P. Piot, PHYS 571 – Fall 2007. Power radiated in linear accelerators 1. • In linear
accelerators. • We need to evaluate the acceleration. Start from the momentum.
Power 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 loss
P. Piot, PHYS 571 – Fall 2007
Power 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 Pext is comparable to the radiated power:
P. Piot, PHYS 571 – Fall 2007
Power 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 2007
Power 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 is
P. Piot, PHYS 571 – Fall 2007
Power radiated in circular accelerators 2 • That is
• For an e- synchrotron this becomes
• Take E=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 2007
Angular distribution of radiation emitted by an accelerated charge • Starting from the radiation field, we have
where we used
P. Piot, PHYS 571 – Fall 2007
Angular distribution for linear motion 1 • Introducing θ, we have:
• And the numerator becomes
• So the radiated power writes
P. Piot, PHYS 571 – Fall 2007
Angular distribution for linear motion 2 • The power distribution has maxima given by
• With solutions • Only cos θ+ is possible so:
P. Piot, PHYS 571 – Fall 2007
Angular distribution for linear motion 3
P. Piot, PHYS 571 – Fall 2007
Angular distribution for linear motion 4 • Small angle approximation for ultra-relativistic case:
JDJ equation 14.41
P. Piot, PHYS 571 – Fall 2007
Angular distribution for circular motion 1 • We have:
• That is
• Which gives:
• In the ultra-relativistic limit (small angle approximation):
P. Piot, PHYS 571 – Fall 2007
Angular distribution for circular motion 2 • Note that
