Doppler Broadening in Beam Experiments

The distribution in center‐of‐mass energy caused by the thermal motion of the target gas molecules has been rigorously derived for the case of a monoenergetic particle beam interacting with target molecules having an isotropic Maxwellian velocity distribution corresponding to temperature T°K. Provided the nominal c.m. energy E₀ exceeds a few kT, the distribution has a full width at half‐maximum (FWHM) of $W_{\text{1/2}}{\text{=(11.1γk T\hspace{0.17em}E}}_0{)}^{\text{1/2}}$ . where $\mathrm{\gamma =m/(m+M)}$ , m and M being the projectile and target masses. This is identical to the width derived previously in a one‐dimensional approximate treatment by Bethe and Placzek. The exact and approximate distributions differ significantly, however, in shape and mean energy, particularly at low values of $E_0\mathrm{/\gamma kT}$ . The Doppler width, $W_{\text{1/2}}$ , is shown to significantly affect the appearance curve of the products of endothermic reactions involving heavy particles. Convolution integrals are derived for a number of idealized forms of the cross section for such reactions. In the extreme case of a step‐function cross section the estimation of the threshold E_T by the usual linear extrapolation technique gives a value which is too low by approximately ${\text{0.6W}}_{\text{1/2}}{\mathrm{(E}}_T)$ , where $W_{\text{1/2}}{\mathrm{(E}}_T)$ is the FWHM evaluated at $E_0{\mathrm{=E}}_T$ . The effect of the thermal velocities of the parent molecules from which the primary beam is formed is considered in the case of an accelerated beam with no kinetic energy imparted to the beam particles by the formation process, and in the case where the energy of the interacting particles is controlled only by the kinetic energy released in a resonant dissociative formation process. The latter situation is shown to be described exactly by the equations derived for the monoenergetic beam case provided an appropriately defined equivalent temperature is used in place of the true temperature of the target gas. The results of the present theory are applied to the experimental data of Berkowitz, Chupka, and Gutman on the reactions ${\mathrm{I}}^{-}+{\mathrm{O}}_2\to {\mathrm{O}}_2^{-}+\mathrm{I}\hspace{0.17em}\mathrm{and}\hspace{0.17em}{\mathrm{I}}^{-}+\mathrm{NO}\to {\mathrm{NO}}^{-}+\mathrm{I}$ . In addition data on the reaction ${\mathrm{O}}^{-}+{\mathrm{O}}_2\to {\mathrm{O}}_2^{-}+\mathrm{O}$ are presented and analyzed. The derived thresholds are consistent with $\mathrm{A\; (}{\mathrm{O}}_2\text{)≥0.56±0.10\hspace{0.17em}}\mathrm{eV}\mathrm{,\hspace{0.17em}A\; (}\mathrm{NO}\text{)≥0.06±0.1\hspace{0.17em}}\mathrm{eV,\; and}\mathrm{\hspace{0.17em}A\; (}{\mathrm{O}}_2\text{)≥0.50±0.1\hspace{0.17em}}\mathrm{eV}$ , respectively. The experimental data of Maier on the reaction ${\mathrm{C}}^{+}+{\mathrm{D}}_2\to {\mathrm{CD}}^{+}+\mathrm{D}$ is found to be consistent with a cross section which rises from threshold with approximately twice the slope of the cross section computed by Truhlar from statistical phase space theory.