Download Accelerators

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.
Efield = V / D
heated
filament
With electron charge q:
F = q . Efield
distance D
Potential diffence V
electron kinetic energy:
Ee- = ∫ F dD = q.V
Ee- 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: Ekin= ½ m0 v2 Ł v = sqrt(2 Ekin/ m0) = 94,000
km/s = 0.31 x c
Relativistic effects! So v is smaller, and m is larger than m0
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:
E2 = mo2 c4 + p2c2
≈ p2c2
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:
E2 = mo2 c4 + p2c2
With:
β = v / c, and the Lorentz factor γ:
relativistic mass mr = γ m0
γ = 1 / sqrt(1- β 2), and β = sqrt(γ2 -1) / γ
So: total energy E = m0 c2 sqrt(1+ β 2 γ2) [= rest mass + kinetic energy]
= γ m0 c2 = mr c2
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
Pe
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: E2 = m2c4 + p2 c2 = (511 + 0.012) 2 (keV) 2
p2 c2 = (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
= q v ×B
dt
The speed of a charged particle, and therefore its γ, does not change by a static
magnetic field:
dv
= q v × B (1)
γm
dt
If s is equal to the distance along the particle trajectory: ds = v dt and if x is the position
vector of the particle:
2
d2x
d 2 x q dx
dx 1 dx v
2d x
=
v
=
× B (2)
and
using
(1):
=
= and:
Then:
ds v dt v
dt 2
ds2
ds 2 p ds
(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
Color TV in Earth magnetic field
B┴ ~ 10 µT (varies with latitude!)
E2 = mo2 c4 + p2c2 = {511 keV + 25 keV) 2 = (536 keV) 2
p2c2 = (536 keV) 2 - (511 keV) 2
pe = 162 keV/c
(note: not E/c, not very relativistic!)
ρ = pe / (0.2998 B┴ ) = 53 m
Shift Sh after D = 0.2 m: Sh = D2 / (2 ρ) = 0.4 mm
(p in GeV/c, B in T, ρ in m,
for 1 elementary charge
unit = 1.602177x10-19 C,
and obtained using
1 eV/c2 = 1.782663x10-36 kg
and c = 299792458 m/s )
D
Sh
ρ
The cyclotron
"Dee": conducting,
non-magnetic box
Top view
Constant
magnetic field
Side view
~
r.f. voltage
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".
Focussing magnet
r.f. cavity
Vacuum beam line
Bending magnet
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
At time of operation of LEP
Fast extraction
of remainder of beam
SPS used as
injector for LEP
The cycle of the SPS magnets, typical for synchrotrons.
For LHC related
studies
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
N = L σ cross-section
number of events /s
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 = 4πe2β3γ4/(3ρ) = 4πe2p4/ (3m4βρ), where ρ is the radius of the orbit
The time per turn is 2πρ/(βc), so the loss per second is 2e2p4c/(3m4ρ2)
-> For the same energy and orbit radius electrons and positrons lose about
20004 more energy then protons -> reason for large radius of LEP
For high-energy electrons (β=1): ∆E = 4πe2β3γ4/(3ρ) =4πremeE4/(3ρme4)
(re = classical electron radius = 2.82 fm)
∆E = 8.85 10-2 E4/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.
pp collisions
2) heavy collisions:
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