Accelerators I

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Accelerators I Powered By Docstoc
					        Transverse Phase Space
1.   Under linear forces, any                                           x´                                    x´
     particle moves on an ellipse in
     phase space (x,x´).
2.   Ellipse rotates in magnets and
     shears between magnets, but                                                      x                            x
     its area is preserved:
                                                                 General equation of ellipse is
     Emittance
                                                                        β x′2 + 2α x x′ + γ x 2 = ε
                                                                 α, β, γ are functions of
                                                                 distance (Twiss parameters),
                                                                 and ε is a constant. Area = πε.
                                                                 For non-linear beams can use
                                                                 95% emittance ellipse or RMS
                                                                 emittance
                                                                 ε rms =                     x′2 − xx ′
                                                                                                          2
                                                                                    x2
                                                                 (statistical definition)
                     R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
 Circular Acceleration: Synchrotron
             Principle of frequency modulation but in addition variation in
                 time of B-field to match increase in energy and keep
                 revolution radius constant.

    p        Magnetic field produced by several bending magnets
ρ=
   qB            (dipoles), increases linearly with momentum. For q=e
                 and high energies:
f = nω             p E
               Bρ = ≈ so E [GeV] ≈ 0.3 B [T] ρ [m] per unit charge
             .     e ce
             Practical limitations for magnetic fields => high energies
   qBc  2
         v
ω=     =         only at large radius
    E    ρ
                 e.g. LHC: B = 8.36 T, ρ = 2.7 km => E ~ 7 TeV

                                                                                              2
                  R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
                             Synchrotrons
Storage rings: accumulate particles and keep
circulating for long periods; used for high intensity
beams to inject into more powerful machines or
synchrotron radiation factories.
Colliders: two beams circulating in opposite
directions, made to intersect; maximises energy in
centre of mass frame.


   Variation of
parameters with
time in the ISIS
  synchrotron:
B=B0-B1 cos(2πft)
                                        B
                                        E
                                        ω

                                                                                                    3   3
                        R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
               Stability Conditions
   Recall, the tune is the number of betatron oscillations per turn
   Provided integer values of the tune are avoided the machine operates reliably
   Synchrotron motion during acceleration cycle: p=0.3 B R
   Bending dipoles and focussing quadrupoles carry same (increasing) current
   The optics – (magnets effectively perform the functions of prisms and lenses) –
   looks the same
   Tune stays constant
   Setting the tune to something sensibly non-resonant means it stays there?
   Imperfections:
Errors in position, current etc, in a magnet means a particle gets the wrong ‘kick’.
Over many turns this smears out – if the particle is at different points on its betatron
oscillation each time


                                                                                                     4
                         R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
        Synchrotron Oscillations
In principle, synchrotrons
or linacs are designed such       RF accelerating cavity
that the synchronous
particle is accelerated
continuously.                                                                            RF accelerating cavities
In practice, non-
synchronous particles will
require acceleration too!
Thus, the question of phase       Series of accelerating cavities in:
stability, or do particles
                                  (a) synchrotron and (b) a linac
with different energies and
phases remain close to the
synchronous phase?


In order to derive an equation for the passage of non-synchronous particle through the
accelerator we consider the time interval τ between passages of two successive modules of
accelerating cavities (or sometimes called accelerating stations):
  τ = L/v
                                                                                                                    5
where L is the distance between stations and v the particle velocity v:
                              R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Synchrotron Oscillations -cont
The fractional change from small changes in L and v is:
Δτ ΔL Δv
   =   −
 τ   L   v
Thus, a particle moving faster than the synchronous (ideal) particle will take less time
between accelerating stations and a small error in the circumference will increase the transit
time (not appropriate to linacs)
The second term is readily expressed as:
Δv 1 ⎛ Δp ⎞
  = 2⎜    ⎟
 v γ ⎝ p ⎠
In a circular accelerator the orbit circumference is slightly larger for those particles having a
slightly larger momentum and thus:
ΔL 1 ⎛ Δp ⎞
    = 2⎜        ⎟
 L γt ⎝ p ⎠
Here γt is determined according to the device design (in linacs we expect 1/ γt = 0).
We introduce a slip factor (which changes sign at the transition energy γ = γt):
     1 1
η= 2 − 2                                                                                          6
    γt γ                R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Synchrotron Oscillations –cont.
We will investigate the equations
                                                                 Station n                                    Station n+1
of the particle with arbitary phase
and energy.                                              En                                              En+1
At the n+1 station the phase is:                         ψn                                              ψn+1

ψ n +1 = ψ n + ωrf ( τ + Δτ )n +1                          Passage of particle through n and n+1
                                    ⎛ Δτ ⎞                 accelerating stations (enters nth station
      = ψ n + ωrf τn +1 + ωrf τn +1 ⎜ ⎟                    with energy En and phase ψn)
                                    ⎝ τ ⎠ n +1
As the synchronous particle always arrives at the nth station at the same phase, we switch to a
variable which is relative to the rf phase (Tn being the time at the entrance to the nth station)
 φn ≡ ψ n − ωrf Tn
Thus in terms of this variable:
                                                        ⎛ Δτ ⎞
φn +1 + ωrf Tn +1 = φn + ωrf Tn + ωrf τn +1 + ωrf τn +1 ⎜ ⎟ , and as Tn +1 = Tn + τn +1
                                                        ⎝ τ ⎠n +1 Where we have used:
                                                                                               Δτ/τ = ηΔp/p

                       ⎛ Δτ ⎞                ⎛ Δp ⎞
φn +1 = φn + ωrf τn +1 ⎜ ⎟ = φn + ηωrf τn +1 ⎜    ⎟
                                                                                             We will drop the suffix on
                       ⎝ τ ⎠n +1             ⎝ p ⎠n +1                                       (ωrfτ)n as is designed into the
                                                                                             geometry of the accelerating
                                                                                             structure.                   7
                             R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Synchrotron Oscillations –cont.
We now consider the deviation in the energy from that of the synchronous energy:
 ΔE n +1 = ΔE n + eV ( sin φn − sin φs ) , where ΔE n = E n − E s
Also as E=γmc 2 and p=γmv it is straightforward to show:
Δp c 2 ΔE
   = 2
 p v E
Recall, the phase equation is of the form:
                         ⎛ Δp ⎞
 φn +1 = φn + ηωrf τn +1 ⎜    ⎟
                         ⎝ p ⎠n +1
Thus the phase equation and energy equations are:
             ηωrf τc 2
 φn +1 = φn + 2        ΔE n +1
              v Es
 ΔE n +1 = ΔE n + eV ( sin φn − sin φs )
It remains to solve these difference equations numerically………..
                                                                                                      8
                          R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
                               Phase Stability
     A few iterations of the difference equations, for initial
     phase equal to the synchronous phase (φ1 = φs) and for
     various initial value energy differences ΔE.
      Particles close to the ideal, synchronous particle,
     remain close to it.                                                           Solution of difference eqns for
     Particles with larger ΔE depart from the neighbourhood                        synchrotron motion, with φ1=φs
     of the ideal particle –and are lost from acceleration




     Solution of difference eqns for synchrotron motion with various initial φ1 conditions


     Similarly for a whole range of initial phase (φ1) conditions
     It is notable that there is a boundary between stable and unstable motion –the separatrix.
     Two qualitatively different points:                                                    See Mathematica notebook for details on how these and
                                                                                            other plots are obtained and to explore other regimes
1.   Stable fixed point at φ=φs with ΔE = 0
2.   Outer edge of separatrix –unstable stable region
                                                                                                                                      9
     In general, for a circular accelerator, there are many stable fixed points (spaced by 2π) –three shown.
                                   R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
            Phase Stability –cont.
The area in phase defined by the separatrix is known as a “bucket”
It is not necessary to fully populate all buckets –particles occupying a bucket are known as bunches
For example, when LEP, the electron-positron collider at CERN was in operation, 4 bucket were
populated by each species of particle. As the circumference of LEP is 27 km and the bunch spacing is
approx 1m, just over 0.1% of buckets are populated.
Another interesting circumstance consists of phase space for φs = 0 or π.
Here the ideal particle remains un-accelerated and the phase stable region is 2π in extent.
These are known as stationary buckets.
A particle will be un-accelerated and will remain in these buckets forever –unstable though, as it will
continue to undulate in energy and wander away in phase from the ideal particle phase.
C.f. the accelerated buckets, in which in this case particles outside the separatrix diverge in both energy
and phase and eventually leave the accelerator (into the walls or similar obstacles)




                             Stationary Buckets –for φs = 0 or π
                                                                                                       10
                           R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
           Phase Stability –diff. eq.
   We now seek a closed form analytical solution and we replace the small difference changes by
   differentials –acceptable as the energy changes per station are usually small
   dφ ηωrf τc 2     dΔE
      = 2       ΔE,     = eV ( sin φ − sin φs )
   dn   v Es         dn
   Taking the derivative of the first equation allows them to be turned into one differential eq.

   d 2 φ ηωrf τeVc 2                                        - Horizontal

       2
         =    2      ( sin φ − sin φs )                     lines denote T
   dn      v Es                                             -Curve is V

  -here we have assumed dEs/dn, dV/dn ~ 0 .                 -horizontal lines
                                                            give T=0
   Multiplying by dφ/dn and integrating over dn

⌠ d φ dφ dn − ηωrf τeVc ⌠ sin φ − sin φ dφ dn = 0
   2                   2

⎮ 2                      ⎮(            s)
⌡ dn dn         v2 Es ⌡                   dn
                                                                                               This is in the form: “U =T+U”

 1 ⎛ dφ ⎞ ηωrf τeVc 2
             2                                                                                 -where the total “energy” is U
⇒ ⎜ ⎟ +               ( cos φ + φ sin φs ) = const                                             -the first term is the KE
 2 ⎝ dn ⎠      2
            v Es                               U                                               -the second is the potential energy
       T                           V                                                                                       11
                              R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
            Phase Stability –diff. eq.
  Combining these two equations:

  1 ⎛ dφ ⎞ ηωrf τeVc2
            2
                                                                                         dφ ηωrf τc 2
    ⎜ ⎟ +             ( cos φ + φ sin φs ) = const.                                         = 2       ΔE
  2 ⎝ dn ⎠      2
             v Es                                                                        dn   v Es

       2v 2 E s eV
                 2 (
⇒ ΔE +  2
                     cos φ + φ sin φs ) = const .
       ηωrf τc

  Implicitly, in deriving this equation we have employed dEs/dn ~0 (not used in the difference equations)
  and so it remains to be seen that this is a faithful representation of the physics…




                                                                                                           12
                             R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Phase Stability –contour sol.
                                                              Contour of particle motion in
                                                              longitudinal phase space for
                                                              φs = π
                                                              The form obtained earlier, namely,
                                                              that obtained through solving the
                                                              difference equations, is verified –
                                                              here we have stationary buckets



                                                              Contour of particle motion in
                                                              longitudinal phase space for
                                                              φs = 5π/6


                                                            Colours represent different contours of
                                                            various values of the constant:
                                                                  2v 2 E s eV
                                                            ΔE +
                                                              2
                                                                              ( cos φ + φ sin φs ) = const
                                                                  ηωrf τc2


                                                                                                   13
       R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Phase Stability –contour sol.
                                                              Contour of particle motion in
                                                              longitudinal phase space for
                                                              φs = 2π/3




                                                              φs = 5π/4




                                                                                              14
       R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
            Phase Stability –tune
 To investigate the tune we return to the phase equations and consider small
 excursions about the synchronous phase: φ = Δφ + φs
d 2 φ ηωrf τeVc 2
    2
      =    2      ( sin φ − sin φs )                          Here νs is number of synchrotron oscillations, or the
dn      v Es                                                “tune”.
                                                              For η<0 (i.e. γ<γt) the motion is stable provided
  ηωrf τeVc 2                                               cosφs>0
       2
    v Es
              ( sin ( φs + Δφ ) − sin φs )                    For η>0 (i.e. γ>γt) the motion is stable provided
                                                            cosφs<0
                                                              For circular accelerators that cross the transition
   d 2 Δφ ηωrf τeVc 2                                       energy during acceleration the rf system must
⇒       2
          −    2
                      cos φs Δφ = 0                         perform a phase jump to maintain phase stability
    dn      v Es                                              At the transition (γ=γt) the synchrotron period
                                                            becomes infinite and no phase focusing occurs.
or
 d 2 Δφ                                 ηωrf τc 2 eV cos φs
        + ( 2πν s ) Δφ = 0 where νs = −
                   2

  dn  2
                                            4π 2 v 2 E s
                             2πνs     ηωrf c 2 eV cos φs
and the angular frequency is      = −
                              τ            τv 2 E s
                                                                                                           15
                         R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
    Phase Stability –linac tune
 For linacs, we replace the slip factor η with -1/γ2
 This requires cosφs>0
 Hence, as the energy increases the synchrotron frequency approaches zero
     1 ωrf τc 2 eV cos φs
νs =
     γs      4π 2 v 2 E s
 Thus for particles initially γ>>1, the solution to
  dφ ηωrf τc 2
     = 2        ΔE
  dn      v Es
 is of the form Δφ ~ const –electrons start out at a particular phase and stay in phase.




                                                                                                     16
                         R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
          Fixed Field Alternating
         Gradient Machines (FFAG)
An old idea, dating from 1950’s,
given a new lease of life with the
development of new magnetic alloy
cavities.
Field constant in time, varies with
radius (scaling FFAG has a specific
slowly varying, B-field B ~ Rk)
Wide aperture magnets and stable
orbits.
High gradient accelerating cavities
combine with fixed field for rapid
acceleration.                       Prototype FFAG, accelerating protons from
Good for particles with short half-  50 keV to 500 keV, was successfully built
lives (e.g. muons).                 and tested at the KEK laboratory in Japan,
                                                       2000.
                                                                                               17   17
                   R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
                  Fixed Field Alternating
                 Gradient Machines (FFAG)
  nsFFAGs (non scaling FFAGS)

  Drop scaling requirement –ensures simplicty
and more compact!

  Do not worry about going through an integer
tune –just ensure you do not linger!

  Electron Machine with Many Applications
(EMMA)
10-20 MeV electron accelerator
42 cells. 19 RF cavities.
Accelerates in ~16 turns




                                                                                                          18
                              R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
            Synchrotrons
Principle of operation: This is a constant radius accelerator where both the
frequency and the magnetic field vary with time in order to maintain the
synchronism condition.
                     qB
Now         ω=                      and it is straightforward to show that:
                     qBR 2       2       2

               m0 [1 +                       ]1 / 2
                         m0 c2       2




                qB       ⎛ ∂ω ⎞    q
          ω=         and ⎜    ⎟ =
               m0γ 3     ⎝ ∂B ⎠ R m0γ
                                      3



Therefore, angular frequency increases at B-dependent rate, initially fast
but then more and more slowly, as g>>1.
Energy range for p beams:
Conventional magnets give fields of up to 2T. There is a min. field of
about 10-2 T, hence Emin@5x10-3 Emax
Hence, the beam has to be injected with Emin, which is provided by a linac
or another synchrotron (‘booster’) or a linac-booster combination.
Superconducting magnets provide Bmax@8T and Bmin=?
                                                                   19
                 Synchrotrons
4.   Energy range for (e-,e+) beams:
•    At medium energies, previous comments apply.
•    Above about 10GeV, SR loss is severe. Accelerator cost
     optimization requires that maximum machine radius Rmax~(Emax)2.
     Hence, Bmax ~1/Emax. Typically, LEP had Bmax @0.1T.
5.   Synchrotrons do not need a continuous magnet around the orbit but
     operate with a ring of magnets. Therefore, they have achieved major
     savings in the magnet iron.
6.   These are the dominant machine for high energy physics.
     Although to limit SR, linacs are envisaged for future e-e+
     colliders –ILC (Low-loss SC) and CLIC (High gradient NC).




                                                                     20
Synchrotron Radiation
Accelerated charges emit radiation whose power (P) is given by:

          P=
             2 e2 6
             3c
                     [
                 γ β .β − ( β × β ) 2   ]   (Lienard, 1898)


For linear motion,
                                  dβ   1 dp
                β × β = 0 and        =          , hence
                                  dt m0 cγ 3 dt
                   2 e 2 dp 2 2 e 2 dE 2
                P=        ( ) =        ( )
                   3 m0 c dt
                      2 3
                                3 m0 c dx
                                   2 3




For a linear accelerator dE/dx must be of the order of 1014 MeV/m
before this becomes significant compared to power supplied by
external sources.
For circular machines
              2 e2 2 2 2 2 e2c 4 4
           P=            γ ω p =     β γ , where ρ = radius.
              3 m 02 c 3         3 ρ
           Note extra γ 2 term.                                   21
       Synchrotron Radiation
For b@1, the energy dE lost per turn has a numerical value given by:
                             E4
δE ( MeV ) = 8.85 x10   −2
                                  , ( E in GeV, ρ in m)
                             ρ
Some numbers:
For LEP at 86GeV, dE@1.37GeV/electron.
There are about 6x1012 electrons per beam and the hence power
required to make up for this loss is @20MW.
Power needed for RF is about 96MW.
SR was originally perceived as a nuisance!
Now it is a field with many important applications and has a large
user community (both university and industry), with large, dedicated
accelerators (ESRF at Grenoble, ALS at Brookhaven, USA and
DIAMOND at RAL,UK and SSRL at SLAC, USA etc).


                                                                       22
      Synchrotron Radiation
 Radiation is produced within a light cone
 of angle
      1         511
θ ≈       =            for speeds close to c
      γ       E [keV ]
 For electrons in the range 90 MeV to 1
 GeV, θ is in the range 10-4 - 10-5 degs.

 Such collimated beams can be directed
 with high precision to a target - many
 applications, for example, in industry.




                                                                                                  23
                      R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
             Diamond Light Source, RAL: April 2006
R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Beam Transport and Delivery
  The term is meant to cover the optics of the charged-particle beam
  and the systems that focus and steer the beam (together with
  associated diagnostic devices). These devices are both internal and
  external to the accelerating structure.
  The optics of beams is a broad field covered by a number of
  specialist books.
  Key concept is that of ‘emittance’: each particle of the beam can be
  represented by a single point in the 6-dimensional phase space (x, y,
  z, px, py, pz). The volume occupied by these points is the 6-D
  emittance (e) of the beam. Emittance must be as small as possible
  and the density of points in phase space (‘brightness’) as high as
  possible.
  We normally use the 3 projections of the hyper-volume (x,px) (y,py)
  and (z,pz). For a beam drifting in a field-free space in the z-direction,
  pz= const. and px,py<<pz. Introduce        p      p
                                        x' =    x
                                                    y' =    y

                                               pz          pz

                                                                         25
Beam Transport and Delivery
  The transverse emittances are usually plotted as (x,x’) and (y,y’)
  contours. For the third dimension (i.e. along the acceleration) we
  normally plot (Dz,Dpz), i.e. relative to a ‘reference’ particle.
  Equivalent phase spaces are (Dt,DE) or (f,DE).
  Emittances usually shown as ellipses in phase space, although they
  are frequently distorted. In reality, not all particles are contained
  within a single, elliptical contour. The usual convention is the quote
  the area of the ellipse occupied by 90% of the beam and, also, to
  include the factor p explicitly, e.g. 0.1 pmm.mrad.
  Normalized emittance en=bge, to take account of the fact that during
  acceleration pz increases and the measured emittance in, say, the
  (x,x’) plane decreases.




                                                                      26
                         Liouville
Liouville’s theorem:
‘In the absence of collisions or dissipative processes, the local density in
      phase space remains constant’.
Alternatively:
‘For a system of non-interacting particles that are in a conservative
      system, the density in phase space along the trajectory is invariant’.

There are a number of apparent violations of Liouville’s theorem, most of
     them leading to an increase of the beam emittance, e.g.
     Space charge forces between beam particles increase emittance.
     Beam-beam and beam-gas scattering do the same.
     Foil stripping of H- ions into H+, at the point of injection into a
     synchrotron, also increases emittance.
     Synchrotron radiation (see later) can lead to a reduction in phase-
     space volume (‘damping’).
                                                                          27
          Beam Cooling
The principle: the emittance is a measure of ‘disorder’ in the beam,
hence it is analogous to entropy. For a closed system DS>=0 but if
system is not closed, it is possible to reduce S (or emittance).
The beam ‘temperature’ is determined by transverse motion of
particles relative to the central orbit.
Methods available for cooling:
1. Electron cooling.
2. Stochastic cooling.
Electron cooling relies on bringing a p beam in contact with a low-
temperature e- beam, of the same b. Very effective for low energies,
up to about 100MeV.




                                                                 28
           Beam Cooling
Stochastic cooling relies on a pick-up electrode which detects large
amplitude excursions of a particle from its central orbit (‘betatron
oscillation’), sends a message to a ‘kicker’ downstream which
applies a correction field to reduce this amplitude. If g is gain of
detection system, W its bandwidth and N the number of circulating
particles, then cooling time t is given by:
                                2N
                       τ=
                            W (2 g − g 2 )
Ionization cooling. All previous methods are ‘slow’. The neutrino
factory will require fast cooling of the muons, before they decay.
Ionization cooling relies on successive reduction of momenta and,
then, addition of longitudinal momentum by acceleration. Net result
is the reduction of transverse/longitudinal ratio, i.e. cooling. MICE
experiment planned at RAL...


                                                                   29
Targets and Kinematics
Historically, fixed-target experiments came first.
Still in use now for nuclear physics, radioactive beam facilities,
neutron production, cancer therapy and neutrino factories.
However, the really important quantity is the ‘centre-of-mass’ (or,
better, the ‘centre-of-momentum’) energy T*, which depends on
mass of target and its motion.
In the relativistic case, m1 hitting stationary target m2, the CM
energy of the system is:
E * = c 2 m12 + m22 + 2γ 1m1m2
If m1=m2=m (e.g. proton-proton collision), then E* = mc 2γ + 2 and
                                                       2
                                                            1

the useful kinetic energy T* is only
T * = E * −2m0 c 2 = m0 c 2   [   2γ 1 + 2 − 2   ]
Use head-on colliding beams E* ≅ 2 E1E2.
For two colliding proton beams, each 28GeV: E* = 56GeV
To get this from a stationary target, one would require ~1.8TeV.
                                                                   30
                     Targets
Fixed target experiments are not finished yet!
A neutrino factory will be based on an intense p beam impinging on
fixed target, capture of pions and then of muons, and then
acceleration of the latter.
Similar problems for the design of neutron spallation sources and of
radioactive beam facilities.
 All the above areas will require considerable R/D in target design.




                                                                  31
             Luminosity
                               Measures interaction rate per unit cross
                               section - an important concept for colliders.
                               Simple model: Two cylindrical bunches of
                               area A. Any particle in one bunch sees a
                               fraction Nσ /A of the other bunch.
                               (σ=interaction cross section). Number of
                               interactions between the two bunches is
Area, A                         N2σ /A.
                               Interaction rate is R = f N2σ /A, and

                                                                  N2
                                      Luminosity             L= f
                                                                  A
                                      LHC and Fermilab p-pbar colliders have L ~
                                      1030 cm-2s-1. SSC was aiming for L ~ 1033
                                      cm-2s-1 and CLIC aims for L>1034

                                                                                      32
          R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
                                                Luminosity
Exercise 3: The luminosity L is a measure of the probability rate of
particle encounters in a collision process. Thus the total counting
rate of a physics event is R = σphysL, where σphys is the cross-
section of a physics process. For two beams colliding head-on the
luminosity is:

     L = 2 fN1 N 2 ∫ ρ1 ( x,z,s1 ) ρ 2 ( x,z,s2 )dx ⋅ dz ⋅ ds ⋅ d(βct)

Where s1=s+βct, s2=s-βct, f is the collision frequency N1 and N2
are the number of particles per bunch and ρ1 and ρ2 are the
nomalised distribution functions

1.      Using a Gaussian bunch distribution,

                              1        ⎛ x2   x2  x2 ⎞
     ρ ( x,z,s ) =                 exp ⎜ − 2 − 2 − 2 ⎟
                   ( 2π ) σ xσ zσ s ⎝ 2σ x 2σ z 2σ s ⎠
                         3/ 2


        where σx , σz, and σ are the horizontal vertical rms bunch
        widths and rms bunch length, show that the luminosity for two
        bunches with identical distribution profiles is:
                 fN1 N 2
           L=
                4πσ xσ z
2.      Also show that when the two beams are offset by a horizontal
        distance b, the luminosity is reduced by a factor
        exp ( −b 2 / 4σ x )
                        2

                                             R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
Accelerator Categories




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    R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
 Summary of Circular Accelerators
Machine            RF frequency                Magnetic Field Orbit Radius ρ Comment
                       ω/2π                         B
Cyclotron         constant                     constant                        increases with            Particles out of synch
                                                                               energy                    with RF; low energy
                                                                                                         beam or heavy ions

Isochronous       constant                     varies                          increases with            Particles in synch,
Cyclotron                                                                      energy                    but difficult to create
                                                                                                         stable orbits
Synchro-          varies                       constant                        increases with            Stable oscillations
cyclotron                                                                      energy
Synchrotron       varies                       varies                          constant                  Flexible machine,
                                                                                                         high energies
                                                                                                         possible
FFAG              varies                       constant in time,               increases with            Increasingly
                                               varies with radius              energy                    attraction option for
                                                                                                         21st century designs
                                   In General…
                                    Dipole magnets are used to bend the beam into a circular path
    p            qBc 2 v
ρ=            ω=      =             Quadrupole to focus the beam and maintain stability
   qB             E     ρ           (AG focusing)
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                             R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
              Summary of Linacs
Linacs            Periodic         Space Charge Wakefields                                          Comments

Low Energy    No                  Yes                            No                        For intense beams, at low energy
                                                                                           both space charge and
                                                                                           wakefields are issues. Cavities
                                                                                           must be modified in order to
                                                                                           remain in phase with changing
                                                                                           velocity of beam. At injection,
                                                                                           or close to the particle source,
                                                                                           space charge often dominates.

Medium-       ?                   ?                              Yes (1/γ2)                Once the beam is sufficiently
                                                                                           relativistic, cavities are periodic
High Energy                                                                                and wakefields are the main
                                                                                           issue. Electron cloud is another
                                                                                           serious concern.




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                        R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.
                            Linacs
Next week Linacs!




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            R.M. Jones, Physics of Particle Accelerators, PHYS 4722, Lecture 1, 2010.