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Courant–Snyder Parameters
In accelerator physics, the Courant–Snyder parameters (frequently referred to as Twiss parameters or CS parameters) are a set of quantities used to describe the distribution of positions and velocities of the particles in a beam. When the positions along a single dimension and velocities (or momenta) along that dimension of every particle in a beam are plotted on a phase space diagram, an ellipse enclosing the particles can be given by the equation: :\gamma x^2 + 2 \alpha x x' + \beta x'^2 = \epsilon where x is the position axis and x' is the velocity axis. In this formulation, \alpha, \beta, and \gamma are the Courant–Snyder parameters for the beam along the given axis, and \epsilon is the emittance. Three sets of parameters can be calculated for a beam, one for each orthogonal direction, x, y, and z. History The use of these parameters to describe the phase space properties of particle beams was popularized in the accelerator physics community by Ernest Courant and Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Courant Snyder Ellipse
Courant may refer to: * '' Hexham Courant'', a weekly newspaper in Northumberland, England * ''The New-England Courant'', an American newspaper, founded in Boston in 1721 * ''Hartford Courant'', a newspaper in the United States, founded in 1764 *Courant (surname) *Courant, Charente-Maritime, a commune in France *Courant, in heraldry, signifying a running animal with all four paws raised - see Attitude (heraldry)#Courant * The Courant Institute of Mathematical Sciences at New York University * Courant, an alternative spelling for the Baroque dance form, courante * The Courant–Friedrichs–Lewy condition (CFL condition) in mathematics * Richard Courant, German mathematician See also * Corante ''Corante: or, Newes from Italy, Germany, Hungarie, Spaine and France'' was the first newspaper printed in England. The earliest of the seven known surviving copies is dated 24 September 1621 (although John Chamberlain is on record as having c ... {{disambiguation ru:Курант [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position And Momentum Space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional space, three-dimensional but in general of any finite dimension. Position space (also real space or coordinate system, coordinate space) is the set of all ''position vectors'' r in space, and has dimensional analysis, dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all ''momentum vectors'' p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of [mass][length][time]−1. Mathematically, the duality between position and momentum is an example of ''Pontryagin duality''. In particular, if a function (mathematics), function is given in position space, ''f''(r), then its Fourier transform obtains the function in momentum space, ''φ''(p). Conversely, the inverse Fourier transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray Transfer Matrix Analysis
Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, interface, mirror, or beam travel) is described by a 2×2 ''ray transfer matrix'' which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system. The same mathematics is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see electron optics. This technique, as described below, is derived using the ''paraxial approximation'', which requires that all ray directions (directions normal to the wavefronts) are at small angles ''θ'' relative to the optical axis of the system, such that the approximation \sin \theta \approx \theta remains v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive coefficient, constant. If ''F'' is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude). If a frictional force (Damping ratio, damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can: * Oscillate with a frequency lower than in the Damping ratio, undamped case, and an amplitude decreasing with time (Damping ratio, underdamped oscillator). * Decay to the equilibrium p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hill Differential Equation
In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function by minimal period \pi . By these we mean that for all t :f(t+\pi)=f(t), and :\int_0^\pi f(t) \,dt=0, and if p is a number with 0 < p < \pi , the equation must fail for some . It is named after , who introduced it in 1886. Because has period , the Hill equation can be rewritten using the of : : |
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity. Mathematical definition The quadrupole moment tensor ''Q'' is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. Q_ + Q_ + Q_ = 0). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent. For a discrete system of \ell point charges or masses in the case of a gravitational quadrupole, each with charge q_\ell, or mass m_\ell, and position \vec_\ell = \left(r_, r_, r_\right) relative to the coordinate system origin, the components of the ''Q'' matrix are defined by: : Q_ = \sum_\ell q_\ell\left(3r_ r_ - \left\, \vec_\ell \ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Focusing
In accelerator physics strong focusing or alternating-gradient focusing 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 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 was first conceived by Nicholas Christofilos in 1949 but not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995). The radian is defined in the SI as being a dimensionless unit, with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing. Definition One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, \theta = \frac, where is the subtended angle in radians, is arc length, and is radius. A right angle is exactly \frac radians. The rotation angle (360°) corresponding to one complete revolution is the length of the circumference divided by the radius, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Angle Approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians: : \begin \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \frac \approx 1\\ \tan \theta &\approx \theta \end These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision. There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation, \textstyle \cos \theta is approximated as either 1 or as 1-\frac. Justifications Graphic The accuracy of the approximations can be seen belo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beam Emittance
In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space. Each particle in a beam can be described by its position and momentum along each of three orthogonal axes, for a total of six position and momentum coordinates. When the position and momentum for a single axis are plotted on a two dimensional graph, the average spread of the coordinates on this plot are the emittance. As such, a beam will have three emittances, one along each axis, which can be described independently. As particle momentum along an axis is usually described as an angle relative to that axis, an area on a position-momentum plot will have dimensions of length × angle (for example, millimeters × milliradian). Emittance is important for analysis of particle beams. As long as the beam is only subjected to conservative forces, Liouville's Theorem shows that emittance is a conserved quantity. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems. There are extensions of Liouville's theorem to stochastic systems. Liouville equations The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the impo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
