An accelerator system comprising a plurality of cells, each cell including a focus magnet and a defocus magnet set up in a FDF or DFD magnet configuration, each focus magnet and defocus magnet being configured to create a magnetic field so as to confine and accelerate a particle beam, the focus magnet being configured to focus the particle beam in a horizontal direction and defocus the particle beam in a vertical direction, and the defocus magnet being configured to focus the particle beam in a vertical direction and defocus the particle beam in a horizontal direction, wherein each of the plurality of cells is configured to confine the particle beam in an isochronous orbit during acceleration, wherein the accelerator system is a non-scaling fixed field alternating gradient (FFAG) particle accelerator; wherein the focus magnet is specified by the following focus parameters: a magnetic field of the focus magnet at injection B if, a magnetic field of the focus magnet at extraction B ef, a half-length of the focus magnet at injection L if, a half-length of the focus magnet at extraction L ef, a linear edge angle for the focus magnet η f, and a distance from the injection orbit to the extraction orbit in the focus magnet δx if; wherein the defocus magnet is specified by the following defocus parameters: a magnetic field of the defocus magnet at injection B id, a magnetic field of the defocus magnet at extraction B ed, a half-length of the defocus magnet at injection L id, a half-length of the defocus magnet at extraction L ed, a linear edge angle for the defocus magnet η d, and a distance from the injection orbit to the extraction orbit in the defocus magnet δx id wherein the accelerator system is specified by the focus parameters, the defocus parameters, and the following additional parameters: De, a drift distance between the focus and defocus magnets at extraction, Di, a drift distance between the focus and defocus magnets at injection, Dl, a half length of a drift inserted between focus magnets of neighboring cells in case of a FDF magnet configuration or defocus magnets of neighboring cells in case of a DFD magnet configuration, Nsectors, a number of periodic cells, a horizontal half-cell tune and a vertical half-cell tune φH and φV; and wherein the injection drift ( Di) and the injection radii R i (avg) and extraction radii R e (avg), can be expressed in terms of De and Nsectors; the accelerator system characterized in that the focus parameters and the defocus parameters are related, in a thin lens approximation, by the following equations to confine the particles of the particle beam in a dynamically stable isochronous orbit during acceleration: k if l if + θ if ρ if + θ ef − θ if + η if ρ if = 1 f if k id L id + θ if + η id ρ id = 1 f id k ef L ef + θ ef ρ ef + η ef ρ ef = 1 f ef k ed L ed + θ ef + η ed ρ ed = 1 f ed θ if + θ id = θ ef + θ ed = θ halfcell L if cos θ if + sin θ if tan θ ef + η ef = L ef cos θ ef − δ x if − L ef sin θ ef tan θ ef + η ef L ed cos θ ef = L id cos θ if + δ x id sin θ id + θ if + δ x if + L if + D i sin θ if − L ef + D e sin θ ef tan η ed D i cos θ if − sin θ if tan η ed = L ef + D e cos θ ef − L if cos θ if + δ x if − L if sin θ if − L ef + D e sin θ ef tan η ed δ x id cos θ id + θ if = δ x if + L ihalf sin θ if − L ehalf sin θ ef R e avg = β e β i R i avg R i avg = 2 L i halfcell 2 π = N sector L if + L id + D i + D l π R e avg = 2 L e halfcell 2 π = N sector L ef + L ed + D e + D l π, and wherein the horizontal half-cell tune and the vertical half-cell tune φH and φV are provided by: 2 cos φ H = Tr cosh k d l d sinh k d l d / k d − k d sinh k d l d cosh k d l d 1 0 tan η d / ρ d 1 1 D 0 1 x 1 0 − tan η f / ρ f 1 cos k f l f sin k f l f / k f − k f sin k f l f cos k f l f 1 D l 0 1 2 cos φ V = Tr cos k d l d sin k d l d / k d − k d sin k d l d cos k d l d 1 0 − tan η d / ρ d 1 1 D 0 1 x 1 0 tan η f / ρ f 1 cosh k f l f sin k f l f / k f − k f sinh k f l f cos k f l f 1 D l 0 1. a magnetic field of the focus magnet at injection B if, a magnetic field of the focus magnet at extraction B ef, a half-length of the focus magnet at injection L if, a half-length of the focus magnet at extraction L ef, a linear edge angle for the focus magnet η f, and a distance from the injection orbit to the extraction orbit in the focus magnet δx if; a magnetic field of the defocus magnet at injection B id, a magnetic field of the defocus magnet at extraction B ed, a half-length of the defocus magnet at injection L id, a half-length of the defocus magnet at extraction L ed, a linear edge angle for the defocus magnet η d, and a distance from the injection orbit to the extraction orbit in the defocus magnet δx id De, a drift distance between the focus and defocus magnets at extraction, Di, a drift distance between the focus and defocus magnets at injection, Dl, a half length of a drift inserted between focus magnets of neighboring cells in case of a FDF magnet configuration or defocus magnets of neighboring cells in case of a DFD magnet configuration, Nsectors, a number of periodic cells, a horizontal half-cell tune and a vertical half-cell tune φH and φV; and