P. Piot, PHYS 571 – Fall 2007Scattering - 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 2007Energy 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 2007Energy 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)incidenttargetP. Piot, PHYS 571 – Fall 2007Energy 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 isP. Piot, PHYS 571 – Fall 2007Energy transfer: the impulse approximation• The associated kinetic energy change• The electrons• For nuclei• So we haveNRP. Piot, PHYS 571 – Fall 2007Energy transfer: the impulse approximation• Let’s what are the implication of the IA– Deflection angle is given byso – 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 isP. Piot, PHYS 571 – Fall 2007Energy 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 1stcondition is fulfilled and IA is legitimate• So for IA to be valid we need• Which can also be written (β<1)P. Piot, PHYS 571 – Fall 2007NR approximation• NR approximation implies• This is the SAME condition as for IA to be valid but just with γ →1P. Piot, PHYS 571 – Fall 2007Passage 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 neto this block of matter• The total number of electronin a cylindrical shell or radius b isP. Piot, PHYS 571 – Fall 2007Passage through a bulk of matter• The differential energy loss by the charge q is • Integrate over bP. Piot, PHYS 571 – Fall 2007Passage 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 sureP. Piot, PHYS 571 – Fall 2007Passage through a bulk of matter• So finally•Sincewe have• Compare to Beth