The REMPI spectra of scattered CO show a distinct rotational structure. Each vibra-tional band consists of a P-, Q- and R-branch. The population of individual rotavibra-tional states and the mean rotational energy of scattered molecules can be derived from these spectra. This section describes the underlying analysis, in which a J-dependent popu-lation factor is determined in a fitting procedure such that the experimental spectrum can be accurately reproduced.
Fitting rotational spectra requires the knowledge of precise rotational line positions.
In order to assign the observed rotational lines, A1Π(v0, J0)←X1Σ+(v00, J00) transition frequencies are calculated using the computer program PGOPHER [154, 155]. Spec-troscopic constants published by Schneider et al. [102] and Simmons et al. [103] are used to calculate term energies of the X1Σ+ and A1Π state, respectively. However, due to experimental limitations, the constants provided by Simmons are only accurate for rotational excitations up to J ≈ 20. Exact line positions for transitions probing J00>20 are calculated from transition frequencies reported by Gerö [156]. Since Gerö made a careful analysis of perturbations in the A1Π → X1Σ+ system, the list is also used to identify perturbed levels in the A1Π state and to recalculate their term energies.
Manually replacing transition frequencies from the PGOPHER output with frequencies calculated from Gerö’s data provides an accurate linelist for the following analysis.
The 7-17 and 7-16 vibrational bands are good candidates for the rotational analysis of vibrationally elastically and inelastically scattered molecules. v00 = 16 and v00 = 17 are probed via the A1Π(v0 = 7) state. Note that in A1Π(v0 = 7), the rotational states J0= 26, 29, 32, 33, 35, 38, 39, 42 and 43 are perturbed [156]. As the mixing of states may strongly affect transition intensities, rotational transitions involving perturbed states are omitted from the rotational analysis. A list of rotational transition wavenumbers of the 7-17 and the 7-16 band, including transitions to perturbed levels, can be found in Appendix A.1. Figure 4.7 shows assigned rotational transitions for both the 7-16 band and the 7-17 band of surface-scattered CO.
The population factors PJ00 of the rotational states J00 are determined by fitting a Lorentzian line profile to each rotational line in the observed spectrum. A single rotational line can be labeled byJ00and the respective rotational branch. The Lorentzian line shape is given by
I(˜ν) =I0 γ/2π
(˜ν−ν˜0)2+ (γ/2)2, (4.16) where the natural linewidthγ is the full width at half maximum,I is the intensity, ˜ν is the wavenumber, and ˜ν0 is the resonant wavenumber of the transition.
41200 41250 41300 41350 41400 41450 41500 41550
42920 42970 43020 43070 43120 43170 43220 43270
REMPIsignal/a.u.
R Q P (A) 7-16 vibrational band
41400 41420 41440 41460 41480 41500
(C) 7-17 vibrational band (zoomed in)
REMPIsignal/a.u. rota-tional transitions for CO scattered from Au(111) athEinci= 0.57 eV. The R-branch exhibits a band head around R(7)–R(8). Note that transitions to perturbed states are not marked. Panel (A) shows the 7-16 (v0-v00) vi-brational band. Panel (B) shows the 7-17 vivi-brational band. Panel (C) shows an enlargement of the spectrum displayed in Panel (B) in order to illustrate the quality of the assignment.
The total intensity I0 of a rovibronic transition is proportional to the population factorPJ00 and to the rotational transition strengthSJbranch00 for a certainJ00 and branch.
SJbranch00 corresponds to the Hönl-London factor as long as the transition is not saturated.
In the case of complete saturation,SJbranch00 equals the number of possible transitions per J00. In the absence of external fields, each J00 state is 2J00+ 1 degenerate. Thus, the number of possible transitions is 2J00+ 1 for the R-branch, 2J00 for the Q-branch as MJ0 = 0 ← MJ00 = 0 is forbidden, and 2J00 −1 for the P-branch as transitions from MJ00 =±J00 do not exist. Since the observed REMPI intensity depends linearly on the REMPI laser power (see Figure 4.4), saturation plays a significant role in the resonant excitation step of the two-photon REMPI process. The effect of saturation on the rotational transition strength is treated as described in the PGOPHER manual [154].
In order to account for saturation, SJbranch00 is expressed in terms of
SJbranch00 = 1−exph−SsatζJbranch00 i, (4.17) whereSsat is the saturation factor andζJbranch00 is the Hönl-London factor for a certainJ00 and rotational branch. For highSsat, the transition is saturated and SJbranch00 converges to the saturation limit. If Ssat approaches 0, SJbranch00 equals the product of the Hönl-London factor and the saturation factor, which scales with laser power. Hönl-Hönl-London factors can be calculated from the rotational matrix elements (see Reference [101] for more details) Here, J is the total angular momentum, Λ is the projection of the electronic orbital angular momentum on the molecular axis, MJ is the projection of J in the laboratory frame, ω = 1 is the photon angular momentum, σ0 is the polarization of the light in the molecular frame, and σ= 0 is the polarization of the light in the laboratory frame.
ζJbranch00 is obtained from Equation 4.18 by summing over MJ00 from −J00+ 1 to J00−1 for the P-branch (∆J =J0−J00=−1) and from −J00 toJ00 for the Q-branch (∆J = 0) and R-branch (∆J = 1). Note that ∆MJ =MJ00−MJ0 = 0.
Ssat is determined in an iterative way by fitting the rotational spectrum for various Ssat and calculating the population factor PJbranch00 separately for the P-, Q- and R-branch. The sum of PJbranch00 over allJ should give the same value for each branch. Ssat
is chosen such that the deviation between summed populations of individual branches is minimized.
Since the A←X transitions are partially saturated, the observed absorption lines are power broadened. Thus, the natural linewidthγ in Equation 4.16 needs to be replaced by a linewidth that accounts for power broadening. According to Equation 4.18, the rotational transition strength decreases with increasing J00. Following the description by Demtröder [157], the power broadened Lorentzian line widthγJbranch00 is calculated for P-, Q- and R-transitions originating fromJ00 as
γJbranch00 = ∆˜ν0
s
1 +Ssat
ζJbranch00
gbranch. (4.19)
The degeneracy factor gbranch is 2J00−1 for the P-branch, 2J00 for the Q-branch, and 2J00+ 1 for the R-branch. ∆˜ν0 is limited by the spectral width of the frequency doubled dye laser and is estimated to be 0.16 cm−1.
Finally, the rotational population factor PJ00 is determined by fitting the function IJbranch00 (˜ν)∝PJ00SJbranch00 γJbranch00 /2π
˜
ν−˜νJbranch00
2
+ γJbranch00 /22 (4.20) to each rotational line observed in the REMPI spectrum. The only fit parameter PJ00 is restrained to be the same for all rotational branches and varies only with J00. Line positions ˜νJbranch00 are listed in Appendix A.1.
A Boltzmann plot of rotational statesJ00is obtained by plotting ln (PJ00) as a function of the rotational energy of the scattered molecule. Note thatPJ00 need not to be divided by the degeneracy of rotational states due to the definition of SJbranch00 . The rotational energy of a molecule in the highly vibrationally excited X1Σ+(v00) state is given by
E v00, J00−E v00, J00 = 0,
whereE(v00, J00) can be expressed in terms of the standard Dunham expansion [101]
E v00, J00=
The mean rotational energy of scattered molecules is calculated as hErot v00i=
This chapter presents and discusses the results obtained from state-to-state surface scat-tering experiments using molecular beams of highly vibrationally excited CO. The exper-imental data comprises time-of-flight distributions, angular distributions, and REMPI spectra of surface scattered CO, which are used to extract final vibrational state and ro-tational state distributions. Final vibrational state distributions of highly vibrationally excited CO scattered from Au(111) are published in Reference [31]. For the scatter-ing from Ag(111), final vibrational state distributions and angular distributions are published in Reference [32].