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Simple Expression For Minimum Emittance With Linearly Varied Bending Radius In Dipole Magnets

We study the theoretical minimum emittance for a non-uniform bending magnet with the bending radius linearly ramped from the dipole center to its end. We derive the expression for the minimum emittance as a function of the bending angle and expand it into a power series with respect to a small angle. The first term of the expansion gives the TME minimum emittance while the high-order terms are responsible for its modification. On the contrary of the vague and entangled closed-form solution, the coefficients of the power series are simple and clearly indicate conditions and limitations for emittance reduction below the TME value. With the help of analytical predictions we design a lattice cell with longitudinally varied bends demonstrating the emittance less than that for the TME structure of the same bending angle.