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22-449: The Alternating Gradient Synchrotron was built on the innovative concept of the alternating gradient, or strong-focusing principle , developed by Brookhaven physicists. This new concept in accelerator design allowed scientists to accelerate protons to energies that were previously unachievable. The AGS became the world's premiere accelerator when it reached its design energy of 33 billion electron volts (GeV) on July 29, 1960. Until 1968,

44-437: A defocusing effect that can be countered with a convergent magnet 'lens'. This can be shown schematically as a sequence of divergent and convergent lenses. The quadrupoles are often laid out in what are called FODO patterns (where F focusses vertically and defocusses horizontally, and D focusses horizontally and defocusses vertically and O is a space or deflection magnet). Following the beam particles in their trajectories through

66-403: A quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity. For the proofs of these results, Archimedes used the method of exhaustion attributed to Eudoxus . In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles

88-424: A square with the side x = a b {\displaystyle x={\sqrt {ab}}} (the geometric mean of a and b ). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b , then the height ( BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to

110-438: A variety of methods, depending on the phenomena of interest. Quadrature (mathematics) In mathematics , particularly in geometry , quadrature (also called squaring ) is a historical process of drawing a square with the same area as a given plane figure or computing the numerical value of that area . A classical example is the quadrature of the circle (or squaring the circle). Quadrature problems served as one of

132-408: Is now called the definite integral , and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals . Christiaan Huygens successfully performed a quadrature of the surface area of some solids of revolution . The quadrature of the hyperbola by Gregoire de Saint-Vincent and A. A. de Sarasa provided a new function ,

154-447: Is the principle that, using sets of multiple electromagnets , it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction of travel. By contrast, weak focusing is the principle that nearby circles, described by charged particles moving in a uniform magnetic field, only intersect once per revolution. Earnshaw's theorem shows that simultaneous focusing in two directions transverse to

176-583: The Alternating Gradient Synchrotron . Courant and Snyder found that the net effect of alternating the field gradient was that both the vertical and horizontal focusing of protons could be made strong at the same time, allowing tight control of proton paths in the machine. This increased beam intensity while reducing the overall construction cost of a more powerful accelerator. The theory revolutionised cyclotron design and permitted very high field strengths to be employed, while massively reducing

198-586: The AGS Booster. The AGS Booster then accelerates these particles for injection into the AGS. The AGS Booster also provides particle beams to the NASA Space Radiation Laboratory . It became increasingly clear that if further progress was to be made in high energy nuclear physics by experiments using artificially accelerated particles some new principle must be found that would cheapen the cost per GeV. It

220-533: The AGS was the highest energy accelerator in the world, slightly higher than its 28 GeV sister machine, the Proton Synchrotron at CERN , the European laboratory for high-energy physics. While 21st century accelerators can reach energies in the trillion electron volt region, the AGS earned researchers three Nobel Prizes and today serves as the injector for Brookhaven's Relativistic Heavy Ion Collider ; it remains

242-439: The amplitude of the free oscillations is to increase the frequency by increasing the restoring force, and although this is easy to achieve in the vertical direction by increasing the magnetic field gradient, the condition for horizontal stability is violated if n exceeds unity. The new principle discovered by Christofilos and Courant , Livingston and Snyder increases the frequency of the betatron oscillations by alternating

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264-428: The beam axis at once by a single magnet is impossible - a magnet which focuses in one direction will defocus in the perpendicular direction. However, iron "poles" of a cyclotron or two or more spaced quadrupole magnets (arranged in quadrature ) can alternately focus horizontally and vertically, and the net overall effect of a combination of these can be adjusted to focus the beam in both directions. Strong focusing

286-472: The circle ), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lune of Hippocrates and the parabola . By a certain Greek tradition, these constructions had to be performed using only a compass and straightedge , though not all Greek mathematicians adhered to this dictum. For a quadrature of a rectangle with the sides a and b it is necessary to construct

308-413: The focusing arrangement, an oscillating pattern would be seen. The action upon a set of charged particles by a set of linear magnets (i.e. only dipoles, quadrupoles and the field-free drift regions between them) can be expressed as matrices which can be multiplied together to give their net effect, using ray transfer matrix analysis . Higher-order terms such as sextupoles, octupoles etc. may be treated by

330-399: The main sources of problems in the development of calculus . They introduce important topics in mathematical analysis . Greek mathematicians understood the determination of an area of a figure as the process of geometrically constructing a square having the same area ( squaring ), thus the name quadrature for this process. The Greek geometers were not always successful (see squaring

352-406: The point where it crosses the circle, equals the geometric mean of a and b . A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle. Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. Nevertheless, for some figures

374-559: The sign of the gradient of the magnetic field. The structure of the magnet is no longer uniform round the ring with a constant gradient but is broken up into sectors whose gradient is alternatively positive and negative. The work performed at the accelerator led to three Nobel Prizes in Physics : This particle physics –related article is a stub . You can help Misplaced Pages by expanding it . Strong focusing In accelerator physics strong focusing or alternating-gradient focusing

396-466: The size of the magnets needed by minimising the size of the beam. Most particle accelerators today use the strong-focusing principle. Modern systems often use multipole magnets, such as quadrupole and sextupole magnets , to focus the beam down, as magnets give a more powerful deflection effect than earlier electrostatic systems at high beam kinetic energies. The multipole magnets refocus the beam after each deflection section, as deflection sections have

418-525: The world's highest intensity high-energy proton accelerator. The AGS Booster , constructed in 1991, further augments the capabilities of the AGS, enabling it to accelerate more intense proton beams and heavy ions such as Gold . Brookhaven's linear particle accelerator (LINAC) provides 200 million electron volt (MeV) protons to the AGS Booster, and the Electron Beam Ion Source (EBIS) and Tandem Van de Graaff accelerators provide other ions to

440-445: Was first conceived by Nicholas Christofilos in 1949 but not published (Christofilos opted instead to patent his idea). In 1952, the strong focusing principle was independently developed by Ernest Courant , M. Stanley Livingston , Hartland Snyder and J. Blewett at Brookhaven National Laboratory , who later acknowledged the priority of Christofilos' idea. The advantages of strong focusing were then quickly realised, and deployed on

462-412: Was lucky for CERN that just at the time a European machine was being considered this new principle was discovered. The problem was simple enough. A cheaper machine could be built if the amplitudes of the free and forced oscillations of the accelerating particles could be decreased in some way so that the vacuum chamber size and the cross-section of the magnet ring could be reduced. The simplest way to reduce

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484-617: Was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola ( Opus Geometricum , 1647), and Alphonse Antonio de Sarasa , de Saint-Vincent's pupil and commentator, noted the relation of this area to logarithms . John Wallis algebrised this method; he wrote in his Arithmetica Infinitorum (1656) some series which are equivalent to what

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