Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 1 PHYS 3446 – Lecture #4 Monday, Sep Dr. Brandt 1.Differential Cross Section of Rutherford Scattering 2.Measurement of Cross Sections 3.Lab Frame and Center of Mass Frame 4.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

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 2 Assignment #2 Due Weds. Sep Plot the differential cross section of the Rutherford scattering as a function of the scattering angle  for three sensible choices of the lower limit of the angle. (use Z Au =79, Z he =2, E=10keV). 2.Compute the total cross section of the Rutherford scattering in unit of barns for your cut-off angles. 3.Find a plot of a cross section from a current HEP experiment, and write a few sentences about what is being measured. 4.Book problem 1.10 *New Problem* 5 points extra credit if you are one of first 10 people to an electronic version of a figure showing Rutherford 1/sin^4(x/2) angular dependence

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 3 Scattering Cross Section For a central potential, measuring the yield as a function of , 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:

Monday, Sep. 20, 2010PHYS 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  : What is the result of this integration? –I–Infinity!! Does this make sense? –Y–Yes Why? –S–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? –I–Integrate to a cut-off angle since after certain distance the force is too weak to impact the scattering. (  0 >0); note this is sensible since alpha particles far away don’t even see charge of nucleus due to screening effects.

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 5 Measuring Cross Sections With the flux of N0 N0 per unit area per second Any  particles in range b to b+db will be scattered into  to  -d  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? 2  /d 

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 6 Measuring Cross Sections Fraction of incident particles (N 0 ) approaching the target in the small area  =bd  db at impact parameter b is dn/N 0. –so dn particles scatter into R 2 d , the aperture of the telescope This fraction is the same as –The sum of  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

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 7 Measuring Cross Sections For a foil with thickness t, mass density , atomic weight A: Since from what we have learned previously The number of  scattered into the detector angle (  ) is A 0 : Avogadro’s number of atoms per mole Eq. 1.40

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 8 Measuring Cross Sections This is a general expression for any scattering process, independent of the theory This gives an observed counts per second Number of detected particles/sec Projectile particle flux Density of the target particles Scattering cross section Detector acceptance

Monday, Sep. 20, 2010PHYS 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)

Monday, Sep. 20, 2010PHYS 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

Monday, Sep. 20, 2010PHYS 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

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 12 Lab Frame and CM Frame From the equations in previous slides and 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 Thus Reduced Mass quiz!

Monday, Sep. 20, 2010PHYS 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  (the reduced mass) and coordinate r.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.

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 14 Relationship of variables in Lab and 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!! CM and

Monday, Sep. 20, 2010PHYS 3446, Fall 2010 Andrew Brandt 15  CM 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, .. Z components of the velocities of scattered particle with m1m1 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: Scattering angles in Lab and CM