The equations of beamtracer.solver.diagnostics
Return the mismatch parameter M of a beam (alpha, beta).
mismatch M = 1 + 1 2 ( Δ + Δ ( Δ + 4 ) ) − 1 , Δ = ( α − α 0 ) 2 − ( γ − γ 0 ) ( β − β 0 ) , γ = 1 + α 2 β M = \sqrt{1 + \tfrac12 \left(\Delta + \sqrt{\Delta (\Delta + 4)}\right)} - 1, \qquad \Delta = (\alpha - \alpha_0)^2 - (\gamma - \gamma_0)(\beta - \beta_0), \qquad \gamma = \frac{1 + \alpha^2}{\beta} M = 1 + 2 1 ( Δ + Δ ( Δ + 4 ) ) − 1 , Δ = ( α − α 0 ) 2 − ( γ − γ 0 ) ( β − β 0 ) , γ = β 1 + α 2 Assumes ( α 0 , β 0 ) (\alpha_0, \beta_0) ( α 0 , β 0 ) M = 0 M = 0 M = 0 References Return the BeamMoments of the trajectories crossing a plane.
mean ⟨ ⋅ ⟩ = ∑ k ∣ I k ∣ ( ⋅ ) ∑ k ∣ I k ∣ over the forward crossings of the plane \langle \cdot \rangle = \frac{\sum_k |I_k|\, (\cdot)}{\sum_k |I_k|} \quad \text{over the forward crossings of the plane} ⟨ ⋅ ⟩ = ∑ k ∣ I k ∣ ∑ k ∣ I k ∣ ( ⋅ ) over the forward crossings of the plane rings ⟨ x 2 ⟩ = 1 2 ⟨ r 2 ⟩ , ⟨ x ′ 2 ⟩ = 1 2 ⟨ r ′ 2 + a ′ 2 ⟩ , ⟨ x x ′ ⟩ = 1 2 ⟨ r r ′ ⟩ , r ′ = v r v x , a ′ = v θ v x \langle x^2 \rangle = \tfrac12 \langle r^2 \rangle, \quad \langle x'^2 \rangle = \tfrac12 \langle r'^2 + a'^2 \rangle, \quad \langle x x' \rangle = \tfrac12 \langle r r' \rangle, \qquad r' = \frac{v_r}{v_x}, \ a' = \frac{v_\theta}{v_x} ⟨ x 2 ⟩ = 2 1 ⟨ r 2 ⟩ , ⟨ x ′2 ⟩ = 2 1 ⟨ r ′2 + a ′2 ⟩ , ⟨ x x ′ ⟩ = 2 1 ⟨ r r ′ ⟩ , r ′ = v x v r , a ′ = v x v θ planar ⟨ y 2 ⟩ , ⟨ y ′ 2 ⟩ , ⟨ y y ′ ⟩ about their means , y ′ = v y v x \langle y^2 \rangle, \ \langle y'^2 \rangle, \ \langle y y' \rangle \ \text{about their means}, \qquad y' = \frac{v_y}{v_x} ⟨ y 2 ⟩ , ⟨ y ′2 ⟩ , ⟨ y y ′ ⟩ about their means , y ′ = v x v y emittance ε = ⟨ x 2 ⟩ ⟨ x ′ 2 ⟩ − ⟨ x x ′ ⟩ 2 , ε n = ε ⟨ v x ⟩ c \varepsilon = \sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle x x' \rangle^2}, \qquad \varepsilon_n = \varepsilon \frac{\langle v_x \rangle}{c} ε = ⟨ x 2 ⟩ ⟨ x ′2 ⟩ − ⟨ x x ′ ⟩ 2 , ε n = ε c ⟨ v x ⟩ twiss α = − ⟨ x x ′ ⟩ ε , β = ⟨ x 2 ⟩ ε , γ = ⟨ x ′ 2 ⟩ ε \alpha = -\frac{\langle x x' \rangle}{\varepsilon}, \qquad \beta = \frac{\langle x^2 \rangle}{\varepsilon}, \qquad \gamma = \frac{\langle x'^2 \rangle}{\varepsilon} α = − ε ⟨ x x ′ ⟩ , β = ε ⟨ x 2 ⟩ , γ = ε ⟨ x ′2 ⟩ Assumes cylindrical: each trajectory a ring, projected on one transverse plane and averaged over the azimuth (no mean subtracted), which removes the cross term r a ′ r a' r a ′ non-relativistic: ε n \varepsilon_n ε n β = ⟨ v x ⟩ / c \beta = \langle v_x \rangle / c β = ⟨ v x ⟩ / c no crossings or zero current give NaN moments; ε = 0 \varepsilon = 0 ε = 0 ε 2 \varepsilon^2 ε 2