Why π and mrad?

Becker, R.; Herrmannsfeldt, W. B.
March 2006
Review of Scientific Instruments;Mar2006, Vol. 77 Issue 3, p03B907
Academic Journal
Today it is generally accepted in the ion source and accelerator communities to communicate emittance data with units of π mm mrad or π cm mrad. The emittance of a beam with 2 cm diameter or width and an opening angle of 200 mrad then is written as E=100 π cm mrad. This example shows that π has not been involved when calculating the half-axis’ product which is the numerical definition for the emittance. Why should we add π to the dimensions? We will explain that this convention is not at all a logical one and is confusing to all those who are entering the field as freshmen or just do not want to become experts by adjusting to strange rules. We therefore propose to skip the “π” in the dimension and to add characters as sub- or superscripts to the variable E, which describe the specific kind of emittance formulation used, e.g., Er,r′, Ex,x′, Erms, E4*rms, Enorm, Earea, and Eellipse. In a real case, such an emittance naming could appear as Ex,x′,4rmsnorm. Additionally—to be consistent with cleaning up—the dimension of emittances should be given in centimeters or meters, because rad are ratios and free of dimension. Therefore, instead of using cm mrad it is much more logical to communicate emittance data by writing [variant_greek_epsilon] m, where [variant_greek_epsilon] is the product of the maximum value for half of the beam thickness in meters and its opening angle in rad.


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