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RACE-TRACK MICROTRON ON 50 МЭВ WITH SMALL NUMBER OF ORBITS А.М . The Gromov Institute of Nuclear researches of the Russian Academy of Science. Modeling of phase movement Time of flyover through the bending magnets and the drift paths

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## RACE-TRACK MICROTRON ON 50 МЭВ WITH SMALL NUMBER OF ORBITS А.М . The Gromov Institute of Nuclear researches of the Russ

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**RACE-TRACK MICROTRON ON 50 МЭВWITH SMALL NUMBER OF**ORBITSА.М. The GromovInstitute of Nuclear researches of the Russian Academy of Science Modeling of phase movement Time of flyover through the bending magnets and the drift paths Acceleration of particles in the race-track microtron Focusing of the beam by system of the recirculation Matching of the beam in the horizontal median plane Matching of the beam in the vertical plane Разрезной микротрон на 50 МэВ**The scheme of a race-track microtron**• 1 - the pole of the bending magnet. 2 - the pole of the reverse field. 3 - the single quadrupoles. • 4 - the cryostat with the accelerating section. • Frequency of accelerating voltage - f = 1500 ∙ 106 Hz. A multiplicity of a cycle time - ΔТturn= 4Тgen. • Equilibrium increase of energy - ΔEs = 10,0 MeV. Strength of magnetic field - 2695 Gs (*). • Energy of injection (complete) - Ео = 0,875 MeV (*). Strength of reverse field Bmin = - 650 Gs (*). • Length of the drift paths - Ldr = 226,5 cm (*). The quadrupole lenses Δl = 10 cm, g = 2,5 Gs / cm (*). • (*) - the values providing the optimal mode of acceleration are marked. Разрезной микротрон на 50 МэВ**The scheme of the poles and distribution of the fringing**field • 1 - distribution of the magnetic field for edge of the entry gap of bending magnets. • 2 - distribution of the first derivative of a magnetic field for edge of the gap. • The field created by the pole of reverse field, is entered for creation of regulated vertical focusing. Разрезной микротрон на 50 МэВ**The equations of movement of the electrons in the field of**the bending magnets • The equations of movement of electrons were integrated by the numerical method with a take into account of the nonlinearity brought by the fringing field of the gap of the magnet. During integration the distortion of the orbit and caused by it nonlinear dependence of length of the orbit on energy was determined. The influence of nonlinearity brought by the reverse field was taken into account at research of the phase movement. • The reverse field is entered for support of stability of the beam in a vertical direction at its movement on the orbit. Разрезной микротрон на 50 МэВ**Equilibrium orbits in a median plane**• Entry conditions of integration for a median plane: • At s = 0 • x = 0, dx/ds = 0, • y = 0, dy/ds = 0, • z = 0, dz/ds = 1. Разрезной микротрон на 50 МэВ**Change of the derivative of increase of the phase depending**on energy at passing by the particles of two magnets. • Change of the derivative of increase of the phase depending on energy at passing by particles of the path of drift. Разрезной микротрон на 50 МэВ**Acceleration of particles in the race-track microtron**• The scheme of the accelerating section. • The equations of movement of the electrons in the accelerating section Разрезной микротрон на 50 МэВ**Isolines of energies and phases of 5-th orbit**• Isolines of output energies • of 5-th orbit. • -------------------------------- • If the injection beam is in the area selectedin figures,with energy (0,87±0,05) MeV and phase extent of 0,1 rad, on exit of 5-th orbit the energy disper-sion will be (50 ± 0,25) MeV and phase extent - 0,25 rad. • --------------------------------Isolines output phases of 5-th orbit. Разрезной микротрон на 50 МэВ**Focusing of the beam by system of a recirculation**• The beam moving on orbits pass thru the same recirculation system. Thus, to a movement of the beam it is possible to apply a design procedure to systems with periodic coefficients. Solutions of the equations of movement on period of such system have the first integral: • γ∙ y2 + 2α∙ yy′ + β ∙ y′2 = ε The equality is fulfilled:βγ - α2 = 1 • This expression on the phase plane y, y ′ describes the ellipse matched with system. Owing to periodicity, ellipses on an input and on an exit of system are identical. The matrix of transition Mc, in this case, (Twiss matrix) also expresses through units of the matched ellipses. Разрезной микротрон на 50 МэВ**Calculation of the matrix of the recirculation system**• For definition of units of the matched emittance for horizontal and vertical movements the matrix of transition of the path of the recirculation is calculated. By matching units of the received matrix with the units of Тwiss matrix Mc the parameters of the matched emittance are defined. • Condition of steady movement is: |cos μ| ≤ 1 Разрезной микротрон на 50 МэВ**Parameters of matched emittancesfor horizontal movement**Energy of an injection beam - 0,87 MeV. Horizontal emittance of an injection beam - Sin = π 7,1 . 10-3cm . rad, Bin = 1,98 см, A′in = 3,6 . 10-3rad. Ellipses have canonical position- α0 =0 with semiaxesA′ andB. Разрезной микротрон на 50 МэВ**Parameters of matched emittancesfor vertical movement**Energy of an injection beam - 0,87 MeV. Vertical emittance of an injection beam - Sin = π 7,1 . 10-3cm . rad, Bin = 1,62cm, A′in= 4,4. 10-3rad. Ellipses have canonical position- α0 =0 with semiaxesA′ andB. Разрезной микротрон на 50 МэВ**Согласованные эмиттансы**• On the data of the table the matched emittances are constructed. • Horizontal emittancesVertical emittances Разрезной микротрон на 50 МэВ**The conclusion**• With the help of the created mathematicalmodel research of the race-track microtron with five orbits and with a multiplicity equalto four is carried out. • It is shown, that particles can be injected in accelerating section at rather low energy - (0,8 - 1) MeV. • Application of a reverse field in a combination to the single quadrupoles allows to receive transversal stability of the beam for all orbits. • At research of the mode of creation of transversal stability of the beam at movement on the path of the recirculation the method of research of the equations of movement with periodic coefficients is applied. Разрезной микротрон на 50 МэВ

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