The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading

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

This item fixes the cross-section vocabulary of the central scattering models 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 relative to the uniform-incident preparation of that definition.

(i) The differential cross-section of a branch. Let a central scattering model be given, let $\theta :(a_1,a_2)\to (0,\pi )$ be a deflection branch in the sense of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density — an interval $(a_1,a_2)$ of impact parameters with $\theta$ of class $C^1$ and ${\theta }^{\mathrm{\prime }}(b)\mathrm{\ne }0$ — and suppose the branch records the deflection of the model: for every $b\in (a_1,a_2)$ the scattering motion of the model with impact parameter $b$, fixed energy $E$ and angular momentum magnitude $J(b)=\sqrt{2\mu E}b$ has scattering angle ${\theta }_{\mathrm{s}}=\theta (b)$. Write $(a_1,a_2)\ni b\mapsto \theta \in ({\theta }_1,{\theta }_2)$ for this bijection and $b(\theta ):={\theta }^{-1}(\theta )$ for its inverse (Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density). The differential cross-section of the branch is the function $\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}(\theta ):=\frac{b(\theta )\mid b^{\mathrm{\prime }}(\theta )\mid }{\sin \theta }=\frac{b(\theta )}{\sin \theta \mid {\theta }^{\mathrm{\prime }}(b(\theta ))\mid }({\theta }_1<\theta <{\theta }_2),$ the cross-section density of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density. Its dimension is $[\mathrm{d}\sigma \mathrm{/}\mathrm{d}\mathrm{\Omega }]={\mathrm{\ell }}^2$ (solid angle dimensionless in the chosen convention, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), and one says "per steradian" for the unit of solid angle.

(ii) Branch sums. Let $\mathcal{B}=\{(a_1^{(j)},a_2^{(j)},{\theta }_j){\}}_{j=1}^N$ be a finite family of deflection branches of the same model whose incident intervals $(a_1^{(j)},a_2^{(j)})$ are pairwise disjoint, with each branch recording the model's deflections as in clause (i). Write $A_j:={\theta }_j((a_1^{(j)},a_2^{(j)}))$ for its image and $b_j:={\theta }_j^{-1}$ and $q_j(\theta ):=b_j(\theta )\mathrm{\mid }{b}_j^{\mathrm{\prime }}(\theta )\mathrm{\mid }\mathrm{/}\sin \theta$ for its branch density from clause (i), defined on $A_j$. Angular images are not required to be disjoint. The differential cross-section of the model for this family is, for $\theta \in A:={\bigcup }_{j=1}^N{A}_j$, $\frac{\mathrm{d}{\sigma }_{\mathcal{B}}}{\mathrm{d}\mathrm{\Omega }}(\theta ):={\sum }_{\{j:\theta \in A_j\}}{q}_j(\theta )\mathrm{.}$ Thus $q_j$ is the contribution of branch $j$, and contributions add at every angle belonging to more than one image. The total density for this family is left undefined at angles belonging to no branch image. No claim is made that the branch images cover $(0,\pi )$ or that the family includes every deflection of the model.

(iii) Cross-section measure and truncated totals. For one branch $j$ with image $A_j$, and a Borel angular window $Y\subseteq A_j$, define ${\sigma }_j(Y):={\int }_Y{q}_j(\theta )2\pi \sin \theta d{\lambda }_1(\theta )={\int }_{Y\times (0,2\pi )}{b}_j(\theta )\mathrm{\mid }{b}_j^{\mathrm{\prime }}(\theta )\mathrm{\mid }d{\lambda }_2\mathrm{.}$ When only one branch is under discussion, write $\sigma (Y)$ for this branch-specific measure. For the finite family $\mathcal{B}$ of clause (ii), define ${\sigma }_{\mathcal{B}}(Y):={\sum }_{j=1}^N{\sigma }_j(Y\cap A_j)={\int }_{Y\cap A}\frac{\mathrm{d}{\sigma }_{\mathcal{B}}}{\mathrm{d}\mathrm{\Omega }}(\theta )2\pi \sin \theta d{\lambda }_1(\theta )$ for every Borel $Y\subseteq (0,\pi )$. These are nonnegative Lebesgue integrals over measurable sets in the sense of Integral over a measurable subset, with extended value $+\mathrm{\infty }$ allowed. The measure-level identity of One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b\mathrm{\mid }\mathrm{d}b\mathrm{/}\mathrm{d}\theta \mathrm{\mid }$, hence cross-section density $b\mathrm{\mid }\mathrm{d}b\mathrm{/}\mathrm{d}\theta \mathrm{\mid }\mathrm{/}\sin \theta$ identifies each branch integral with the area of that branch's incident preimage; finite additivity gives the family formula, including where angular images overlap. On a compact interior interval each branch integral equals the ordinary Riemann integral; on an open interval its value is the monotone limit over compact interior truncations, equivalently the nonnegative improper integral when finite. General Borel windows use the Lebesgue integral, not a two-piece interval improper integral. The ideal total cross-section of a branch is ${\sigma }_j(A_j)$; for a family it is ${\sigma }_{\mathcal{B}}(A)$. If infinite, there is no finite ideal total. Every compact interior angular window in a branch image has finite branch cross-section; more generally a cutoff whose incident preimage is bounded has finite cross-section. An arbitrary positive angular cutoff need not suffice for an arbitrary branch.

(iv) The uniform-incident reading. Under the uniform-incident preparation 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 (iv) with cutoff $b_{\max }$ and flux $\mathrm{\Phi }$, the model predicts the expected rate of scattered labels into a Borel angular window $Y$ from one branch to be the flux times its cutoff cross-section for the portion of that branch with $b<b_{\max }$. Writing $T_j(b,\phi )=({\theta }_j(b),\phi )$ on the branch rectangle and $D_{\max }=(0,b_{\max })\times (0,2\pi )$, the branch cutoff measure is ${\sigma }_{j,b_{\max }}(Y):=\nu \text{\hspace{0.17em}⁣}(T_j^{-1}(Y\times (0,2\pi ))\cap ((a_1^{(j)},a_2^{(j)})\times (0,2\pi ))\cap D_{\max }),$ and its expected rate is $\mathrm{\Phi }{\sigma }_{j,b_{\max }}(Y)$. For a finite family with pairwise disjoint incident intervals, define ${\sigma }_{\mathcal{B},b_{\max }}(Y):={\sum }_{j=1}^N{\sigma }_{j,b_{\max }}(Y)$; the expected rate is $\mathrm{\Phi }{\sigma }_{\mathcal{B},b_{\max }}(Y)$, including when the angular images overlap. If the entire incident preimage of $Y$ lies below the cutoff, this sum is $\mathrm{\Phi }{\sigma }_{\mathcal{B}}(Y)$. This is a prediction of the model under the adopted preparation. In particular:

• ${\sigma }_j(Y)$ and ${\sigma }_{\mathcal{B}}(Y)$ are not probabilities per incident label, not counts of detected events, and not apparatus properties;
• a comparison with data requires a separately specified experiment: an apparatus model, a detector with finite angular resolution and acceptance, a sampling model and uncertainties, and a statistical analysis, none of which is supplied by this page;
• an ideal branch cross-section may diverge at a forward angle, while the adopted finite incident cutoff gives ${\sigma }_{\mathcal{B},b_{\max }}(Y)\le \pi b_{\max }^{}$ for every window $Y$, because the branch incident intervals are disjoint. Thus these cutoff rates are finite even if the ideal window cross-section is infinite. Without an incident cutoff, the rate $\mathrm{\Phi }{\sigma }_{\mathcal{B}}(Y)$ is finite exactly when that ideal window cross-section is finite. A finite measured rate constrains the model jointly with its incident preparation and apparatus acceptance;
• finite samples fluctuate: an observed angular distribution can miss a branch's features with positive probability under ordinary sampling models, and agreement is evidence about the model-plus-apparatus assumptions, not a proof of the model.

(v) Status and exclusions. The branch hypothesis includes ${\theta }^{\mathrm{\prime }}\mathrm{\ne }0$: the definition excludes angles at which ${\theta }^{\mathrm{\prime }}(b)=0$ (rainbow-type stationary deflections) and leaves them to separate limiting measures. It also excludes, at the level of this page, friction and tangential contact laws, attractive capture, screening, radiation, relativistic and quantum modifications, and any identification of the model parameters with measured electromagnetic quantities; those require separate models or cross-framework items. The functions $\theta$, $b$ and the data $E$, $J$, $u$, $b_{\max }$, $\mathrm{\Phi }$ are model data (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 (vi)).

Why the objects are defined. The inverse $b$ is $C^1$ with $b^{\mathrm{\prime }}(\theta )=1\mathrm{/}{\theta }^{\mathrm{\prime }}(b(\theta ))$ because a $C^1$ function on an interval with nowhere vanishing derivative is strictly monotone and has a differentiable inverse (Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density); $\sin \theta >0$ on $(0,\pi )$; hence the denominators do not vanish and the density is finite at every interior point of the branch image. The pushforward identity that justifies the reading "density carried by the branch" is One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b\mathrm{\mid }\mathrm{d}b\mathrm{/}\mathrm{d}\theta \mathrm{\mid }$, hence cross-section density $b\mathrm{\mid }\mathrm{d}b\mathrm{/}\mathrm{d}\theta \mathrm{\mid }\mathrm{/}\sin \theta$, a theorem of this page's mathematical suppliers.