P. Piot, PHYS 571 – Fall 2007Power radiated in linear accelerators 1• In linear accelerators• We need to evaluate the acceleration. Start from the momentum • Thus the radiated power is Lighter particles are subject to higher lossP. Piot, PHYS 571 – Fall 2007Power radiated in linear accelerators 2• One important question is how does the emission of radiation influence the charge particle dynamics. • The accelerator induce a momentum change of the form(where we assumed the acceleration is along the z-axis)• Let be the power associated to the external force. The particle dynamics is affected when Pextis comparable to the radiated power:P. Piot, PHYS 571 – Fall 2007Power radiated in linear accelerators 3• Consider a relativistic electron then•…..• So the effect seems to be negligible. • This is actually part of the story some coherent effect can kick in an induce some distortion when considering highly charged electron bunches for instance…P. Piot, PHYS 571 – Fall 2007Power radiated in circular accelerators 1• Now and • The radiated power is where E is the energy. Let’s introduce• So radiative energy loss per turn isP. Piot, PHYS 571 – Fall 2007Power radiated in circular accelerators 2• That is • For an e- synchrotron this becomes•TakeE=1 TeV, R=2 km we have• Conclusion:– bad idea to build electron circular accelerator for HEP – but good as copious radiation sources (e.g. APS in Argonne).P. Piot, PHYS 571 – Fall 2007Angular distribution of radiation emitted by an accelerated charge• Starting from the radiation field, we havewhere we usedP. Piot, PHYS 571 – Fall 2007Angular distribution for linear motion 1• Introducing θ, we have:• And the numerator becomes• So the radiated power writesP. Piot, PHYS 571 – Fall 2007Angular distribution for linear motion 2• The power distribution has maxima given by•With solutions•Only cosθ+is possible so:P. Piot, PHYS 571 – Fall 2007Angular distribution for linear motion 3P. Piot, PHYS 571 – Fall 2007Angular distribution for linear motion 4• Small angle approxi-mation for ultra-relativisticcase:JDJ equation 14.41P. Piot, PHYS 571 – Fall 2007Angular distribution for circular motion 1• We have:• That is• Which gives:• In the ultra-relativistic limit (small angle approximation):P. Piot, PHYS 571 – Fall 2007Angular distribution for circular motion 2• Note that • So we
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