In a collider, particles are collided head-on at high energy. The particle energy is determined by the size of the collider. The following table shows the energy of various colliders along with the rest energy of various particles.
Energy Collider Year Collider where the (TeV) size (km) particle was discovered Proton .000938 Stanford Linear Collider .045 3.2 1966 W boson .080 Large Electron Positron Collider Z boson .091 Large Electron Positron Collider LEP Collider .104 27 1989 Higgs Boson .125 Large Hadron Collider Top quark .171 Fermilab Fermilab collider 1 6 1989 Compact Linear Collider 3 30 Near future Large Hadron Collider 7 27 2011 China ring collider 50 150 Near future Continent linear collider 25 250 Far future Continent ring collider 1000 6000 Far future Cosmic rays 109 Big bang particle energy 1016"Collider size" is the length for linear colliders and the circumference for ring colliders.
Particle diameter is proportional to Mass1/3.
The electron is exaggerated otherwise it would be invisible.
Blue particles represent the heaviest particle that can be produced by each accelerator.
At this scale, a Planck-mass particle has a diameter of 10 km.
Photons, Gluons, and Gravitons are massless.
The Stanford Linear Accelerator (SLAC) is the largest linear collider and larger ones are planned. All linear colliders collide electrons with antielectrons.
The energy of a linear collider is determined by its length and acceleration gradient.
Accelerator length = X = 3.2 km for SLAC Acceleration gradient = G = = .014 Tev/km for SLAC Electron energy = E = X G = .045 TeV for SLACSLAC uses radiofrequency aceleration, which has a limit of .06 TeV/km. The proposed Compact Linear Collider will use 2-beam acceleration and reach a gradient of at least .1 TeV/km.
In "Large Electron Positron" ring collider, magnets steer electrons one way around the ring and antielectrons the other way. Ring colliders have an edge over linear colliders because they can accelerate particles each pass around the ring, and so they are not limited by the acceleration gradient. They are limited instead by the strength of the bending magnets or by synchrotron radiation.
Particles are steered around the ring by superconducting magnets. The maximum particle energy is determined by the magnetic field strength and by the collider size.
Magnetic field strength = B = 5.4 Tesla for the LHC (average field) Radius of the ring = R = 4300 meters for the LHC Maximum particle energy = E = .00030 B R = 7.0 TeV for the LHC
An accelerating charged particle emits photons (synchrotron radiation). Particles traveling around a ring emit photons and lose energy. The larger the collider, the larger the particle energy and the larger the synchrotron loss rate. A ring collider can only be made so large before synchrotron loss exceeds energy gain from acceleration.
The synchrotron radiation power is
Particle energy = E Particle rest energy = Erest Ring radius = R Synchrotron radiation power = P ~ (E/Erest)4 / R2 Erest (Proton) ------------------- = 1836 Erest (Electron)Electrons have a smaller rest energy than protons and emit vastly more synchrotron radiation. An electron ring collider is limited by synchrotron radiation rather than by magnet strength. The maximum energy is .15 TeV. Generating electron energies larger than this requires a linear collider.
Protons emit negligible synchrotron radiation and the radiation is never a concern. Proton ring colliders are limited by magnet strength rather than by synchtrotron radiation.
Future colliders will consist of proton ring colliders and linear electron colliders.
Particle energy = E (TeV) Particle velocity = V Speed of light = C Particle charge = 1.602e-19 Coulombs Magnetic field = B (Teslas) Magnet strength averaged around the ring Collider radius = R Particle rest mass = m Particle relativistic mass = M Electric force constant = K = 8.988⋅109 Newton meter2 / Coulomb2 Electric force on a charge = Fe = q E Magnetic force on a charge = Fm = q V B Centripetal force = Fc = M V2 / RParticles in a collider are "ultrarelativistic" and so we may assume:
M >> m V ~ C E ~ M C2Particles are steered around the ring by magnets
Magnetic force = Centripetal force q B V = M V2 / R q B = E / (C R) E = q C B R E = .00030 B R (Energy in TeV)The maximum energy of a ring collider is determied by the magnetic field and the ring radius.
Magnets are a finicky technology and thus far the LHC magnets are not at full strength. The current proton energy is 4 TeV and an energy of 6.5 TeV is expected in 2015.
The Lorentz factor of a 7 TeV proton is   γ = E/(mC) = 7460 = 7 TeV / .000938 TeV
If a charged particle is accelelerated it emits photons (synchrotron radiation). Particles in a ring collider emit synchrotron radiation when they are bent by the magnetic field.
Particle energy = E Particle rest energy = Eo Particle charge = q Particle velocity = V Speed of light = C Electric force constant = K = 8.988⋅109 Newton meter2 / Coulomb2 Collider radius = R Synchrotron power = P = (2C/3) K q2 (V/C)4 (E/Eo)4 R-2 Decelerating synchrotron force= Fs = P/V (Joules/meter) Synch. energy loss / cycle = Es = (2πR/V) P Energy lost to synch. each ring trip = (4πC/3V) K q2 (V/C)4 (E/Eo)4 R-1 ~ (4π/3) K q2 (E/Eo)4 / R (Using V ~ C) Fraction energy lost per cycle= Q = Es / E Collider maximum energy = EmaxIn order for a ring to be effective it must have Q << 1. If Q >= 1 then you might as well build a linear collider. For a given collider radius R the synchrotron energy limit is
Emax = Es / Q = (4/3 πKq2/Q) (Emax/Eo)4 R-1 Q-1 = (4/3 πKq2/Q)-1/3 Eo4/3 R1/3The CERN ring has R = 4243 meters. If we set Q=1 we get the following energies:
Eo Emax (TeV) (TeV) Electron .000511 .27 Muon .106 325 Proton .938 5961When the CERN ring housed the Large Electron Positron collider (LEP) it operated at an energy of .104 TeV. The ring was subsequently converted to a proton collider with an energy of 7 TeV. Proton synchrotron radiation is negligible at this energy.
The energy limit from synchrotron radiation has the form: E ~ R1/3 The energy limit from the bending magnets has the form: E ~ B RWhen these are equal, the energy and radius are "Er" and "Rr".
If R < Rr then E is limited by the strength of the magnets If R > Rr then E is limited by synchrotron radiationIf we set B = 10 Tesla and Q = 1 we get
Er (TeV) Rr Electron .061 20 meters Muon 2630 880 km Proton 207000 69000 km = 11 Earth radiiAn electron collider larger than 20 meters is limited by synchrotron radiation.
Any proton or muon collider that could conceivably be built on the Earth is limited by the bending magnets rather than by synchrotron radiation.
In a collider, a particle "X" collides with its antiparticle "x" and they annihilate to create a new particle-antiparticle pair Y and y.
X + x --> Y + yThe maximum rest mass of the Y particle is equal to the energy of the X particle. Hence, a collider that collides 1 TeV electrons with 1 TeV antielectrons can produce particles with a rest mass of up to 1 TeV.
Protons consist of three quarks. In a proton collision, new particles are produced by collisions between individual quarks. Since each quark carries only a fraction of the proton's total energy, not all of the proton's energy can be used to make new particles. At Fermilab, protons have an energ of 1 TeV and the heaviest particles it can create are the .171 TeV top quark and the .125 TeV Higgs boson. This gives electron and muon colliders an energy edge over proton colliders.
Electron and muon colliders can also tune the beam energy to be exactly equal to the rest energy of the particle they're trying to create, whereas the energy of a quark-quark collision can't be tuned. This means that electron and muon colliders can produce particles with greater rates than a proton collider.
Electron and muon collisions also produce less background particles than proton collisions, making the events easier to analyze.
Particle physics is starved for data. The Large Hadron Collider has discovered all particles up to an energy of 1 TeV and now we need a larger-energy collider. The largest Earth-based collider that can conceivably be built is a linear electron collider, which can reach an energy of 100 TeV. Going beyond this requires going into space.
The largest force that a particle collider can deliver is .01 TeV/km, using the dual-beam technique. An electron collider with this force following the curvature of the Earth is limited by synchrotron radiation to an energy of 23 TeV. Going beyond this requires a collider straighter than the curvature of the Earth, which can be done using mountains and tunnels. Such a collider can reach an energy of 100 TeV.
There's no reason to not start building now. The energy can be increased incrementally by building out from the collision point in the two opposite beam directions. Upon reaching the energy of the Higgs particle (.125 TeV) the collider becomes useful as a Higgs factory to refine measurements of the Higgs. Upon reaching 1 TeV, new physics is being explored and it's all gravy after that. For a proton ring collider, you can't start using the collider until you build the entire ring.
An electron collider is better than a proton collider because:
*) In an electron-electron collision, the collision energy is exactly equal to the sum of the energy of the electrons. In a proton-proton collision the collision energy is 1/5 this because collisions occur between quarks.
*) The energy of an electron-electron collision is precise whereas the energy of a proton-proton collision is variable. This allows precise energy measurements.
*) Electron-electron collisions are easier to analyze than proton-proton collisions.
The deepest mines have a depth of 4 km and the tallest mountain ranges have a height of 8 km, and this allows one to build a straight collider where each beam has a length of 1000 km and the electron energy is 100 TeV.
Suppose you are on a ship in the middle of the ocean. The distance that you can see a bouy floating on the surface as a function of your height about the surface is
Radius of the Earth = R = 6371 km Height above the ocean surface = H (In km) Distance for which you can see a buoy = D = 113 H½ kmIf you have two mountains separated by some distance with a valley between them, you can run a straight collider beam between them that starts from the peak of one mountain, runs underground in the valley, and ends at the other mountain. If we assume that the mountains are 4 km above the valley and that the beam is 4 km underground in the valley, then the total bream length is of order 800 km. If the collider uses a dual-beam system with an acceleration of .1 Tev/km then the energy gain in the straight section is 80 TeV. Previous to the straight section, a section at lower energy that follows the curvature of the Earth can accelerate the electrons up to of order 23 TeV. The final electron energy at the collision point is of order 100 TeV.
The average thermal gradient of the Earth is 25 Celsius per km of depth. The deepest mines are:
Depth Rock temperature km Celsius Mponeng Gold Mine 4.0 66 South Africa TauTona Gold Mine 3.9 60 South Africa Savuka Gold Mine 3.7 South Africa Driefontein Mine 3.4 South Africa
If a collider follows the curvature of the Earth then the synchrotron energy limit is:
Collider radius = R = 6.371⋅106 meters (For the Earth) Particle energy = E = 23 TeV (For electrons limited by the Earth's curvature) Particle rest energy = Eo = .00000051 TeV (Electrons) Forward accelerating force = Fc = .0001 TeV/m (For the Compact Linear Accelerator) Synchrotron decelerating force = Fs = (2/3) K q2 (E/Eo)4 / R2If the collider force is equal to the synchrotron force,
E = Eo Fc1/4 R1/2 [(2/3) K q2]-1/4The following table gives the synchrotron energy limit for electrons, protons, and muons.
Radius Electron Muon Proton TeV TeV TeV Earth surface 6371 km 23 4800 43000 Orbit around Earth 10000 km 29 6000 53000 Earth orbit around sun 1.2 AU 6000 850000 7500000 Kuiper Belt orbit around sun 40 AU 53000 4700000 41000000
Muons have a halflife of 2.2 microseconds. Muons in a collider live longer because of time dilation, and so they can traverse the ring many times before decaying.
Muon halflife = T = 2.2e-6 seconds Collider radius = R Magnetic field = B Muon energy in TeV = E = .00030 B R (from the "ring colliders" section) Muon Lorentz factor = Z = E / .000106 = E / Restenergy Speed of light = C Ring cycles = N = Z T C / (2 π R) (Trips around the ring in one halflife) = (.00030 B R / .000106) T C / (2 π R) = 297 BA future muon collider would have B ~ 10 Tesla, and so the muons will traverse the ring ~ 3000 times before decaying.
The obstacle to building a muon collider is the need for a muon colling technology. Such a technology is being researched at Fermilab. If this can be done, then muons could be used at the Large Hadron Collider.
Muons emit neutrino radiation, which is a hazard to nearby humans. A set of sample radiation parameters can be found in http://arxiv.org/pdf/hep-ex/0005006v1.pdf
Muon energy Radiation (TeV) (mSieverts/yr) 2 .0005 5 2.3 50 10Earth average background radiation is 3.5 mSieverts/yr. A muon collider with energy > 5 TeV would have to be located at a remote site away from civilization.
These are hypothetical colliders that could be built in the near or distant future. We assume a gradient of .1 TeV/km for linear colliders and a magnetic field of 10 Tesla for ring colliders.
Particle Shape Length or Energy Radius (km) (TeV) Electron Linear 30 3 Compact Linear Collider Proton Ring 4.2 7 LHC at full strength Muon Ring 4.2 7 Using the LHC ring Electron Linear 250 25 Limited by synchrotron radiation from the Earth's curvature Muon Ring 100 60 At a remote location because of radiation Muon Linear 1000 100 Continent-sized Proton Ring 1000 600 Continent-sized Proton Ring 10000 6000 Orbiting the Earth Muon Linear 10000 6000 Orbiting the Earth. Magnet limit Electron Linear 2e8 4200 Orbiting between Earth and Mars. Synchrotron limit Muon Linear 2e8 1000000 Orbiting between Earth and Mars. Magnet limit Proton Linear 2e8 10000000 Orbiting between Earth and Mars. Magnet limitA Planck particle has an energy of 10^16 TeV. If we assume an acceleration gradient of .1 TeV/km then the collider length is 10^17 km = 10000 light years. This is 1/3 of the distance between the sun and the center of the Milky Way. Such a collider would have to be built in intergalactic space to escape the gravity of galaxies.
Collider Particle Type Energy Size Lum Power BeamX BeamY Particles/ (TeV) (km) (MW) (nm) (nm) second (1024) Stanford Linear Accel. e- Linear .045 3.2 .0003 64 2000 2000 .12 Large Electron Positron e- Ring .104 4.3 .01 18 200000 2000 3600. Fermilab Proton Ring 1.0 1.0 .04 30000 30000 Large Hadron Collider Proton Ring 7 27 1 17000 17000 29000. Int. Linear Collider e- Linear .5 16 2 230 5.7 Compact Linear Collider e- Linear .5 5 6 240 40 1 .024 Circular Electron Positron e- Ring .12 8.5 1.8 50"Size" corresponds to radius for a ring collider and length for a linear collider.
"Lum" is the luminosity in 10^34 particles / cm2 / s
BeamX is the horizontal beam size
BeamY is the vertical beam size
Linear colliders tend to have a smaller beam cross section than ring colliders.
The "Circular Electron Positron" collider is a future collider that is optimized for generating Higgs particles.
Suppose we compare the tunnel lengths of a linear and a ring collider.
In an electron-positron collision, all of the particle energy can be harnessed for creating new particles. In a proton-proton collision, only a fraction of the energy is available for creating new particles.
Re = Length over which an electron is accelerated in a linear collider (km) G = Acceleration gradient of a linear collider (TeV/km) = .1 for the Compact Linear Collider Ee = Electron energy in a linear collider (TeV) = G Re Le = Tunnel length of a linear collider = 2 Re Rp = Radius of a ring collider B = Mean magnetic field in a ring collider (Tesla) = 10 Tesla for a hypothetical future proton collider Ep = Proton energy in a ring collider (TeV) = .3 B Rp (see earlier section) f = Maximum fraction of the proton energy that can be harnessed for creating a new particle = Approximately 0.2 Lp = Tunnel length of a ring collider = 2 Pi RpIf a linear electron collider and a ring proton collider have equivalent prowess for producing new particles,
Ee = f EpThe tunnel lengths for a linear and ring collider are
Le = 2 Ee / G Lp = 2 Pi Ep / (.3 B) = 2 Pi Ee / (.3 B f)The ratio of tunnel lengths is
Le / Lp = .095 f B / G = 1.9 (using values from the table)In terms of tunnel length, a linear electron collider and a ring proton collider are on nearly equal footing.
What is the interplanetary magnetic field as a function of distance from the sun? What would you estimate is the lowest-energy cosmic ray that can reach the Earth from interstellar space?