PHYS 3446 – Lecture #4

1. PHYS 3446 – Lecture #4 Monday, Sep. 20 2010 Dr. Brandt • Differential Cross Section of Rutherford Scattering • Measurement of Cross Sections • Lab Frame and Center of Mass Frame • Relativistic Variables • ***HW now due Weds • ***Labs due at next lab • ***Everyone should go to lab Friday 24 and Fri Oct. 1 weeks as we will have a half lecture before each lab PHYS 3446, Fall 2010 Andrew Brandt

2. Assignment #2 Due Weds. Sep. 22 • Plot the differential cross section of the Rutherford scattering as a function of the scattering angle q for three sensible choices of the lower limit of the angle. (use ZAu=79, Zhe=2, E=10keV). • Compute the total cross section of the Rutherford scattering in unit of barns for your cut-off angles. • Find a plot of a cross section from a current HEP experiment, and write a few sentences about what is being measured. • Book problem 1.10 *New Problem* • 5 points extra credit if you are one of first 10 people to email an electronic version of a figure showing Rutherford 1/sin^4(x/2) angular dependence PHYS 3446, Fall 2010 Andrew Brandt

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? • 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 2010 Andrew Brandt

4. Total X-Section of Rutherford Scattering • To obtain the total cross section of Rutherford scattering, one integrates the differential cross section over all q: • What is the result of this integration? • Infinity!! • Does this make sense? • Yes • Why? • Since the Coulomb force’s range is infinite (particle with very large impact parameter still contributes to integral through very small scattering angle) • What would be the sensible thing to do? • Integrate to a cut-off angle since after certain distance the force is too weak to impact the scattering. (q=q0>0); note this is sensible since alpha particles far away don’t even see charge of nucleus due to screening effects. PHYS 3446, Fall 2010 Andrew Brandt

5. Measuring Cross Sections • With the flux of N0 per unit area per second • Any a particles in range b to b+db will be scattered into q to q-dq • The telescope aperture limits the measurable area to • How could they have increased the rate of measurement? • By constructing an annular telescope • By how much would it increase? 2p/df PHYS 3446, Fall 2010 Andrew Brandt

6. Measuring Cross Sections • Fraction of incident particles (N0) approaching the target in the small area Ds=bdfdb at impact parameter b is dn/N0. • so dn particles scatter into R2dW, the aperture of the telescope • This fraction is the same as • The sum of Ds over all N nuclear centers throughout the foil divided by the total area (S) of the foil. • Or, in other words, the probability for incident particles to enter within the N areas divided by the probability of hitting the foil. This ratio can be expressed as Eq. 1.39 PHYS 3446, Fall 2010 Andrew Brandt

7. Measuring Cross Sections A0: Avogadro’s number of atoms per mole • For a foil with thickness t, mass density r, atomic weight A: • Since from what we have learned previously • The number of a scattered into the detector angle (q,f) is Eq. 1.40 PHYS 3446, Fall 2010 Andrew Brandt

8. Detector acceptance Measuring Cross Sections Number of detected particles/sec Projectile particle flux Density of the target particles Scattering cross section • This is a general expression for any scattering process, independent of the theory • This gives an observed counts per second PHYS 3446, Fall 2010 Andrew Brandt

9. 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 2010 Andrew Brandt

10. 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 2010 Andrew Brandt

11. 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 2010 Andrew Brandt

12. 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 quiz! PHYS 3446, Fall 2010 Andrew Brandt

13. 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. • 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 2010 Andrew Brandt

14. Relationship of variables in Lab and CM CM • The speed of CM in lab frame is • Speeds of the particles in CM frame are • The momenta of the two particles are equal and opposite!! and PHYS 3446, Fall 2010 Andrew Brandt

15. 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 scattered particle with m1 in lab and CM are: • The perpendicular components of the velocities are: (boost is only in the z direction) • Thus, the angles are related (for elastic scattering only) as: PHYS 3446, Fall 2010 Andrew Brandt