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PHYS 3446 – Lecture #4

PHYS 3446 – Lecture #4. Monday, Sept. 11, 2006 Dr. Jae Yu. Lab Frame and Center of Mass Frame Relativistic Treatment Feynman Diagram Invariant kinematic variables. Announcements. I now have 9 on the distribution list. Only three more to go!! Whoopie!!

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PHYS 3446 – Lecture #4

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  1. PHYS 3446 – Lecture #4 Monday, Sept. 11, 2006 Dr. JaeYu • Lab Frame and Center of Mass Frame • Relativistic Treatment • Feynman Diagram • Invariant kinematic variables PHYS 3446, Fall 2006 Jae Yu

  2. Announcements • I now have 9 on the distribution list. • Only three more to go!! Whoopie!! • Please come and check to see if your request has been successful • I will send a test message this week • No lecture this Wednesday but • The class time should be devoted for your project group meetings in preparation for the workshop PHYS 3446, Fall 2006 Jae Yu

  3. 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? • Measurement of yield as a function of specific experimental variable • This is equivalent to measuring the probability of certain process in a specific kinematic phase space • Cross sections are measured in the unit of barns: PHYS 3446, Fall 2006 Jae Yu

  4. Lab Frame and Center of Mass Frame • We assumed that the target nuclei do not move through the collision in Rutherford Scattering • In reality, they recoil as a result of scattering • Sometimes we use two beams of particles for scattering experiments (target is moving) • This situation could be complicated but.. • If the motion can be described in the Center of Mass frame under a central potential, it can be simplified PHYS 3446, Fall 2006 Jae Yu

  5. 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 2006 Jae Yu

  6. 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 2006 Jae Yu

  7. Now with some simple arithmatics • From the equations of motion, we obtain • Since the momentum of the system is conserved: • Rearranging the terms, we obtain PHYS 3446, Fall 2006 Jae Yu

  8. 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 as if that of a fictitious particle with mass m (the reduced mass) and coordinate r. • In the frame where 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 2006 Jae Yu

  9. Relationship of variables in Lab and CMS CMS • The speed of CM is • Speeds of the particles in CMS are • The momenta of the two particles are equal and opposite!! and PHYS 3446, Fall 2006 Jae Yu

  10. Scattering angles in Lab and CMS • qCM represents the changes in the direction of the relative position vector r as a result of the collision in CMS • Thus, it must be identical to the scattering angle for the particle with the reduced mass, m. • Z components of the velocities of particle with m1 in lab and CMS are: • The perpendicular components of the velocities are: • Thus, the angles are related, for elastic scattering only, as: PHYS 3446, Fall 2006 Jae Yu

  11. Differential cross sections in Lab and CMS • 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: reorganize Using Eq. 1.53 PHYS 3446, Fall 2006 Jae Yu

  12. 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 2006 Jae Yu

  13. 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 2006 Jae Yu

  14. 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 2006 Jae Yu

  15. Relativistic Variables • For high energies, if m1~m2, • gCM becomes: • And • Thus gCM becomes Invariant Scalar: s PHYS 3446, Fall 2006 Jae Yu

  16. Relativistic Variables • The invariant scalar, s, is defined as: • So what is this the CMS frame? • Thus, represents the total available energy in the CMS 0 PHYS 3446, Fall 2006 Jae Yu

  17. Useful Invariant Scalar Variables • Another invariant scalar, t, the momentum transfer (differences in four momenta), is useful for scattering: • Since momentum and total energy are conserved in all collisions, t can be expressed in terms of target variables • In CMS frame for an elastic scattering, where PiCM=PfCM=PCM and EiCM=EfCM: PHYS 3446, Fall 2006 Jae Yu

  18. Feynman Diagram • The variable t is always negative for an elastic scattering • The variable t could be viewed as the square of the invariant mass of a particle with and exchanged in the scattering • While the virtual particle cannot be detected in the scattering, the consequence of its exchange can be calculated and observed!!! • A virtual particle is a particle whose mass is different than the rest mass t-channel diagram Momentum of the carrier is the difference between the two particles. Time PHYS 3446, Fall 2006 Jae Yu

  19. Useful Invariant Scalar Variables • For convenience we define a variable q2, • In the lab frame, , thus we obtain: • In the non-relativistic limit: • q2 represents “hardness of the collision”. Small qCM corresponds to small q2. PHYS 3446, Fall 2006 Jae Yu

  20. Relativistic Scattering Angles in Lab and CMS • For a relativistic scattering, the relationship between the scattering angles in Lab and CMS is: • For Rutherford scattering (m=m1<<m2, v~v0<<c): • Divergence at q2~0, a characteristics of a Coulomb field Resulting in a cross section PHYS 3446, Fall 2006 Jae Yu

  21. Assignments • Extension of previous w/ the following (due 9/18): • Plot the differential cross section of the Rutherford scattering as a function of the scattering angle q with some sensible lower limit of the angle + express your opinion on the sensibility of the cross section, along with good physical reasons • Derive Eq. 1.55 starting from 1.48 and 1.49 • Derive the formulae for the available CMS energy ( ) for • Fixed target experiment with masses m1 and m2 with incoming energy E1. • Collider experiment with masses m1 and m2 with incoming energies E1 and E2. • Reading assignment: Section 1.7 • End of chapter problem 1.7 • These assignments are due next Wednesday, Sept. 20. PHYS 3446, Fall 2006 Jae Yu

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