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The Rutherford deflection map, its inverse, the cross-section density $(κ / 4 E)^{2} \csc^{4} (θ / 2)$ and the divergence of the total cross-section

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the cross-section measures. Fix $μ > 0$ , $κ > 0$ and $E > 0$ . For every $b > 0$ put $J (b) = \sqrt{2 μ E} b$ and use the Coulomb trajectory family of Repulsive Coulomb scattering: the unique outer turning point, global non-colliding motion, the asymptotic impact distance and $Δ = π / 2 - \arctan (κ / (2 E b))$ clause 4. Its impact distance is $b$ and its deflection map $θ_{d e f} : (0, \infty) \to (0, π), θ_{d e f} (b) : = 2 \arctan \frac{κ}{2 E b} .$ Then:

$θ_{d e f}$ is $C^{1}$ and strictly decreasing, with $θ_{d e f} (b) \to π$ as $b ↓ 0$ and $θ_{d e f} (b) \to 0$ as $b \to \infty$ ; it is a $C^{1}$ bijection $(0, \infty) \to (0, π)$ with inverse $b (θ) = \frac{κ}{2 E} \cot \frac{θ}{2} (0 < θ < π) .$
For every interval $(θ_{1}, θ_{2}) \subseteq (0, π)$ the pair $(θ, b)$ is a deflection branch with cross-section density $\frac{d σ}{d Ω} (θ) = b (θ) ∣ \frac{d b}{d θ} (θ) ∣ \frac{1}{\sin θ} = (\frac{κ}{4 E})^{2} \csc^{4} \frac{θ}{2},$ finite and positive on $(θ_{1}, θ_{2})$ ; and the pushforward of the incident flux measure under the branch is described by One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b ∣ d b / d θ ∣$ , hence cross-section density $b ∣ d b / d θ ∣ / \sin θ$ .
For every $θ_{0} \in (0, π)$ the truncated total cross-section $\int_{0}^{θ_{0}} ⁣ ⁣ \int_{0}^{2 π} (\frac{κ}{4 E})^{2} \csc^{4} \frac{θ}{2} \sin θ d φ d θ = + \infty,$ while for $0 < θ_{1} < θ_{2} < π$ the integral over $(θ_{1}, θ_{2}) \times (0, 2 π)$ is finite; in this sense the total cross-section of the ideal Coulomb model diverges at forward angles and every compact angular acceptance away from $0$ has finite cross-section.

Facts & Assumptions

Given: Countable choice, the fixed data $μ, κ, E$ , the family $J (b) = \sqrt{2 μ E} b$ , the deflection map $θ_{d e f}$ and its asserted inverse $b (θ)$ .

[L1]

$\arctan : R \to (- π / 2, π / 2)$ is the inverse of $\tan$ , $\arctan$ is $C^{1}$ and strictly increasing, and $\tan (\arctan y) = y$ for every real $y$ ; on $0 < θ / 2 < π / 2$ one has $\arctan (\tan (θ / 2)) = θ / 2$ (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ , Tangent is a continuous strictly increasing bijection from $(- π / 2, π / 2)$ onto $R$ , Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series).

[L2]

Trigonometric calculus: derivatives of $\tan, \cot, \sec, \csc$ and their Pythagorean and parity identities, the cotangent double-angle form $\sin θ = 2 \sin (θ / 2) \cos (θ / 2)$ , and $\cos (θ / 2) > 0$ for $0 < θ < π$ (Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant, Pythagorean and parity identities for all six trigonometric functions on their natural domains, The addition formulas for sine and cosine, The subtraction formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains).

[L3]

The half-angle advance for the Coulomb branch is $Δ = π / 2 - \arctan (κ / (2 E b))$ and the deflection magnitude is $π - 2 Δ = 2 \arctan (κ / (2 E b))$ (Repulsive Coulomb scattering: the unique outer turning point, global non-colliding motion, the asymptotic impact distance and $Δ = π / 2 - \arctan (κ / (2 E b))$ clauses 3-4).

[L4]

Chain, quotient and inverse-function rules: $θ_{d e f}$ and $b$ are $C^{1}$ with the derivatives computed below; the inverse of a continuous injective $C^{1}$ function with nonvanishing derivative has reciprocal derivative (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c \in I$ with $f^{'} (c) \neq 0$ , then the inverse $g$ is differentiable at $f (c)$ with $g^{'} (f (c)) = 1 / f^{'} (c)$ ; and if $f^{'} (c) = 0$ then $g$ is not differentiable at $f (c)$ , $C^{k}$ Euclidean maps and diffeomorphisms, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L5]

One-dimensional substitution/change of variables with the absolute derivative and the branch density identity $d σ / d Ω = b ∣ b^{'} ∣ / \sin θ$ hold on every deflection branch (One-dimensional substitution on a deflection branch: the outgoing angular measure has density $b ∣ d b / d θ ∣$ , hence cross-section density $b ∣ d b / d θ ∣ / \sin θ$ , Scattering data: the incident flux measure, the solid-angle element, deflection branches and the cross-section density).

[L6]

A continuous nonnegative function with a positive value has positive integral, the integral is monotone, and for $0 < ε < θ_{0}$ the computed integral $\int_{ε}^{θ_{0}} θ^{- 3} d θ = \frac{1}{2} (ε^{- 2} - θ_{0}^{- 2})$ diverges as $ε ↓ 0$ ; $\sin x \leq x$ for $x \geq 0$ (mean value theorem applied to $\sin$ on $[0, x]$ , using $\cos \leq 1$ ) and $\cos (θ / 2) \geq \cos (θ_{0} / 2) > 0$ for $0 < θ \leq θ_{0} \leq π / 2$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ , 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 $f (b) - f (a) = f^{'} (c) (b - a)$ , Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ , Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

Proof

technique · direct

1.1givenL1L2L4algebra

Differentiation of the branch: by [L4] and [L2], $θ_{d e f}^{'} (b) = 2 \cdot \frac{1}{1 + (\frac{κ}{2 E b})^{2}} \cdot (- \frac{κ}{2 E b^{2}}) < 0 (b > 0),$ so $θ_{d e f}$ is $C^{1}$ with strictly negative derivative, hence strictly decreasing; as $b ↓ 0$ , $κ / (2 E b) \to \infty$ and $\arctan$ tends to $π / 2$ , so $θ_{d e f} \to π$ ; as $b \to \infty$ , $κ / (2 E b) \to 0$ and $θ_{d e f} \to 0$ . The image is therefore $(0, π)$ and the inverse exists, is $C^{1}$ by [L4], and satisfies the claimed formula: for $0 < θ < π$ , $\arctan (\tan (θ / 2)) = θ / 2$ by [L1] (since $0 < θ / 2 < π / 2$ ) and $\tan (θ / 2) = κ / (2 E b (θ))$ with $b (θ) = \frac{κ}{2 E} \cot (θ / 2)$ , because $\tan (θ / 2) \cot (θ / 2) = 1$ by [L2]; substituting shows $θ_{d e f} (b (θ)) = θ$ .

2.1step 1.1L3algebra

The half-angle and deflection relation of [L3] applied for each $b > 0$ to the corresponding trajectory with $J = J (b)$ give exactly the deflection value $π - 2 Δ = 2 \arctan (κ / (2 E b))$ ; thus the abstract branch formula is the computed scattering deflection of the Coulomb problem, and clause 1 is complete.

2.2step 1.1L2L5algebra

Cross-section density: by [L2], $b^{'} (θ) = - \frac{κ}{4 E} \csc^{2} (θ / 2)$ , so $∣ b^{'} (θ) ∣ = \frac{κ}{4 E} \csc^{2} (θ / 2)$ ; combining with [L5], $\frac{d σ}{d Ω} (θ) = \frac{b (θ) ∣ b^{'} (θ) ∣}{\sin θ} = \frac{κ^{2}}{8 E^{2}} \cdot \frac{\cot (θ / 2) \csc^{2} (θ / 2)}{\sin θ} .$ Using $\sin θ = 2 \sin (θ / 2) \cos (θ / 2)$ from [L2], $\cot \frac{θ}{2} \csc^{2} \frac{θ}{2} / \sin θ = \frac{\cos (θ / 2)}{\sin^{3} (θ / 2)} \cdot \frac{1}{2 \sin (θ / 2) \cos (θ / 2)} = \frac{1}{2} \csc^{4} \frac{θ}{2},$ whence $d σ / d Ω = \frac{κ^{2}}{16 E^{2}} \csc^{4} (θ / 2) = (κ / (4 E))^{2} \csc^{4} (θ / 2)$ , finite and positive on $(θ_{1}, θ_{2})$ ; this is clause 2.

3.1step 2.2L6algebra

Divergence: for $0 < θ \leq θ_{0}$ with $θ_{0} \leq π / 2$ , the integrand of clause 3 equals $(κ / (4 E))^{2} \sin θ \csc^{4} (θ / 2) = (κ / (4 E))^{2} \cdot 2 \cos (θ / 2) / \sin^{3} (θ / 2)$ by the same identities; since $\sin (x) \leq x$ for $x \geq 0$ and $\cos (θ / 2) \geq \cos (θ_{0} / 2) > 0$ by [L6], the integrand is at least $C θ^{- 3}$ with $C = 16 (κ / (4 E))^{2} \cos (θ_{0} / 2) > 0$ . Hence for $0 < ε < θ_{0}$ , by monotonicity of the integral and the $φ$ -integration of a constant, $\int_{ε}^{θ_{0}} ⁣ ⁣ \int_{0}^{2 π} (κ / (4 E))^{2} \csc^{4} (θ / 2) \sin θ d φ d θ \geq 2 π C \int_{ε}^{θ_{0}} θ^{- 3} d θ = π C (ε^{- 2} - θ_{0}^{- 2}) ⟶ + \infty$ as $ε ↓ 0$ , by the fundamental theorem and [L6]; hence the truncated integral over $(0, θ_{0})$ diverges. If instead $π / 2 < θ_{0} < π$ , cut the interval at $π / 2$ and use the divergence on $(0, π / 2]$ together with additivity.

4.1step 3.1L6algebra∎

Finiteness on compact angular intervals: the integrand $(κ / (4 E))^{2} \csc^{4} (θ / 2) \sin θ$ is continuous on the compact rectangle $[θ_{1}, θ_{2}] \times [0, 2 π]$ because $\sin (θ / 2) > 0$ there, so its integral is finite by [L6]; this completes clause 3.

The Rutherford deflection map, its inverse, the cross-section density $(κ / 4 E)^{2} \csc^{4} (θ / 2)$ and the divergence of the total cross-section

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The Rutherford deflection map, its inverse, the cross-section density (κ/4E)2csc⁡4(θ/2)(κ/4E)2csc4(θ/2) and the divergence of the total cross-section

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

The Rutherford deflection map, its inverse, the cross-section density $(κ / 4 E)^{2} \csc^{4} (θ / 2)$ and the divergence of the total cross-section