Scattering data of the central model class: the scattering motion, the impact parameter, the asymptotic directions, the deflection conventions and the uniform-incident preparation

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.