Abstract
This discussion revisits two articles on synchrotron radiation damping published in 1958, one by this author and Evgeny K. Tarasov [Zh. Eksp. Teor. Fiz. 34, 651 (1958); Sov. Phys. JETP 34, 449 (1958)], and one by Kenneth W. Robinson [Phys. Rev. 111, 373 (1958)]. The latter is the source of what is known as ``Robinson's sum rule.'' Both present the familiar rule, but with very different proofs and calculations of concrete damping decrements. Comparative analysis of these differences reveals serious flaws in Robinson's proof and calculations.
Highlights
AND OVERVIEWWe revisit here two articles on synchrotron radiation damping published in 1958
Because he does not correctly take into account the off-diagonal term in (3), he fails to prove his central claim [ [2], Eq (6)] that D 1⁄4 1 À 4hP =EiT, where À4ðP =EÞdt is the sum of the diagonal terms in (3) and (4). (Whether this formula for D is true is irrelevant to the validity of his proof.). Another problem with his proof turns on the incorrect idea that one can ignore off-diagonal terms when multiplying an infinite number of elementary matrices. This idea automatically excludes from consideration the physically possible parametric resonances caused by the variation of radiation power along the orbit in a general, nonisomagnetic ring
We see first that the final matrix contains a diagonal term of the first order in V; second, this term appears despite the fact that none of the elementary matrices contains it at the diagonal
Summary
We revisit here two articles on synchrotron radiation damping published in 1958. The first was written by myself with the late Evgeny K. Another problem with his proof turns on the incorrect idea that one can ignore off-diagonal terms when multiplying an infinite number of elementary matrices This idea automatically excludes from consideration the physically possible parametric resonances caused by the variation of radiation power along the orbit in a general, nonisomagnetic ring. With respect to nonisomagnetic rings Robinson errs in calculating changes of the x; z coupling necessary to cancel antidamping (in the absence of x; y coupling) He fails to notice a correlation—a sort of resonance, again—between radiation power and the ring parameters in nonisomagnetic rings, separately averaging one of those parameters—namely, x=R in (3). As [1] proves, this cannot be done [see Eq (23) below]
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