Effective-range function methods for charged particle collisions

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ [Ramírez Suárez and Sparenberg, Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton ${}^1\mathrm{S}_0$ collision. The resonant state of the proton-proton collision is successfully computed from the extrapolation of ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ instead of the standard function. It is shown that interpolating ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (${\displaystyle 12.5\mathrm{keV}}$ for the proton-proton system). This property is due to the connection between ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because ${\displaystyle {\mathrm{\Delta }}_{\ell }}$ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.

Figure 1

Representations of the function ${\displaystyle \psi }$ (a) in the complex ${\displaystyle z}$ plane and (b) along the real ${\displaystyle z}$ axis. Crosses and circles represent poles and zeros respectively. The color legend is described in Appendix pp1. The poles occur when ${\displaystyle z}$ reaches a negative integer or zero. The unique zero on the positive real axis lies at ${\displaystyle z=1.4616...}$ [13].

Figure 2

Plots of the regularized Coulomb wave function ${\displaystyle I_{\eta \ell }\left(x\right)}$ for the ${\displaystyle s}$ wave ${\displaystyle (\ell =0)}$ and different values of ${\displaystyle \eta }$. The inflection points ${\displaystyle x_{\mathrm{ip}}}$ are marked with a black dot. They lie at the same abscissa as for the usual Coulomb functions ${\displaystyle F_{\eta \ell }\left(x\right)}$ and ${\displaystyle G_{\eta \ell }\left(x\right)}$.

Figure 3

Representations of the principal branch of ${\displaystyle g_{\ell }\left(\eta \right)}$ for ${\displaystyle \ell =0}$ (a) in the complex plane of the energy ${\displaystyle E}$ and (b) along the real ${\displaystyle E}$ axis. In (a), the conventional branch cut lies along the negative real ${\displaystyle E}$ axis. Throughout this paper, real functions are depicted in solid black. Where functions are complex, the real part is shown in solid red (dark gray) and the imaginary part in dashed blue. In (b), the imaginary part is either ${\displaystyle +i\pi /2}$ (if ${\displaystyle argE=+\pi }$) or ${\displaystyle -i\pi /2}$ (if ${\displaystyle argE=-\pi }$).

Figure 4

Plots of the principal branch of ${\displaystyle h_{\ell }^{+}\left(\eta \right)}$ for ${\displaystyle \ell =0}$ (b) in the complex ${\displaystyle E}$ plane, (a) along the real ${\displaystyle E}$ axis in the physical sheet (${\displaystyle argE=0,+\pi }$) and (c) along the real axis in the unphysical sheet (${\displaystyle argE=0,-\pi }$). The function is complex valued for ${\displaystyle argE=0}$ but real for ${\displaystyle argE=+\pi }$. In (c), ${\displaystyle h_0^{+}\left(\eta \right)}$ is complex valued and shows the Coulomb poles.

Figure 5

Energy intervals where the relative error of the truncated Stirling expansion (73) of ${\displaystyle h_{\ell }^{+}\left(\eta \right)}$ (for ${\displaystyle \ell =0}$) does not exceed ${\displaystyle 5\%}$ and ${\displaystyle 1\%}$. The expected optimum order ${\displaystyle n_{\mathrm{opt}}}$ of Eq. (76) is shown in solid black.

Figure 6

Function ${\displaystyle {\mathrm{\Delta }}_0\left(E\right)}$ for the proton-proton ${\displaystyle {}^1{S}_0}$ scattering (a) in the complex ${\displaystyle E}$ plane and (b) on the real ${\displaystyle E}$ axis. The function is real and positive at ${\displaystyle E>0}$ but complex-valued at ${\displaystyle E<0}$. In (a), the branch cut along the negative ${\displaystyle E}$ axis is depicted by a black line. The imaginary part around the cut is either ${\displaystyle -i\pi /2}$ (if ${\displaystyle argE=+\pi }$) or ${\displaystyle +i\pi /2}$ (if ${\displaystyle argE=-\pi }$).

Figure 7

Same as Fig. 6 but at higher energies. At this scale, the Coulomb poles look merged at ${\displaystyle E=0}$. The two zeros are located at about ${\displaystyle E=(-1.75\pm 1.33i)\mathrm{MeV}}$.

Figure 8

Proton-proton function ${\displaystyle {\mathrm{\Delta }}_0^{+}\left(E\right)}$ (b) in the complex ${\displaystyle E}$ plane, (a) on the real ${\displaystyle E}$ axis in the physical sheet and (c) in the unphysical sheet. The function is real for ${\displaystyle argE=+\pi }$. Its real part reduces to ${\displaystyle {\mathrm{\Delta }}_0\left(E\right)}$ at ${\displaystyle E>0}$. In (c), the Coulomb poles at ${\displaystyle argE=-\pi }$ are indistinguishable at this scale.

Figure 9

Standard ERF ${\displaystyle {\varkappa }_0\left(E\right)}$ for the proton-proton ${\displaystyle {}^1{S}_0}$ collision (a) in the complex ${\displaystyle E}$ plane and (b) on the real ${\displaystyle E}$ axis. The zero is located at about ${\displaystyle E\simeq -3.8\mathrm{MeV}}$.

Figure 10

Interpolations of ${\displaystyle {\mathrm{\Delta }}_0\left(E\right)}$ at ${\displaystyle E>0}$ for the proton-proton ${\displaystyle {}^1{S}_0}$ wave with Padé approximants of orders ${\displaystyle [2/1]}, {\displaystyle [4/3]}$, and ${\displaystyle [6/5]}$. ${\displaystyle {\mathrm{\Delta }}_0\left(E\right)}$ is sampled logarithmically in 120 points (not shown) on different ranges (a) ${\displaystyle [10,{10}^3]\mathrm{Ry}}$, (b) ${\displaystyle [{10}^{-2},{10}^3]\mathrm{Ry}}$, and (c) ${\displaystyle [{10}^{-2},{10}^{-1}]\mathrm{Ry}}$. The Coulomb range ${\displaystyle [-1,1]\mathrm{Ry}}$ is surrounded by vertical dashed lines. The real part of ${\displaystyle {\mathrm{\Delta }}_0^{+}\left(E\right)}$ (for ${\displaystyle argE=+\pi }$) is shown in solid black.

Figure 11

Identity function in the complex plane with the color code used in this paper. The complex argument is represented with a hue: red color at ${\displaystyle 0^{\circ }}$ (for ${\displaystyle {\mathbb{R}}^{+}}$), chartreuse green at ${\displaystyle {90}^{\circ }}$, cyan at ${\displaystyle {180}^{\circ }}$ (for ${\displaystyle {\mathbb{R}}^{-}}$), and violet at ${\displaystyle -{90}^{\circ }}$. The log-polar grid highlights both the modulus and the argument.