Energy loss in a dielectric Expliciting the E and B field in the latter equation gives First derived by Enrico Fermi Energy loss occurs if either or are complex P Piot PHYS 571 Fall 2007 Energy 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 External field Damping term Natural oscillation frequency Then the polarization is defined as P Piot PHYS 571 Fall 2007 Energy loss in a dielectric So the electric permittivity can be written as Where is the plasma frequency If we explicit this form of in the energy loss equation and perform the integral Not trivial need make a narrow band resonance approximation P Piot PHYS 571 Fall 2007 Energy loss in a dielectric Which leads to Also assume b 1 that is b atomic radius Using the small argument approximation for the modified Bessel functions gives b where P Piot PHYS 571 Fall 2007 Energy loss in a dielectric The energy loss for our model for is where Explicit gives P Piot PHYS 571 Fall 2007 Energy loss in a dielectric We need to perform the integral This is done in the Complex plane Two sources of poles C Consider the path integral along the contour C we have I1 I2 I3 0 Note that P Piot PHYS 571 Fall 2007 Energy loss in a dielectric Start with evaluating the integral C Introduce then P Piot PHYS 571 Fall 2007 Energy loss in a dielectric The brackets simplifies to And finally I3 is real so iI3 is imaginary so this integral has NO contribution to the energy loss P Piot PHYS 571 Fall 2007 Energy loss in a dielectric Start with evaluating the integral I2 introduce C Then Taking the limit R gives P Piot PHYS 571 Fall 2007 Energy 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 two folds folds It removes the energy loss dependence on atomic structure 0 replaced by p It reduces the dependence on in the ln argument is gone P Piot PHYS 571 Fall 2007