Power radiated in linear accelerators 1

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

• 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 approxi-mation for ultra-relativisticcase:

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

• So we have