Understanding Rutherford Scattering: Cross Sections & Differential Analysis

1. PHYS 3446 – Lecture #3 Wednesday, Sept. 3, 2008 Dr. Andrew Brandt • Rutherford Scattering with Coulomb force • Scattering Cross Section • Differential Cross Section of Rutherford Scattering • Measurement of Cross Sections • ***Don’t forget radiation training!!! • ***Please turn in HW PHYS 3446, Fall 2008 Andrew Brandt

2. Rutherford Scattering • From the solution for b, we can learn the following • For fixed b, E and Z’ • The scattering is larger for a larger value of Z. • Since Coulomb potential is stronger with larger Z. • For fixed b, Z and Z’ • The scattering angle is larger when E is smaller. • Since the speed of the low energy particle is smaller • The particle spends more time in the potential, suffering greater deflection • For fixed Z, Z’, and E • The scattering angle is larger for smaller impact parameter b • Since the closer the incident particle is to the nucleus, the stronger Coulomb force it feels PHYS 3446, Fall 2008 Andrew Brandt

3. What do we learn from scattering? • Scattering of a particle in a potentialis completely determined when we know both • The impact parameter, b, and • The energy of the incident particle, E • For a fixed energy, the deflection is defined by • The impact parameter, b. • What do we need to perform a scattering experiment? • Incident flux of beam particles with known E • Device that can measure number of scattered particles at various angle, q. • Measurements of the number of scattered particles reflect • Impact parameters of the incident particles • The effective size of the scattering center • By measuring the scattering angle q, we can learn about the potential or the forces between the target and the projectile PHYS 3446, Fall 2008 Andrew Brandt

4. Scattering Cross Section All these land here • N0: The number of particles incident on the target foil per unit area per unit time. • Any incident particles entering with impact parameter b and b+db will scatter to the angle q and q-dq. • In other words, they scatter into the solid angle dW (=2psinqdq). • So the number of particles scattered into the solid angle dW per unit time is 2pN0bdb. • Note:have assumed thin foil, and large separation between nuclei—why? PHYS 3446, Fall 2008 Andrew Brandt

5. Scattering Cross Section • For a central potential • Such as Coulomb potential • Which has spherical symmetry • The scattering center presents an effective transverse cross-sectional area of • For the particles to scatter into q and q+dq PHYS 3446, Fall 2008 Andrew Brandt

6. Scattering Cross Section • In more generalized cases, Ds depends on both q & f. • With a spherical symmetry, f can be integrated out: Why negative? Since the deflection and change of b are in opposite direction!! What is the dimension of the differential cross section? Differential Cross Section reorganize Area!! PHYS 3446, Fall 2008 Andrew Brandt

7. femptobarn Scattering Cross Section • For a central potential, measuring the yield as a function of q (the 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 variables • This is equivalent to measuring the probability of occurrence of a physical process in a specific kinematic phase space • Cross sections are measured in the unit of barns: Where does this come from? picobarn nanobarn Cross sectional area of a typical nucleus! PHYS 3446, Fall 2008 Andrew Brandt

8. Total Cross Section • Total cross section is the integration of the differential cross section over the entire solid angle, W: • Total cross section represents the effective size of the scattering center integrated over all possible impact parameters (and consequently all possible scattering angles) PHYS 3446, Fall 2008 Andrew Brandt

9. Cross Section of Rutherford Scattering • The impact parameter in Rutherford scattering is • Thus, • Differential cross section of Rutherford scattering is what happened? can you say “trig identity?” sin(2x)=2sin(x) cos(x) plot/applets PHYS 3446, Fall 2008 Andrew Brandt

10. Rutherford Scattering Cross Section • Let’s plug in the numbers • ZAu=79 • ZHe=2 • For E=10keV • Differential cross section of Rutherford scattering PHYS 3446, Fall 2008 Andrew Brandt

11. 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 2008 Andrew Brandt

12. Measuring Cross Sections • Rutherford scattering experiment • Used a collimated beam of a particles emitted from Radon • A thin Au foil target • A scintillating glass screen with ZnS phosphor deposit • Telescope to view limited area of solid angle • Telescope only needs to move along q not f. Why? • Due to the spherical symmetry, scattering only depends on q not f. PHYS 3446, Fall 2008 Andrew Brandt

13. 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 2008 Andrew Brandt

14. 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 2008 Andrew Brandt

15. Measuring Cross Sections A0: Avogadro’s number of atoms per mol • 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 2008 Andrew Brandt

16. Detector acceptance Measuring Cross Sections Number of detected particles 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 2008 Andrew Brandt

17. Example Cross Section: Jet +X • Inclusive jet production cross section as a function of transverse energy triumph of QCD what if there were an excess at high energy? PHYS 3446, Fall 2008 Andrew Brandt

18. Assignment #2 • 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 • Compute the total cross section of the Rutherford scattering in unit of barns for your cut-off angles • 5 points extra credit if you are one of first three people to email me an electronic version of a figure showing Rutherford 1/sin^4(x/2) angular dependence. • Assignment # 2 is due Wednesday, Sept. 10 • Must have rad training done by then as well PHYS 3446, Fall 2008 Andrew Brandt