| Summary: | To overcome the numerical difficulties inherent in the Maxwell–Boltzmann integral of the velocity-weighted cross section that gives the radiative attachment rate coefficient <inline-formula> <math display="inline"> <semantics> <msub> <mi>α</mi> <mrow> <mi>R</mi> <mi>A</mi> </mrow> </msub> </semantics> </math> </inline-formula> for producing the negative hydrogen ion H<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>−</mo> </msup> </semantics> </math> </inline-formula> or its antimatter equivalent, the positive antihydrogen ion <inline-formula> <math display="inline"> <semantics> <msup> <mover> <mi mathvariant="normal">H</mi> <mo>¯</mo> </mover> <mo>+</mo> </msup> </semantics> </math> </inline-formula>, we found the analytic form for this integral. This procedure is useful for temperatures below 700 K, the region for which the production of <inline-formula> <math display="inline"> <semantics> <msup> <mover> <mi mathvariant="normal">H</mi> <mo>¯</mo> </mover> <mstyle displaystyle="false" scriptlevel="1"> <mo>+</mo> </mstyle> </msup> </semantics> </math> </inline-formula> has potential use as an intermediate stage in the cooling of antihydrogen to ultra-cold (sub-mK) temperatures for spectroscopic studies and probing the gravitational interaction of the anti-atom. Our results, utilizing a 50-term explicitly correlated exponential wave function, confirm our prior numerical results.
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