1 / 13

PHYS 3446 – Lecture #4

PHYS 3446 – Lecture #4. Monday, Sept. 8, 2006 Dr. Andrew Brandt. Lab Frame and Center of Mass Frame Relativistic Variables ***HW2 due Weds ***Radiation training. Scattering Cross Section.

lark
Download Presentation

PHYS 3446 – Lecture #4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PHYS 3446 – Lecture #4 Monday, Sept. 8, 2006 Dr. Andrew Brandt • Lab Frame and Center of Mass Frame • Relativistic Variables • ***HW2 due Weds • ***Radiation training PHYS 3446, Fall 2008 Andrew Brandt

  2. Scattering Cross Section • For a central potential, measuring the yield as a function of q, or differential cross section, is equivalent to measuring the entire effect of the scattering • So what is the physical meaning of the differential cross section? • This is equivalent to the probability of certain process in a specific kinematic phase space • Cross sections are measured in the unit of barns: PHYS 3446, Fall 2008 Andrew Brandt

  3. Lab Frame and Center of Mass Frame • So far, we have neglected the motion of target nuclei in Rutherford Scattering • In reality, they recoil as a result of scattering • This complication can best be handled using the Center of Mass frame under a central potential • This description is also useful for scattering experiments with two beams of particles (moving target) PHYS 3446, Fall 2008 Andrew Brandt

  4. Lab Frame and CM Frame • The equations of motion can be written where Since the potential depends only on relative separation of the particles, we redefine new variables, relative coordinates & coordinate of CM and PHYS 3446, Fall 2008 Andrew Brandt

  5. Lab Frame and CM Frame • From the equations in previous slides Reduced Mass and Thus • What do we learn from this exercise? • For a central potential, the motion of the two particles can be decoupled when re-written in terms of • a relative coordinate • The coordinate of center of mass PHYS 3446, Fall 2008 Andrew Brandt

  6. Now some simple arithmetic • From the equations of motion, we obtain • Since the momentum of the system is conserved: • Rearranging the terms, we obtain PHYS 3446, Fall 2008 Andrew Brandt

  7. Lab Frame and CM Frame • The CM is moving at a constant velocity in the lab frame independent of the form of the central potential • The motion is that of a fictitious particle with mass m (the reduced mass) and coordinate r. • In the frame where the CM is stationary, the dynamics becomes equivalent to that of a single particle of mass m scattering off of a fixed scattering center. • Frequently we define the Center of Mass frame as the frame where the sum of the momenta of all the interacting particles is 0. PHYS 3446, Fall 2008 Andrew Brandt

  8. Relationship of variables in Lab and CM CM • The speed of CM is • Speeds of the particles in CM frame are • The momenta of the two particles are equal and opposite!! and PHYS 3446, Fall 2008 Andrew Brandt

  9. Scattering angles in Lab and CM • qCM represents the change in the direction of the relative position vector r as a result of the collision in CM frame • Thus, it must be identical to the scattering angle for a particle with reduced mass, m. • Z components of the velocities of particle with m1 in lab and CM are: • The perpendicular components of the velocities are: • Thus, the angles are related (for elastic scattering only) as: PHYS 3446, Fall 2008 Andrew Brandt

  10. Differential cross sections in Lab and CM • The particles that scatter in lab at an angle qLab into solid angle dWLab scatter at qCM into solid angle dWCM in CM. • Since f is invariant, dfLab = dfCM. • Why? • f is perpendicular to the direction of boost, thus is invariant. • Thus, the differential cross section becomes: quiz! reorganize Using Eq. 1.53 PHYS 3446, Fall 2008 Andrew Brandt

  11. Some Quantities in Special Relativity • Fractional velocity • Lorentz g factor • Relative momentum and the total energy of the particle moving at a velocity is • Square of four momentum P=(E,pc), rest mass E PHYS 3446, Fall 2008 Andrew Brandt

  12. Relativistic Variables • Velocity of CM in the scattering of two particles with rest mass m1 and m2 is: • If m1 is the mass of the projectile and m2 is that of the target, for a fixed target we obtain PHYS 3446, Fall 2008 Andrew Brandt

  13. Expansion Relativistic Variables • At very low energies where m1c2>>P1c, the velocity reduces to: • At very high energies where m1c2<<P1c and m2c2<<P1c , the velocity can be written as: PHYS 3446, Fall 2008 Andrew Brandt

More Related