Presentation on theme: "Lecture 5: Electron Scattering, continued... 18/9/2003 1"— Presentation transcript:
1 16.451 Lecture 5: Electron Scattering, continued... 18/9/2003 1 Details, Part III: KinematicsprotonElectrons are relativistic kinetic energy K ≠ p2/2m ...Einstein mass-energy relations:(total energy E, rest mass m)Problem: units are awkward, too many factors of c ...Notice that if c=1 then (E, m, p, K) all have the same units!
2 right”, the factor c has to be absorbed in the units of p and m: High Energy Units:2If we set c = 1 in Einstein’s mass-energy relations, then in order to “get the answerright”, the factor c has to be absorbed in the units of p and m:Let the symbol “[ ]” mean “the units of”, and then it follows that:(Frequently, physicists set c = 1 and quote mass and/or momentum in “GeV” units, asin the graph of the proton electric form factor, lecture 4. This is just a form ofshorthand – they really mean GeV/c for momentum and GeV/c2 for mass.... numericallythese have the same value because the value of c is in the unit – we don’t divide bythe numerical value 3.00x108 m/s or the answer would be ridiculously small (wrong!)
3 From lecture 4: proton electric form factor data 2a) 4 – momentum transfer: Q2Ref: Arnold et al., Phys. Rev. Lett. 57, 174 (1986)(Inverse Fourier transform gives the electric charge density (r))
4 write down wave functions to describe the initial and final states.... More units....3When we want to describe a scattering problem in quantum mechanics, we have towrite down wave functions to describe the initial and final states....For example, the incoming electron isa free particle of momentum:The electron wave function is:where V is a normalization volume.protonIf we set ħ = 1, then momentum p and wave number k have the same units, e.g. fm-1;to convert, use the factor:Example:An electron beam with total energy E = 5 GeV has momentum p = 5 GeV/c (m << E) ...the same momentum is equivalent to 5 GeV/(0.197 GeV.fm) or p = 25 fm-1.So, p = 5 GeV, 5 GeV/c, and 25 fm-1 all refer to the same momentum!
5 Analysis: Kinematics of electron scattering 4 Note: Elastic scattering is the relevant case for our purposes here. This means thatthe beam interacts with the target proton with no internal energy transfer.protonSpecify total energy and momentum for the incoming and outgoing particles as shown.Electron mass m << Eo. Proton mass is M.Conserve total energy and momentum:Next steps: find the scattered electron momentum p’ in terms of the incidentmomentum and the scattering angle. Also, find the momentum transfer q2 as afunction of scattering angle, because q2 will turn out to be an important variablethat our analysis of the scattering depends on ...
6 use kinematic relations to substitute for K = (po – p’) and q2: Details ...5protonconserve momentum:now use conservation of energy with W = M + K for the proton; E = p for electrons:use kinematic relations to substitute for K = (po – p’) and q2:Note: these solutions 1), 2) are perfectlygeneral as long as the electron isrelativistic. The target can be anything!
8 Relativistic 4 - momentum: 7It is often convenient to use 4 – vector quantities to work out reaction kinematics.There are several conventions in this business, all of them giving the same answerbut via a slightly different calculation. We will follow the treatment outlined inPerkins, `Introduction to High Energy Physics’, Addison-Wesley (3rd Ed., 1987).Define the relativistic 4-momentum:The length of any 4 – vector is the same in all reference frames:`length’ squaredFor completeness, a Lorentz boostcorresponding to a relative velocity along the x-axis is accomplished bythe 4x4 “rotation” matrix , with:
9 Analysis via 4 – momentum: 8 protonDefine: momentum transfer Q between incoming and outgoing electrons:Since Q is a 4 – vector, the square of its length is invariant:Expand, simplify, remembering to use p2 – E2 = - m2 and m << E,p ...
10 Four-momentum versus 3-momentum, or Q2 versus q2 for e-p scattering 94–momentum transfer squaredis the variable used to plot highenergy electron scattering data:For a nonrelativistic quantum treatment of the scattering process (next topic!) the form factor is expressed in terms of the 3-momentum transfer (squared):p’Q2q2GeV orGeV25 GeV beamhuge difference for the proton!significant variationwith angle.
11 q2 Q2 p’ What happens if the target is a nucleus? (Krane, ch. 3) 10 The difference between numerical values of Q2 and q2 decreases as the mass of thetarget increases we can “get away” with a 3-momentum description (easier) toderive the cross section for scattering from a nucleus. Note also the simplificationthat p’ po and becomes essentially independent of as the mass of the targetincreases (Why? as M ∞, the electron beam just ‘reflects’ off the target –just like an elastic collision of a ping pong ball with the floor – !!)M = 16M = 1005 GeV electron beam in both cases, as before, but the target mass increasedfrom a proton to a nucleus (e.g., 16O, 100Ru )p’q2Q2
12 Details, Part IV: electron scattering cross section and form factor 11 Recall from lecture 4:protondetectorBeforeAfterExperimenters detect elastically scattered electrons and measure the cross section:point charge result (known)“Form factor” gives the Fouriertransform of the extended targetcharge distribution. Strictly correct forheavy nuclei: same idea but slightly morecomplicated expression for the proton...Last job: we want to work out anexpression for the scattering crosssection to see how it relates to thestructure of the target object.
13 Scattering formalism: (nonrel. Q.M.) 12Basic idea:The scattering process involves a transition between an initial quantum state:|i = incoming e-, target p and a final state |f = scattered e-, recoil p .The transition rate if can be calculated from“Fermi’s Golden Rule”, a basic prescription inquantum mechanics: (ch. 2)Units: s-1where the `matrix element Mif’ is given by:The potential V(r) represents the interaction responsible for the transition,in this case electromagnetism (Coulomb’s law!),and the `density of states’ f is a measure of the number ofequivalent final states per unit energy interval – the more statesavailable at the same energy, the faster the transition occurs.This formalism applies equally well to scattering and decay processes!We will also use it to analyze and decay later in the course...
14 V = A c dt Relation of transition rate if to d/d: 13 Recall: A beam particle will scatter from the target particle into solid angle d at (, ) if it approaches within the corresponding area d = (d/d) d centered on the target.V = A c dtElectron (speed c) is in a plane wave state normalized in volume V as shown.Probability of scattering at angle is given by the ratio of areas:Transition rate = (electrons/Volume) x (Volume/time) x P()Next time: we will put this all together and calculate the scattering cross section...