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For slowly accelerated paraxial motion, $z^{\prime }\simeq p_z/p_s$ decreases as longitudinal momentum grows while transverse canonical phase-space area is conserved. The normalized emittance stays constant in the ideal limit, so the geometric emittance scales inversely with longitudinal momentum. This is kinematic damping rather than dissipation of canonical area.

The area occupied by a beam in transverse position-momentum or position-angle phase space, with a stated area or rms convention. Geometric emittance uses trajectory angle, while normalized emittance removes the ideal kinematic damping due to acceleration.

The transverse transport of a charged-particle beam relative to a reference trajectory. Linear focusing gives betatron oscillations, described by Hill's equation and Courant-Snyder parameters; the beam envelope depends on its geometric emittance.

A lattice cell with two bending magnets whose dispersion and dispersion slope vanish at its boundaries. In an ideal identical-uniform-bend model with horizontal damping partition one, its minimum geometric emittance is $C_q{\gamma }^2{\theta }^3/(4\sqrt{15})$, larger than the unconstrained theoretical minimum because of the achromatic boundary conditions.

The geometric emittance multiplied by the relativistic velocity and Lorentz factor. For paraxial acceleration without dilution, this is proportional to transverse canonical phase-space area and remains constant. Thus geometric emittance decreases approximately as inverse beam momentum.

Write $z=\sqrt{ϵ}\omega \cos \psi$, with $\omega =\sqrt{\beta }>0$ and $\psi =\varphi -{\varphi }_0$. Differentiating twice givesThe coefficients must vanish for arbitrary initial phase ${\varphi }_0$. The sine coefficient givesThe conventional Courant-Snyder parameters normalize the phase so that $C=1$; any positive constant $C$ can be absorbed into the scale assigned to $\beta$ and $ϵ$. Explicitly, $\tilde{\beta }=\beta /C$ and $\widetilde{ϵ}=Cϵ$ preserve the orbit while making $\tilde{\beta }{\varphi }^{\prime }=1$. With this convention,The second expression is the Ermakov-Pinney equation for the beam envelope. It follows by substituting ${\varphi }^{\prime }=1/{\omega }^2$ into the cosine coefficient.

The printed equation omits the factor $\omega$ multiplying $k(s)$. For a concrete counterexample, take the constant periodic focusing strength $k=1/16$ and the constant envelope $\omega =2$. Then ${\varphi }^{\prime }=1/4$ and $z=2\sqrt{ϵ}\cos (s/4-{\varphi }_0)$ solve $z^{\prime \prime }+z/16=0$. The correct envelope equation is $0-1/8+(1/16)2=0$, whereas the printed equation gives $-1/8+1/16=-1/16$.

The associated invariant isThus $ϵ$ fixes the phase-space ellipse of a trajectory; a beam's rms geometric emittance uses the corresponding second-moment convention. Factors of $\pi$ depend on whether emittance means ellipse area or area divided by $\pi$ and do not alter the envelope equation.

For a beam with rms geometric emittance ${ϵ}_y$, its vertical size at the collision point is ${\sigma }_y^{\star }=\sqrt{{ϵ}_y{\beta }_y^{\star }}$. Reducing ${\beta }_y^{\star }$ therefore concentrates the beam, increasing the number of particles encountering the opposite beam per transverse area. For identical Gaussian beams colliding head-on,Keep bunch populations, horizontal size, collision frequency and emittances fixed. If both vertical beta values are decreased by a factor $f>1$, meaning ${\beta }_{y,\mathrm{new}}^{\star }={\beta }_{y,\mathrm{old}}^{\star }/f$, both vertical beam sizes decrease by $1/\sqrt{f}$, andThe luminosity is inversely proportional to the common vertical size, not its square, because only one transverse dimension has changed. The same ratio follows from the unequal-beam overlap denominator $\sqrt{{\sigma }_{y1}^2+{\sigma }_{y2}^2}$.

A final-focus quadrupole system on either side of the interaction point forms a waist there. Strong quadrupole doublets or triplets, with suitable matching sections, focus both planes. At a symmetric waist ${\beta }^{\prime }=0$ and the nearby drift gives $\beta (s)={\beta }^{\star }+s^2/{\beta }^{\star }$, so small ${\beta }^{\star }$ requires large beam envelopes and strong optical gradients in the adjacent magnets. Chromatic correction is needed for off-momentum particles. The formula assumes negligible crossing-angle and hourglass reductions; a bunch length comparable to ${\beta }^{\star }$ changes the effective overlap.

For a paraxial orbit $z^{\prime }\simeq p_z/p_s$, where $p_z$ is transverse momentum and $p_s$ longitudinal momentum. Acceleration mainly increases $p_s$, so a given transverse kick produces a smaller angular deviation. In an ideal matched accelerator without scattering or radiation, the conserved transverse canonical phase-space area becomes a smaller area in the coordinates $(z,z^{\prime })$. This is adiabatic damping of geometric emittance, not loss of canonical phase-space volume.

The invariant normalized emittance isHere ${\beta }_{\mathrm{rel}}$ is speed in units of $c$, distinct from the lattice beta function. Neglecting dilution mechanisms, each acceleration stage thus supplies a beam with smaller angular spread and geometric emittance for injection into the next stage.

A chain of increasingly energetic particle accelerators lets each ring operate over a manageable range of magnetic rigidity and radio-frequency settings. It also avoids injecting a large-emittance low-energy beam directly into the small-aperture high-field main ring. Higher injection energy reduces space-charge effects and improves capture and transmission. Emittance dilution, mismatching and collective instabilities can oppose the ideal damping; acceleration alone does not reduce normalized emittance.

At fixed flux and bandwidth, brightness is inversely proportional to ${\Sigma }_x{\Sigma }_x^{\prime }{\Sigma }_y{\Sigma }_y^{\prime }$. Lowering both geometric emittances by ten makes the electron rms widths and divergences smaller by $\sqrt{10}$. Add the unchanged radiation widths in quadrature again and divide the old phase-space product by the new one. The resulting brightness factor is approximately $\boxed{3.87}$, an increase of about $287\%$, rather than the factor 100 that would follow if diffraction were negligible in both planes. The exact value under the adopted width convention is given by the accompanying check and plot.

For the same lattice radiation-integral ratio, natural geometric emittance scales as ${\gamma }^2$. A factor-ten decrease therefore requires $\boxed{E^{\prime }=3/\sqrt{10}\simeq 0.949\mathrm{GeV}}$. Keeping coupling ratio fixed reduces the vertical emittance by the same factor, within that ideal scaling model.