P. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• Expliciting the E and B field in the latter equation gives• First derived by Enrico Fermi. Energy loss occurs if eitherλ or ε are complexP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• We now introduce a simple model for the dielectric permittivity• Consider the electron to be bounded to the nuclei via a damped harmonic oscillator type force• Then the polarization is defined asExternal fieldDamping term“Natural oscillation” frequencyP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• So the electric permittivity can be written as• Where is the plasma frequency• If we explicit this form of ε(ω)in the energy loss equation and perform the integral…• Not trivial, need make a “narrow band resonance” approximationP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• Which leads to• Also assume bλ<<1 that is b< atomic radius• Using the small argument approximation for the modified Bessel functions gives•wherebλP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• The energy loss for our model for ε(ω)is•where•Explicit ε(ω)givesP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• We need to perform the integral. This is done in the Complex plane• Two sources of poles• Consider the path integral along the contour C, we have:I1+I2+I3=0• Note that CP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• Start with evaluating the integral• Introduce• then CP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• The brackets simplifies to•And finally • I3is real so iI3is imaginary so this integral has NO contribution to the energy lossP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• Start with evaluating the integral I2• introduce• Then•Taking the limit R→∞ givesCP. Piot, PHYS 571 – Fall 2007Energy loss in a dielectric• So finally the energy loss is• Compare with our initial derivation without dielectric screening and use the impulse approximation••Influence of dielectric screening is twoInfluence of dielectric screening is two--foldsfolds:– It removes the energy loss dependence on atomic structure ω0replaced by ωp– It reduces the dependence on γ (γ in the ln argument is