Russell
Cooper
^{a},
Igor
Rahinov‡
^{a},
Zhisheng
Li
^{a},
Daniel
Matsiev
^{a},
Daniel J.
Auerbach
^{b} and
Alec M.
Wodtke
*^{a}
^{a}Dept. of Chemistry and Biochemistry, University of California Santa Barbara, Santa Barbara, CA 93106-9510, USA. E-mail: wodtke@chem.ucsb.edu
^{b}GRT Inc., 861 Ward Drive, Santa Barbara, CA 93111, USA
First published on 24th May 2010
Vibrational overtone excitation is, in general, inefficiently stimulated by photons, but can under some circumstances be efficiently stimulated by electrons. Here, we demonstrate electron mediated vibrational overtone excitation in molecular collisions with a metal surface. Specifically, we report absolute vibrational excitation probabilities to ν = 1 and 2 for collisions of NO(ν = 0) with a Au(111) surface as a function of surface temperature from 300 to 985 K. In all cases, the observed populations of vibrationally excited NO are near those expected for complete thermalization with the surface, despite the fact that the scattering occurs through a direct “single bounce” mechanism of sub-ps duration. We present a state-to-state kinetic model, which accurately describes the case of near complete thermalization (a regime we call the strong coupling case) and use this model to extract state-to-state rate constants. This analysis unambiguously shows that direct vibrational overtone excitation dominates the production of ν = 2 and that, within the context of our model, the intrinsic strength of the overtone transition is of the same order as the single quantum transition, suggesting a possible way to circumvent optical selection rules in vibrational pumping of molecules. This result also suggests that previous measurements of vibrational relaxation of highly vibrationally excited NO exhibiting highly efficient multi-quantum jumps (Δν ∼ −8) are mechanistically similar to vibrational excitation of NO(ν = 0).
Developing better understanding of interfacial energy conversion at the atomic level is particularly promising in studies at the gas–solid interface. Here, sophisticated dynamics methods including molecular beams^{8} and laser spectroscopy^{9} can be combined with surface analytical probes to create well defined experiments, the observations of which can be directly compared to first principles theory.^{10–12} This approach has led to significant insights which have been reviewed on several occasions.^{13–15}
The interconversion of energy between molecular vibration and solid excitation is of particular interest due to the close relationship between molecular vibration and bond dissociation. Furthermore, it is here that some of the clearest evidence has become available that molecular interactions at surfaces are not purely mechanical in nature,^{16,17} but can also interact strongly with electrons in the solid,^{18} even resulting in electron emission when the transferred vibrational energy is greater than the solid's work function.^{19–21} In contrast, when this quantity is less than the solid's band gap, vibrational energy transfer to or from the solid is less efficient.^{22–24}
A recent theoretical treatment of electronically non-adiabatic interactions^{10} has been remarkably successful at reproducing experimental observations including rotational cooling with vibrational relaxation^{25} and vibrational insensitivity to trapping.^{26}
Perhaps the most interesting agreement between experiment and theory involves multi-quantum (Δν ∼ −8) vibrational relaxation of highly vibrationally excited NO(ν = 15) in collisions with a Au(111) surface. Here, sub-ps time-scale electron hopping between NO and Au results in efficient high overtone vibrational relaxation and creation of a single correspondingly high energy electron hole pair (EHP).^{10,18} By comparison, photon emission from NO(ν = 15) is dominated by Δν ∼ −1 transitions.^{27} These results demonstrate a crucial difference between electron mediated and photon mediated vibrational transitions; namely, the probability of “high overtone” transitions can be much larger in the former than in the latter. This begs the question to what extent electron mediated processes might be used to circumvent optical selection rules for vibrational overtone excitation, providing an avenue for direct pumping of many quanta into molecular vibration on the sub ps time-scale.
In principle, the mechanism of multi-quantum vibrational relaxation^{10} just described is consistent with one originally discussed^{28} in connection to single quantum vibrational excitation of small molecules on metal surfaces;^{16,29} however, the direct observation of multi-quantum vibrational excitation has never been reported within the context of an electron mediated energy transfer mechanism.^{30}
In this edge article, we present results of new experiments where electron mediated multi-quantum vibrational excitation is seen for the first time. Absolute excitation probabilities result in populations of ν = 1 and 2 close to those expected for complete vibrational thermalization with the surface (strong coupling case) despite the fact that the excitation happens on a sub-ps time-scale. We present a new kinetic model capable of describing the strong (as well as the weak) coupling case and, through detailed comparison with experiment, we show that electron mediated vibrational overtone excitation proceeds with similar intrinsic efficiency to single quantum excitation, limited only by the population of suitably energetic EHPs.
(1) |
For example, eqn (1) breaks down in the strong coupling case – i.e. where A is large or where T_{S} is high – as it cannot describe thermal equilibrium with the surface,^{31} which is clearly non-Arrhenius.
(2) |
Another limitation of the Arrhenius analysis represented by eqn (1) is the difficulty in distinguishing between direct vibrational overtone excitation, which is expected to have the Arrhenius form:
(3) |
(4) |
To address these issues, we have developed a kinetic model for vibrational state changing events occurring in collisions with a metal surface that incorporates microscopic reversibility and a statistical treatment of the thermal energy distribution of metallic EHPs modelled as a Fermi gas of electrons in a three dimensional box. The model includes rate processes that describe one and two quantum excitation and de-excitation between the three lowest vibrational states: ν = 0, 1 and 2. In addition to describing deviations from eqn (1) associated with the transition to equilibrium with the surface in the strong coupling case, this model also predicts conditions where molecule–surface collisions result in multi-quantum excitation. Furthermore, the model suggests under what condition the efficiency of multi-quantum excitation will result from direct vibrational overtone up-pumping as opposed to sequential single quantum up-pumping events.
To test these ideas, we have carried out temperature dependent measurements from T_{S} = 300–985 K of EHP mediated vibrational excitation probabilities for NO(ν = 1 and 2) in collisions of NO(ν = 0) with a Au(111) surface. We show that application of eqn (1) to these data results in an ambiguous physical interpretation concerning the importance of overtone vibrational excitation. In contrast, not only are the data well described by the new model but clear physical insights emerge. Specifically, the absolute temperature dependent excitation probabilities of NO(ν = 0) to NO(ν = 1 and 2) are correctly captured by the model when the intrinsic coupling is roughly equal for both fundamental and overtone transitions. Furthermore, if the model is adjusted to eliminate direct overtone transitions, the data cannot be accurately reproduced.
We use a simple rate equation method to describe EHP mediated energy transfer. The model includes six rate processes:
n_{0} ⇌ n_{1} |
n_{0} ⇌ n_{2} |
n_{1} ⇌ n_{2} |
k_{0,1} = ξ_{0,1}*f_{1}(T_{S}) |
k_{1,2} = ξ_{1,2}*f_{1}(T_{S}) |
k_{0,2} = ξ_{0,2}*f_{2}(T_{S}) |
k_{1,0} = ξ_{1,0}*f_{−1}(T_{S}) |
k_{2,1} = ξ_{2,1}*f_{−1}(T_{S}) |
k_{2,0} = ξ_{2,0}*f_{−2}(T_{S}) |
The electronic statistics are calculated as follows. We first assume that the Fermi function:
p_{e}(ε;T_{S}) = F(ε,T_{S})*ρ(ε) |
p_{h}(ε;T_{S}) = ρ(ε)*(1 − F(ε,T_{S})). |
f_{Δv}(T_{S}) = ∫^{∞}_{ε=0}dε∫^{∞}_{ε′=0}dε′p_{e}(ε;T_{S})*p_{h}(ε′;T_{S})*δ(ε − ε′ − Δv*hν) |
The function f_{Δν}(T_{S}) represents the sum over all states where a thermally excited EHP is present whose relaxation energy is equal to the vibrational excitation, Δν > 0. Similarly, it represents the sum over all electronic states where an electron and a higher energy hole is present capable of accepting the vibrational energy, Δν < 0.^{32}
With these definitions of the rate constants, we solve the system of differential equations analytically using the Laplace transform method,^{33,34} assuming initial conditions where population is found only in ν = 0.
n_{0}(0) = 1 and n_{1}(0) = n_{2}(0) = 0 |
Fig. 1 shows an example of solutions for the populations, n_{1}(t) and n_{2}(t), derived from the rate model. Here we have assumed certain reasonable values (see figure caption) for the intrinsic coupling strengths, ξ_{i,j}, in order to examine the short and long time behaviour of the model. While the precise solution is sensitive to the choice of coupling strengths, the qualitative fact that the populations for ν = 1 and 2 are initially zero and grow to a steady-state thermal population distribution – shown as dot-dashed lines – is not.
Fig. 1 The approach to thermal equilibrium at T_{S} = 1000 K. Upper panel: The solution to the kinetic equations for ξ_{0,2} = 0.69*ξ_{1,2} and ξ_{1,2} = ξ_{0,1} = 1 for the population, n, of ν = 1 (dashed) and ν = 2 (solid). The dot-dashed lines show the thermal populations for the two vibrational states. The vertical line at τ_{eff} = 0.96 represents the effective collision time derived from comparison to the NO/Au scattering data. See text. Although not shown, the population of ν = 0 also approaches the thermal value on this time-scale. Lower panel: for the case of ξ_{0,2} = 0.69*ξ_{1,2} and ξ_{1,2} = ξ_{0,1} = 0.01. |
It is useful to envision a physical picture to accompany the mathematical solutions obtained from the model; perhaps most importantly as this helps illuminate the assumptions of the model. We envision a molecule colliding with a surface in a direct scattering process. The model assumes that non-adiabatic coupling occurs with a constant magnitude. Thus as the molecule approaches the surface, there is some distance where the coupling turns on like a step function. It is then further assumed that the coupling remains constant during an “effective” collision time, τ_{eff}. We will show below that the vertical line in the upper panel of Fig. 1 represents the NO/Au scattering in this work and corresponds to an effective collision time of about τ_{eff} = 400 fs, within which the system comes approximately halfway to the equilibrium limit. We dub this scattering regime the strong coupling case.
Fig. 1 also shows the behaviour of the model when the coupling strength is reduced by 100 fold (lower panel), conditions that could be achieved for example by reducing the incidence energy of translation. Here, one obtains analogous growth curves that terminate in a steady state thermal population; however, the time-scale required to reach the steady state is 100 times longer. For a similar value of τ_{eff}, only about one-100th of the vibrational excitation would be observed, a regime we describe as the weak coupling case.
One may note from this comparison that when we arbitrarily fix ξ_{0,1}, we may vary the population transfer by adjusting τ_{eff}.^{35} Thus in all of the comparisons with real data that we will make below, ξ_{0,1} = 1, as in Fig. 1 (upper panel). We then adjust the effective collision time, τ_{eff}, to describe the overall strength of coupling. We can then vary the magnitudes of ξ_{1,2} and ξ_{0,2} with respect to ξ_{0,1} to explore the relative importance of single and multi-quantum processes.
Fig. 2 shows angular distributions for scattered NO molecules that have undergone vibrational excitation in collisions with Au(111). The angular distributions peak at the specular angle (1.6° from the surface normal) and exhibit a FWHM of less than 40°. They are much narrower than a cosθ distribution characteristic of trapping/desorption and indicate a direct, single-bounce vibrational excitation mechanism.
Fig. 2 Angular distribution of the scattered beams, ν = 2 (solid circles), ν = 1 (open circles). The solid line is a cos^{7.3}(θ − 1.6°) function and the dashed line is a cosθ function, expected if trapping/desorption was the predominant process. The absolute intensities of the two angular distributions are normalized with respect to each other to emphasize the shape of the scattered angular distribution. |
Fig. 3 shows rotationally resolved REMPI spectra of scattered NO(ν = 1 and 2)^{25} at two surface temperatures. The scattered molecules are rotationally hot (T_{rot} ∼ 700 K) compared to the incident beam (T_{rot} ∼ 10 K)^{46} and T_{rot} is found to be only weakly dependent on T_{S}, consistent with a direct single-bounce scattering mechanism. A glance at the figure reveals that the increase in REMPI signal with surface temperature is strong for both ν = 1 and 2 but stronger for ν = 2.
To derive the absolute vibrational excitation probabilities, data like those shown in Fig. 3 probing ν = 0, 1^{48} and 2 were recorded between T_{S} = 300 and 985 K, integrating the entire spectral intensity in each vibrational band. These integrated intensities were corrected for differences in temporal and angular distributions and normalized to differences in the laser power and MCP voltage.^{49} We also corrected for differences between each band's Franck–Condon factor.^{50} Ratios of these corrected signal intensities (S_{ν}) yield absolute excitation probabilities, P_{ν}:^{51}
The derived values of P_{ν} are shown in Fig. 4. At all surface temperatures in this work, P_{1} is at least 10× larger than P_{2}. At T_{S} = 985 K, P_{1} = 0.025, which compares well with the result P_{1} = 0.07 ± 0.05 reported for NO scattering from Ag(111).^{16,53} Also shown in Fig. 4 are the thermal limits – eqn (2) – as well as Arrhenius functions – eqn (1) – obtained by fitting in two different ways. See below. The Arrhenius activation energies (Table 1) are close to ΔE_{vib} in all cases; however, as discussed above in relation to eqn (3), this is not sufficient to establish the importance of direct vibrational overtone excitation.
Fig. 4 Vibrational excitation probabilities vs. surface temperature. ν = 2 (closed circles), ν = 1 (open circles). The thick lines are the canonical Arrhenius fits and the thin lines are the floating Arrhenius fits, constants for which are presented in Table 1. See text. The dot-dashed lines are the thermal limit. The data have been culled to eliminate statistical outliers.^{52} |
Fitting parameters | Canonical Arrhenius fit | Floating Arrhenius fit | Kinetic model Arrhenius fit^{a} | |
---|---|---|---|---|
a The coupling strength was τ_{eff} = 0.96 and the state-to-state coupling was ξ_{0,2} = 0.69*ξ_{1,2} and ξ_{1,2} = ξ_{0,1} = 1. b The RMS is defined as follows: | ||||
A | 0.39 | 0.34 | 0.41 | |
ν = 1 | E _{a}/cm^{−1} | 1904 | 1814 | 1942 |
RMS^{b} | 0.0014 | 0.0014 | — | |
A | 0.46 | 0.83 | 0.45 | |
ν = 2 | E _{a} | 3808 | 4189 | 3798 |
RMS^{b} | 0.00028 | 0.00028 | — |
Fig. 5 shows the best fit^{54} to the data. Comparing to the thermal limit (dot-dashed curves) and noting the derived value of τ_{eff} = 0.96, one can clearly see that NO scattering from Au(111) at this incidence energy is an example of strong electronically non-adiabatic coupling. Referring to Fig. 1, one can see that this value of τ_{eff} represents a coupling strength that brings the system roughly halfway to the thermal limit.
Fig. 5 Comparison of experimental NO vibrational excitation probabilities (P_{ν}) to the kinetic model. The effective collision time was τ_{eff} = 0.96, and the state-to-state coupling was ξ_{0,2} = 0.69*ξ_{1,2} and ξ_{1,2} = ξ_{0,1} = 1. The closed and open circles are experimental data for ν = 2 and 1, respectively. The solid and dashed lines are the results of the kinetic model. The dot-dashed lines are the thermal expectation. |
In passing, we note that one may estimate τ_{eff} and hence the absolute time-scale of the plot in the upper panel of Fig. 1. A recent paper of ours identified a critical distance for vibrationally promoted electron emission,^{21}Z_{C} ∼ 5 Å, which despite several important differences we take to be characteristic of the interaction distance for vibrational excitation in this work at least for the purpose of estimation. From the measured velocity of the NO molecular beam, v_{NO}, we estimate the round trip time of the collision, τ_{eff}, within a constant velocity approximation:
In characterizing the excitation to NO(ν = 2) it is important to distinguish between direct and sequential excitation. Since production of NO(ν = 1) is always at least 10× more efficient than production of NO(ν = 2) under our conditions, it is reasonable to think that NO(ν = 2) might be produced by two single-quantum excitation steps and not by a vibrational overtone mechanism.
Fig. 6 shows the impact on the fit if vibrational overtone pumping is neglected, that is setting ξ_{0,2} = 0. While this change has no noticeable effect on the predicted ν = 1 populations, the ν = 2 excitation probability is dramatically underestimated. To obtain a reasonable fit when overtone pumping is neglected ξ_{1,2} has to become unreasonably large, ξ_{1,2} > 20*ξ_{0,1}. There is no reason to think that ξ_{1,2} should be much larger than ξ_{0,1}, as indicated by the full fit. In contrast, one may start with the best fit parameters of Fig. 5 and change ξ_{1,2} to 0, with no detectable effect on the fit.
Fig. 6 The importance of vibrational overtone excitation. This is the resulting fit if ξ_{0,2} = 0. Otherwise, the fitting parameters are identical to those of Fig. 5. The Arrhenius parameters obtained for the 0-2 excitation in the sequential mechanism (solid line) are A = 0.11 and E_{a} = 3931 cm^{−1}. |
Therefore, we conclude that ν = 2 is predominantly populated by direct vibrational overtone excitation. Indeed, the best fit results – ξ_{0,2} = 0.69*ξ_{0,1} – suggest that the multi-quantum vibrational excitation is about as strong as single quantum excitation. This contrasts starkly with optical overtone transition probabilities. For example for NO, the 1st optical overtone strength is 0.015 that of the fundamental.^{56}
Prior to the kinetic model presented in this edge article, it has been customary to compare the T_{S}-dependence of vibrational excitation data to an Arrhenius relation:
Table 1 shows derived results for A and E_{a} using both the canonical and floating methods as well as the root-mean-square (RMS) deviations of the fits. Also see Fig. 4. It is immediately obvious that one cannot distinguish the canonical from the floating fit based on a goodness of fit criterion. This ambiguity results from the finite noise associated with the data and the resulting statistical correlation between A and E_{a}.
This ambiguity limits our ability to interpret the meaning of the derived parameters, A and E_{a}. According to the canonical fit results, the ratio of the A-factors is 0.46/0.39 = 1.2, while the floating fit gives a larger ratio of ∼2.5. Interestingly, by fitting the results of the kinetic model to an Arrhenius form, letting both A and E_{a} be optimized, we find excellent agreement with the canonical results. See the last column of Table 1. This reflects the underlying correctness of the canonical Arrhenius picture, which can, in fact, be written in terms of the kinetic model of this edge article, a topic that will be treated in detail in a future publication.^{57}
Footnotes |
† Electronic supplementary information (ESI) available: The Mathematica Notebook used to analyze the data in this edge article using the multi-quantum kinetic model. See DOI: 10.1039/c0sc00141d |
‡ Present address: Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel. |
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