Picosecond Vi bra tional Spec tro scopy of Intermolecular Energy Transfer and Overtone Re lax ation in Liquid Bromoform

Transcription

1 Jour nal of the Chi nese Chem i cal So ci ety, 2000, 47, Picosecond Vi bra tional Spec tro scopy of Intermolecular Energy Transfer and Overtone Re lax ation in Liquid Bromoform G. Seifert*, T. Patzlaff and H. Graener Mar tin-luther-university Halle-Wittenberg, Phys ics De part ment, Hoher Weg 8, D Halle/Saale, Ger many Intermolecular en ergy trans fer in liq uid bromoform and d-bromoform was in ves ti gated by help of pi co sec - ond IR and Raman spec tros copy. For a com pre hen sive in ter pre ta tion of the cor re spond ing re sults the study of C-H bend ing over tone (2 4) re lax ation was nec es sary; the ef fec tive vi bra tional life time of this vi bra tional mode was de ter mined to T 1 = 3 1 ps. For the intermolecular trans fer a dom i nat ing pro cess be tween the intramolecularly equil i brated C-Br stretch ing modes 2 and 5 of CHBr 3 and CDBr 3 with a rate con stant of s -1 was rec og nized. Other pos si ble chan nels, par tic u larly the one be tween the quasi res o nant modes 1(C-D) and 2 4(C-H) were found to be neg li gi ble. IN TRO DUC TION The ap pli ca tion of pico- and femto second la ser pulses in stud ies of ultrafast re lax ation phe nom ena in con densed mat ter has yielded a va ri ety of new and of ten quite de tailed in for ma - tion about time scales and chan nels of var i ous en ergy trans fer pro cesses on the mo lec u lar scale. In par tic u lar, stud ies of vi - bra tional en ergy re lax ation in liq uids us ing, e.g., in fra red dou - ble res o nance or spon ta ne ous Raman scat ter ing af ter IR ex ci - ta tion, pro vided quite com pre hen sive in sight into the re lax - ation path ways of vi bra tional ex cess en ergy in weakly in ter - act ing liq uids, 1,2,3 while the tran sient spec tral changes of hy - dro gen-bonded liq uid sys tems of ten turned out to be dom i - nated by ef fects like tran sient hole-burning, 4,5 vi bra tional predissociation 6 or the ther mal re ac tion of the hy dro gen bond sys tem to the la ser in duced tem per a ture in crease. In gen eral, how ever, these stud ies do not re veal an an swer to the ques tion if the vi bra tional ex cess pop u la tion re mains dur ing re lax ation mainly on the orig i nally ex cited mol e cules (and its sur round - ings can be treated as ther mal bath), or if quasi res o nant intermolecular en ergy trans fer is im por tant. If how ever ap pro - pri ate iso to pic mix tures are stud ied, a dis tinc tion may be come pos si ble; 7 the ba sic idea of this method is to probe vi bra tional pop u la tion on (low-lying) states of one iso tope af ter ex ci ta - tion of the other via iso tope se lec tive com bi na tion tran si tions. In a very re cent work 8 we used this method ap ply ing pi co sec - ond time re solved vi bra tional spec tros copy to liq uid CHBr 3, CDBr 3 and mix tures of both, and found ev i dence for an ef fi - cient pro cess of near res o nant intermolecular trans fer of vi - bra tional en ergy be tween lower-lying modes of the in ves ti - gated mol e cules af ter C-H stretch ex ci ta tion. In the pres ent pa per the cor re spond ing sit u a tion af ter C-D stretch ex ci ta tion will be an a lyzed in de tail. As this mode ap pears at nearly the same fre quency as the first over tone of the C-H bend ing mode of bromoform, the re lax ation of this mode was stud ied sep a - rately in pure CHBr 3; the cor re spond ing re sults are pre sented and con clu sions about the intermolecular trans fer are dis - cussed. EX PER I MEN TAL SEC TION Con ven tional FTIR and Raman spec tra of the sam ples were taken in the fun da men tal and over tone re gion. Most of the tran sient data were ob tained per form ing in fra red dou ble res o nance spec tros copy in a setup which was de scribed in de - tail re cently. 9 Briefly sum ma rized, the ex per i men tal sys tem com prises a Kerr-lens mode-locked, flashlamp-pumped, 70 Hz Nd:YLF-laser sys tem ( = 1047 nm), op ti cal para met ric gen er a tion and am pli fi ca tion stages, and a de tec tion setup in - clud ing com pre hen sive po lar iza tion anal y sis. Sep a rately tun - able in fra red pulses of 2.5 ps tem po ral and 8-10cm -1 spec tral width (tun ing range m) are pro duced in two iden ti cal fre quency con ver sion de vices. One path pro vides ex ci ta tion pulses of up to 30 J en ergy per pulse, while the out put of the sec ond path (en er gies clearly be low 1 J) is used for the probe pro cess; these probe pulses can be de layed up to 4 ns with re - spect to the pump pulses. The two beams are fo cused on the same spot of a sam ple, and the trans mis sion changes caused by the pump pulse are mea sured with time, fre quency and po - lar iza tion res o lu tion. Usually sig nals pro por tional to vi bra - tional pop u la tion changes are de fined by, = ln(t, /T 0); the in di ces re fer to the reg is tered probe po lar iza tion with re - spect to that of the pump pulses. In par tic u lar, a ro ta tion free com bi na tion rf = ( + 2 ) / 3 can be de fined, which ex cludes all in flu ences of mo lec u lar re ori en ta tion and is in

2 668 J. Chin. Chem. Soc., Vol. 47, No. 4A, 2000 Seifert et al. sim ple cases di rectly pro por tional to tran sient pop u la tion dif - fer ences. Due to the de pos ited la ser en ergy the ex cited sam ple vol ume will end up at higher tem per a ture when the vi bra tional en ergy is re laxed. Ac cord ingly the sam ple is con tin u ously moved within the fo cal plane to have a fresh por tion of the sam ple for each la ser shot. Some of the re sults ref er enced be low were ob tained mea sur ing anti-stokes spec tra af ter IR ex ci ta tion with pi co - sec ond time res o lu tion. The cor re spond ing ex per i men tal sys - tem, which was also de scribed in de tail be fore, 3,10 is very sim i - lar to the IR dou ble res o nance ex per i ment, only the probe pro - cess is dif fer ent: in stead of mea sur ing trans mis sion changes in the IR, the sec ond har monic of the Nd:YLF fun da men tal ( = nm) is used to pro duce anti-stokes Raman pho tons, which are reg is tered as com plete spec tra by help of a CCD cam era be hind a mono chro ma tor. These data can be nor mal - ized yield ing the time de pend ence of real ex cess pop u la tion on Raman ac tive vi bra tional modes. 3 RE SULTS AND DIS CUS SION All ex per i ments de scribed in the fol low ing were per - formed on liq uid sam ples at room tem per a ture. The bromoform mol e cule is known to have six dif fer ent nor mal vi - bra tions, three of which are dou bly de gen er ate. Eval u ating the first mo ments of the cor re spond ing bands from mea sured IR and Raman spec tra, fre quency po si tions as given in Ta ble 1 are found for the nor mal modes of CHBr 3 and CDBr 3, show ing that upon deuteration not only C-H stretch ing ( 1 ) and bend - ing ( 4) mode are shifted to lower fre quen cies by ap prox i - mately 25%, but also the C-Br stretch ing modes 2 and 5 are slightly af fected (3-4% red-shift), whereas the C-Br bend ing lev els 3 and 6 are iden ti cal for both iso topes. Some more de - tails can be seen in Fig. 1: the bold solid curve rep re sents a Raman spec trum of CDBr 3, the thin solid line is a mag ni fied ver sion of its 4 re gion, and the dashed curve gives the IR absorbance of neat CHBr 3 in the re gion of the CH bend ing over tone 2 4 (up-shifted for better vis i bil ity). In the Raman spec trum the de gen er ate modes 4 and 5 are slightly asym - met ric and broader than 1 and 2; a sim i lar be hav ior was ob - served in liq uid chlo ro form and dis cussed to be due to intermolecular in ter ac tions in the liq uid. 11 The C-D bend ing band even seems to be com posed of two in di vid ual lines; how - ever, in this case also the close ly ing com bi na tion mode could reach the same Raman cross sec tion as the rather weak 4 band. In the plot ted range (2 4 band) of the IR spec trum there is def i nitely a main peak at 2255cm -1 and a sig nif i cant shoul der at about 2280cm -1 ; these two fea tures have been ex - plained as the two dif fer ent sym me try com po nents of the C-H bend ing over tone, 12 where the ad di tional red-shift is caused by Fermi res o nance with the C-H stretch ing fundmental. This dis cus sion will be re sumed be low to gether with the anal y sis of tran sient data. At this point it is only im por tant to no tice that the C-D stretch ing band of d-bromoform and (the stron gest peak of) the C-H bend ing over tone of CHBr 3 dif fer by only 3cm -1, so that in an iso to pic mix ture they can not be ex cited sep a rately re gard ing the spec tral width of the used in fra red pulses. Be fore the novel re sults ob tained on neat CHBr 3 and the 1:1 iso to pic mix ture with an ex ci ta tion fre quency around 2255cm -1 can be dis cussed, it is nec es sary to sum ma rize the find ings of our pre vi ous in ves ti ga tion. 8 To find ev i dence for the intermolecular trans fer pro cess, it was a nec es sary pre req - ui site to de ter mine the re lax ation times and path ways of the Fig. 1. Con ven tional Raman spec trum of CDBr 3 (solid curve, left or di nate scale; ar bi trary units); thin solid line: mag ni fied (by a fac tor of 10) plot of C-D bend ing re gion of the same spec trum; dashed curve: IR ab sorp tion of 2 4 over tone of CHBr 3 (right or di nate scale). Table 1. Vibrational Frequencies of the Fundamental Modes of CHBr 3 and CDBr 3 in Liquid Phase at Room Temperature Vibr. mode (symmetry) 1 (a 1) 2 (a 1) 3 (a 1) 4 (e) 5 (e) 6 (e) Frequency, CHBr 3 [cm -1 ] Frequency, CDBr 3 [cm -1 ]

3 Intermolecular En ergy Trans fer in Bromoform J. Chin. Chem. Soc., Vol. 47, No. 4A, Table 2. Relaxation Time Constants Obtained Fitting Raman and IR Double Resonance Data on CHBr 3 and CDBr 3. The Indices Denote Start and End Level of a Relaxation Step; the Numbering Refers to Fig. 2 Time Constant CHBr 3 CDBr 3 12 / ps / ps / ps / ps pure liq uids in as much de tail as pos si ble. This in for ma tion was ob tained an a lyz ing tran sient IR and Raman data of the two iso topes by help of both an a lyt i cal and nu mer i cal so lu - tions of ap pro pri ate rate equa tions. This anal y sis re vealed the re lax ation chan nels as de picted in Fig. 2; the cor re spond ing time con stants are given in Ta ble 2. In short, in both iso topes the first re lax ation step af ter ex ci ta tion of the C-H or C-D stretch ing vi bra tion is most prob a bly a four quanta step in - volv ing at least one of the lev els (2 4+ 2) and (2 4+ 5); this as sump tion eas ily ex plains the dif fer ent C-H(D) stretch ing life times of 50 (25) ps by the dif fer ent en ergy mis matches be tween stretch ing mode and com bi na tion tones (com pare Fig. 2). These in ter me di ate states then ob vi ously de cay very quickly (<5 ps) pop u lat ing 2, 4 and 5 (or ap pro pri ate com - bi na tion modes of these) nearly si mul ta neously with the time con stant of C-H (or C-D) stretch ing de cay. Af ter wards the pop u la tion of 4 shows a rel a tively fast de cay com po nent ( 50 ps), be fore the C-H(D) bend ing and sym met ric and antisymmetric C-Br stretch ing modes re lax with a com mon, very slow de cay time of 600 ps. In ter est ingly the pop u la tions of the low est modes 3 and 6 (C-Br bend ing), which were also de ter mined di rectly from the anti-stokes Raman ex per i ments, grow very slowly (on a time scale of 1 ns), and the cor re spond - ing num ber of vi bra tional quanta re mains be low that of 2, 4 and 5 up to 1 ns de lay time. Mea suring now for an 1:1 iso to pic mix ture of CHBr 3 and CDBr 3 the C-H and C-D stretch ing life times (e.g. us ing the 1 2 tran si tion in the IR dou ble res o nance ex per i ment), the same val ues of T 1 as in the neat sub stances (50 and 25 ps, re spec tively) were found within the ex per i men tal ac cu racy of typ i cally 2 ps. This al lows the con clu sion that no inter - molecular trans fer (IET) pro cess with an ef fec tive time con - stant shorter than 100 ps start ing from the C-H(D) stretch ing mode can be pres ent, be cause oth er wise the ob served re lax - ation times should be di min ished by at least 20%. This is an im por tant find ing par tic u larly in case of the C-D stretch, where a nearly per fectly en ergy matched level of the CHBr 3 mol e cules (C-H bend ing over tone 2 4 ) is avail able. For the lower-lying modes, how ever, a quite ef fi cient IET was ob - served. Fig. 3 gives a char ac ter is tic ex am ple for data, from which the IET can be con cluded di rectly: the two dif fer ent curves of tran sient trans mis sion change were mea sured at the same probe fre quency of = 2240cm -1, where con tri bu tions of in duced ab sorp tion due to pop u la tion of 2, 4 and 5 of the CDBr 3 mol e cules must be con sid ered. The pump fre quen cies re fer to ex ci ta tion of the C-D ( Pu = 2252cm -1, open squares) and C-H ( Pu = 3020 cm -1, solid cir cles) stretch, re spec tively. In the lat ter case the rather slowly ris ing ab sorp tion in crease di rectly rep re sents the pop u la tion on the low-lying modes of CDBr 3, which can only oc cur if there is an IET pro cess. Also the rise time of this in duced ab sorp tion cor re sponds very well to the C-H re lax ation, whereas af ter C-D stretch ex ci ta tion (up per curve) its sig nif i cantly faster re lax ation is re flected in a Fig. 2. En ergy level scheme (rel a tive dis tances of vi - bra tional states true to scale) of bromoform (left side) and d-bromoform (right side) in clud ing de - ter mined re lax ation chan nels (ar rows); num bers in cir cles de note the lev els used for rate equa tion cal cu la tions and also re fer to the in di ces of the re lax ation time con stants given in Ta ble 2. Fig. 3. Time re solved tran sient ab sorp tion change rf of 1:1 iso to pic mix ture CHBr 3:CDBr 3, probe fre quency pr = 2240 cm 1 ; ex ci ta tion of C-H ( = 3020 cm 1, full points) or C-D stretch ing mode ( = 2252 cm 1, open squares).

4 670 J. Chin. Chem. Soc., Vol. 47, No. 4A, 2000 Seifert et al. clearly ear lier max i mum of ab sorp tion in crease. The solid curves are the re sult of nu mer i cal sim u la tions us ing the re lax - ation time con stants given in Ta ble 2. In or der to study in de tail the ef fects of this IET pro cess in the iso to pic mix ture in prin ci ple 4 types of ex per i ments us - ing stretch ing ex ci ta tion and prob ing pop u la tion of low-lying modes are pos si ble: (i) pump ing and prob ing of CHBr 3, (ii) pump ing and prob ing of CDBr 3, (iii) CH pump, CD probe, and (iv) CD pump, CH probe. A va ri ety of such data was mea sured and an a lyzed nu mer i cally fix ing the pre vi ously ob tained dy - nam ics of the pure liq uids and add ing an ad di tional re lax ation chan nel at trib uted to the IET pro cess (see Fig. 2). All data of types (i) to (iii) could be ex plained with good ac cu racy as sum - ing a sin gle intermolecular chan nel be tween the C-Br stretch - ing modes with a char ac ter is tic time con stant of ps (only such data were dis cussed in our pre vi ous work 8 ). How - ever, data of type (iv) could not be ex plained with the same as - sump tions. As il lus trated by the re sults pre sented as up per curve (open cir cles) in Fig. 4, a very rapid in crease of the ab - sorp tion, i.e., the pop u la tion of the low-lying modes of CHBr 3 is ob served, which is even con sid er ably faster than the re lax - ation of CDBr 3 it self. This ob ser va tion can be un der stood if the lower curve (solid squares) is taken into ac count, where the same pump and probe fre quen cies were ap plied to pure CHBr 3, and a very sim i lar rise time and po si tion of max i mum ab sorp tion in crease is found. In ter est ingly the two curves dif - fer by an am pli tude fac tor of 3 at early times, but are iden ti cal within ex per i men tal ac cu racy at later de lay times (> 200 ps). The only plau si ble ex pla na tion for this ob ser va tion is the as - sump tion that in the case of neat CHBr 3 the C-H bend ing over - tone is quite ef fi ciently di rectly ex cited, and it takes only the life time of this level to trans fer en ergy intramolecularly down to modes ob serv able via the tran si tion to the ap pro pri ate com - Fig. 4. Time re solved tran sient ab sorp tion change rf of 1:1 iso to pic mix ture CHBr 3:CDBr 3 (cir cles) and neat CHBr 3 (tri an gles); in both cases ex ci ta - tion fre quency = 2255 cm 1, probe fre quency pr = 3007 cm 1. bi na tion mode with a C-H stretch ing quan tum. This would also ex plain the agree ment of the curves at later times, be - cause in the iso to pic mix ture of course the C-D stretch is pop - u lated more ef fi ciently, and the intermolecular trans fer at later times will in crease the pop u la tion on the C-Br stretch ing modes of CHBr 3 mol e cules. Turn ing the ar gu ment around, the dif fer ence be tween the curves at early de lay times con firms that no ef fec tive IET be tween 1 (C-D) and 2 4 (C-H) is pres - ent. In gen eral the above de fined type (iv) of data is well com - pat i ble with the re sults ob tained by the nu mer i cal anal y sis of the first three types, and it can ac cord ingly be con cluded that the as sumed (sin gle) intermolecular pro cess trans fer ring vi - bra tional en ergy be tween the C-Br stretch ing modes of bromoform and d-bromoform is in deed the most im por tant one. The other in ter est ing con clu sion from the data of Fig. 4 is that ob vi ously the C-H bend ing over tone is strong enough to be sig nif i cantly pop u lated by the used pump pulses. So one can try to de ter mine the vi bra tional life time of 2 4 of CHBr 3 and the con sec u tive re lax ation pro cesses by IR dou ble res o - nance spec tros copy (the Raman ac tiv ity of this mode is too weak to al low anal o gous ex per i ments us ing anti-stokes Raman prob ing). Cor re sponding re sults are pre sented in Fig. 5: Fig. 5a gives four tran sient spec tra of trans mis sion changes taken at de lay times of -2, +1, 6 and 13 ps af ter ex ci ta - tion at the max i mum ab sorp tion of the band at 2255 cm -1. For better vis i bil ity, the in di vid ual spec tra are shifted with re spect to each other. Be sides bleach ing close to the pump fre quency three bands of in duced ab sorp tion are vis i ble at 2180, 2200 and 2230 cm -1. While the band around 2230cm -1 is not pres ent at the ear li est de lay time, the one at 2200cm -1 seems to have van ished al ready af ter 13 ps. The time de pend ence at these four po si tions can be seen in Fig. 5b, where the tran sient trans - mis sion changes of bleach ing and in duced ab sorp tion at 2230 cm -1 (two up per curves) and those at 2200 cm -1 and 2180 cm -1 are plot ted (the two lat ter curves are down-shifted for better vis i bil ity by an ar bi trary value). In ter est ingly both the time de pend ence of bleach ing and that of the 2 lower in - duced ab sorp tions show a fast rise of the sig nal to a max i mum around zero de lay, and af ter wards one fast and (at least) one slow com po nent of de cay, though with dif fer ent in di vid ual am pli tudes. In con trast to this, at the probe fre quency of 2230 cm -1 sig nal rise and max i mum ab sorp tion in crease are clearly de layed and a dif fer ent, rel a tively slow de cay of the or - der of 50 ps is ob served. These ob ser va tions can be ex plained us ing a re lax ation scheme, which as sumes fast de cay of one 4 quan tum of the ex cited 2 4 mode into a 2 and/or a 5 quan - tum. In this view, the fol low ing spec tral as sign ments are plau - si ble: at 2255 cm -1, the bleach ing is due to both pop u la tion of 2 4 (fast com po nent) and de layed ground state re cov ery (slow

5 Intermolecular En ergy Trans fer in Bromoform J. Chin. Chem. Soc., Vol. 47, No. 4A, com po nent); the fast com po nents at 2180 and 2200 cm -1 are due to ex cited state tran si tions start ing from the two com po nents of 2 4, while the slow com po nents mon i tor pop u - la tion on the lower fun da men tal modes 2, 4 and 5. Only these lev els are as sumed to be probed via 2 4 quanta at 2230 cm -1. This is of course not an un am big u ous in ter pre ta - tion, be cause the Fermi res o nance sit u a tion is rather com pli - cated to dis cuss and the ob served fre quen cies re quire some - what larger anharmonicities than found in the gas phase; 12 on the other hand a trend to con sid er ably larger anharmonicity in the liq uid com pared to the gas phase has been ob served for high over tones of bromoform. 13 If an ap pro pri ate rate equa - tion sys tem is set up and solved nu mer i cally, all avail able data can be fit ted with suf fi cient ac cu racy; vary ing the best fits around the least mean squared er ror, time con stants of 3 1 ps for the 2 4 de cay, 45 8 ps for the de cay of 4 it self and ps for the C-Br stretch ing modes are found. The solid curves in Fig. 5b (and also in Fig. 4) were cal cu lated us ing these best pa ram e ters. The lat ter two time con stants agree very well with the val ues found from the above de scribed ex - per i ments af ter C-H and C-D ex ci ta tion. It should be men - tioned that for the solid curve rep re sent ing the iso to pic mix - Fig. 5. Tran sient data on C-H bend ing over tone re lax - ation of CHBr 3, pump fre quency = 2255 cm 1 ; (a) tran sient spec tra at four dif fer ent de lay times, (b) time re solved at four dif fer ent probe fre quen - cies. ture in Fig. 4 the ra tio of fast ( 4) and slow (, 5) com po nent had to be changed fa vor ing the slow com po nent with re spect to the data ob tained on neat bromoform. This is ba si cally the same ar gu ment as the one found above from the qual i ta tive dis cus sion of am pli tudes in Fig. 4 and thus again con firms the pres ence of the IET pro cess. The mea sured ef fec tive life time of the 2 4 mode of CHBr 3 is of the same or der of mag ni tude as the one of the C-H stretch ing over tone, which was found to be 5 ps sev eral years ago. 14 CON CLU SION The most im por tant new re sult of the pre sented in ves ti - ga tion is that the life time T 1 = 3 1 ps of the C-H bend over tone of bromoform could be mea sured di rectly from time re solved ex per i ments. Al though a de fin i tive de tailed as sign ment of the ob served tran sient spec tral changes was not pos si ble, it could be shown that both the 4 fun da men tal level and the two C-Br stretch ing modes 2 and 5 are pop u lated in the course of re - lax ation, and then show within ex per i men tal ac cu racy the same life times as in the case of C-H or C-D stretch ex ci ta tion. This find ing al lows char ac ter iza tion in more de tail of the pro - cess of near res o nant intermolecular en ergy trans fer, which was found to oc cur be tween the sym met ric and antisymmetric C-Br stretch ing fun da men tals 2 and 5 of CHBr 3 and CDBr 3 with a rate con stant of ap prox i mately s 1 in the liq uid phase at room tem per a ture: first of all, the data re fer ring to C-D stretch ex ci ta tion could be in cluded in the anal y sis con - firm ing the pre vi ously ob tained rate con stant. Sec ond, hav ing de ter mined the quite short life time of the C-H bend over tone, a no tice able con tri bu tion of the en er get i cally well matched pos si ble intermolecular chan nel 1 (C-D) 2 4 (C-H) to the IET can def i nitely be ex cluded. This re sult agrees with sim ple the o ret i cal de scrip tions: dis cuss ing the en ergy trans fer pro - cess in terms of the model of bi nary col li sions, 15 the trans fer rate de creases (for large mis matches ex po nen tially) with in - creas ing en ergy mis match, and each quan tum num ber change nec es sary for a tran si tion re duces its prob a bil ity by roughly one or der of mag ni tude. Re gard ing the en ergy mis match of ap prox i mately 20cm -1 be tween the C-Br stretch ing modes of the two iso topes, it is thus plau si ble that no in di ca tion of ad di - tional intermolecular chan nels was found: the dis cussed pro - cess 1(C-D) 2 4(C-H) re quires the change of three vi bra - tional quanta, and the other ob vi ous path way 4(C-H) 4(C-D) has a sig nif i cantly larger en ergy mis match of 300cm -1. The ques tion if both or only one of the two C-Br stretch ing lev els are im por tant for the IET can not be re solved due the fast intramolecular re dis tri bu tion be tween sym met ric and antisymmetric mode (within 10 ps 8 ).

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