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Particle Colliders
Dr. Jay Maron

Particle collisions
A proton and antiproton collide to form a top and an antitop particle

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 energy

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.

Linear colliders

Stanford Linear Accelerator

Typical linear collider

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 SLAC
SLAC 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.
Ring colliders

A small ring collider

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.

SLAC tunnel
Fermilab bending magnet
Large Hadron Collider (LHC) at CERN

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

Synchrotron radiation

Particles moving straight don't emit synchrotron radiation
Curving electrons emit lots of synchrotron radiation
Curving protons emit negligible synchrotron radiation

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.

Maximum energy of a ring collider
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 / R
Particles in a collider are "ultrarelativistic" and so we may assume:
M  >>  m
V  ~  C
E  ~  M C2
Particles 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

Ring colliders and synchrotron radiation

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       =  Emax
In 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 R1/3
The 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       5961
When 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.
Ring collider size limit
The energy limit from synchrotron radiation has the form:    E  ~  R1/3
The energy limit from the bending magnets has the form:      E  ~  B R
When 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 radiation
If 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 radii
An 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.

Proton collider

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  +  y
The 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.

Synchrotron limit from the Earth's curvature

If an electron linear collider has sufficiently high energy then its energy is limited by synchrotron radiation from the curvature of the Earth's surface.

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 / R2
If the collider force is equal to the synchrotron force,
E  =  Eo  Fc1/4  R1/2  [(2/3) K q2]-1/4
Curvature limit:
            Earth      Orbit around   Orbit around     Kuiper Belt
            surface       Earth           sun

Radius      6371 km      10000 km          1.2 AU          40 AU
Electron      23 TeV        29 TeV      4100 TeV        22000 TeV
Muon        4800 TeV      6000 TeV    850000 TeV      4700000 TeV
Proton     43000 TeV     53000 TeV   7500000 TeV     41000000 TeV

Colossal electron linear collider

The highest-energy collider that can be built on the Earth is an electron linear collider.

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 km

If 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

Muon ring collider

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 B
A 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

Muon energy    Radiation
  (TeV)      (mSieverts/yr)
    2            .0005
    5           2.3
   50          10
Earth 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.
Future colliders

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 limit
A 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.
Luminosity, power, and beam cross section

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.

Linear electron collider vs. proton ring collider

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 Rp
If a linear electron collider and a ring proton collider have equivalent prowess for producing new particles,
Ee  =  f Ep
The 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?

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