Repulsive Coulomb scattering in the model class: global non-colliding motion, the deflection map $\theta =2\arctan (\kappa \mathrm{/}(2Eb))$, the Rutherford density $(\kappa \mathrm{/}4E{)}^2{\csc }^4(\theta \mathrm{/}2)$, finite angular cutoffs and the divergent ideal total

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the cross-section measures.

Adopt a central scattering model of Scattering data of the central model class: the scattering motion, the impact parameter, the asymptotic directions, the deflection conventions and the uniform-incident preparation clause (i) with masses $m_1,m_2>0$, reduced mass $\mu$, and the repulsive Coulomb potential $u(r)=\frac{\kappa }{r}(r>0),\kappa >0,$ where $\kappa$ is a model datum of the mechanical model (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework: $[\kappa ]=\mu {\mathrm{\ell }}^3{\tau }^{-2}$), so that $V(r)=\frac{\kappa }{r}+\frac{J^2}{2\mu r^2}(r>0)\mathrm{.}$ Let the data satisfy $E>0$, $J>0$ and put $b:=\frac{J}{\sqrt{2\mu E}}>0,\alpha :=\frac{\kappa }{2Eb}>0.$ Then:

1. (Turning data.) $V$ is strictly decreasing on $(0,\mathrm{\infty })$, with $V(r)\to +\mathrm{\infty }$ as $r\downarrow 0$ and $V(r)\to 0$ as $r\to \mathrm{\infty }$; the equation $V(r)=E$ has exactly one solution $r_{\ast }>0$, it is simple ($V^{\mathrm{\prime }}(r_{\ast })<0$), one has $V(r)<E$ for every $r>r_{\ast }$ and $V(r)>E$ for every $0<r<r_{\ast }$, and $V=U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}$ is of class $C^2$ with no other crossing of the level $E$.
2. (Global non-colliding scattering motion.) Let $I\subseteq I_0$ be a time interval with nonempty interior and $r:I\to {\mathbb{R}}^3$ be the relative coordinate of a motion of the model with attached angular momentum $j=\mu r\times r^{\mathrm{\prime }}=J{e}_3$ and energy $E$, with $\mathrm{\mid }r(t)\mathrm{\mid }\ge r_{\ast }$ for every $t\in I$ and $\mathrm{\mid }r(t_{\ast })\mathrm{\mid }=r_{\ast }$ for some $t_{\ast }\in I$. Then $r$ extends uniquely to a central-force trajectory $\bar{r}:\mathbb{R}\to {\mathbb{R}}^3$ whose extension is the relative coordinate of a motion of the two-label model with its autonomous force law extended to time window $\mathbb{R}$; $\rho :=\mathrm{\mid }\bar{r}\mathrm{\mid }$ satisfies $\rho \ge r_{\ast }>0$ (no collision), $\rho (t)>r_{\ast }$ for $t\mathrm{\ne }{t}_{\ast }$, and $\rho$ is strictly decreasing on $(-\mathrm{\infty },t_{\ast }]$ and strictly increasing on $[t_{\ast },\mathrm{\infty })$; the asymptotic velocities $v_{\pm }={\lim }_{t\to \pm \mathrm{\infty }}{\bar{r}}^{\mathrm{\prime }}(t)$ exist with $\mathrm{\mid }{v}_{+}\mathrm{\mid }=\mathrm{\mid }{v}_{-}\mathrm{\mid }=\sqrt{2E\mathrm{/}\mu }=:v_{\mathrm{\infty }}$, and the asymptotic perpendicular distance of each branch to its limiting direction equals $b=J\mathrm{/}(\mu v_{\mathrm{\infty }})=J\mathrm{/}\sqrt{2\mu E}$, so that $J=\mu b{v}_{\mathrm{\infty }}$. Unlike the finite-range case, there is no radius beyond which the motion is exactly free: the limits above are asymptotic.
3. (Deflection and its single branch.) With polar data $(\rho ,\theta )$ in a triad with $e_3=j\mathrm{/}J$, the half-angle of the motion is $\mathrm{\Delta }=\frac{\pi }{2}-\arctan \frac{\kappa }{2Eb}\in (0,\frac{\pi }{2}),$ its scattering angle is ${\theta }_{\mathrm{s}}=\pi -2\mathrm{\Delta }=2\arctan \frac{\kappa }{2Eb}\in (0,\pi ),$ and its signed velocity rotation is $2\mathrm{\Delta }-\pi =-2\arctan (\kappa \mathrm{/}(2Eb))$. For the family at fixed $\mu ,\kappa ,E$ and varying $J(b)=\sqrt{2\mu E}b$, realised by the apsis initial data of Repulsive Coulomb scattering: the unique outer turning point, global non-colliding motion, the asymptotic impact distance and $\mathrm{\Delta }=\pi \mathrm{/}2-\arctan (\kappa \mathrm{/}(2Eb))$ clause 4, the deflection map ${\theta }_{\mathrm{d}\mathrm{e}\mathrm{f}}:(0,\mathrm{\infty })\to (0,\pi ),{\theta }_{\mathrm{d}\mathrm{e}\mathrm{f}}(b):=2\arctan \frac{\kappa }{2Eb},$ is $C^1$ and strictly decreasing, hence a $C^1$ bijection with inverse $b(\theta )={\displaystyle \frac{\kappa }{2E}}\cot {\displaystyle \frac{\theta }{2}}$; it is a single deflection branch of the model on $(0,\mathrm{\infty })$ with ${\theta }_{\mathrm{d}\mathrm{e}\mathrm{f}}^{\mathrm{\prime }}(b)\mathrm{\ne }0$ everywhere and no wrapping ($0<\mathrm{\Delta }<\pi \mathrm{/}2$ for every $b>0$).
4. (Cross-section, angular cutoffs and divergence.) The differential cross-section of that branch is $\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}(\theta )=(\frac{\kappa }{4E}{)}^2{\csc }^4\frac{\theta }{2}(0<\theta <\pi ),$ finite and positive; for $0<{\theta }_1<{\theta }_2<\pi$ the truncated total cross-section over the angular band is the finite number $\sigma (({\theta }_1,{\theta }_2))=\pi (b({\theta }_1{)}^2-b({\theta }_2{)}^2),$ and by monotonicity the band $\theta \ge {\theta }_1$ has $\sigma (({\theta }_1,\pi ))=\pi b({\theta }_1{)}^2=\pi (\frac{\kappa }{2E}{)}^2{\cot }^2\frac{{\theta }_1}{2}$; whereas the truncated integral over $(0,{\theta }_0)$ diverges for every ${\theta }_0\in (0,\pi )$, so the model has no finite ideal total cross-section. Every positive angular cutoff therefore yields a finite truncated ideal cross-section. Under the declared uniform-incident preparation with $b_{\max }>0$ and flux $\mathrm{\Phi }>0$, the cutoff cross-section satisfies ${\sigma }_{b_{\max }}(Y)\le \pi b_{\max }^{}$ for every Borel angular window $Y\subseteq (0,\pi )$, and the expected scattered rate is $\mathrm{\Phi }{\sigma }_{b_{\max }}(Y)<\mathrm{\infty }$ (The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading clause (iv)). Thus ideal cross-section divergence does not imply an infinite rate under the finite incident cutoff; comparison with detected rates additionally requires a declared apparatus acceptance and its uncertainties.
5. (Status of the parameter and exclusions.) The coupling $\kappa$ is a parameter of the mechanical model. This item does not identify it with $q_1{q}_2\mathrm{/}(4\pi {\epsilon }_0)$ or with any electromagnetic quantity, states no measured value, and asserts no screening law for a medium of scatterers. Attractive Coulomb scattering (for which capture is possible for suitable data), inelastic, radiative, relativistic and quantum modifications are outside the adopted model and this statement.

• Dimensions. $[\kappa ]=\mu {\mathrm{\ell }}^3{\tau }^{-2}$, $\alpha$ and $\mathrm{\Delta }$ dimensionless, $[\mathrm{d}\sigma \mathrm{/}\mathrm{d}\mathrm{\Omega }]={\mathrm{\ell }}^2$; the identity $\sigma =\pi b({\theta }_1{)}^2$ at a cutoff shows that the truncated total is the area of the disk of impact parameters feeding the accepted angles.

• The divergence is a feature of the ideal model. The ideal Coulomb branch has no total cross-section because the full-azimuth angular density $2\pi \sin \theta \mathrm{d}\sigma \mathrm{/}\mathrm{d}\mathrm{\Omega }$ grows like ${\theta }^{-3}$ (the solid-angle density grows like ${\theta }^{-4}$). An angular cutoff gives the displayed truncated ideal value, while the declared finite incident cutoff gives finite predicted rates even for the entire angular interval. Comparing either prediction with observations requires the specified preparation and apparatus model; no experimental acceptance or rate is reported here.

• What is not claimed. No attractive branch, no capture, no screening of a medium of scatterers, no radiation loss, no relativistic or quantum correction, and no identification of $\kappa$ with an electromagnetic coupling; those require separate models and, for the identification, a cross-framework item.