Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the Borel/Lebesgue measure foundation below.

Data. Let $\mu >0$, let $U:(0,\mathrm{\infty })\to \mathbb{R}$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), let $R>0$ satisfy $U(r)=0$ for every $r\ge R$ (a finite-range potential), and let $E>0$ and $J>0$ be reals. Write

$V(r):=U(r)+\frac{J^2}{2\mu r^2}(r>0)\mathrm{.}$

Incident flux measure. Let $\nu$ be the Borel measure on the open rectangle $(0,\mathrm{\infty })\times (0,2\pi )$ given by the density $b$ against Lebesgue measure ${\lambda }_2$,

$\nu (A):={\int }_{A}bd{\lambda }_2(b,\phi )(A\subseteq (0,\mathrm{\infty })\times (0,2\pi )\text{ Borel}),$

the integral being that of Integral over a measurable subset; the nonnegative Borel function $(b,\phi )\mapsto b$ defines a measure by The indefinite integral of a nonnegative measurable function is a measure (Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$). In the model this measure records a uniform incident beam: an annulus of impact parameters $[b_1,b_2)$ contributes $\pi (b_2^2-b_1^2)$ over the full azimuth range, or $\frac{1}{2}(b_2^2-b_1^2)$ per unit azimuth.

Solid-angle element. Let $\mathrm{\Omega }$ be the Borel measure on $(0,\pi )\times (0,2\pi )$ with density $\sin$ against ${\lambda }_2$,

$\mathrm{\Omega }(B):={\int }_B\sin \theta d{\lambda }_2(\theta ,\phi )(B\subseteq (0,\pi )\times (0,2\pi )\text{ Borel}),$

where $\sin$ is the function of Sine and cosine defined by their real power series. It is continuous, being differentiable by The derivatives of sine and cosine are cosine and minus sine. Also $\sin 0=\sin \pi =0$ by The zero sets of sine and cosine and the least positive common period 2 pi; strict increase on $[0,\pi \mathrm{/}2]$ and strict decrease on $[\pi \mathrm{/}2,\pi ]$ (Signs, monotonicity intervals, and ranges of sine and cosine) give $\sin \theta >0$ for $0<\theta <\pi$. Thus $(\theta ,\phi )\mapsto \sin \theta$ is a nonnegative Borel function (continuous preimages of Borel sets are Borel, A continuous map has Borel preimages of Borel sets), and The indefinite integral of a nonnegative measurable function is a measure establishes that $\mathrm{\Omega }$ is a measure. This is the standard solid-angle element written $\sin \theta d\theta d\phi$ (the surface measure of the unit sphere in spherical coordinates, restricted to the open chart).

Deflection branch. A deflection branch is an interval $(a_1,a_2)\subseteq (0,\mathrm{\infty })$ with at least two points together with a function $\theta :(a_1,a_2)\to (0,\pi )$ that is of class $C^1$ and satisfies ${\theta }^{\mathrm{\prime }}(b)\mathrm{\ne }0$ for every $b\in (a_1,a_2)$. Such a $\theta$ is strictly monotone (the continuous nonvanishing derivative has constant sign, and The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$ gives strict monotonicity) and its image is an interval with endpoints ${\theta }_1<{\theta }_2$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); its inverse $b:({\theta }_1,{\theta }_2)\to (a_1,a_2)$ is differentiable with $b^{\mathrm{\prime }}(\theta )=1\mathrm{/}{\theta }^{\mathrm{\prime }}(b(\theta ))$, by the derivative-of-an-inverse theorem Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c\in I$ with $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, then the inverse $g$ is differentiable at $f(c)$ with $g^{\mathrm{\prime }}(f(c))=1\mathrm{/}{f}^{\mathrm{\prime }}(c)$; and if $f^{\mathrm{\prime }}(c)=0$ then $g$ is not differentiable at $f(c)$ applied to the continuous injective $\theta$ (every point of $(a_1,a_2)$ is a limit point of it, Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set).

Cross-section density. The cross-section density of the branch is the function

$\frac{d\sigma }{d\mathrm{\Omega }}(\theta ):=\frac{b(\theta )}{\sin \theta \mid {\theta }^{\mathrm{\prime }}(b(\theta ))\mid }({\theta }_1<\theta <{\theta }_2)\mathrm{.}$

Mathematical status. The measures, branches and density are abstract data independent of the finite-range trajectory data $(\mu ,U,R,E,J)$. A branch is not asserted to arise from any trajectory; the scattering theorems of this page identify the deflection angle computed from the central-force equation with the values of such a function under their hypotheses, and they compute the pushforward of $\nu$ under the induced map. No counting of detected particles, and no empirical claim, is part of this definition.