Question

A beam of particles is moving along an accelerator pipe in the $z$ direction. The particles are uniformly distributed in a cylindrical volume of length $L_{\mathrm{o}}$ (in the $z$ direction) and radius $R_{\mathrm{o}}.$ The particles have momenta uniformly distributed with $p_z$ in an interval $p_{\mathrm{o}}\pm \mathrm{\Delta }{p}_z$ and the transverse momentum $p_{\perp }$ inside a circle of radius $\mathrm{\Delta }{p}_{\perp }.$ To increase the particles' spatial density, the beam is focused by electric and magnetic fields, so that the radius shrinks to a smaller value $R$. What does Liouville's theorem tell you about the spread in the transverse momentum $p_{\perp }$ and the subsequent behavior of the radius $R?$ (Assume that the focusing does not affect either $L_{\mathrm{o}}$ or $\mathrm{\Delta }{p}_z$.)

Short answer

The transverse momentum spread increases as the radius decreases.

Step-by-step solution

Understanding Liouville's Theorem

Liouville's theorem states that the phase space volume occupied by a set of particles remains constant over time in a Hamiltonian system. This means even though the distribution of particles in coordinate space (like position) evolves, the distribution in momentum space evolves accordingly to keep the overall phase space volume constant.

Initial Phase Space Volume

Initially, the phase space volume for the transverse direction can be described by the cylindrical area $\pi R_{\mathrm{o}}^2$ for position and the momentum circle $\pi (\mathrm{\Delta }{p}_{\perp }{)}^2$, giving a volume of $(\pi R_{\mathrm{o}}^2)\cdot (\pi (\mathrm{\Delta }{p}_{\perp }{)}^2)$."

Final Phase Space Volume

After focusing, the spatial distribution becomes $\pi R^2$ for position. According to Liouville's theorem, the phase space volume must remain constant, so $\pi R^2\cdot (\mathrm{\Delta }{p}_{\perp ,\text{final}}{)}^2=(\pi R_{\mathrm{o}}^2)\cdot (\mathrm{\Delta }{p}_{\perp }{)}^2$.

Solving for Final Transverse Momentum

Solving $\pi R^2(\mathrm{\Delta }{p}_{\perp ,\text{final}}{)}^2=\pi R_{\mathrm{o}}^2(\mathrm{\Delta }{p}_{\perp }{)}^2$ yields $\mathrm{\Delta }{p}_{\perp ,\text{final}}=\mathrm{\Delta }{p}_{\perp }\frac{R_{\mathrm{o}}}{R}$. This shows that as the radius $R$ decreases, the transverse momentum spread $\mathrm{\Delta }{p}_{\perp ,\text{final}}$ increases to conserve the phase space volume.

Conclusion on the Behavior of R and Momentum

Liouville's theorem indicates that shrinking the radius $R$ due to focusing must result in an increased spread in transverse momentum $\mathrm{\Delta }{p}_{\perp ,\text{final}}$. As the physical size of the beam decreases, energy (spread in momentum) must increase proportionally.

Key concepts

Phase Space Volume

In the realm of classical mechanics, Liouville's theorem plays a pivotal role in explaining how particles behave within a given space. This is particularly focused on the concept of phase space volume. Phase space is a mathematical construct that combines position and momentum coordinates into a unified space. For a particle beam, each particle has its own position and momentum, and plotting these creates a point in phase space. A collection of particles forms a "volume" in this space.

According to Liouville's theorem, in a Hamiltonian system, this phase space volume remains constant over time. Even as particles are subject to various forces and constraints, the overall volume of their positions and momenta does not change. This principle applies to the cylindrical volume of particles moving through an accelerator, where both position (defined by the radius) and momentum (defined by momentum intervals) create a distinct phase space profile. When particles are focused, changes to the radius alter the configuration, but not the volume, requiring adjustments to transverse momentum.

• Phase space combines position and momentum.
• Liouville's theorem maintains volume constancy.
• Changes in one (position or momentum) necessitate adjustments in the other.

Transverse Momentum

Transverse momentum refers to the component of momentum perpendicular to the primary direction of motion. In the case of particles moving through an accelerator, the transverse momentum is the momentum perpendicular to the primary motion along the z-direction. Changes in transverse momentum are crucial when considering how focusing in particle beams works.

When the radius of the beam is reduced using electric and magnetic fields, the focus alters the configuration, shrinking the beam's physical size. However, due to Liouville's theorem, even as the radius decreases, the transverse momentum must spread out more to keep the phase space volume constant. This spread in transverse momentum ensures that the system is conserving its overall energy distribution, a crucial aspect for maintaining stability in accelerators.

• Transverse momentum acts perpendicular to motion direction.
• Focusing affects beam radius and momentum distribution.
• Momentum spread adjusts to conserve phase space volume.

Hamiltonian System

Hamiltonian systems are a cornerstone of classical mechanics, describing systems that are conservative, meaning they conserve energy over time. In the context of particle physics, a Hamiltonian system allows for the precise modeling of particles moving through an accelerator.

The beauty of Hamiltonian systems lies in their ability to elegantly track how energy and momentum interact under physical constraints. Liouville's theorem is directly applicable, as it assumes the system is Hamiltonian. In such systems, focusing a beam using external fields does not add or subtract energy from the overall system; rather, it redistributes energy across different components, like altering the transverse momentum when the beam's radius changes.

• Hamiltonian systems are energy-conservative.
• Essential for modeling particle physics scenarios.
• Energy redistribution occurs without changing total energy.