Two critical quantities in muon-catalyzed d-t fusion are the d-t fusion rate 〈ψ‖${\delta }^{\mathrm{}\left(3\right)\mathrm{}}$${(\mathrm{r}}_{\mathrm{dt}}$)‖ψ〉 and the α-μ sticking fraction mU〈cphi(r)${\mathrm{e}}^{\mathrm{i}}$q⋅r‖${\mathrm{\delta }}^{\mathrm{}\left(3\right)\mathrm{}}$(r ${}_{\mathit{d}\mathit{t}}{}^{}{}_{}{}^{}$)‖ψ(${\mathrm{r}}_{\mathit{t}}$,r)〉${\mathrm{\Vert }}^2$/ 〈ψ‖${\mathrm{\delta }}^{\mathrm{}\left(3\right)\mathrm{}}$(${\mathrm{r}}_{\mathit{t}}$)‖ψ〉, where ψ is a (dtμ${)}^{+}$ wave function and cphi is a hydrogenic α-μ wave function. It is explained why conventional approaches to calculate these quantities directly are slowly convergent. It is suggested that the use of a basis that explicitly includes terms that appear in the Fock expansion will lead to more rapid convergence. Furthermore, an identity relating 〈ψ‖${\delta }^{\mathrm{}\left(3\right)\mathrm{}}$(r)‖ψ〉 to expectation values of more diffuse operators, which was first derived by Hiller, Sucher, and Feinberg [Phys. Rev. A 18, 2399 (1978)] and then extended by Drake [Nucl. Instrum. Methods Phys. Res. B 31, 7 (1988)] in the context of atomic calculations, is generalized to the calculation of fusion rates and sticking fractions. It is anticipated that these relations will facilitate the accurate calculation of fusion rates and of sticking fractions.

• Received 21 July 1989

DOI:https://doi.org/10.1103/PhysRevA.40.6857