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Newtonian gravitational constant measurement. All atomic variables become extreme when using a source mass consisting of 3 or more parts

Atomic interferometry methods used to measure the Newtonian gravitational constant. To improve the accuracy, one should measure the phase of an atomic interferometer at extreme values of atomic vertical velocities and coordinates. Owing to symmetry, the horizontal components of atomic velocities and coordinates are also extreme. We propose using a source mass consisting of 3 or more parts, since only in this case one can find such an arrangement of parts that all atomic variables become extreme. Nonlinear dependences of the phase on the uncertainties of atomic positions and velocities near those extreme values required us to modify the expression for the phase relative standard deviation (RSD). Moreover, taking into account nonlinear terms in the phase dependence on the atomic coordinates and velocities leads to a phase shift. In the last experiment to measure the Newtonian gravitational constant by atomic interferometry, this shift was not included. We took the shift into account, got a value of 199ppm for it, and this leads to a decrease in the value of the Newton constant by 0.02%. In addition, we showed that at equal sizes of the atomic cloud in the vertical and horizontal directions, as well as at equal atomic vertical and transverse temperatures, systematic errors due to the finite size and temperature of the cloud disappear. The calculation also showed that when using the 13-ton source mass proposed recently, the measurement accuracy can reach 17ppm for a source mass consisting of 4 quarters. We assumed that the source mass consisting of a set of cylinders is used for measurements. We have obtained a new analytical expression for the gravitational field of a homogeneous cylinder.