aa r X i v : . [ phy s i c s . a t m - c l u s ] J u l The C − thermal electron emission rate K. Hansen ∗ Center for Joint Quantum Studies and Department of Physics,School of Science, Tianjin University, 92 Weijin Road, Tianjin 300072, China (Dated: Monday 13 th July, 2020, 00:31)The thermal electron emission rate constant for C − has been deduced over a range of 4 eVinternal energy from storage ring measurements of the decays of ions reheated with single photonsabsorption. The thermal radiation from the ions is quantified with respect to continuous coolingand discrete photon quenching. I. INTRODUCTION
Measurements of rate constants in molecular beamswith standard approaches require very good control overthe excitation energy. A width in the internal energydistribution in a decaying particle or molecule will alsointroduce a width in the distribution of rate constants inthe beam molecules, and due to the strong dependence ofrate constants on excitation energy, any spread in energyis strongly amplified for the rate constants. This makesdirect measurement of rate constants very difficult evenfor fairly narrow internal energy distributions. The prob-lem is not solved by extracting molecules from canonicalthermal distributions into molecular beams, as demon-strated in [1] with a calculation of a numerical examplefor C − .When the energy distribution of the molecules in abeam is sufficiently broad and in the absence of compet-ing channels, molecular decay will occur with a rate witha time dependence close to 1 /t [2]. In the presence of thefrequently occurring phenomenon of thermally radiation,this power law will be suppressed at long times with analmost exponential time dependence [3].Situations with broad energy distributions arise par-ticular frequently for large molecules and clusters, be-cause for these, excitation to internal energies where re-actions occur on measurable time scales will require largeamounts of energy, up to several tens of eV. Depositionof precise amounts of energies of such magnitudes is avery challenging experimental task. Photo excitation ex-periments with a single high energy photon, for exam-ple, will often lead to direct (first or secondary) ioniza-tion of the molecules or to electron detachment from an-ions. The alternative strategy of multiple absorption ofsmaller energy photons suffers from the inherent spreadsin absorption statistics. Collisional excitation is possible,as demonstrated with electron collisions with fullerenes[4, 5], but these suffer from a broad energy transfer ef-ficiency, requiring a detailed quantitative analysis of thereaction products with a number of highly non-trivial as-sumptions.The origin of the power law behavior is the loss of a ∗ [email protected] well defined energy scale in the excitation energy distri-bution caused either by such post-production excitationor by the use of hot sources, which almost unavoidablyproduce clusters with broad energy distributions. For aunimolecular decay in vacuum, loss of an energy scale isequivalent to loss of a time scale. This is reflected in theabsence of a characteristic time scale in the 1 /t depen-dence of the decay rate. If one wants to measure absoluteenergies under this kind of conditions, it is therefore nec-essary to introduce an energy scale by hand.This was done for C − in the experiments reported in[6]. In these experiments, the anions were created hotfrom the source and injected into an electrostatic storagering, where they decayed by spontaneous electron emis-sion. At a variable time after production, a small frac-tion of the un-decayed ions were reheated by one-photonexcitation. The photon energy absorbed and dissipatedcaused a heating of the molecule that lead to an enhanceddelayed thermal electron emission. The time profile ofthe enhanced decay was used to locate the equivalentbackshifted time, i.e. the time where the spontaneouslydecaying ions decayed with the same time dependence asthe laser excited ions. An overall multiplicative constanton the enhanced decay, which reflects quantities such aslaser fluence, beam overlap, and photon absorption crosssection was only relevant for the amplitude of the laserenhanced signal and did not enter the analysis.Together with the instrumental laser firing time, thedetermination of this apparent shift of the zero time ofthe power law decay due to the reheating provides thetime interval during which one photon energy was lost.The procedure can therefore be used to determine theabsolute cooling rate of the ions. The results obtainedwere in very good agreement with the known facts ofC − , such as the electron affinity and also with the modelfor the radiative cooling developed in [7].The data from this experiment somewhat surprisinglyalso allow for the determination of the parameters thatdetermine the energy resolved rate constants. Further-more, they provide a measure of the relative importanceof continuous and discrete cooling. These two types ofthermal photon emission differ only by the magnitude ofthe energies of the photons emitted, and thereby by theeffect they have on the measured decay dynamics of theions.The demonstration of this is the purpose of this paper.The outcome of the analysis of the C − data will providethe absolute decay rate, parametrized by the product ofactivation energy and heat capacity, the frequency factorof the rate constant, and a binary spectral distributionof the thermally emitted photons.The remainder of the paper is divided into a sectionwhere the theory behind the experimental data and thepresent analysis is described in some detail. This is fol-lowed by a section where the experiments are described,after which a section presents the data analysis and theresults. Finally, the procedure and the results are sum-marized and discussed. II. THEORETICAL BACKGROUND
The spontaneous statistical decay of an ensemble ofparticles in a molecular beam is given by the decay rateaveraged over all excitation energies present in the en-semble; R ( t ) ∝ Z ∞ g ( E ) k ( E ) exp( − k ( E ) t )d E. (1)where R is the measured decay rate, i.e. the number ofdecays per time unit, and k ( E ) is the decay rate constantof an ion with excitation energy E . The quantity g ( E )is the ensemble density of excitation energy at the timeit was created in the source, and t is the time elapsedfrom the creation of g in the source to the measurement.The constant of proportionality is the combined trans-mission and detection efficiency. When g ( E ) is broad,the integrand peaks at the rate constant k m for whichdd E k m ( E ) exp( − k m ( E ) t ) = 0 ⇒ k m ( E ) t = 1 , (2)corresponding to a peak value of exp( − /t for k ( E ) exp( − k ( E ) t ). This result is derived without speci-fying the expression for k ( E ) and holds generally, insofaras Eq.(2) has solutions, which may not be the case forultrafast processes but will be the case for measurementtimes relevant here. The equation only has one solutionif k ( E ) is a monotonically increasing function of E , whichcan also be safely assumed here.It is worth demonstrating the generality of the result inEq.(2) with some different expressions for rate constants.Fig.(1) shows a few examples. The expression for thedecay constant, which will also be used in the analysis,is of the simple form k = ω exp (cid:18) − φC v E + E ′ (cid:19) . (3)Here C v is the canonical heat capacity in units of k B , lessone ( k B = 1 will be used throughout), and φ is the decaychannel activation energy. For thermionic emission fromC − this parameter is to a first approximation expectedto be the electron affinity of 2.67-2.68 eV [8, 9]. In spiteof its simplicity, Eq.(3) is very accurate for our purpose because only an energy interval of ca. 4 eV is coveredin the experiments here. This question is discussed inthe appendix and corrections to parameters made in thediscussion section.The rate constants used in Fig.(1) are all variations ofthe rate constant in Eq.(3). Other examples with differ-ent functional forms are given in [3], with an identicalconclusion. Figure 1. The product of rate constants and survival proba-bility for a broad energy distribution after 1 ms for a few dif-ferent parameters for the rate constant given in the main text.The line is the calculated value of 1000 / exp(1) s − The pa-rameters are, from low to high peak energies: ( ω, φC v , E ′ ) =(10 Hz ,
434 eV ,
10 eV), (10 Hz ,
434 eV ,
10 eV),(10 Hz ,
300 eV , Hz ,
434 eV , The decay rate is the integral of the peaks in Fig.(1).The width of the peaks are on the order of 1 / d ln k/ d e .Given the rapid variation of k with energy, this will re-main fairly constant over a wide range of times. Thissuggests the possibility that also the decay rate may varyapproximately as 1 /t . The decay rate is most easily cal-culated by considering the time dependence of the energydistributions. Ignoring the variation of g with energy, theenergy distribution of the surviving ions, exp( − k ( E ) t ),is essentially constant up to an energy close to the valuedefined by k ( E ) = t , at which point it crosses over andrapidly reaches zero. The motion of this cross-over en-ergy with time represents the decay rate. Solving Eq.(3)for E and deriving with respect to time then gives thedecay rate R ( t ) = − c ′ g d E ( k = 1 /t )d t = c ′ g φC v (ln( ωt )) t , (4)where c ′ is a constant that includes the detection, trans-mission efficiency and other instrumental parameters,and g ( E ) is set to a constant, g = g ( E ( k = 1 /t )) ≈ g ( E ( k = 1 /t )). Absorbing g into the constant, c ≡ c ′ g ,and rewriting gives1 t = k peak = R ( t ) (ln( ωt )) cφC v , (5)where k peak is the value for which the decay peaks. Thedifference between the time dependence of the decay rateand the rate constant at peak decay rate is the time vari-ation of the width of the decaying peak considered a func-tion of excitation energy, and is summarized by the factor(ln( ωt )) .In the presence of thermal radiation, which will bepresent for C − in the experimental data used here, therelation must be reconsidered. In principle also the C emission is a possible channel. However, this has an acti-vation energy which is close to four times that of electronemission from the anion and can safely be ignored. Theonly channel competing with electron emission is there-fore thermal radiation.In the context of ensembles there are two categories ofthermal radiation, defined by the magnitude of the ener-gies of the emitted photons. When the emission is by suf-ficiently low energy photons, the radiation is effectively acontinuous cooling. This means that the energy distribu-tion shifts down with time similarly to the non-radiativesituation, just faster. The shape of the cross-over re-gion of the energy distribution is virtually unchanged inthis small photon energy limit. When only this type ofradiation is present, its effect can be determined fromthe observed decay rate with an expression analogous toEq.(5) where t ′ is given by R ( t ) = cgφC v t ′ ln( ωt ′ ) , (6)from which the peak rate constant is identified as k ( t ) = 1 t ′ (7)where t ′ is a fictitious time which is equal to the timeneeded to wait to have an identical decay rate in the ab-sence of radiation. The decay at short times which is notinfluenced by any radiative cooling can be used to deter-mine the constants of proportionality. In the logarithmthe difference between the physical time and t ′ can oftenbe ignored.When large energy photons are emitted, the simplepowerlaw relation needs to be modified once more. Pho-ton energies are considered large if the emission of a singlephoton will quench the decay on a time scale correspond-ing to the rate constant after emission. The precise en-ergy where this shift from continuous cooling to quench-ing happens was analyzed in [10], and will be discussedhere after the presence of these photons is quantified.For the fullerenes, the largest part of the radiation iswell understood as being carried by the broad surfaceplasmon resonance [7]. Although centered at 20 eV, itreaches into the near infrared which allows the low energytail to be excited thermally with an oscillator strengthwhich gives a radiative energy emission rate which is twoorders of magnitude higher than the contribution fromthe vibrational transitions [11]. The calculated magni-tude is consistent with both the anion cooling and theoriginal observation of the strong radiative cooling of themuch hotter fullerene cation fragments [12]. The distri-bution of photon energies generated by the plasmon res-onance emission covers both the small and large values, and both types of channels therefore need to be consid-ered in the analysis.Whereas for small photon energies the emitted poweris the relevant quantity, for large photon energies it isthe emission rate constant. As photon emission rate con-stants only vary slowly with the excitation energy of themolecule when compared to the thermionic emission, wecan here set the discrete energy emission rate constantto a single value, k p . Its presence means that the abun-dances, and hence also the decay rates, are reduced bythe factor exp( − k p t ). Together with the effect of the con-tinuous cooling, R n , and after normalization to the shorttime behavior of 1 /t the observed rate is then equal to R n ( t ) = 1 t ′ e − k p t = k ( t )e − k p t , (8)or k ( t ) = R n ( t )e k p t . (9)The fitted curve from the experimentally measured spon-taneous decay rate of C − from a hot source gives thefunction [6] R n ( t ) = 1 t exp (cid:0) − − t + 1320s − t ) (cid:1) , (10)and hence k ( t ) = e k p t t exp (cid:0) − − t + 1320s − t ) (cid:1) . (11)The analysis so far has only dealt with the spontaneousdecay. If the molecule is exposed to a laser pulse sometime after production, the absorbing fraction of the en-ergy distribution will be shifted up by the photon energy.The situation is illustrated schematically in Fig.(2). The
12 13 14 15 16 17 180.00.20.40.60.81.01.2 h n E n e r gy d i s t r i bu ti on s E (eV) Figure 2. A schematic view of the energy distributions imme-diately before (dotted line) and after (full line) a photon withenergy hν has been absorbed at t las . small fraction of the distribution that has been shiftedup in energy has almost the same shape as the unshifteddistribution had at some earlier time, apart from the ab-solute height. This has been shown in [13] to which thereader is referred for details of the calculation. After pho-ton absorption at t las the decay rate is therefore given by R las ( t ) = p e − k p ( t las − t ) R ( t − t las + t )+(1 − p ) R ( t ) , (12)where t is a backshifted time, and p is the photon ab-sorption probability. The backshifted time, t , can bedetermined by a fit of the first term on the right handside of Eq.(12) to the decay rate at earlier times. Thefraction of absorbing ions, p , was so low in the experi-ments that is practically unobservable in the second termin the equation. This facilitated the analysis although itis not an essential requirement. The non-zero value of k p has no effect on this part of the analysis. It was not ex-plicitly considered in [6], but the cooling rates obtainedthere remain unchanged, although it is clear that theyonly refer to the small photon energy cooling power.The non-exponential decay is essential to deter-mine the cooling with this procedure because for non-exponential decays the value of both p (or more precisely p e − k p ( t las − t ) ) and t can both be determined, a possibil-ity which is not present for an exponential decay.As shown, decay rates are proportional to decay con-stants and Eq.(12) therefore also holds for the peakdistribution values k m ( t ) with the substitution R ( t ) → k m ( t )e − k p t . The values of t depend on the photon en-ergy and t las but are independent of the absorption crosssection and instrumental parameters. Keeping the laserfiring time t las fixed and varying the photon energy, itis therefore possible to obtain the variation of the rateconstant with photon energy as k m ( E ( t las ) + hν ) = e k p t R n ( t ( t las , hν )) , (13)and similarly k m ( E ( t las )) = e k p t las R n ( t las ) . (14)The energy E ( t las ) is the energy where the decay ratepeaks at time t las . It is unknown but both hν and t areknown. When considering decay rates in the following,the term energy will always refer to this particular energyor the corresponding peak rate energy for the shifted dis-tributions. In statements about the rate constant, theenergy will refer to the argument in Eq.(3). Eqs.(13,14)are the basic equations for the analysis of the experimen-tal data.It should be noted that although a number of measuredvalues of t correspond to times before the mass selectionhas been completed in these experiments, this causes noproblem for the analysis, because other experiments onC − have established the short time behavior as a wellbehaved power law, see e.g. [11], and this behavior iswell established as a general phenomenon, see e.g. theexamples listed in [3]. III. EXPERIMENTS
The data used for the analysis were recorded at theTokyo Metropolitan electrostatic storage ring, TMU e- ring. The analysis of the absolute cooling rates derivedfrom these data was published in [6], and the descriptionof the experiment here will be limited to the pertinentpoints. For a detailed description of electrostatic storagerings and their use for decay measurements, the readeris referred to the rich literature on the subject, see e.g.[14–19].The C − anions were produced in a laser ablationsource without any cooling gas and injected into thering together with some amount of other anionic car-bon species produced during the ablation, mainly otherfullerenes. The circulation time of C − in the ring was 122 µs . A set of pulsed deflection plates was used to eject theunwanted species, based on their mass dependent circula-tion time. This beam purification process was completedwithin 1 ms after production of the ions in the source.After a variable storage time, the C − beam was ex-posed to a laser pulse from a tunable optical parametricoscillator (OPO) laser, with photon energies which werevaried between 1.9 eV and 2.7 eV in steps of 0.1 eV or0.2 eV. Pulse energies were kept low, typically a few mJor lower, to ensure single photon absorption conditions.Spectra were recorded with laser firing times between 4ms and 35 ms.Fig.(3) shows two example spectra that were recordedwith laser firing time 12.5 ms and photon energies 2.0and 2.7 eV. Reference spectra without exposure to laser
10 12 14 160200400600800
I(t) h n = 2.0 eV I(t) t (ms) h n = 2.7 eV Figure 3. Two spectra with photo-enhanced decays. light were recorded under identical source and ring condi-tions, with a timewise interleaving of laser-on and laser-off spectra. The ion source was found to be very stable,with reproducible spontaneous decay rates as a functionof time, with variations restricted to minor and slow fluc-tuations in the absolute overall ion intensity. Such sourceintensity variations were accounted for by a normaliza-tion using pre-laser time counts of the laser-on and thelaser-off spectra.The main result of the experiments were the back-shifted times of the photo-induced decays. As illustratedwith a couple of examples in [6], the photon enhancedsignal can be represented well by the expression R p ( t ) ∝ t − t las + t , (15)where t is the time after production of the ions in thesource, t las is the laser firing time, and t the backshiftedtime. This simple expression only works for situationswhere, like here, the backshifted time is located in thepure power law sector before radiative cooling modifiesthe decay. Irrespective of which sector the backshiftedtime is located, its interpretation is the same, viz. asthe reciprocal of the rate constant of the molecule at theenergy E ( t las ) + hν , modified with k p as given above.Fig.(4) shows examples of the fitted t for experimentswith two different photon energies and a range of differentlaser firing times. t ( s ) t las (s) Figure 4. Traces of t as a function of laser firing time mea-sured with the photon energies hν = 2 . IV. DATA ANALYSIS
The data analysis proceeds from the data set com-prising associated values of laser firing times, t las , back-shifted times, t , and photon energies, hν , together withthe rate constants for these times, k ( E ( t )), derived fromthe measured decay rates, as explained above.The first part of the analysis is initiated by assigninga zero energy arbitrarily to the edge energy, E ( t las ), atsome given laser time. In this case it was chosen to be t las = 0 . t ’s are close, ideally identical, for differentlaser firing times and photon energies. The criterion fortwo t ’s being identical was chosen to be a difference of nomore than 10 % in value. The computational procedureis illustrated in Fig.(5). All 62 measured combinations of Figure 5. The computational flow in the calculation linkingthe relative energies, illustrated with three laser firing times,two of which are identical. The energy assignment begins atthe top right corner by choosing this as the zero of energy,and flows in the direction of the arrows. As described in themain text, all assigned points can be assigned a known rateconstant, modulo the value of k p . laser firing times and photon energies were linked to thecommon energy reference this way. The linked energiesare independent of the values of k p , but the rate constantsfor each time are not. They need to be calculated withEq.(9).As the value of k p is not known at this point, curves ofthe thermionic emission rate constant k ( E ) were calcu-lated for different assumed values of k p , varying it from10 to 100 s − in steps of 10 s − . For each of these, thelogarithmic slope was fitted. The logarithmic slope takesthe form d ln k d E = φC v E = ln (cid:0) ω/k (cid:1) φC v , (16)where k is the logarithmic midpoint of the data range forwhich the derivative is fitted.The second step in the analysis is taken by consideringthe variation of the rate constants when the laser time ischanged and the photon energy is kept constant. Takingthe ratio of the rate constant at the backshifted time tothe rate constant at the laser firing time one gets, with E las denoting the energy edge at the laser firing time, k ( t ) k ( t las ) = exp (cid:18) − φC v E las + hν + φC v E las (cid:19) (17) ≈ exp (cid:18) φC v hνE las − φ ( hν ) E las (cid:19) , or ln (cid:18) k ( t ) k ( t las ) (cid:19) ≈ φC v hνE las (cid:18) − hνE las (cid:19) . (18)The value of E las can be expressed in terms of the rateconstant as k ( t las ) = ω exp (cid:18) − φC v E las (cid:19) ⇒ E las = φC v ln ( ω/k ( t las )) . (19)Inserting this and taking the square root gives the quasilinear relation (cid:18) ln (cid:18) k ( t ) k ( t las ) (cid:19)(cid:19) / (20)= (cid:18) hνφC v (cid:19) / (ln( ωt ref ) − ln( k ( t las ) t ref )) × (cid:18) − hν ln ( ω/k ( t las )) φC v (cid:19) / , where t ref is a reference time that can conveniently betaken as 1 s.Repeating this procedure with t las replaced by t onthe right hand side gives a similar result apart from theexchange t las → t , and a change of sign on the last termin the last bracket. Averaging the two and dividing by √ hν gives 1 √ hν (cid:18) ln (cid:18) k ( t ) k ( t las ) (cid:19)(cid:19) / (21) ≈ (cid:18) φC v (cid:19) / (cid:18) ln( ωt ref ) −
12 ln( k ( t las ) k ( t ) t ref ) (cid:19) . When evaluating the quality of this approximation, it wascompared with that the ratio of rate constants expressedas ln (cid:18) k ( t ) k ( t las ) (cid:19) = − φC v E las + hν + φC v E las (22)= φC v E las ( E las + hν ) = hνφC v ln (cid:18) ωk ( t ) (cid:19) ln (cid:18) ωk ( t las ) (cid:19) . This is inconvenient for graphical representation, but atest using it (not shown) confirms the validity of theabove approximation.Eq.(21) defines a straight line. The value of k p entersinto the values of k ( t ) and k ( t las ) and hence also of theslope and the intercept of the straight line. The interceptsquared allows a comparison with the value obtained withEq.(16) after a correction for the difference between timescales used in the factor ln( ωt ) in the two equations. The Figure 6. The values of ln( ω /φC v vs. k p calculated withEq.(16) (circles) and Eq.(21) (triangles). comparison of the two values is shown in Fig.(6) vs k p .Consistency requires identical values for the two curves,yielding the value k p = 60 s − . This value inserted intoEq.(21) gives the line in Fig.(7). Figure 7. Plot of the data calculated with Eq.(21) and k p = 60s − . The points are grouped in bunches 0.1 wide and the errorbars are calculated as the statistical average on the mean. Fora few points where there is only one datum in the bunch, theerror is set to 0.05. Another possible contribution to the analysis shown inFig.(7) should be mentioned. It is obtained by replacingthe rate constant at the laser firing time with one for adifferent photon energy, i.e. using two different photonenergies and hence two different backshifted times fromthe same laser firing time. The equation then reads1 √ hν − hν (cid:18) ln (cid:18) k ( t (1)) k ( t (2)) (cid:19)(cid:19) / (23) ≈ (cid:18) φC v (cid:19) / (cid:18) ln( ωt ref ) −
12 ln( k ( t (1)) k ( t (2)) t ref ) (cid:19) , where the arguments (1) and (2) refer to different pho-ton energies at the same laser firing time. The presentdata (not shown) are too scattered to provide any strongconfirmation of the analysis, but are consistent with it.The parameters of the line in Fig.(7) gives the valuesln( ω . ± . , φC v = 510 ±
180 eV , (24)corresponding to a frequency factor of ω = 4 . × s − with a 1- σ uncertainty of a factor 400.The above results can be used to verify the procedureby applying them to the rate constants found with thelinking procedure illustrated in Fig.(5). As k p is known,also these rate constants are known, apart from the off-set in energy. The expression for the rate constant isrewritten, reintroducing the offset energy E ′ , as1ln( ω/k ( E )) = E + E ′ φC v . (25)Using the value of ω fitted above, the left hand side isplotted vs. E in Fig.(8). The expected straight line be- Figure 8. Plot of Eq.(25) with rate constants calculated withthe linking procedure explained in the main text, and thevalue k p = 60 s − . The line is a straight line fit. The pa-rameters of the line give the values φC v = 546 ±
12 eV, and E ′ = 18 . ± . ω is no included in these twostandard deviations. havior is observed, and the fitted value of φC v is con-sistent with the previously fitted values, although theuncertainty is significant larger than the fitted value in-dicates. The rate constant calculated with the two fitparameters from Fig.(8) and the previously determined ω is shown in Fig.(9). V. DISCUSSION
The analysis has been based on experimental data andthe result in Fig.(9) gives the rate constant from exper-imental data alone. The different determinations of theparameters can be summarized as a value for the fre-quency factor of ω = 5 × s − with an uncertaintyof a factor 400; two values of φC v of which 510 ±
180 eVmust be considered the primary. The second is consis-tent with this value but is derived assuming the abovevalue of the frequency factor. Finally, the energy offsetfor the arbitrarily chosen zero of energy has been fittedto a value of 18.6 eV. The first two parameters in this listhave obvious interpretations, but also the energy offsetcontains information on the reacting species.
Figure 9. The thermionic rate constant of C − vs. energy.Error bars can be taken as the average point-to-point fluctu-ation. The full line are the values calculated with the param-eters ω = 4 . × s − , φC v = 546 eV, and E ′ = 18 . E . The parameters extracted from the fits differ from thevalues measured in other experiments because approxi-mating a microcanonical rate constant, which is essen-tially a ratio of level densities with an exponential, willgenerate some finite size corrections. These correctionswere calculated in [20] and can be summarized as C v = s − ln( ωt ) s , (26) φ = E a + E r − E t , (27)where s is the average of the number of thermally acti-vated oscillators of precursor and product, and the twoenergies E r , E t are the offsets in the canonical caloriccurves for the anion ( E r ) and the neutral molecule ( E t ),defined as: E = s i T − E i , (28)where i represents either r or t . E a is the adiabatic elec-tron affinity with the previously cited value of 2.67 or2.68 eV.The correction to the heat capacity is very minor forC , on the order on 1, and can be ignored here. Alsothe slight variation in the heat capacity due to its tem-perature dependence will be ignored (see the appendixfor this discussion).The correction to the activation energy is the mostimportant. It vanishes for a harmonic oscillator system ifthe number and frequencies are identical in the precursorand product, because for harmonic oscillators the offsetsare just the sum of their zero point energies. Althoughthe number of oscillators is identical for the anion andthe neutral molecule and the oscillators can be consideredharmonic because the degree of excitation is very low, acorrection arises because the frequencies differ.The entire sets of frequencies of the neutral and the an-ion are not known. The two anion infrared active modesreported in [21] of 570 cm − and 1374 cm − are shiftedslightly relative to the neutral values of 570 cm − and1411 cm − [22]. If the reduction of the highest frequencyis used as the scaling for all frequencies, the correspond-ing reduction in total zero point energy of the anion com-pared to the neutral is 0.26 eV. For this estimate theset of vibrational frequencies of [23] was used. Althoughthese frequencies refer to fullerite and not to gas phasemolecules, the values are sufficient for the purpose. Thenet result is to reduce the effective activation by 10 %. Atthe same time the reduced vibrational quantum energiespush the heat capacity up toward the classical canonicallimit of 3 N −
6. The combined effect is therefore lessthan the 10 % reduction of the activation energy alone.As the anion spectrum is by and large unknown, a moreaccurate estimate of the expected value of φC v will notbe attempted.In the definition of an emission temperature that isused here, some offsets are included into the energy con-tent of the decaying anions [20]. To a sufficient precisionthe emission temperature is, in terms of the physical ex-citation energy E equal to T e = 1 C v (cid:18) E − E a E r (cid:19) . (29)The quantity in the bracket is the energy that appears onthe abscissa in Fig.(9), i.e. the offset energy E is equalto E r − E a /
2. With the reduced frequencies for the anion,this amounts to E = 7 . k d E hν > ⇒ hν > φC v (ln( ω/k )) (30)= 510 eVln(4 . × / = 0 .
66 eV . Photons of this magnitude are within thermal reach. Themicrocanonical temperature of the anion is ( E + E + E a / / E + E is the fitted effective energycontent, and E a / k photon ( hν )d hν ∝ ( hν ) e − hν/T − e hν/T d hν. (31)The total emitted power is bounded from below by0.66 eV ×
60 s − = 40 eV/s. This should be comparedwith the radiative energy loss of approximately 100 eV/sreported in [6]. As discussed, this emitted power refers tothe radiation emitted as continuous cooling exclusively.We can use this value to normalize Eq.(31) and find thetotal emitted power as well as the distribution on lowand high energy photons. Using the temperature to 0.66eV/5.5, the low energy photon determines the constant c as 100 eV / s = c Z .
66 eV0 ( hν ) e − hν/T − e − hν/T d hν. (32)The corresponding high photon energy emission rate con-stant is k p = c Z ∞ .
66 eV ( hν ) e − hν/T − e − hν/T d hν. (33)The value is calculated to 120 s − , i.e. a factor 2 higherthan the fitted value. The value decreases to 90 s − forthe temperature 0.11 eV. Considering that the spectrumin Eq.(31) is somewhat schematic, the agreement is rea-sonable. In any case, the data suggest that a considerablefraction of the radiative energy is emitted as high energyphotons. This is remarkable. both because the systemsis as large as it is, and that the electron affinity, whichacts as the activation energy and therefore sets the tem-perature scale, is not particularly large compared withactivation energies for unimolecular fragmentation, forexample. VI. SUMMARY AND PERSPECTIVES
The rate constant for thermal electron emission fromC − has been determined over a 4 eV energy range. Thedetermination applies a simplified rate constant but doesnot rely on any modeling. The experimental input isthe set of associated values of backshifted times, photonenergies and laser firing times in a reheating experiment.The experiment was performed in a storage ring, whichis an ion storage device which allows to probe a widerange of times and thereby to cover a reasonable internalenergy range.The analysis provided the absolute value and the log-arithmic derivative of the rate constant with respect toenergy, and the product of activation energy and heatcapacity, together with the frequency factor for the rateconstant. The values were found to be in the range ofexpected and physical reasonable, although the uncer-tainties were not negligible. The main problem of theanalysis of the data is the presence of betatron oscilla-tions. Although these are inherent to the operation ofstorage rings, their magnitude decreases in smaller rings,for simple geometrical reasons related to relative detectorsize. The analysis presented here is a proof of principlefor the method which provides rate constants for largesystems that are otherwise in practice beyond reach of ex- perimental measurement, and the commissioning of stillsmaller storage rings promise the possibility for still moreaccurate measurements. VII. AKNOWLEDGEMENTS
My coauthors of [6] are gratefully acknowledged for theproductive collaboration which provided the experimen-tal material of this paper. [1] J. U. Andersen, E. Bonderup, and K. Hansen, J. Phys.B At. Mol. Opt. Phys. , R1 (2002).[2] K. Hansen, J. U. Andersen, P. Hvelplund, S. Møller,U. Pedersen, and V.V. Petrunin, Phys. Rev. Lett. ,123401 (2001).[3] K. Hansen, Rev. Mass Spectrom. xx , xx (2020).[4] S. Matt, O. Echt, T. Rauth, B. D¨unser, M. Lezius,A. Stamatovic, P. Scheier, and T. D. M¨ark, Z. Phys. D , 389 (1997).[5] S. Matt, P. Scheier, A. Stamatovic, H. Deutsch,K. Becker, and T. D. M¨ark, Phil. Trans. R. Soc. Lond.A , 1201 (1999).[6] A. E. K. Sund´en, M. Goto, J. Matsumoto, H. Shiromaru,H. Tanuma, T. Azuma, J. U. Andersen, S. E. Canton,and K. Hansen, Phys. Rev. Lett. , 143001 (2009).[7] J. Andersen and E. Bonderup, Eur. Phys. J. D , 413(2000).[8] C. Brink, L. H. Andersen, P. Hvelplund, D. Mathur, andJ. D. Voldstad, Chem. Phys. Lett. , 52 (1995).[9] D.-L. Huang, P. D. Dau, H.-T. Liu, and L.-S. Wang, J.Chem. Phys. , 224315 (2014).[10] P. Ferrari, E. Janssens, P. Lievens, and K. Hansen, Int.Rev. Phys. Chem. , 405 (2019).[11] J. U. Andersen, C. Brink, P. Hvelplund, M. O. Larsson,B. Bech Nielsen, and H. Shen, Phys. Rev. Lett. , 3991(1996).[12] K. Hansen and E. Campbell, J. Chem. Phys. , 5012(1996).[13] K. Hansen, Int. J. Mass. Spectrom. , 14 (2018).[14] S. P. Møller, Nucl. Instr. Meth. A , 281 (1997).[15] S. Jinno, T. Takao, Y. Omata, A. Satou, H. Tanuma,T. Azuma, H. Shiromaru, K. Okuno, N. Kobayashi, andI. Watanabe, Nucl. Instrum. Methods Phys. Res. A ,477 (2004).[16] L. H. Andersen, O. Heber, and D. Zajfman, J. Phys. B.:At. Mol. Opt. , 57 (2004).[17] S. Martin, J. Bernard, R. Br´edy, B. Concina, C. Joblin,M. Ji, C. Ortega, and L. Chen, Phys. Rev. Lett. ,063003 (2013).[18] H. T. Schmidt, R. D. Thomas, M. Gatchell, S. Ros´en,P. Reinhed, P. L¨ofgren, L. Br¨annholm, M. Blom,M. Bj¨orkhage, E. B¨ackstr¨om, et al., Rev. Sci. Instrum. , 055115 (2013), URL http://scitation.aip.org/content/aip/journal/rsi/84/5/10.1063/1.4807702 .[19] Y. Nakano, Y. Enomoto, T. Masunaga, S. Menk,P. Bertier, and T. Azuma, Rev. Sci. Instrum. , 033110 (2017).[20] K. Hansen, Chem. Phys. Lett. , 43 (2015).[21] P. Kupser, J. D. Steill, J. Oomens, G. Meijer, and G. vonHelden, Phys. Chem. Chem. Phys. , 6862 (2008).[22] L. Nemes, R. S. Ram, P. F. Bernath, F. A. Tinker, M. C.Zumwalt, L. D. Lamb, and D. R. Huffman, Chem. Phys.Lett. , 295 (1994).[23] J. Men´endez and J. B. Page, Light Scattering in Solids (Springer, Berlin, 2000), vol. VIII, chap. VibrationalSpectroscopy of C , pp. 27–95. VIII. APPENDIX: THE APPROXIMATION OFTHE RATE CONSTANT
The use of Eq.(3) requires that parameters extractedfrom the experiments need corrections before they canbe compared with parameter values from other types ofexperiments. The corrections are known [20] and willapplied after the analysis.The energy in the denominator, E + E ′ , is the sum ofthe true thermal energy, E , and an offset, E ′ , which isrequired to account for situations where the thermal en-ergy is not simply proportional to the temperature. Theoffset includes the zero point energy of the harmonic os-cillators, which provide the largest part of the heat ca-pacity of the molecule, but also accommodates any otherthermal offset that may be present below E , for whateverreasons. In the following this offset will be absorbed intothe energy and will not appear explicitly.The main energy dependence of the electron emissionrate constant is the contribution from the ratio of leveldensities, and the main question therefore concerns theaccuracy of the approximation ρ ( E − φ ) ρ ( E ) = exp (cid:18) − φC v E + E ′ (cid:19) . (34)The quality of this approximation is best seen by plotting E vs. ln ( ρ ( E − φ ) /ρ ( E )). From the rewritten relation E = φC v ln (cid:16) ρ ( E − φ ) ρ ( E ) (cid:17) − E ′ (35)0a straight line is expected. It is indeed also seen inFig.(10). The slope is 434 eV and the offset gives E ′ = 4 .
64 eV,both in good agreement with the expectedvalues. Importantly, the line is straight to a good ap-proximation. The value where the expected abscissa is
Figure 10. The test of the approximation of the rate constantby the expression in Eq.(34). located is centered slightly below -4, with an ±±