Scattering and impulse model boundaries

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

This item fixes the scattering data of the restricted model class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions and the preparation conventions in which the scattering statements of this page are phrased. All objects are model data; nothing here is a measured value.

(i) The scattering model. A central scattering model is a model of class (i) of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions with $N=2$ labels, masses $m_1,m_2>0$ (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses clause (v)), no external forces, and a static central pair potential $u\in C^2(J_0)$ ($C^k$ Euclidean maps and diffeomorphisms) whose collision-excluded interval satisfies $(0,\mathrm{\infty })\subseteq J_0$, so that the only forbidden configuration is the collision $r=0$. Its reduced problem is the one-label problem of class (ii) with reduced mass $\mu :=m_1{m}_2\mathrm{/}(m_1+m_2)$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework: $[\mu ]=\mu$) and the same potential, presented through the relative coordinate $r=x_1-x_2$; by Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection the relative coordinate of every motion of the model is a central-force trajectory for $(\mu ,u)$ and every pair of an affine centre-of-mass motion and a solution of the reduced equation on a common interval reconstructs a motion of the model.

(ii) Scattering motions and their data. Let $r:\mathbb{R}\to {\mathbb{R}}^3$ be a central-force trajectory for $(\mu ,u)$ defined on all of $\mathbb{R}$ with values $\mathrm{\mid }r(t)\mathrm{\mid }\in J_0$ and with constant angular momentum $j=\mu r\times r^{\mathrm{\prime }}=J{e}_3\mathrm{\ne }0$ for $J>0$ and an oriented orthonormal triad $(e_1,e_2,e_3)$ (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Conservation of the angular momentum vector along a central-force trajectory and its orthogonality to position and velocity, Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation, Oriented orthonormal triads in ${\mathbb{R}}^3$: existence over any unit vector, and the orthonormal expansion with no fourth orthogonal direction). Call $r$ a scattering motion when it has a unique apsis $t_{\ast }$ with $\rho =\mathrm{\mid }r\mathrm{\mid }$ decreasing strictly before $t_{\ast }$, increasing strictly after it, $\rho (t)\to \mathrm{\infty }$ as $t\to \pm \mathrm{\infty }$, and the nonzero limits below exist with the stated speed, equal finite impact-distance limits and finite equal half-advances. These are defining conditions, not deductions for arbitrary $C^2$ potentials. The finite-range theorem requires $u=0$ beyond some $R$, $E,J>0$, a simple outer turning point $V(r_{\ast })=E$, $V^{\mathrm{\prime }}(r_{\ast })<0$, $V(r)<E$ for $r>r_{\ast }$, and a trajectory staying above and attaining $r_{\ast }$. The repulsive Coulomb theorem instead requires $u=\kappa \mathrm{/}\rho$, $\kappa ,E,J>0$ and a trajectory attaining its turning radius. Each theorem proves the defining conditions in its own scope. Its energy is the constant $E:=\frac{1}{2}\mu \parallel r^{\mathrm{\prime }}{\parallel }_2^2+u(\mid r\mid )>0$ of Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential (constant by Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation clause 4), its asymptotic velocities are the limits $v_{-}:={\lim }_{t\to -\mathrm{\infty }}{r}^{\mathrm{\prime }}(t),v_{+}:={\lim }_{t\to +\mathrm{\infty }}{r}^{\mathrm{\prime }}(t),$ taken in the sense of Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions (their existence under the hypotheses is a theorem, not a convention), and its scattering speed is $v_{\mathrm{\infty }}:=\parallel v_{-}{\parallel }_2=\parallel v_{+}{\parallel }_2=\sqrt{2E\mathrm{/}\mu },$ the equality being part of the scattering theorems. The incoming branch is the trajectory restricted to times before its apsis. The impact parameter is the asymptotic perpendicular-distance limit $b:={\lim }_{t\to -\mathrm{\infty }}\frac{\parallel r(t)\times v_{-}{\parallel }_2}{\parallel v_{-}{\parallel }_2}={\lim }_{t\to +\mathrm{\infty }}\frac{\parallel r(t)\times v_{+}{\parallel }_2}{\parallel v_{+}{\parallel }_2}=\frac{J}{\mu v_{\mathrm{\infty }}}>0.$ Here the distance at each time is to the line through the origin in the limiting velocity direction, not to an exact affine trajectory line. For a finite-range potential, the sufficiently early incoming tail is a straight half-line whose affine extension has this distance; for Coulomb these are only limits, with no exact free tail or finite longitudinal intercept claimed. The dimension is $[b]=\mathrm{\ell }$.

(iii) Polar advance and deflection conventions. Fix polar data $(\rho ,\theta )$ of the scattering motion in the triad, $r=\rho n(\theta )$, $n(\phi ):=\cos \phi e_1+\sin \phi e_2$ (Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation), and let $t_{\ast }$ be an apsis time, $\mathrm{\mid }r(t_{\ast })\mathrm{\mid }={\min }_t\mathrm{\mid }r(t)\mathrm{\mid }$. The polar advance of one half is $\mathrm{\Delta }:=\theta (+\mathrm{\infty })-\theta (t_{\ast })=\theta (t_{\ast })-\theta (-\mathrm{\infty }),$ defined by the limits asserted in the scattering theorems, and the polar advance of the motion is $2\mathrm{\Delta }$. The scattering angle is the undirected angle between the asymptotic velocity directions, ${\theta }_{\mathrm{s}}:=\arccos \text{\hspace{0.17em}⁣}\left(\frac{⟨v_{-},v_{+}⟩}{\parallel v_{-}{\parallel }_2\parallel v_{+}{\parallel }_2}\right)\in [0,\pi ],$ the incoming and outgoing velocity directions are oriented, so this is not the smaller unoriented angle between two lines; the signed velocity rotation is the continuous rotation angle of $r^{\mathrm{\prime }}(t)$, measured counterclockwise about $e_3$ relative to the incoming direction, so that a positive value is a counterclockwise turn. The scattering theorems give $\cos {\theta }_{\mathrm{s}}=\cos (\pi -2\mathrm{\Delta })$ and the signed rotation $2\mathrm{\Delta }-\pi$; in particular, when $0<\mathrm{\Delta }<\pi \mathrm{/}2$ the scattering angle is ${\theta }_{\mathrm{s}}=\pi -2\mathrm{\Delta }\in (0,\pi ),$ while outside that range ${\theta }_{\mathrm{s}}$ is the wrapped value $\arccos (\cos (\pi -2\mathrm{\Delta }))$. The deflection map at fixed model potential $u$, reduced mass $\mu$ and energy $E>0$ is the partial map $b\mapsto {\theta }_{\mathrm{s}}(b)$ on those $b>0$ for which a scattering motion exists and all such motions have the same angle. Its corresponding angular momentum magnitude varies as $J(b)=\mu b{v}_{\mathrm{\infty }}=\sqrt{2\mu E}b\mathrm{.}$ Time translation and rotation of a motion do not change its angle. Deflection branches are $C^1$ branches of this partial map as defined in Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density.

(iv) The uniform-incident preparation (adopted model datum). The uniform-incident preparation of a central scattering model with cutoff $b_{\max }>0$ and flux $\mathrm{\Phi }>0$ is the following adopted model of a scattering experiment for the reduced problem: independent incident reduced labels with common energy $E>0$ and impact-azimuth probability measure $P(A)=\frac{\nu (A\cap D_{\max })}{\pi b_{\max }^{}},D_{\max }=(0,b_{\max })\times (0,2\pi )\mathrm{.}$ Its density is $b\mathrm{/}(\pi b_{\max }^{})$, and its total mass is one since $\nu (D_{\max })=\pi b_{\max }^{}$. Angular momentum magnitude is $J(b)=\sqrt{2\mu E}b$; its plane orientation may vary with azimuth. The incident flux $\mathrm{\Phi }$ is the expected number of incident labels per unit area per unit time, with $[\mathrm{\Phi }]={\mathrm{\ell }}^{-2}{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The adopted total arrival rate is $\mathrm{\Phi }\pi b_{\max }^{}$, so for a Borel set $A\subseteq D_{\max }$ the expected incident rate is $\dot{N}(A)=\mathrm{\Phi }\pi b_{\max }^{}P(A)=\mathrm{\Phi }\nu (A)\mathrm{.}$ Here $\nu$ is the incident area measure of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density. For a specified deflection branch $\theta :(a_1,a_2)\to ({\theta }_1,{\theta }_2)$ and a Borel scattering-angle window $Y\subseteq ({\theta }_1,{\theta }_2)$, the predicted rate from that branch under this cutoff preparation is ${\dot{N}}_{\mathrm{b}\mathrm{r}}(Y):=\mathrm{\Phi }\nu \text{\hspace{0.17em}⁣}\left(\{(b,\phi )\in D_{\max }:a_1<b<a_2,\theta (b)\in Y\}\right)\mathrm{.}$ The preimage is Borel because $\theta$ is continuous on its branch. For a finite family of branches with pairwise disjoint incident intervals, branch rates add, including when their angular images overlap. Incidences outside recorded scattering branches are not assigned an outgoing scattering angle by this definition. The rate above integrates over the full impact-azimuth range; it is a model prediction, not a detected count. Comparing it with data requires a separately specified experiment, apparatus model, sampling model and statistical analysis.

(v) Reading conventions. All angles and lengths above are data of the relative problem: angles are computed from the relative coordinate $r=x_1-x_2$ and its asymptotic velocities $v_{\pm }$, not from either label's trajectory. The centre of mass $X$ and the reconstruction formulas $x_1=X+\frac{m_2}{M}r$ and $x_2=X-\frac{m_1}{M}r$, with $M=m_1+m_2$, are those of clause (i), Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame and Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection clause 4. They show that individual-label directions generally differ from the relative direction and from each other. "Target", "projectile" and "detector" are names for model components and carry no empirical claim.

(vi) Exclusions and status. The class excludes velocity- or time-dependent, non-central and non-potential forces; constraints and reaction forces; collisions $r=0$; external forces; and models with more than two labels, exactly as in Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions clause (v). The preparation of clause (iv) is one adopted idealisation among many; the scattering theorems state separately which potentials, energies and angular momenta produce the data of clauses (ii)-(iii). The quantities $\mu$, $u$ (with any parameters it carries), $E$, $J$, $b$, $b_{\max }$, $\mathrm{\Phi }$ are model data; quantitative empirical use requires separately stated measured inputs with uncertainties, and no statement of this page asserts that any physical system is exactly a central scattering model or that any beam is exactly uniform.

Why the objects are defined. The relative coordinate is a central-force trajectory by the cited reduction theorem; the attached energy and angular momentum are defined by Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential and are constant on the trajectory by Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation clause 4 and Conservation of the angular momentum vector along a central-force trajectory and its orthogonality to position and velocity; the impact parameter is defined by its limiting distance, with existence and equality proved under the theorem-specific hypotheses; and the preparation measure of clause (iv) is the measure $\nu$ of Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density. The arccosine of clause (iii) is defined because the Cauchy-Schwarz inequality gives $\mathrm{\mid }⟨v_{-},v_{+}⟩\mathrm{\mid }\mathrm{/}(\parallel v_{-}{\parallel }_2\parallel v_{+}{\parallel }_2)\le 1$ (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation).

Mathematical status. The vocabulary is attached to the adopted model class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions; the existence of the limits, the identities for $v_{\mathrm{\infty }}$, $b$, $\mathrm{\Delta }$ and ${\theta }_{\mathrm{s}}$, and the form of the deflection map under explicit potential hypotheses are the content of the mathematics and physics theorems of this page, not of this definition.

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.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for clause 4 and its preparation 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 $\mu =m_1{m}_2\mathrm{/}(m_1+m_2)$, static central pair potential $u\in C^2((0,\mathrm{\infty }))$ ($C^k$ Euclidean maps and diffeomorphisms), and let $E>0$, $J>0$; write $V(r):=u(r)+\frac{J^2}{2\mu r^2}(r>0)\mathrm{.}$ Suppose:

(H1) finite range: there is $R>0$ with $u(r)=0$ for every $r\ge R$;

(H2) simple outer turning point: there is $r_{\ast }>0$ with $V(r_{\ast })=E,V^{\mathrm{\prime }}(r_{\ast })<0,V(r)<E(r>r_{\ast })\mathrm{.}$

Let $r:I\to {\mathbb{R}}^3$ be the relative coordinate of a motion of the model on an interval $I$ with nonempty interior and with attached angular momentum $j=\mu r\times r^{\mathrm{\prime }}=J{e}_3\mathrm{\ne }0$ and attached energy $E$ (Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential, Two-body reduction of a static central pair-potential model: the relative coordinate is a central-force trajectory for the reduced mass, the centre of mass moves uniformly, and the reconstruction is a bijection), and suppose $\mathrm{\mid }r(t)\mathrm{\mid }\ge r_{\ast }$ for every $t\in I$ and $\mathrm{\mid }r(t_{\ast })\mathrm{\mid }=r_{\ast }$ for some $t_{\ast }\in I$. Then:

1. (The scattering motion.) $r$ extends uniquely to a central-force trajectory $\bar{r}:\mathbb{R}\to {\mathbb{R}}^3$ for $(\mu ,u)$; the extension is the relative coordinate of a motion of the two-label model with its autonomous force law extended to time window $\mathbb{R}$, defined on all of $\mathbb{R}$, whose centre of mass is the affine extension of the original one. Along $\bar{r}$, $\rho :=\mathrm{\mid }\bar{r}\mathrm{\mid }$ satisfies $\rho \ge r_{\ast }$, $\rho (t)>r_{\ast }$ for $t\mathrm{\ne }{t}_{\ast }$, and $\rho$ is strictly decreasing on $(-\mathrm{\infty },t_{\ast }]$ and strictly increasing on $[t_{\ast },\mathrm{\infty })$; no collision occurs. Moreover the time quadrature $\tau (r_0):={\int }_{r_{\ast }}^{r_0}\frac{\mathrm{d}s}{\sqrt{(2\mathrm{/}\mu )(E-V(s))}}(r_0\ge r_{\ast })$ with the integral at $r_{\ast }$ interpreted as an improper integral, is finite for every $r_0$ and satisfies $\rho (t_{\ast }\pm \tau (r_0))=r_0$.
2. (Free asymptotes, speed and impact parameter.) The limits $v_{\pm }={\lim }_{t\to \pm \mathrm{\infty }}{\bar{r}}^{\mathrm{\prime }}(t)$ exist and $\mathrm{\mid }{v}_{+}\mathrm{\mid }=\mathrm{\mid }{v}_{-}\mathrm{\mid }=\sqrt{2E\mathrm{/}\mu }=:v_{\mathrm{\infty }}\mathrm{.}$ For every $t_1\ge t_{\ast }$ with $\rho (t_1)\ge R$ the motion is exactly free from $t_1$ on, $\bar{r}(t)=\bar{r}(t_1)+(t-t_1)v_{+}$ for all $t\ge t_1$; symmetrically for the incoming branch with $v_{-}$. The perpendicular distance from the force centre of the affine extensions of the incoming and outgoing free tails is the same number $b=\frac{J}{\mu v_{\mathrm{\infty }}}=\frac{J}{\sqrt{2\mu E}},\text{so that}J=\mu b{v}_{\mathrm{\infty }}\mathrm{.}$
3. (Polar advance and deflection.) Let $(\rho ,\theta )$ be polar data of $\bar{r}:\mathbb{R}\to {\mathbb{R}}^3$ from clause 1 in an oriented orthonormal triad with $e_3=j\mathrm{/}J$ (Polar reduction of a non-radial central-force trajectory: ${\rho }^2{\theta }^{\mathrm{\prime }}=J\mathrm{/}\mu$, the effective-potential equation and energy conservation). Then $\mathrm{\Delta }:=\theta (+\mathrm{\infty })-\theta (t_{\ast })=\theta (t_{\ast })-\theta (-\mathrm{\infty })={\int }_{r_{\ast }}^{\mathrm{\infty }}\frac{J\mathrm{d}r}{r^2\sqrt{2\mu (E-V(r))}}$ is finite and positive, and the asymptotic directions satisfy $\frac{v_{+}}{\mathrm{\mid }{v}_{+}\mathrm{\mid }}=-R_{2\mathrm{\Delta }}\left(\frac{v_{-}}{\mathrm{\mid }{v}_{-}\mathrm{\mid }}\right),\frac{⟨v_{-},v_{+}⟩}{\mathrm{\mid }{v}_{-}\mathrm{\mid }\mathrm{\mid }{v}_{+}\mathrm{\mid }}=\cos (\pi -2\mathrm{\Delta }),$ where for $x\in \mathrm{span}(e_1,e_2)$, $R_{\alpha }x:=\cos \alpha x+\sin \alpha (e_3\times x)$ is the rotation of that plane by $\alpha$. The scattering angle of the motion is ${\theta }_{\mathrm{s}}=\arccos (\cos (\pi -2\mathrm{\Delta }))$; if $0<\mathrm{\Delta }<\pi \mathrm{/}2$ then ${\theta }_{\mathrm{s}}=\pi -2\mathrm{\Delta }\in (0,\pi )$ and the signed velocity rotation is $2\mathrm{\Delta }-\pi$.
4. (Cross-section of a deflection branch.) Let $\theta :(a_1,a_2)\to (0,\pi )$ be a deflection branch of the model in the sense of The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading: $\theta$ is $C^1$ with ${\theta }^{\mathrm{\prime }}(b)\mathrm{\ne }0$, and 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 ${\theta }_{\mathrm{s}}(b)=\theta (b)$, with $0<\mathrm{\Delta }(b)<\pi \mathrm{/}2$. For each member of this family suppose (H2) and the trajectory hypotheses hold with its own $J(b)$ and turning point $r_{\ast }(b)$. Then, for every nonnegative Borel test function on the branch image, or real Borel test function absolutely integrable for its outgoing measure, ${\int }_{(a_1,a_2)\times (0,2\pi )}g(\theta (b),\phi )bd{\lambda }_2(b,\phi )={\int }_{({\theta }_1,{\theta }_2)\times (0,2\pi )}g(\theta ,\phi )b(\theta )\mathrm{\mid }{b}^{\mathrm{\prime }}(\theta )\mathrm{\mid }d{\lambda }_2(\theta ,\phi ),$ the outgoing angular measure of the branch has density $b(\theta )\mathrm{\mid }{b}^{\mathrm{\prime }}(\theta )\mathrm{\mid }$ with respect to $\mathrm{d}\theta \mathrm{d}\phi$, and the differential cross-section of the branch is $\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}(\theta )=\frac{b(\theta )\mid b^{\mathrm{\prime }}(\theta )\mid }{\sin \theta }=\frac{b(\theta )}{\sin \theta \mid {\theta }^{\mathrm{\prime }}(b(\theta ))\mid }({\theta }_1<\theta <{\theta }_2)\mathrm{.}$ 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 $b_{\max }>a_1$ and flux $\mathrm{\Phi }$, put $c=\min (a_2,b_{\max })$ and $Y=\theta ((a_1,c)),{\sigma }_{\mathrm{b}\mathrm{r}}(Y)={\int }_Y\frac{d\sigma }{d\mathrm{\Omega }}(\theta )2\pi \sin \theta d{\lambda }_1(\theta )\mathrm{.}$ Here ${\sigma }_{\mathrm{b}\mathrm{r}}(Y)$ is this branch's contribution as in clause (iii) of The differential and truncated total cross-sections of a deflection branch: densities, units, branch sums and the empirical qualifications of the uniform-incident reading, not the total model cross-section when other branch images overlap $Y$. The window uses the image of the open interval, with one-sided endpoint limits when necessary, and works for either monotonicity. The expected scattered rate from this branch portion is $\mathrm{\Phi }{\sigma }_{\mathrm{b}\mathrm{r}}(Y)=\mathrm{\Phi }\pi (c^2-a_1^2)$, which is finite. Other branch portions, if present, add their own contributions.
5. (Scope of the statement.) The theorem asserts nothing about the deflection map outside the recorded branch: it neither asserts that $b\mapsto {\theta }_{\mathrm{s}}$ is decreasing, injective or free of stationary points, nor that the branch images cover $(0,\pi )$; and it treats finite-range potentials only. The collisions, long-range potentials, velocity-dependent forces and external forces are excluded by the model class of Adopted central-force model class: static central pair potentials, the reduced one-label problem, the fixed-centre idealization, reading conventions and exclusions and by (H1).