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Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds

Physical scope

Conditional cross sections of the adopted ideal harmonic dipole model in a plane wave: the directional and total model cross sections in SI constants, the scalar-polarizability branch, and the explicit fidelity bounds comparing the model with an actual compact smooth source. Excluded: measured cross sections, frequency-domain response laws, magnetic-dipole and higher-multipole content, self-force and radiation reaction, material response, finite-size and every empirical claim.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as carried by Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source and Zonal and quadratic sphere moments for the polar surface measure on $S^{2}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the period-average and cross-section algebra itself uses no choice. Adopt the ideal harmonic electric-dipole radiation model of The ideal harmonic electric-dipole radiation model and its adopted status with frequency $ω > 0$ , wavenumber $k = ω / c$ and amplitude $p_{0} \in C^{3}$ ; let $E_{d}, B_{d}, S_{d}$ be its fields and model flux. Let $κ \in R^{3} ∖ {0}$ with $∣ κ ∣ = ω / c$ , $e \in C^{3} ∖ {0}$ with $κ \cdot e = 0$ , $b : = κ \times e / ω$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and let $(E_{i n c}, B_{i n c})$ be the corresponding plane-wave fields of Real plane-wave fields encoded by complex amplitudes on $R^{3} \times R$ ; by Dispersion, transversality and the two polarization conditions for plane-wave solutions they satisfy the smooth-branch source-free vacuum Maxwell equations, and by Period-averaged energy density, Poynting vector and momentum flux of a plane wave(b) their period-averaged energy flux has magnitude $I_{i n c} = \frac{ε_{0} c}{2} ∣ e ∣^{2} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values). Define the model differential scattering cross section by

$\frac{d σ}{d Ω} (n) : = \frac{r^{2} ⟨ S_{d} (r n, \cdot) \cdot n ⟩}{⟨ S_{i n c} \cdot \hat{κ} ⟩}, n \in S^{2}, r > 0,$

where $\hat{κ} : = κ / ∣ κ ∣$ , $S_{i n c} : = E_{i n c} \times B_{i n c} / μ_{0}$ (Vacuum electromagnetic energy density and energy flux (the Poynting vector)), and $⟨ f ⟩ : = \frac{1}{T} \int_{0}^{T} f (s) d s$ , $T = 2 π / ω$ , is the period average at fixed observation point. Then: (a) [Angular pattern] the ratio is independent of $r$ and of the averaging start, and for every $n \in S^{2}$

$\frac{d σ}{d Ω} (n) = \frac{ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4} ∣ e ∣^{2}} = \frac{μ_{0}^{2} ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ∣ e ∣^{2}} .$

(b) [Total cross section] with $d σ_{S^{2}}$ the surface measure of the unit sphere, $σ : = \int_{S^{2}} \frac{d σ}{d Ω} d σ_{S^{2}}$ equals

$σ = \frac{ω^{4} ∣ p_{0} ∣^{2}}{6 π ε_{0}^{2} c^{4} ∣ e ∣^{2}} = \frac{μ_{0}^{2} ω^{4} ∣ p_{0} ∣^{2}}{6 π ∣ e ∣^{2}};$

if $p_{0} = α e$ for a complex scalar $α$ (the scalar polarizability, in $F m^{2}$ ), then $σ = \frac{ω^{4} ∣ α ∣^{2}}{6 π ε_{0}^{2} c^{4}}$ and $\frac{d σ}{d Ω} (n) = \frac{ω^{4} ∣ α ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4}} ∣ n \times \hat{e} ∣^{2}$ with $\hat{e} : = e / ∣ e ∣$ . (c) [Fidelity for arbitrary actual sources and matched windows] Let an actual compact smooth conserved source satisfy the exact hypotheses of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, with spatial support radius $a > 0$ , source-time interval $[τ_{0}, τ_{1}]$ , moment $p_{a c t} (u)$ , variation bound $D (u)$ and constants $Γ_{E}, Γ_{B}$ . These source data are independent of the prescribed model amplitude $p_{0}$ . Put

$p_{h} (u) : = Re (p_{0} e^{- i ω u}), M (u) : = ∣ p_{a c t}^{''} (u) - p_{h}^{''} (u) ∣ .$

For $n \in S^{2}$ , $u \in [τ_{0}, τ_{1}]$ , $r \geq 2 a$ , define

$H (u, r) : = \frac{Γ_{E}}{r^{2}} + \frac{μ_{0}}{4 π r} [D (u) + M (u)] .$

The actual retarded fields and the model fields satisfy, uniformly in $n$ ,

$∣ E_{a c t} (r n, u + r / c) - E_{d} (r n, u + r / c) ∣ \leq H (u, r),$

$∣ B_{a c t} (r n, u + r / c) - B_{d} (r n, u + r / c) ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}} + \frac{H (u, r)}{c} .$

Also retain the sharper direct source-leading-field consequence

$∣ B_{a c t} - B_{d} ∣ \leq \frac{Γ_{B}}{r^{2}} + \frac{μ_{0} [D (u) + M (u)]}{4 π c r}$

at the same events. If $p_{a c t}^{''} = p_{h}^{''}$ on a specified finite window, $M \equiv 0$ there and these give the full matched dipole bounds; in particular they imply the coarser electric bound $Γ_{E} / r^{2} + μ_{0} D / (2 π r)$ . Neither moment matching nor $D \equiv 0$ is automatic. A sound sufficient matched branch is $p_{a c t} = p_{h} + a_{0} + b_{0} u$ on an open neighbourhood of the window (any fixed real vectors $a_{0}, b_{0}$ ). Such a branch has actual compact realizations over any finite window: choose a smooth time cutoff $η$ equal to $1$ near the window, let $p_{η} : = η (p_{h} + a_{0} + b_{0} u)$ , choose a nonnegative smooth spatial bump $σ$ with support inside $B_{a}$ and integral $1$ , and set $ρ : = - p_{η} \cdot \nabla σ$ , $J : = p_{η}^{'} σ$ . Then continuity holds and the actual moment is $p_{η}$ ; outside the window it is tapered. Apply the actual-source theorem with the resulting $D$ and $Γ$ , retaining every finite-radius and source-variation error. No eternal harmonic moment of a compact-time source is asserted. (d) [Scope] conditional predictions of the adopted ideal model: no measured cross section, no frequency-domain response law (the amplitude $p_{0}$ is prescribed model data), no magnetic-dipole or higher-multipole content, no self-force, material response, finite-size or empirical claim; the scalar- $α$ clause records a declared proportional response, not a derived polarizability. In all complex-amplitude formulas $∣ v ∣^{2} = \sum_{i} ∣ v_{i} ∣^{2}$ is the Hermitian norm square; $n \times v$ is complex-linear for real $n$ . The $p_{0} = 0$ case gives zero scattered fields and cross sections while the nonzero incident intensity still defines the ratios. Actual-source fidelity assumes only the explicitly quantified mismatch and source hypotheses; it does not turn the model cross section into a measured or self-consistent scattering prediction.

Facts & Assumptions

Given: The model data of The ideal harmonic electric-dipole radiation model and its adopted status: $ω > 0$ , $k = ω / c$ , $p_{0} \in C^{3}$ , the model fields $E_{d}, B_{d}$ and flux $S_{d}$ on $(R^{3} ∖ {0}) \times R$ ; the plane-wave data $κ$ , $e \neq 0$ with $κ \cdot e = 0$ , $b = κ \times e / ω$ , and the fields $E_{i n c}, B_{i n c}$ , $S_{i n c}$ ; the constants $c, ε_{0}, μ_{0}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values with $ε_{0} μ_{0} c^{2} = 1$ ; the period $T = 2 π / ω$ and the cross section of the Statement.

[F1]

Model interface: $E_{d} (x, t) = Re (w (x) e^{- i ω t})$ with $w (x) : = - \frac{μ_{0} ω^{2}}{4 π r} n \times (n \times p_{0}) e^{i k r}$ , $r = ∣ x ∣$ , $n = x / r$ ; $B_{d} = n \times E_{d} / c$ ; $S_{d} = E_{d} \times B_{d} / μ_{0}$ is the flux of Vacuum electromagnetic energy density and energy flux (the Poynting vector) evaluated at the model fields (The ideal harmonic electric-dipole radiation model and its adopted status).

[F2]

Incident plane wave: $(E_{i n c}, B_{i n c})$ are the plane-wave fields of Real plane-wave fields encoded by complex amplitudes with amplitude $e$ and $b = κ \times e / ω$ , $∣ κ ∣ = ω / c$ , satisfying the smooth source-free vacuum Maxwell equations by Dispersion, transversality and the two polarization conditions for plane-wave solutions; their period-averaged flux satisfies $⟨ S_{i n c} ⟩ = I_{i n c} \hat{κ}$ with $I_{i n c} = \frac{ε_{0} c}{2} ∣ e ∣^{2} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ (Period-averaged energy density, Poynting vector and momentum flux of a plane wave, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F3]

Period averages: for $a, d \in C$ , $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \overline{d})$ ; in particular the mean of the real square is $\frac{1}{2} ∣ a ∣^{2}$ (Period averages of products of real parts of complex exponentials, Real and imaginary parts, complex conjugation, and modulus).

[F4]

Cross product and norm identities: the cross product is bilinear and alternating; $∣ n \times (n \times v) ∣ = ∣ n \times v ∣ \leq ∣ v ∣$ for $∣ n ∣ = 1$ ; $n \times (u + i v) = n \times u + i n \times v$ for real $n$ ; $∣ n \times v ∣^{2} = \sum_{i} ∣ (n \times v)_{i} ∣^{2}$ is the Hermitian norm square (The cross product is bilinear, alternating, and orthogonal to both factors, The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Real and imaginary parts, complex conjugation, and modulus).

[F5]

Sphere moments: for the surface measure $σ_{S^{2}}$ of the unit sphere, $σ_{S^{2}} (S^{2}) = 4 π$ , $\int_{S^{2}} n d σ_{S^{2}} = 0$ , $\int_{S^{2}} n_{i} n_{j} d σ_{S^{2}} = (4 π / 3) δ_{i j}$ , and for every $a \in R^{3}$ , $\int_{S^{2}} (n \cdot a)^{2} d σ_{S^{2}} = \frac{4 π}{3} ∣ a ∣^{2}$ and $\int_{S^{2}} ∣ n \times a ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} ∣ a ∣^{2}$ (Zonal and quadratic sphere moments for the polar surface measure on $S^{2}$ ).

[F6]

Actual-source theorem: under the hypotheses of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source (jointly $C^{\infty}$ conserved sources vanishing outside ${\overline{B}}_{a} (0) \times [τ_{0}, τ_{1}]$ , $u \in [τ_{0}, τ_{1}]$ ), for $n \in S^{2}$ , $r \geq 2 a$ , $∣ E_{a c t} (r n, u + r / c) - \frac{μ_{0}}{4 π r} n \times (n \times p_{a c t}^{''} (u)) ∣ \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D (u)}{4 π r}$ and $∣ B_{a c t} (r n, u + r / c) - \frac{n \times E_{a c t} (r n, u + r / c)}{c} ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}}$ , and the sharper bound $∣ B_{a c t} (r n, u + r / c) - \frac{n \times E_{l e a d} (n, u, r)}{c} ∣ \leq \frac{Γ_{B}}{r^{2}}$ with $E_{l e a d} = \frac{μ_{0}}{4 π r} n \times (n \times V_{a c t})$ ; the constants $Γ_{E}, Γ_{B}$ depend only on $a, c, μ_{0}$ and the source suprema and the theorem assumes ${A C}_{ω}$ (Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

Window construction toolbox: smooth compactly supported cutoffs and localized bumps exist with the properties used below (Test function cutoffs and euclidean localization, The mollifier family generated by a unit-mass smooth bump); integration by parts holds on a bounding rectangle (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections), and the Riemann-Lebesgue comparison identifies the Lebesgue and Riemann integrals of the continuous integrands used there (Riemann–Lebesgue comparison for distribution test integrands); the first fundamental theorem gives the derivative of an integral over a moving endpoint (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive), and a continuous function on an interval with zero derivative is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant). Rectangles, cubes and their volumes are those of Axis-parallel rectangles in $R^{m}$ and their volume; continuous real functions on nondegenerate closed rectangles are Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable). The integral is monotone (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ), so a nonnegative continuous cutoff equal to $1$ near a nonempty cube has a strictly positive finite integral; dividing by that integral normalizes the spatial bump.

Proof

technique · direct; compute the model flux pointwise from the double cross product, evaluate its period average with the period-average lemma, divide by the incident flux, integrate the pattern against the sphere moments for the total, and obtain the actual-source fidelity bounds by adding the dipole theorem's errors to the moment mismatch through the norm inequalities of the cross product

1.1givenF1F4algebra

[Model flux along the observation direction] Fix $n \in S^{2}$ and $r > 0$ . By [F1], $B_{d} = n \times E_{d} / c$ and $n \cdot E_{d} = 0$ , since $n \times (n \times p_{0})$ is orthogonal to $n$ ; hence with the vector identity $E_{d} \times (n \times E_{d}) = n ∣ E_{d} ∣^{2} - E_{d} (n \cdot E_{d})$ of [F4], $S_{d} \cdot n = \frac{1}{μ_{0}} (E_{d} \times B_{d}) \cdot n = \frac{1}{μ_{0} c} (E_{d} \times (n \times E_{d})) \cdot n = \frac{∣ E_{d} ∣^{2}}{μ_{0} c}$ , because $n \cdot n = 1$ .

1.2givenF1F4algebra

[The model field is the leading field of the harmonic waveform] For $u = t - r / c$ one has $e^{i (k r - ω t)} = e^{- i ω u}$ ; since $p_{h}^{''} (u) = Re (- ω^{2} p_{0} e^{- i ω u})$ by [F1] of the harmonic lemma as recorded in The ideal harmonic electric-dipole radiation model and its adopted status, and $n \times (n \times \cdot)$ is real-linear by [F4], $E_{d} (r n, u + r / c) = \frac{μ_{0}}{4 π r} n \times (n \times p_{h}^{''} (u))$ and $B_{d} (r n, u + r / c) = \frac{1}{c} n \times E_{d} (r n, u + r / c) = - \frac{μ_{0}}{4 π c r} n \times p_{h}^{''} (u)$ , the last identity by the double-cross computation of [F4].

2.1step 1.1F1F3F4algebra

[Period average and the averaged radial power] At fixed $x = r n$ the amplitude $w = w (x)$ of [F1] is independent of $t$ , and $E_{d} (r n, t) = Re (w e^{- i ω t})$ ; by [F3] with $a = d = w$ , $⟨ ∣ E_{d} ∣^{2} ⟩ = \frac{1}{2} ∣ w ∣^{2}$ . Evaluating $∣ w ∣^{2}$ with [F4], $∣ w ∣^{2} = (\frac{μ_{0} ω^{2}}{4 π r})^{2} ∣ n \times (n \times p_{0}) ∣^{2} = \frac{μ_{0}^{2} ω^{4}}{16 π^{2} r^{2}} ∣ n \times p_{0} ∣^{2}$ , because $∣ n \times (n \times p_{0}) ∣ = ∣ n \times p_{0} ∣$ ; therefore $r^{2} ⟨ S_{d} \cdot n ⟩ = \frac{r^{2}}{μ_{0} c} ⟨ ∣ E_{d} ∣^{2} ⟩ = \frac{μ_{0} ω^{4}}{32 π^{2} c} ∣ n \times p_{0} ∣^{2}$ , independent of $r$ .

2.2step 1.2F4F6algebra

[Electric and magnetic fidelity bounds] Let $p_{a c t}$ and $D, Γ_{E}, Γ_{B}$ be the actual-source data of [F6]. Subtracting the two leading fields, $E_{a c t} (r n, u + r / c) - E_{d} (r n, u + r / c) = [E_{a c t} - \frac{μ_{0}}{4 π r} n \times (n \times p_{a c t}^{''})] + \frac{μ_{0}}{4 π r} n \times (n \times (p_{a c t}^{''} - p_{h}^{''}))$ , so taking norms and using $∣ n \times (n \times w) ∣ \leq ∣ w ∣$ of [F4] and $∣ p_{a c t}^{''} - p_{h}^{''} ∣ = M (u)$ , the first bound of [F6] gives $∣ E_{a c t} - E_{d} ∣ \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D (u)}{4 π r} + \frac{μ_{0} M (u)}{4 π r} = H (u, r)$ , uniformly in $n$ at fixed $u$ . For the magnetic field, $∣ B_{a c t} - B_{d} ∣ \leq ∣ B_{a c t} - \frac{1}{c} n \times E_{a c t} ∣ + \frac{1}{c} ∣ n \times (E_{a c t} - E_{d}) ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}} + \frac{H (u, r)}{c}$ by the second bound of [F6], [F4] and the electric bound just proved. For the sharper direct estimate, [F6] gives $∣ B_{a c t} - \frac{1}{c} n \times E_{l e a d} (V_{a c t}) ∣ \leq \frac{Γ_{B}}{r^{2}}$ with $E_{l e a d} = \frac{μ_{0}}{4 π r} n \times (n \times V_{a c t})$ , and by the double-cross computation of [F4] applied to $V_{a c t}$ and $p_{h}^{''}$ , $\frac{1}{c} n \times E_{l e a d} (V_{a c t}) - B_{d} = - \frac{μ_{0}}{4 π c r} n \times (V_{a c t} - p_{h}^{''})$ ; hence $∣ B_{a c t} - B_{d} ∣ \leq \frac{Γ_{B}}{r^{2}} + \frac{μ_{0} (D (u) + M (u))}{4 π c r}$ by the triangle inequality and $∣ V_{a c t} - p_{h}^{''} ∣ \leq D + M$ from the replacement bound of [F6]. When $M \equiv 0$ on the window the bounds reduce to $H = \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D}{4 π r} \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D}{2 π r}$ , and neither $M \equiv 0$ nor $D \equiv 0$ is automatic.

3.1step 2.1F1F7algebra

[Independence of the averaging start] For any real $s$ , define $G (s) : = \int_{s}^{s + T} ∣ E_{d} (r n, t) ∣^{2} d t$ . By [F7] the first fundamental theorem gives $G^{'} (s) = ∣ E_{d} (r n, s + T) ∣^{2} - ∣ E_{d} (r n, s) ∣^{2} = 0$ , because the exponential encoding of [F1] and the addition law give $e^{- i ω (t + T)} = e^{- i ω t} e^{- 2 π i} = e^{- i ω t}$ , so $t \mapsto E_{d} (r n, t)$ is $T$ -periodic; hence $G$ is constant by the zero-derivative corollary of [F7] and the mean $\frac{1}{T} \int_{s}^{s + T} ∣ E_{d} ∣^{2} = \frac{1}{T} \int_{0}^{T} ∣ E_{d} ∣^{2}$ is independent of the start, so the cross-section ratio of step 2.1 is as well.

3.2step 2.2F6F7algebra

[Matched compact-window realization] Let the window be a compact interval $W$ in the source-time line and let $a_{0}, b_{0} \in R^{3}$ . Choose a real $C^{\infty}$ cutoff $η$ in time, equal to $1$ on a neighbourhood of $W$ , supported in a compact interval $(t_{0}, t_{1})$ strictly containing $W$ , as supplied by [F7], and set $p_{η} : = η (p_{h} + a_{0} + b_{0} u)$ . Choose a nonempty closed cube $Q \subset B_{a}$ and a cutoff $χ \in C_{c}^{\infty} (B_{a})$ with $0 \leq χ \leq 1$ and $χ = 1$ near $Q$ , by [F7]. On a bounding rectangle, Fubini and integral monotonicity give $0 < vol (Q) \leq C : = \int χ < \infty$ ; set $σ : = χ / C$ , which is nonnegative, smooth, supported in $B_{a}$ and has $\int σ = 1$ and set $ρ (y, u) : = - p_{η} (u) \cdot \nabla σ (y)$ , $J (y, u) : = p_{η}^{'} (u) σ (y)$ ; both are $C^{\infty}$ and vanish outside ${\overline{B}}_{a} (0) \times [t_{0}, t_{1}]$ , and $\partial_{u} ρ + div J = - p_{η}^{'} \cdot \nabla σ + p_{η}^{'} \cdot \nabla σ = 0$ , so the conservation equation holds. Integrating by parts on a rectangle containing $B_{a}$ and using Fubini as in [F7], $\int_{B_{a}} ρ (y, u) d y = - p_{η} (u) \cdot \int_{B_{a}} \nabla σ = 0$ , so the total charge vanishes, and $\int_{B_{a}} y_{i} ρ (y, u) d y = - \sum_{j} p_{η, j} (u) \int_{B_{a}} y_{i} \partial_{j} σ (y) d y = p_{η, i} (u)$ because $\int_{B_{a}} y_{i} \partial_{j} σ = - δ_{i j} \int_{B_{a}} σ = - δ_{i j}$ by the same integration by parts; hence the actual moment of the source is $p_{a c t} = p_{η}$ . On $W$ one has $η \equiv 1$ , so $p_{a c t}^{''} = p_{h}^{''}$ there and the mismatch $M$ vanishes on the window; applying the fidelity bounds of step 2.2 to this source preserves every finite-radius and source-variation error term $D, Γ_{E}, Γ_{B}$ computed from it, and outside the window the moment is tapered, so no eternal harmonic moment is asserted.

4.1step 2.1step 3.1F2algebra

[Angular pattern] Dividing the averaged radial power of step 2.1 by $I_{i n c} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ of [F2] gives $\frac{d σ}{d Ω} (n) = \frac{μ_{0}^{2} ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ∣ e ∣^{2}}$ , and substituting $μ_{0}^{2} = \frac{1}{ε_{0}^{2} c^{4}}$ from Framework-local SI units, the vacuum constants and the exact/measured status of their values gives the displayed second form $\frac{ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4} ∣ e ∣^{2}}$ ; with step 3.1 this is clause (a).

5.1step 4.1F4F5algebra

[Total cross section] Write $p_{0} = u + i v$ with $u, v \in R^{3}$ ; by [F4], $∣ n \times p_{0} ∣^{2} = ∣ n \times u ∣^{2} + ∣ n \times v ∣^{2}$ for every $n$ , so integrating the pattern of step 4.1 over $S^{2}$ and using $\int_{S^{2}} ∣ n \times a ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} ∣ a ∣^{2}$ of [F5] for the real vectors $u$ and $v$ gives $\int_{S^{2}} ∣ n \times p_{0} ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} (∣ u ∣^{2} + ∣ v ∣^{2}) = \frac{8 π}{3} ∣ p_{0} ∣^{2}$ ; hence $σ = \frac{μ_{0}^{2} ω^{4}}{16 π^{2} ∣ e ∣^{2}} \cdot \frac{8 π}{3} ∣ p_{0} ∣^{2} = \frac{μ_{0}^{2} ω^{4} ∣ p_{0} ∣^{2}}{6 π ∣ e ∣^{2}}$ , and the $ε_{0}$ form follows from $μ_{0}^{2} = \frac{1}{ε_{0}^{2} c^{4}}$ . For the scalar branch $p_{0} = α e$ , [F4] gives $∣ n \times p_{0} ∣^{2} = ∣ α ∣^{2} ∣ n \times e ∣^{2} = ∣ α ∣^{2} ∣ e ∣^{2} ∣ n \times \hat{e} ∣^{2}$ and $∣ p_{0} ∣^{2} = ∣ α ∣^{2} ∣ e ∣^{2}$ , so both displayed substitutions hold; this is clause (b).

6.1step 4.1step 5.1step 2.2step 3.2given∎

[Scope and choice accounting] Clause (d) is the exact scope: the cross sections are conditional predictions of the adopted model, $p_{0}$ is prescribed data rather than a derived response, and no measured, material, multipole, self-force or finite-size claim is made; the $p_{0} = 0$ case gives zero model fields and zero cross sections while the nonzero $I_{i n c}$ still normalizes the ratios. The only choice principle used is the Countable Choice carried by the actual-source theorem and the sphere-moment interface of [F5], both invoked in the finite-window and integration steps; the period averages, the double-cross algebra, the division by the incident flux and the integration of the pattern use only finite sums and the explicit suppliers [F3], [F4], so no further choice is made.

Remarks

The incident intensity used is the framework's own $I_{i n c} = ε_{0} c ∣ e ∣^{2} / 2$ from Period-averaged energy density, Poynting vector and momentum flux of a plane wave(b); no source's printed flux normalisation is used, and the two displayed $ε_{0}$ and $μ_{0}$ forms of the cross sections are related by $ε_{0} μ_{0} c^{2} = 1$ .
The fidelity clause compares the model with actual compact smooth sources through the explicit $D, Γ$ bounds plus the moment mismatch $M$ ; it is not a claim that an actual scatterer's measured cross section equals the model value, and the matched-window branch only shows that the mismatch term is genuinely removable for a realized source class.

Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources