Accelerators Relativistic kinematics basic energy, mass and momentum units, Lorents force, track bending, sagitta Basic static acceleration: First accelerator: cathode ray tube Cathode C consist of a filament, heated by power from A. The energy of some electrons in the cathode will exceed the bounding energy at the cathode surface, and will evaporate as free electrons in the vacuum. The applied potential difference (source B) will create an electric field, accelerating the electron towards anode P. The local fluorescent material may emit due to the absorption of the fast electrons.
heated filament E field = V / D With electron charge q: F = q. E field distance D Potential diffence V electron kinetic energy: E e- = F dd = q.v E e- independent of: - distance D - particle mass Energy unit: ElectronVolt: ev 1000 ev = 1 kev 1000 MeV = 1 GeV 1000 GeV = 1 TeV 1 ev = q Joules = 1.6 x 10-19 Joules Nota Bene: here, q is just a real value, and has NO unit like charge!! An ev is a convenient unit for energy, used for particles with elementary charge.
So the energy of the electrons in a (cathode ray) color TV with a screen HV of 25 kv is 25 kev. If protons would be used, their energy would also be 25 kev. Due to their higher mass, they would travel much slower. Example: an old-fashioned color TV has a cathode ray tube operating at 25 kv electron energy: 25 kev = 25 k x 1.6 x 10-19 J = 4.0 x 10-15 J electron speed: E kin = ½ m 0 v 2 Ł v = sqrt(2 E kin / m 0 ) = 94,000 km/s = 0.31 x c Relativistic effects! So v is smaller, and m is larger than m 0 Now assume very high-energetic particles with a speed close to c: the energy associated with their rest mass is small with respect to the kinematic part: E 2 = m o 2 c 4 + p 2 c 2 p 2 c 2 So E = p c, and p = E/c. So, a proton of 100 GeV has a momentum p of 100 GeV/c Note that this is also (stronger: always) true for photons (gammas, X-rays (rest mass = 0).
From Einstein s Special Theory on Relativity: E 2 = m o2 c 4 + p 2 c 2 With: β = v / c, and the Lorentz factor γ: relativistic mass m r = γ m 0 γ = 1 / sqrt(1- β 2 ), and β = sqrt(γ 2-1) / γ So: total energy E = m 0 c 2 sqrt(1+ β 2 γ 2 ) [= rest mass + kinetic energy] = γ m 0 c 2 = m r c 2 Another example of relativistic kinematics (exercise!) A positron and an electron annihilate. Just before the annihilation the electron had a kinetic energy of 12 ev and the positron was at rest. The two emitted photons were measured to have precisely an equal energy. - What is the energy of the photons? - What is the angle between the directions of the photons? Assume the rest mass of an electron to be exactly 511 kev. Solution: the total energy of the system equals 2 x 511 kev + 12 ev = 1022.012 kev. This is equally distributed ovet two photons, so their energy is 511.006 kev.
Pγ α P e Pγ The energy of the gammas is equal, so the momentum of the electron is equally shared by the photons: see figure in which the momenta are shown. Total energy of the electron before annihilation: E 2 = m 2 c 4 + p 2 c 2 = (511 + 0.012) 2 (kev) 2 p 2 c 2 = (511+0.012) 2 (511) 2 => p = 3.5 kev/c Angle α follows from the ratio of Pe and Pγ: Α ~ Pe/(2 Pγ) = 3.4 mrad = 0.19 deg. The angle between the photons is 180 2 x 0.19 = 179.6 deg. First applied accelerator: the X-ray tube. Accelerate electrons from a heated filament. At the anode, they generate X-rays by means of Bremstrahlung. Note that the maximum energy of these X-rays in (kev) equals the voltage of the tube (in kv). Essential for a static accelerator is a large potential difference V. This could be made with a Wimshurst generator. See Wikipedia for a correct (and not trivial) explanation. Around 1910, rather high potentials could be made with transformers (Ruhmkorff Induction coil). These devices were limited in their maximum potential difference due to internal discharge: they were not large enough for the voltage that they created. This problem was solved in the Van de Graaff Generator (1931), in which charge, applied on an insulating running belt, is transported against the electric field onto a metal sphere. See Wikipedia for a correct explanation: essential is that the charge is brought into the centre of the sphere, in which there is no electric field due to the net charge on the sphere.
With Van de Graaff Genrators, potentials of several MVs are possible. Practical limit to transformers Cockcroft-Walton high-voltage generator Sir John Douglas Cockroft Ernest Walton Nobel Prize 1951 From: Principles of Charged Particle Acceleration Stanley Humphries, Jr., on-line edition, p. 210 http://www.fieldp.com/cpa/cpa.html The Cockcroft-Walton generator (1937) piles up the potential of (many) charged capacitors Cockcroft-Walton generator. As introduction to the Cyclotrons first the effect of a magnetic field on moving charges particles is analysed (Lorentz Force).
Motion of charged particle in magnetic field Lorentz force: dp dt = q v B The speed of a charged particle, and therefore its γ, does not change by a static magnetic field: γ m dv dt = q v B (1) If s is equal to the distance along the particle trajectory: ds = v dt and if x is the position vector of the particle: dx ds = 1 dx v dt = v d 2 x v dt 2 = v2 d 2 x d 2 x ds 2 ds 2 = q dx Then: and: and using (1): p ds B (2) (2) describes a helix in a uniform field Motion of charged particle in magnetic field If magnetic field direction perpendicular to the velocity: γ mv 2 ρ = q v B which can be written as : p = ρ q B p = 0.2998 B ρ radius of curvature (p in GeV/c, B in T, ρ in m, for 1 elementary charge unit = 1.602177x10-19 C, and obtained using 1 ev/c 2 = 1.782663x10-36 kg and c = 299792458 m/s ) Color TV in Earth magnetic field B ~ 10 µt (varies with latitude!) E 2 = m o2 c 4 + p 2 c 2 = {511 kev + 25 kev) 2 = (536 kev) 2 p 2 c 2 = (536 kev) 2 - (511 kev) 2 p e = 162 kev/c (note: not E/c, not very relativistic!) ρ = p e / (0.2998 B ) = 53 m Shift Sh after D = 0.2 m: Sh = D 2 / (2 ρ) = 0.4 mm D ρ Sh
The cyclotron Top view "Dee": conducting, non-magnetic box Side view ~ r.f. voltage Constant magnetic field Ernest O.Lawrence at the controls of the 37" cyclotron in 1938, University of California at Berkeley. 1939 Nobel prize for "the invention and development of the cyclotron, and for the results thereby attained, especially with regard to artificial radioelements." (the 37" cyclotron could accelerate deuterons to 8 MeV) Speed increase smaller if particles become relativistic: special field configuration or synchro-cyclotron (uses particle bunches, frequency reduced at end of acceleration cycle) http://www.lbl.gov/science-articles/archive/early-years.html http://www.aip.org/history/lawrence/ The cyclotron consists of two Dee shaped vacuum chambers, mutually insulated. Preaccelerated articles are injected in the centre between the Dee s in a direction perpendicular to the plane between the Dee s. Due to the Lorentz force the particle will follow a half circle path. An AC voltage is applied between the Dee s, such that the particle is accelerated when crossing the gap between the Dee s. This process continues until the particle reaches the edge of the magnet field, and is extracted.
Linear Drift Tube accelerator, invented by R. Wideröe ~ r.f. voltage: frequency matched to velocity particles, so that these are accelerated for each gap crossed Particles move through hollow metal cylinders in evacuated tube In a linear accelerator the charged particles pass a number of tubes onto which an RF AC voltage is applied, common for the odd and even tubes. The particles are accelerated when crossing from one tube to the next. Linear Drift Tube accelerator, Alvarez type Metal tank ~ small antenna injects e.m. energy Particles move through into resonator, e.m. wave in tank hollow metal cylinders in accelerates particles when they cross evacuated tube gaps, particles are screened from e.m. wave when electric field would decelerate Luis Walter Alvarez Nobel prize 1968, but not for his work on accelerators: "for his decisive contributions to elementary particle physics, in particular the discovery of a large number of resonance states, made possible through his development of the technique of using hydrogen bubble chamber and data analysis"
Instead of an AC RF voltage, usually EM waves are applied in order to obtain a high energy increase per unit length of the accelerator. R.f. cavity with drift tubes as used in the SPS (Super Proton Synchrotron) at CERN NB: traveling e.m. waves are used Frequency = 200.2 MHz Max. 790 kw 8MV accelerating voltage Synchrotron : circular accelerator with r.f. cavities for accelerating the particles and with separate magnets for keeping the particles on track. All large circular accelerators are of this type. Injection During acceleration the magnetic field needs to be "ramped up". r.f. cavity Focussing magnet Bending magnet Vacuum beam line Extracted beam
The Synchrotron consists of a number of bending magnets and one or more (linear) accelerators. The initial magnetic field is quite low, and pre-accelerated particles are injected. The particles are accelerated, and the magnetic field is increased synchronously. During this ramp up, the particles are accelerated to their final energy. Aereal view of accelerators at CERN, Geneva, Swiss. Note the scale: the airport is just visible at the right-hand side.
Typical view of the SPS accelerator: curved tunnel, bending magnets, focusing magnets, vacuum beam pipe, vacuum equipment. During acceleration the magnetic field needs to be "ramped up". Slow extraction Fast extraction of part of beam Fast extraction of remainder of beam At time of operation of LEP SPS used as injector for LEP For LHC related studies The cycle of the SPS magnets, typical for synchrotrons.
Collider: two beams are collided to obtain a high Centre of Mass (CM) energy. Colliders are usually synchrotrons (exception: SLAC). In a synchrotron particles and anti-particles can be accelerated and stored in the same machine (e.g. LEP (e + e - ), SppS and Tevatron (proton - anti-proton). This is not possible for e.g. a proton-proton collider or an electron-proton collider. Important parameter for colliders : Luminosity L number of events /s N = L σcross-section Unit L: barn -1 s -1 or cm -2 s -1
CERN accelerator complex to Gran-Sasso (730 km) The complex of acellerators at CERN. Note the neutrino beam, pointing to the target in GranSasso. The neutrinos travel 730 km through the earth crush
Charged particles inside accelerators and in external beamlines need to be steered by magnetic fields. A requirement is that small deviations from the design orbit should not grow without limit. Proper choice of the steering and focusing fields makes this possible. Consider first a charged particle moving in a uniform field and in a plane perpendicular to the field: design orbit displaced orbit In the plane a deviation from the design orbit does not grow beyond a certain limit: it exhibits oscillatory behavior. However, a deviation in the direction perpendicular to the plane grows in proportion to the number of revolutions made and leads to loss of the particle after some time. Synchrotron radiation. Synchrotron radiation Particles may radiate when changing direction in a magnetic field, the radiation is called synchrotron radiation and can be in the form of UV light or of soft X-rays, emitted at high energy in a cone with opening angle 1/γ around the direction of the particle. The energy loss per turn is: E = 4πe 2 β 3 γ 4 /(3ρ) = 4πe 2 p 4 / (3m 4 βρ), where ρ is the radius of the orbit The time per turn is 2πρ/(βc), so the loss per second is 2e 2 p 4 c/(3m 4 ρ 2 ) -> For the same energy and orbit radius electrons and positrons lose about 2000 4 more energy then protons -> reason for large radius of LEP For high-energy electrons (β=1): E = 4πe 2 β 3 γ 4 /(3ρ) =4πr e m e E 4 /(3ρm e 4 ) (r e = classical electron radius = 2.82 fm) E = 8.85 10-2 E 4 /ρ MeV with E specified in GeV and ρ in m.
ESRF: European Synchrotron Radiation Facility, Grenoble, France 300 m circumference booster synchrotron, 6 GeV 16 m linac, 200 MeV When deflected by a magnetic field (vertical field lines), electrons emit synchrotron radiation in the form of X-rays. At ESRF, there are 32 channels providing X-ray facilities, mainly for material research.
The Large Hadron Collider (LHC) at CERN, which is due to start operation in July 2008. The four experiments (ATLAS, LHC-b, CMS and ALICE), are indicated. Large Hadron Collider LHC: proton-proton collider Interaction point Bunch size squeezed near interaction point Crossing angle to avoid long range beam beam interaction R ~4 km, E ~ 7 TeV (2x!) Ł B ~ 7 T! The bunch structure of the colliding beams. One of the aspects of this structure is that the timing of a collision is well known. This information is essential for operating detectors in experiments (drift chambers, calorimeters).
Superconducting magnets: no pole shoes Current distributions Superconducting coils: the magnetic field is determined by the coil configuration. Magnets waiting for installation at LHC.
2) heavy collisions: pp collisions A proton is a bag filled with quarks en gluonen Proton-proton collisions at LHC: a complicated affair. A proton is a complex particle with internal structure. Electron-positron colliders have much cleaner events, but a high CM energy can only be reached for a high price