- #1

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$$\nu = \nu_0+F<r^2>+G(<r^2>)^2+...$$

where ##\nu_0## is the transition assuming a point nucleus, ##<r^2>## is the mean square nuclear radius and F and G are some parameters that depend on the electronic transition. Usually in literature (except for the super sensitive measurements), the higher order terms are dropped and only the F term is kept (this is how the so called King linearity of the isotope shift is obtained). I am not sure why we can drop the higher order terms. Let's say that ##<r^2> = 5fm^2##, which is a reasonable value. Then ##<r^2>^2 = 25fm^4##. I guess I am confused about the units. If we write this in terms of meters, the second term is smaller by a factor of ##10^{-30}##, but can't we redefine units and treat 1 fm as the unit and in this case the ##<r^2>^2## term would be actually bigger? In QED, for example, the expansion is in terms of ##\alpha##, which is unitless, so there is no confusion there. But here I am a bit confused. Can someone help me understand it? Thank you!