Scattering Introduction We will consider two types of scattering Scattering on electron q e me 9 110 31 kg which results in high energy losses but small deflections Scattering on nuclei q Ze mn me which are associated to low energy losses but large deflections Naively since matter is composed of much more electron that nuclei a factor Z we may conjecture that electron scattering is the dominant type of scattering P Piot PHYS 571 Fall 2007 Energy transfer We now are going to compute the energy transfer between two particle during scattering The technique is imply to consider the matter particle at rest and the particle to be scattered moving and penetrating the matter block Scattering is not a point like collision it occur via long range electromagnetic interaction considering the e m field of the moving particle P Piot PHYS 571 Fall 2007 Energy transfer the impulse approximation Calculation of energy transfer in the most general case can be tedious So we make some simplifying assumptions Incident particle is NOT locally deflected by collision rather a momentum kick is imparted and as the particle drifts away might be deflected Target particle is stationary during collision These two assumptions are part of the impulse approximation IA incident target P Piot PHYS 571 Fall 2007 Energy transfer the impulse approximation The E field generated by the incident particle at the location of the target particle is The momentum transfer from q to e is P Piot PHYS 571 Fall 2007 Energy transfer the impulse approximation The associated kinetic energy change NR The electrons For nuclei So we have P Piot PHYS 571 Fall 2007 Energy transfer the impulse approximation Let s what are the implication of the IA Deflection angle is given by so Target is stationary means that recoil of the target during collision is much smaller than impact parameter d b Interaction time during collision given by The corresponding recoil is P Piot PHYS 571 Fall 2007 Energy transfer the impulse approximation This is a stronger condition than the small deflection angle condition by a factor M m so if the latter condition is fulfilled then the 1st condition is fulfilled and IA is legitimate So for IA to be valid we need Which can also be written 1 P Piot PHYS 571 Fall 2007 NR approximation NR approximation implies This is the SAME condition as for IA to be valid but just with 1 P Piot PHYS 571 Fall 2007 Passage through a bulk of matter Now we generalize our treatment to the case of a particle passing through a bulk of matter many electrons We associate a electronic density ne to this block of matter The total number of electron in a cylindrical shell or radius b is P Piot PHYS 571 Fall 2007 Passage through a bulk of matter The differential energy loss by the charge q is Integrate over b P Piot PHYS 571 Fall 2007 Passage through a bulk of matter Limit of the integral When b goes to zero IA is no more valid so we must limit our integral to values such that that is When b goes to infinity the stationary condition breaks electron orbit with an angular frequency so we must make sure P Piot PHYS 571 Fall 2007 Passage through a bulk of matter So finally Since we have Compare to Beth 1915 P Piot PHYS 571 Fall 2007