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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 Lo (in the z direction) and radius Ro. The particles have momenta uniformly distributed with pz in an interval po±Δpz and the transverse momentum p inside a circle of radius Δp. 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 and the subsequent behavior of the radius R? (Assume that the focusing does not affect either Lo or Δpz.)

Short Answer

Expert verified
The transverse momentum spread increases as the radius decreases.

Step by step solution

01

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.
02

Initial Phase Space Volume

Initially, the phase space volume for the transverse direction can be described by the cylindrical area πRo2 for position and the momentum circle π(Δp)2, giving a volume of (πRo2)(π(Δp)2)."
03

Final Phase Space Volume

After focusing, the spatial distribution becomes πR2 for position. According to Liouville's theorem, the phase space volume must remain constant, so πR2(Δp,final)2=(πRo2)(Δp)2.
04

Solving for Final Transverse Momentum

Solving πR2(Δp,final)2=πRo2(Δp)2 yields Δp,final=ΔpRoR. This shows that as the radius R decreases, the transverse momentum spread Δp,final increases to conserve the phase space volume.
05

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 Δp,final. As the physical size of the beam decreases, energy (spread in momentum) must increase proportionally.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A roller coaster of mass m moves along a frictionless track that lies in the xy plane (x horizontal and y vertically up). The height of the track above the ground is given by y=h(x). (a) Using x as your generalized coordinate, write down the Lagrangian, the generalized momentum p, and the Hamiltonian H=px˙L (as a function of x and p ). (b) Find Hamilton's equations and show that they agree with what you would get from the Newtonian approach. [Hint: You know from Section 4.7 that Newton's second law takes the form Ftang =ms¨, where s is the distance measured along the track. Rewrite this as an equation for x¨ and show that you get the same result from Hamilton's equations.]

Find the Hamiltonian H for a mass m confined to the x axis and subject to a force Fx=kx3 where k>0. Sketch and describe the phase-space orbits.

Consider a mass m moving in two dimensions, subject to a single force F that is independent of r and t. (a) Find the potential energy U(r) and the Hamiltonian H. (b) Show that if you use rectangular coordinates x,y with the x axis in the direction of F, then y is ignorable. (c) Show that if you use rectangular coordinates x,y with neither axis in the direction of F, then neither coordinate is ignorable. (Moral: Choose generalized coordinates carefully!)

Find the Lagrangian, the generalized momentum, and the Hamiltonian for a free particle (no forces at all) confined to move along the x axis. (Use x as your generalized coordinate.) Find and solve Hamilton's equations.

Here is a simple example of a canonical transformation that illustrates how the Hamiltonian formalism lets one mix up the q s and the p 's. Consider a system with one degree of freedom and Hamiltonian H=H(q,p). The equations of motion are, of course, the usual Hamiltonian equations q˙=H/p and p˙=H/q. Now consider new coordinates in phase space defined as Q=p and P=q. Show that the equations of motion for the new coordinates Q and P are Q˙=H/P and P˙=H/Q; that is, the Hamiltonian formalism applies equally to the new choice of coordinates where we have exchanged the roles of position and momentum.

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