State-to-State Rate Coefficients for NH₃–NH₃ Collisions from Pump–Probe Chirped Pulse Experiments

Abstract

The kinetics of rotational inelastic NH₃–NH₃ collisions are recorded using pump–probe experiments, carried out with a K-band waveguide chirped pulse Fourier transform microwave spectrometer, in which the population of one inversion doublet is altered by the pump pulse. Due to self-collisions, the resulting deviation from equilibrium propagates to other states and, thus, can be interrogated by probe pulses as a function of the pump–probe delay time. A clear hierarchy of the state-to-state collision processes is found and subsequently translated into propensity rules. State-to-state rate coefficients are estimated, first via an analysis of the kinetics, and then more robustly and accurately derived from the pressure-dependent measurements using a global fitting procedure.

Reliable data on state-to-state rate coefficients are essential for accurately modeling rarefied gases. Experimental studies on these rate coefficients are often either missing or limited, and thus, the overwhelming majority of such values currently available are the result of numerical calculations performed using a variety of different levels of theory. Due to its importance, ammonia (NH₃) and its isotopologues might, at first glance, seem to be exceptions to this general trend because they have been the subjects of a number of previous collision studies. The first of these experiments were carried out by Oka using steady-state double-resonance techniques to derive propensity rules and to quantify population transfer between rotational states. Oka and co-workers subsequently investigated not only NH₃–NH₃ collisions¹ but also collisions of NH₃ with rare gases² as well as with polar³ and nonpolar molecules.⁴ Additional time-resolved measurements based on Stark-switching techniques enabled direct observations of rotational relaxation in ammonia, including its M dependence, and the application of double-resonance techniques in a beam-maser chamber allowed Kukolich,⁵ and later Klaassen et al.,⁵⁻⁸ to measure scattering cross sections and compare their experimental values with ones calculated using a modified version of Anderson’s theory.^9,10 The comparison of experimentally derived pressure-broadening and pressure-shift parameters obtained from an observation of the line shapes with those predicted by theory has been the most commonly used technique to study collisions to date. Since the early work of Bleaney and Penrose,¹¹⁻¹³ the line shapes of ammonia have been studied extensively at microwave and infrared frequencies with a number of different collision partners. The most recent works have focused on collisions at low temperatures with He or H₂ due to their astronomical importance.¹⁴⁻¹⁷ There is a large body of work on the use of double resonance techniques that, in particlar, employ time-resolved spectroscopy to derive rate coefficients for collisional processes, e.g., for vibrationally excited ammonia.¹⁸ However, no individual experimental state-to-state rate coefficients for vibrational ground-state ammonia and many other collision systems are available, a deficiency we seek to remedy by the method we propose here.

Specifically, in order to better study NH₃–NH₃ collisions, we performed pump–probe experiments that use a K-band chirped pulse waveguide spectrometer at the Center for Astrochemical Studies at MPE. In our experiments, resonant radiation (the pump) is applied, thereby putting the system out of thermal equilibrium by strongly altering the population of one inversion doublet of ammonia, corresponding to one (J,K) rotational state. Figure 1 schematically shows the energy term diagram of this process, with the bold arrow in the center indicating the pump excitation process between the ammonia states of different parity (±).

Figure 1.

Schematic energy levels of the (J,K) rotational states of ammonia including the tunneling doublet (+/−). Inversion transitions (+ ↔ −) that connect the levels of a doublet are observed at microwave frequencies in this work. Black arrows indicate pump (bold) and probe (thin) transitions. Different types of collision-induced transitions are depicted by vertical arrows: transitions within one inversion doublet (ΔJ = ΔK = 0, green), parity-changing collisions between inversion doublets (blue, ΔJ = 1,ΔK = 0), parity-conserving transitions (orange, ΔJ = 1,ΔK = 0), and others (gray).

Once the microwave is turned off, the ensemble returns to thermal equilibrium via collisions. The respective rates, K, for these inelastic collision processes are indicated in Figure 1 by arrows connecting the associated states of ammonia. In contrast to previous studies, chirped pulse spectroscopy (the probe) allows one to follow this relaxation process over time, and many states can be monitored at once due to the broad-band radiation probing characteristic of the this technique, indicated in Figure 1 by thin black arrows. Normally, unravelling the individual rates, K [1/s], from a single time evolution is a formidable task, given the large number of microscopic processes involved, only a very limited number of which are shown in Figure 1 for the sake of clarity. However, thanks to both the hierarchy of collisional rates as well as the time-resolved decay curves obtainable from chirped pulse experiments, the determination of state-to-state rate coefficients can easily be done, as will be shown in this Letter. We like to note here that, although |M_J| resolved measurements should be possible by applying static electric fields, as shown, e.g., by Oka,¹⁹ Mäder,²⁰ and Vogelsanger,²¹ only the (2J + 1) degeneracy is taken into account here, and averaged rate coefficients are obtained.

Ammonia inversion transitions are observed in a rather narrow frequency band of only a few GHz close to 23 GHz. Many of these are revealed during the submicrosecond chirped probe pulse, called the probe chirp, which excites all of the inversion doublets in a 5 GHz range, thereby resulting in a probe spectrum that shows signals from many individual (J,K) states at once. The overall sequence of events in the experiment is shown in Figure 2. Following excitation during the probe chirp, the free induction decay (FID) is recorded for about 10 μs, represented by the second trace in Figure 2, which shows an actual measurement. The exponential decay of the envelope of the recorded signal is related to the loss of polarization created during the coherent probe excitation. The time constant for this process, the decoherence time (T₂), has been previously measured²² and will therefore not be discussed in great detail here. Decoherence rates, T₂^–1, are both linearly proportional to pressure and, thus, related to the rate coefficients of the collisions involved. Shown in Table 1 is a comparison between the slopes of these values determined from both our experiments and previous studies, as well as more conventional pressure-broadening parameters as derived from T₂. We stress, however, that T₂ is an integral measure of many detailed processes, thus necessitating more specific measurements. In our experiment, these are obtained by addressing a single state (J,K), as indicated in Figure 2a, where a monochromatic pump pulse of duration t₁ and a delay of t₂ with respect to the subsequent probe pulse are shown. The resulting FID signals are shown in Figure 2b, which are the actual measurements. When one compares the FID signal curves before and after the pump, the signal change due to the pump pulse appears, at first glance, to be small. In fact, though, a sizable signal becomes apparent when the FID signal is subtracted from the one without the pump, as indicated in Figure 2c. There, one can observe that the first probe signal leads to a vanishing signal (no corresponding pump) but the second probe leads to the desired pump–probe difference signal. In fact, each pump–probe cycle has been recorded with four-step phase cycles (0°, 90°, 180°, and 270°) for sideband separation and removal of dependencies on coherences remaining from the pump pulse. In so doing, we thus optimally cancel out uncorrelated signals and background in order to best recover the desired difference signal.

Figure 2.

Pump–probe pulse sequence and signal processing. (a) A sequence of pump pulses followed by 5 GHz broad chirped pulses (200 ns) are applied, and the subsequent FIDs are recorded. Each sequence starts with only a probe pulse to obtain a reference signal at thermal equilibrium conditions, followed by a number of pump–probe combinations (of which only one is shown in the figure) using different pump pulse lengths and delay times between the pump and probe pulse. Every combination, including the reference probe pulse, is repeated four times with 90° phase shifted probe pulses (ϕ = 0°, 90°, 180°, 270°). (b) The quadrature phase-averaged signal S_t1,t2(t) with the pump pulse length t₁ and delay t₂ is calculated from the FIDs recorded after the four phase-shifted probe pulses for each combination. (c) Difference signal ΔS_t1,t2(t) as explained in the text. (d) Fourier transform to derive the fractional change in signal intensity ΔS_t1,t2^J,K(f) for each individual transition. (e) s^J,K(t₁,t₂) plotted as a function of the delay t₂ between pump and probe pulses.

Table 1. Comparison of 1/T₁ and 1/T₂ in μs^–1 mbar^–1 for Inversion Doublets (J,K) = (2,1) and (1,1) and Respective Pressure-Broadening Parameters Δν_p in MHz mbar^–1.

	(J,K)	T1–1	T2–1	Δνp
present work	(1,1)	111 ± 5
Hokea,b,23	(1,1)	150 ± 7	106 ± 2	16.8 ± 0.4
Amanoa,24	(1,1)	113 ± 7	102 ± 2	16.2 ± 0.4
present work	(2,1)	75 ± 5	62.5 ± 5.0	9.9 ± 0.8
Tanaka22	(2,1)		61.7 ± 0.7	9.82 ± 0.11
Hokea,b,23	(2,1)	82 ± 7	72.7 ± 2.3	11.6 ± 0.4
Amanoa,24	(2,1)	84 ± 4	68.8 ± 1.4	11.0 ± 0.2
present work	(3,1)	66 ± 5
present work	(4,1)	59 ± 5

^aValue for ¹⁵NH₃.

^bValue for |M| = J states. It is important to note that each level is (2J + 1)-fold degenerate with the quantum number M_J and that rates in this work are in fact average rates in contrast to the Stark-switching-type experiments where individual |M_J| states are resolved. The M dependency of T₁ as discussed by Hoke et al.¹⁰ explains their significantly higher value of T₁^–1 compared to our and Amano’s work. In addition, our T₂ contains also contributions from reorientation collisions (only ΔM_J ≠ 0).

In principle, this signal still contains information about all transitions that have been addressed in the probe excitation and are recorded in the FID measurement. The Fourier transform of the difference signal shown in Figure 2d carries information on the two (3,1) tunneling doublet states (+/−), which have been pumped as shown in the top trace. Recording the intensity of the (3,1) peak in the spectrum as a function of time for varying delay times, t₂, is shown in the right part of the lower trace in Figure 2e. There, one can see the temporal behavior of the (3,1) difference signal in which, first, the population is practically inverted by an effective π-pulse reached at t = t₁ and then drops back exponentially to zero (difference signal). The time constant of this relaxation, T₁, has also been recorded in previous studies, and the pressure coefficients are summarized in Table 1, showing reasonable agreement. We again stress, though, that T₁ is determined by the net effect of all inelastic processes resulting in a population loss from the pumped doublet of the (3,1) state; therefore, it is not at all obvious which states are involved in the transfer or to what extent.

This uncertainty is the motivation for further studying the temporal behavior of the state-resolved signal difference as a function of the pump–probe delay, t₂. Figure 3 shows the result of a series of such measurements for a pumped (2,1) state and different probe states all recorded simultaneously. The previously noted initial inversion of the pumped (2,1) state is once again observed and is reached after 380 ns at t = 0, at which time s(t₁,t₂) reaches approximately unity before dropping ca. 3 orders of magnitude. This behavior is very well described by the theoretical curve obtained from a simulation that includes the desired rates and that is described in more detail below. Also shown in that graph is the temporal evolution of the neighboring (3,1) and (1,1) states. The signal of these states likewise shows the expected initial growth, due to the population difference transferred into these states via collisions, before dropping to zero at later times. Similarly, as with the pumped state, the measured data are nicely reproduced by the simulation using one consistent set of rate coefficients. The observed slopes of the three curves show that the relaxation rates for the three different states, despite being comparable in size, are nevertheless significantly different. Because all of the population differences of the neighboring (3,1) and (1,1) states stem from the fast decaying (2,1) state, resolving individual rates from these observations would be rather difficult.

Figure 3.

Fractional changes of signal intensities for selected transitions. The population of the (2,1) state, altered by an effective π-pump pulse, is transferred to states (1,1) and (3,1) (ΔJ = 0, +1, −1,ΔK = 0). The time origin is set to the end of the pump pulse t₁^max, and the time axis corresponds to t₂, the delay time between pump and probe pulses. The upper plot shows the temporal evolution on a logarithmic scale, whereas the lower plot shows the evolution of the doublets (1,1), (3,1), and (2,2) on the normal scale. Population is still transferred from the pumped doublet to the doublets (1,1) and (3,1) after the end of the pump pulse. No population transfer is observed to the doublet (2,2). The experimental data is shown as error bars, and the solid lines correspond to simulations based on the global fit including measurements at different pressures and altering different doublets.

Shown in the lower part of Figure 3 are the measured signal differences, but on a linear scale to illustrate the disparity between the two in more detail. There, one can clearly see that population differences in neighboring states have the opposite sign, which is associated with an ordering of specific rates in size. More importantly, the signals increase in magnitude even when the pump light has been switched off at t = 0 (see the vertical dotted line). In fact, the maximum signal for the (3,1) state is only reached at t ≈ 0.4 μs. This is an important finding because it shows that, although the difference signal of the pumped (2,1) state is already decaying, the signal for the neighboring states is still increasing. As it turns out, this increase is directly related to the rates that connect the states (2,1) and (3,1) as well as (2,1) and (1,1) for the other trace. More important than even these, however, is our finding that practically no other state obtains population difference from the pumped (2,1) state. This is shown, e.g., for the (2,2) state, which, apart from some small signal created for short times (t < 100 ns) due to off-resonant radiative excitation, stays zero at all times. Therefore, the presumption that the (J,K) states only “talk” to very few and neighboring states is clearly demonstrated by this experiment. This finding implies that one can thus infer state-dependent rates and rate coefficients from the sorts of data that we have obtained in this study.

This simple inspection of the experimental data without further analysis points to the same hierarchy of rates for inelastic ammonia collisions that had previously been inferred by Oka,¹ based on intensity changes due to optical pumping, unlike in this study where the temporal behavior of individual states is unfolded in a coherent experiment. Most rates are extremely small, as seen above, such that the following propensity rules for inelastic collisions can be formulated, which Oka related to dominant long-range dipole–dipole interactions:^4,19 (i) ΔK ≠ 0 transitions are extremely rare; (ii) for the observed ΔK = 0 transitions, only transitions with ΔJ = ±1 are found; and (iii) collisions only changing parity, (ΔJ = ΔK = 0), are the most likely processes.

Neglecting all but these dominant collisions allows one to deduce an approximate value for the rate connecting the levels within one doublet: 2K_+–^pp ≈ (T₁)⁻¹; see Figure 1. Here, the factor of 2 arises because a population change counts twice in the measured population difference. The superscript ^pp on K_+– denotes the pumped (J,K) state that is not changed, and the subscript _+– indicates the transition within the +– doublet. For a full description of the network of inelastic collisions, the rates (arrows) depicted schematically in Figure 1 have been taken into account. Despite the previously noted hierarchy among these, quite a number still exist; nevertheless, one consistent set of such rate coefficients has been determined, summarized in Table 2, which are based on all measurements, including those made under different experimental conditions, i.e., pressures in the range of p = 5–30 μbar.

Table 2. Rate Coefficients k Obtained by Simulations of All Coupled States Fitted to the Temporal Behavior of the Complete Pump–Probe Experiments^a.

J,K,P	→	J′,K′,P′	k (cm3 s–1)	2k (μs–1 mbar–1)
1,1,+	→	1,1,–	2.41(1) × 10–9	116.5(5)
2,1,+	→	2,1,–	1.48(1) × 10–9	71.3(5)
3,1,+	→	3,1,–	1.27(1) × 10–9	61.1(5)
2,1,+	→	1,1,–	1.01(3) × 10–10
2,1,+	→	1,1,+	4.5(2) × 10–11
3,1,+	→	2,1,–	1.59(8) × 10–10
3,1,+	→	2,1,+	2.3(7) × 10–11

^aRate coefficients for transitions from the lower to upper doublet as well as for opposite parities are obtained via detailed balance. Uncertainties present only the statistical errors derived in the global fit. In total, 29 independent parameters are used to fit the temporal behavior of 218 measurements.

Table 2 also lists, for comparison, the values 2k_+–ⁱⁱ = 2K_+–/p ≈ (T₁)⁻¹, expressed in units of μs^–1 mbar^–1. The rate coefficients, k, given there are related to the measured rates by the corresponding number density, n, of the ammonia collision partner by K = k · n. It is gratifying to see that the 2k value for the (2,1) state agrees reasonably well with the T₁^–1 values listed in Table 1, thereby showing that our systematic investigation has resulted in these rate coefficients having been unambiguously determined with high accuracy. The respective rate coefficients, listed in Table 2, vary over 2 orders of magnitude from 10^–11 to 10^–9 cm³ s^–1, as expected from the observed hierarchy of processes.

With this hierarchy in mind, the negative signals recorded for the neighboring (J,K) states can be easily interpreted. While the pump pulse leads to a population increase of the upper parity state (+) and a lowering of the lower state (−), the negative signal for the (J ± 1,K) states stems from an increase in the lower state (−) and a decrease in the upper state (+). Therefore, the (2,1) to (1,1) population transfer is predominantly associated with a change in parity, and K_+–^pi > K₊₊ still holds. This is also the result of the numerical fitting of the rate coefficients k in Table 2, which, satisfyingly, shows k_+–^pi = 2.2k₊₊ for the transition (2,1) to (1,1). It is this K_+–^pi rate that leads to the initial population change in the (J ± 1,K) states, as seen in Figure 3. Later on, when the population difference of the pumped state (s^2,1(t₁,t₂)) becomes smaller, the fast relaxation rate 2K_+– of the probed state dominates and leads to the final decay. Likewise it is obvious that the population differences for the (1,1) and (3,1) are fed by the (2,1) state, which leads to the rise of the s^1,1 and s^3,1 signals, also shown in Figure 3. The decay of these signals at longer times is determined by the T₁^–1 rates for the individual states. With these two competing processes at work, the minima for the s^1,1 and s^3,1 signals then arise when the products of the rates, times the respective population differences, are equal. Equipped with either 2k_+– ≈ T₁^–1 = 116.5 μs^–1 mbar^–1 from the decay at long times or independent measurements of T₁ (see Table 1), the signal values s^1,1(t₁,t₂) = −0.0137 and s^2,1(t₁,t₂) = 0.835 at the minimum of the (1,1) channel at t₂ = 0.24 μs can be used to infer the missing rate coefficient for the (2,1) → (1,1) transition observed by the early rise of the s^1,1 signal. As noted in the previous discussion, parity-changing collisions (rate coefficients: k_+–^pi and k_–+) are dominant; however, the parity-conserving collisions (k₊₊^pi and k_––) must still be considered. Ratios of these rate coefficients can then be related by detailed balance, based on which the dominant and desired rate coefficient k_+–^pi = 5.9 × 10^–11 cm³ s^–1 has been estimated for the (1,1) state.

The general agreement of these rough estimates with the results from our far more systematic investigations involving an analysis of the full rate equation system further supports the ordering of collision processes and illustrates the successive population transfer into the neighboring states. The most notable difference between the two is the estimated value for k_+–^pi, which, as seen in Table 2, is significantly smaller in our simplistic estimation than the value that we derive using the global fit analysis. This difference, however, is due to our having neglected some terms for the sake of a more straightforward estimate.

In summary, we have found that only a few key processes are involved when thermal equilibrium is disturbed by addressing the inversion doublet of only one rotational state. Therefore, only the rates of these critical processes, as measured in the time-resolved chirped pulse experiment, determine the observed signal, and consequently, the corresponding rate coefficients can thus be robustly and accurately determined. Indeed, we have done exactly this, resulting in a single set of rate coefficients that have been derived from an extended series of measurements at a number of different pressures and other parameters.

Our findings open up the possibility for further interesting experiments in which rate coefficients for collisions with atoms and other molecules can be similarly determined. For example, in astrophysical environments, the dominant collision partners of ammonia are helium, hydrogen, or even pure para-H₂. There currently exists a pressing need for corresponding rate coefficients for these processes in order to accurately interpret signals observed by radio telescopes as well as to test the reliability of existing predictions obtained via ab initio quantum chemical calculations.

Chirped pulse Fourier transform spectroscopy is very well-known for its state-specificity, and here, we have shown that the high temporal resolution of this technique also makes it an ideal tool for studying state-specific kinetics. Moreover, this method can be applied to a large number of molecules thanks to the broad frequency coverage of today’s chirped pulse instruments. Thus, on the basis of the results described here, we conclude that this approach nicely fulfills the long-standing desire for experimental analyses of state-to-state processes.

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01653.

• Experimental methods, including details of the spectrometer, measurements, and brief description of the global fit (PDF)

The authors declare no competing financial interest.