Analytic formulas for the D -mode Robinson instability

The passive superconducting harmonic cavity (PSHC) scheme is adopted by several existing and future synchrotron light source storage rings, as it has a relatively smaller <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>R</a:mi><a:mo>/</a:mo><a:mi>Q</a:mi></a:math> and a relatively larger quality factor (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Q</c:mi></c:math>), which can effectively reduce the beam-loading effect and suppress the mode-one instability. By using a minimum search algorithm to solve the mode-zero Robinson instability equation of uniformly filled rigid bunches, we have revealed that the fundamental mode of PSHC with a large loaded <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Q</e:mi></e:math> possibly triggers the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>D</g:mi></g:math>-mode Robinson instability [T. He , Mode-zero Robinson instability in the presence of passive superconducting harmonic cavities, ]. This instability is a mode-zero coupled bunch instability, with an oscillation frequency close to the PSHC detuning (<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi></i:math>-mode). Uniquely, it is anti-damped by the radiation damping effect. In this paper, we derive analytical formulas for the frequency and growth rate of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>D</k:mi></k:math>-mode Robinson instability by taking several appropriate approximations. These formulas provide crucial insights for analyzing and understanding the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>D</m:mi></m:math>-mode Robinson instability.