t
i
i
I
9/.0
~
I
I
i
\.
I t 950 cm "1
FIG. 8. Absorption cross sections cr (defined in Fig. 2) of SF6 for various laser enerEies.(347) Measurements were made with the multimode TEA CO 2 laser (70ns spike, 3p.s tail) and the waveguidecell (10ram diameter, 1 m length) of.(t ss) The latter was used in double-~r~_~;this makes a very homogeneous distribution of intenmty even for absorption as high as 50~o, so that the evaluation~s)is very simple. SF~ densities n were used such that ~. n. 200cm = 0.2-0.7. The second part of the Fig. shows the effectof density on the spectrum. Other absorption data are compiledin.(s~
The
importance of spectroscopy
for infrared m u l t i p h o t o n excitation
141
change. Such an effectis generated by raising the absorption everywhere by means of rotational hole-filling and by simultaneous shift due to sublevel repopulation. The pressure independence of SF6 absorption at 944.2cm-1 has been a matter of some debate (see the review 5). The explanation presented here does not have to assume an excessivelybroad hole in the rotational contour. At the same frequency, dissociation is strongly diminished by addition of buffer gases. This has been ascribed to V - T relaxation. ¢159~But both V - T relaxation and rotational hole filling increase the absorption, in contrast to observation. In, ~349)the change of the spectrum of SF6 by collisions has been quantitatively taken into account. As a second example consider a molecule which is a linear absorber owing to the following peculiar mechanism: Suppose the molecule has two or more modes of similar oscillator strength, the second one of which comes just into resonance with the pump frequency when the first one moves out. In such a way the molecule can stay in resonance over a considerable number of steps, as explained in Section 4. The molecule will behave as alinear absorber if for all steps e the net absorption is constant:
~
-
a 7 = ~o
(42)
Here a7
= ~ - I "g~-
l/g~
(43)
The observed cross section is then constant, too: O'obs =
~,N,(o: bs - -
O~era) = a o
(44)
where N, is the relative population of the level 8 and Y,,N~ = 1. Condition (42) would also be met by a harmonic oscillator, for which (45) 0"~cm ~
0 o "
8/d
(46)
(d = mode degeneracy). Consider collisional redistribution between (rotational or vibrational) levels of a spacing small compared to the laser quantum. This relaxation will fill up the (rotational) hole in the initial wbrational state, but will shift the molecule out of resonance in the later steps. The latter effect can occur although the spectra can have a width of several crn- 1, since the spectra of vibrational sublevels differ considerably. Rotational and vibrational sublevel redistribution will proceed with collisional efficiency of almost unity, that is to say at a nearly equal rate. Denote by r o the vibrational-rotational sublevel resonant in an individual step of an excitation route like in Fig. 4. r o will be a function of the step number 8. Nonresonant levels will be called r'. Then eq. (30) can be extended to read dN~'°=IA(a,_L,og~_,.,oAN~-I"° I dt g.- 1,ro 1
~, ( k , o ~ r ' N ~ , o - k , , ~ , o N v , , )
(47)
r" • ro
The buffer gas pressure is contained in the rate constants k. The question to ask then is whatwill be their influence on the absorbed number of quanta (e). First sum eq. (47) over r. The relaxation terms will cancel, as expected, as e is not changed by them: dN~ = IA(tr._, ,.g.-t ,o AN~- t"*l dt ' ' g~- 1.,. I
(48)
(N~ := E,N~ and ~r~, = a~, ° 6,, 0 were used). Multiplying by e and summing over it, one obtains by rearrangement
d<8>. d N._t, * dt "= ~ Z , eN, = I E,e~_ 1,,og,_I.,0A g,- I,,.
(49)
This is equivalent to eq. (44) ifone defines :=
(50)
142
W. Fuss and K. L. KOMPA
Thus d
(51)
In general aobs will depend on relaxation rates via N~,o, which have to be calculated from eq. (47). For a linear absorber, however, eq. (42) may be used to get trob.~ = ao,° X~N~, °
(52)
This sum does not depend on relaxation, as may be shown by summation ofeq. (47) over e. (The independence of the k's of e has to be used thereby.) It is equal to the fractional population N~o of the (rotational) state before excitation. Therefore aob~ = ao, oN,o
(53)
The physical interpretation referring to the question why the absorbed energy of a linear absorber is insensitive to sublevel relaxation, is that as many molecules are brought into resonance with the first excitation step as are removed from resonance with the higher steps. For a nonlinear oscillator, however, (for which the net absorption drops with e) hole filling in the initial level must not counteract the "antihole" broadening in the higher levels, so foreign gas usually improves the absorption. Only at high intensities where all the rotational states are excited, the buffer gas effect on absorption will again diminish. It has been observed (17s) that addition of buffer gas to $2F10 has no effect on aobs at low laser energies (where S2F~o behaves like a linear absorber), but increases the absorption at higher energies (where the absorption is less than linear). On the other hand a buffer gas improves the absorption ofSF 6 (nonlinear absorber) at low energies, but has no effect at high energies "sl) (where most of the rotational states are excited). This difference is difficult to understand without the sublevel relaxation mentioned above. What will be the effect of this relaxation on the dissociation probability? Before relaxation, the population distribution of a linear absorber will be: ~2s4'296)
N~,oOCN, og,, ° exp( - e/eo)
(54)
(e) o]. I Another fraction 1--N~o will where eo is related to the mean energy per oscillator eo ~ sN, remain unexcited. Sublevel relaxation increases the excited fraction N,o and possibly also the number of oscillators, but keeps the absorbed energy (e) constant. The corresponding drop of the tail of the exponential eq. (54)--which is responsible for dissociation--will in general not be overcome by the increase of N,o and g~o. Thus sublevel relaxation can reduce the dissociation yield without affecting the absorbed average energy. It must be suspected that a similar phenomenon will occur also in nonlinear absorbers. 9. CONCLUSION It has been pointed out in the introduction that the absorption spectra of highly excited molecules are not wider than the corresponding ground state spectra. Recently this fact has been demonstrated for molecular ions of energies of only 1000 era- 1 below dissociation :(356) CFaBr + and CF3 I+ were generated in an ion source which apparently produced part of them in highly excited states. A continuous-wave CO 2 laser then dissociated them in a singlephoton absorption step, which was inferred from the linear dependence of dissociation probability on laser intensity.(356) The frequency dependence was only 10 cm- 1 wide and the wings were very weak, comparable to infrared spectra of similar molecules; e.g. CFaI has a halfwidth of 20 cm- 1. Whereas evidence of this kind shows that vibrational oscillator strength cluster in a range not much wider than a typical rotational structure, it is even conceivable that under the envelopes even narrower vibrational structures are concealed. Such an assumption allows to explain elegantly why SF6 absorbs ps pulses much more effectively than ns pulses :0 s s) If the gaps (ofa few cm- 1) in the anharmonically split va transitions are not levelled off, a narrow-band radiation (like a ns pulse) will sooner or later experience a poor resonance
The importance of spectroscopy for infrared multiphoton excitation
143
whereas a broad-band radiation (a ps pulse) will not feelit very much; so the latterwill be absorbed nearly likeby alinear absorber, much more effectivelythan a ns pulse.If~however, the gaps would be smoothed out above a certain level,the differentialabsorption cross-section above this energy level would be nearly constant and high in both cases in contrast to the findings.(See also Section 3.).The same experiment also clearlydemonstrates that collisionless relaxation isnot important for excitationofSF~: Ifitwould be necessary (i.e.rate determining) in a certain step, excitation by shorter pulses should be much less effective(Section 6). The opposite was found, however. The similarityof spectra implies that there isno reason to restrictthe study of spectroscopy to the firstthree steps: First of all,the simplicity of spectra even at high energies (at low resolution) is encouraging. Second, the spectra will be of comparable importance in the first few and later steps. Then, however, it has again to be asked how molecules can continue to absorb photon after photon and how they sometimes even ,behave like linear absorbers in spite of their anharmonic shift.W e have shown in the preceding sections,that by considering not only the firsttwo but all the contributing mechanisms of Table 4, a wide variety of phenomena can be understood at leastin principle.These variationscome up not only because spectroscopy is differentin every molecule, but also because every molecule (and even every rotational or vibrational state and every laser wavelength) selectsits own set of contributing mechanisms.
TABLE 4. Mechanisms contributing to multiphoton excitation Mechanism
Important
Function
Section
(1) Spectral intensity borrowing of background states (quasicontinuum)
in large molecules and at high energies
causes energy instead of intensity dependence; compensates small detunings, less well also large detunings by absorption wings
5
(2) Collisionless vibrational relaxation
in very large molecules and close broadens spectrum (due to to dissoc, energy distrib, of anharmonicities and to short lifetime), causes RRKM type dissociation
6
(3) Raising of rotational quantum number J
where rotational constant B is large; for all J
compensates part of anharmonicity; x, ff = x + B
3
(4) P, Q and R type transitions
for rotational states for which J = anharm, detuning/2B
provides three resonances for each vibrational transition
3
(5) Weakly forbidden rotational transition (e.g. AR # 0 in spherical tops or AK d: 0 in parallel transitions of symmetric tops)
in higher states where spherical or cylindrical symmetry is a bad approximation
provides three more resonances for each vibrational transition
3
(6) Multiplet level bands of nearly degenerate modes or anharmonically split degenerate
for very many molecules
reduces anharmoni¢ shift of some sublevels; cooperation with (4) and (5) very effective
3
where multiplet bands are available
converts vibrational into rotational energy
3
where available
can restore resonance with the original mode
4
modes (7)Intermediatestimulated emissionin a (e.g.)P type transition (8)Absorptionvia (anharmonicallyshifted)shorter wavelengthmode (9)Power broadening (10) Direct two- and threephoton transitions
where broadening by (1) and (2) compensates small detunings, is unimportant causes a certain intensity dependence
2, 5
especially if enhanced by near resonance
7
increased number of rotational states can take part; causes sharp resonances and a certain intensity dependence
144
W. Fuss and K. L. K O M P A
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