In frames of a collaboration between IHEP and FNAL, a design of a supercon-ducting quadrupole with a high gradient field is being developed in IHEP. As a current-carrying element of the coil block SC-material, Nb_3Sn possessing a critical current density of 600 A/mm~2 at field 12 T and temperature 4.2 K is chosen. A physical motivation of the choice of the optimal parameters allowing to reach maximal field gradient of 220 T/m in a two-layer quadupole with 70 mm aperture at the nominal temperature of 4.6 K is presented. The cross-sectional geometry is chosen in such a way (i) to minimize the lower field non-linearities at the minimal expenditure of the SC material and (ii) to achieve of the maximal operating temperature. The optimal geometry of end parts reduces the field overshoot to a level pertinent to the central section by means of a certain distribution of turns and the choice of the yoke length. Besides, the lower edge field nonlinearities are suppressed by the optimal geometry of the end parts.
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