Schematic of the experimental apparatus (left) and the anisotropy pattern of decay electrons (right). The incident negative muon (μ –), which is polarized opposite (green arrow) to its momentum direction (black arrow), triggers a “muon counter” and starts a time-to-digital converter (TDC), which is stopped by the detection of a decay electron in an “electron counter” positioned at a fixed angle, θ, with respect to the incident muon beam. In this experiment the energy-averaged “asymmetry’’, a μ , is measured and has a nominal value of a μ = 1/3, and this is the decay pattern shown on the right.
Schematic of the energy/time domains for μ+/μ− thermalization and charge neutralization giving Mu/Heμ formation. For μ+ (left), “surface μ+ beams” are often used, from π+ decay at rest [see Refs. 3 and 4], where the incident energy is 4.1 MeV, which is reduced to ∼2 MeV upon entering the gas target. For μ− (right), this mode is not possible and thus “Backward μ− beams” are used, from π− decay in flight, in which the μ− kinetic energy upon entering the gas target (see Fig. 1) is about 20 MeV, from the M9B channel at TRIUMF. In practice, backward μ+ beams were also utilized in the present study. In both cases most of this incident energy is lost in the “Bethe–Bloch” regime, where the stopping power, S(E) [see Ref. 3], is similar. At energies of ∼100 keV, the μ+ enters into a series of charge exchange cycles, emerging mainly as the neutral Mu atom. In the μ− case, this process is atomic capture, emerging, in the case of helium, as the muonic He ion, (Heμ)+. This is then charge neutralized with a dopant (X) in a manner similar to Mu formation, but at or near thermal energies. It is the neutral Heμ atom that is the basis of the present study.
In a transverse magnetic field (TF), the muon spin in the muonic He atom precesses at a characteristic frequency that depends on the field. Each time the muon spin sweeps past a fixed electron counter one detects its enhanced decay probability in the counter direction (Fig. 1), giving rise to an oscillatory pattern, which can be analyzed by Fourier transform (FT) spectroscopy. The figure shows the FT spectra for Heμ in the presence of about 7 bar NH3 and He up to a total pressure of 300 bar. The upper spectrum shows two split frequencies determined in a transverse magnetic field of 65.63 G, from which the hyperfine coupling constant ν0 can be determined.26,31 The lower spectrum shows a single frequency at the low field of 6.22 G, where the energy splittings become degenerate. It is the observation of the time dependence of this signal (Figs. 4 and 5), and particularly its relaxation rate, upon which the experimental results are largely based.
Asymmetry plots for Heμ + H2 at 295.5 K in a field of 6.8 G: (top) in a gas mixture of 7.2 bar NH3 plus He to 300 bar; (middle) with 25 bar of added H2 to give a concentration [H2] = 5.9×1020 molecules cm−3; (bottom) as in the top curve but with [H2] = 25.7×1020 molecules cm−3 at 500 bar total pressure. The solid curves are fits of the “asymmetry” defined by Eq. (2). The observed signal is mainly characterized by the oscillations seen, due to the Larmor frequency ωHe μ , with initial amplitude A He μ and relaxation rate constant, λHe μ . For [H2] = 0 in the top plot, λHe μ is just the background relaxation rate constant λ b of Eq. (3), and is almost zero (see also Fig. 6). Note though in the middle and bottom plots that the relaxation rate increases and the amplitude decreases with increasing H2 concentration.
Asymmetry plots for Heμ + H2 in 7.2 bar NH3 plus He to 500 bar in a 6.8 G TF, with [H2] = 0.87×1020 molecules cm−3 at 405 K in the top plot, but with [H2] = 0.42×1020 molecules cm−3 at 500 K in the bottom plot. See the caption to Fig. 4. The initial amplitudes A He μ are both 0.005, close to maximum, but the relaxation rates λ He μ are much faster at these temperatures, even at lower concentrations than in Fig. 4.
Relaxation rate constants vs. [H2] from fits of Eq. (2) to the data for the Heμ + H2 reaction at 295.5 K. The data were measured at two different total (mainly He) pressures, ∼300 bar (green data points and fitted line) and 500 bar (blue data points and fitted line), fit to Eq. (3), with the simultaneous fit of both data sets shown by the solid magenta line. The rate constants k He μ are given by the slopes: k He μ = (3.74 ± 0.50) × 10−16 cm3 molecule−1 s−1 at 300 bar, (4.09 ± 0.66) × 10−16 cm3 molecule−1 s−1 at 500 bar, with the combined value, k 295.5K = (3.85 ± 0.40) × 10−16 cm3 molecule−1 s−1. Though errors on the determination of [H2] are shown, these have little impact on the fit. The errors on the plotted points and on the rate constants determined from the slopes are statistical only. Note the value of λ b from the [H2] = 0 intercept, ~0.05 μs−1, close to zero on the plotted scale.
Relaxation rates vs. [H2] for the Heμ + H2 reaction rate at 405 K (lower blue data points and fitted line) and 500 K (upper red data points and fitted line). See also caption of Fig. 6. Accurate H2 partial pressures were measured to 0.5% with a Baratron manometer over the whole concentration range for the data at 500 K and for the lower pressures at 405 K, and these uncertainties give error bar smaller than the plotted points. The total pressure in each case is 500 bar (He). Rate constants, determined from the slopes [Eq. (3)] are: k 405 K = (7.06 ± 0.87) × 10−15 cm3 molecule−1 s−1 and k 500 K = (3.56 ± 0.56) × 10−14 cm3 molecule−1 s−1; the uncertainties are statistical only. The larger intercept giving λ b ∼0.09 μs−1 at 500 K is believed due to an enhanced reaction of Heμ with NH3, added as a dopant to produce muonic He. See the discussion in Sec. III of the text.
Comparison of experimental and theoretical thermal rate constants, k He μ (T), for the Heμ + H2 reaction as a function of temperature. The experimental error bars shown are statistical only, determined from the fits to the relaxation rates shown in Figs. 6 and 7. The temperature uncertainties are ±1.5 K at 295.5 and 500 K and ±1.0 K at 405 K and are too small to plot. See the discussion in Sec. III of the text.
Arrhenius plots of thermal rate constants for the Heμ + H2 and Mu + H2 reactions including accurate QM (solid red lines), ICVT/LAT (dashed black lines), the experimental data from Reid et al. for Mu + H2,9 and the new measurements for Heμ + H2.
Arrhenius plots of the QM thermal rate constants of the D + H2 reaction on the BH surface compared with the ICVT/LAT calculations and with various experimental results.37–40
Kinetic isotope effects. The experimental KIEs are obtained using fitted values for k D and k Mu with uncertainties estimated at 10% and using individual measurements for k He μ ; (a) k D/k He μ and (b) k Mu/k He μ . See Sec. IV E in the text. The large error bars are the result of propagating independent errors in the ratio of rate constants. The experimental point at 405 K in (b) required extrapolation of k Mu and this leads to additional uncertainty not reflected in the plotted error bar.
(a) Cumulative reaction probabilities (CRPs) for 3 isotopes of H reacting with para H2 for the case of J = 0 (the results for ortho H2 are very similar). (b) Density of states (the energy derivative of the CRPs) corresponding to the results of panel (a).
Boltzmann weighted CRPs (summed over all J) for the Heμ + H2 reaction renormalized so that their maximum value on the data grid is 1. These results indicate the energy ranges that contribute to the rate constants at the temperatures of the 3 measurements.
Comparison of total energies (E h) and barrier heights (kcal/mol) of H + H2 and Heμ + H2 as a function of basis set at a near-FCI level of theory. The Heμ + H2 results used a pseudopotential approach as described in the text.
Comparisons between theory and experiment for k Heμ(cm3 molecule−1 s−1).
Rate constants (cm3 molecule−1 s−1) and KIEs for the Heμ + H2, D + H2, and Mu + H2 reactions.
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