PHYS 3446 – Lecture #14
Wednesday, Oct. 25, 2006
Dr. Jae Yu
1. Particle Accelerators
•
•
•
Electro-static Accelerators
Cyclotron Accelerators
Synchrotron Accelerators
2. Elementary Particle Properties
•
•
•
•
•
Forces and their relative magnitudes
Elementary particles
Quantum Numbers
Gell-Mann-Nishijima Relations
Production and Decay of Resonances
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
Jae Yu
1
Announcements
• Quiz in class next Wednesday, Nov. 1
Wednesday, Oct. 25, 2006
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Principles of Calorimeters
Total absorption
calorimeter: See
the entire shower
energy
Sampling calorimeter:
See only some fraction
of shower energy
For EM
Absorber plates
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PHYS 3446, Fall 2006
Jae Yu
X 0vis
E  fEvis  abs
Evis
X 0  X 0vis
0vis
Evis
For HAD E  fEvis  abs
0  30vis
Example Hadronic Shower (20GeV)
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Particle Accelerators
• How can one obtain high energy particles?
– Cosmic ray  Sometimes we observe 1000TeV cosmic rays
• Low flux and cannot control energies too well
• Need to look into small distances to probe the fundamental
constituents with full control of particle energies and fluxes
– Particle accelerators
• Accelerators need not only to accelerate particles but also to
– Track them
– Maneuver them
– Constrain their motions to the order of 1mm
• Why?
– Must correct particle paths and momenta to increase fluxes and control
momenta
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
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Particle Accelerators
• Depending on what the main goals of physics are, one needs
different kinds of accelerator experiments
• Fixed target experiments: Probe the nature of the nucleons 
Structure functions
– Results also can be used for producing secondary particles for further
accelerations  Tevatron anti-proton production
• Colliders: Probes the interactions between fundamental
constituents
– Hadron colliders: Wide kinematic ranges and high discovery potential
• Proton-anti-proton: TeVatron at Fermilab, Sp`pS at CERN
• Proton-Proton: Large Hadron Collider at CERN (late 2007)
– Lepton colliders: Very narrow kinematic reach, so it is used for precision
measurements
• Electron-positron: LEP at CERN, Petra at DESY, PEP at SLAC, Tristan at KEK,
ILC in the med-range future
• Muon-anti-muon: Conceptual accelerator in the far future
– Lepton-hadron colliders: HERA at DESY
Wednesday, Oct. 25, 2006
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Electrostatic Accelerators: Cockcroft-Walton
• Cockcroft-Walton Accelerator
– Pass ions through sets of aligned DC electrodes at successively
increasing fixed potentials
– Consists of ion source (hydrogen gas) and a target with the electrodes
arranged in between
– Acceleration Procedure
• Electrons are either added or striped off of an atom
• Ions of charge q then get accelerated through series of electrodes, gaining kinetic
energy of T=qV through every set of electrodes
• Limited to about 1MeV acceleration due to
voltage breakdown and discharge beyond
voltage of 1MV.
• Available commercially and also used as the
first step high current injector (to ~1mA).
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Electrostatic Accelerators: Van de Graaff
• Energies of particles through DC accelerators are
proportional to the applied voltage
• Robert Van de Graaff developed a clever mechanism to
increase HV
– The charge on any conductor resides on its outermost
surface
– If a conductor carrying additional charge touches another
conductor that surrounds it, all of its charges will transfer to
the outer conductor increasing the charge on the outer
conductor, increasing HV
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
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Electrostatic Accelerators: Van de Graaff
• Sprayer adds positive charge to
the conveyor belt at corona points
• Charge is carried on an insulating
conveyor belt
• The charges get transferred to the
dome via the collector
• The ions in the source then gets
accelerated to about 12MeV
• Tandem Van de Graff can
accelerate particles up to 25 MeV
• This acceleration normally occurs
in high pressure gas that has very
high breakdown voltage
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
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Resonance Accelerators: Cyclotron
• Fixed voltage machines have intrinsic
limitations in their energy due to
breakdown
• Machines using resonance principles
can accelerate particles in higher
energies
• Cyclotron developed by E. Lawrence
is the simplest one
• Accelerator consists of
– Two hallow D shaped metal chambers
connected to alternating HV source
– The entire system is placed under
strong magnetic field
Wednesday, Oct. 25, 2006
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Resonance Accelerators: Cyclotron
• While the D’s are connected to HV sources,
there is no electric field inside the chamber
due to Faraday effect
• Strong electric field exists only in the gap
between the D’s
• An ion source is placed in the gap
• The path is circular due to the
perpendicular magnetic field
• Ion does not feel any acceleration inside a
D but gets bent due to magnetic field
• When the particle exits a D, the direction of
voltage can be changed and the ion gets
accelerated before entering into the D on
the other side
• If the frequency of the alternating voltage is
just right, the charged particle gets
accelerated continuously until it is extracted
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Resonance Accelerators: Cyclotron
• For non-relativistic motion, the frequency appropriate for
alternating voltage can be calculated from the fact that the
magnetic force provides centripetal acceleration for a circular
orbit
v qB
v2
vB
m
r
q
r
c

mc
w
• In a constant angular speed, w=v/r. The frequency of the
motion is

qB
1  q B
f
2

2 mc

2  m  c
• Thus, to continue accelerate the particle the electric field
should alternate in this frequency, cyclotron resonance
frequency
• The maximum kinetic energy achievable for an cyclotron with
2
radius R is
 qBR 
1
1
2
2 2
T

mv

m

Wednesday, Oct. 25, 2006max
2006
maxPHYS 3446, Fall R
2
2Jae Yu
mc 2
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Resonance Accelerators: Linear Accelerator
• Accelerates particles along a linear path using resonance principle
• A series of metal tubes are located in a vacuum vessel and connected
successively to alternating terminals of radio frequency oscillator
• The directions of the electric fields changes before the particles exits the
given tube
• The tube length needs to get longer as the particle gets accelerated to
keep up with the phase
• These accelerators are used for accelerating light particles to very high
energies
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PHYS 3446, Fall 2006
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Synchroton Accelerators
• For very energetic particles, the relativistic effect must be taken
into account
• For relativistic energies, the equation of motion of a charge q
under magnetic field B is
dv
vB
m
dt
 m v    q
• For v ~ c, the resonance frequency becomes
n
c

1 q 1B
  
2 2  m   c
• Thus for high energies, either B or n should increase
• Machines with constant B but variable n are called synchrocyclotrons
• Machines with variable B independent of the change of n is
called synchrotrons
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PHYS 3446, Fall 2006
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Synchroton Accelerators
• Electron synchrotrons, B varies while n is held
constant
• Proton synchrotrons, both B and n varies
• For v ~ c, the frequency of motion can be expressed
• For an electron
1 v
c
f

2 R 2 R
p  GeV / c 
pc
R(m) 

qB 0.3B Tesla) 
• For magnetic field strength of 2Tesla, one needs radius
of 50m to accelerate an electron to 30GeV/c.
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Synchroton Accelerators
• Synchrotons use magnets
arranged in a ring-like
fashion.
• Multiple stages of
accelerations are needed
before reaching over GeV
ranges of energies
• RF power stations are
located through the ring to
pump electric energies into
the particles
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Forewords
• What are elementary particles?
– Particles that make up matter in the universe
• What are the requirements for elementary particles?
– Cannot be broken into smaller pieces
– Cannot have sizes
• The notion of “elementary particles” have changed from
1930’s through present
– In the past, people thought protons, neutrons, pions, kaons, rmesons, etc, as elementary particles
• Why?
– Due to the increasing energies of accelerators that allows us to
probe smaller distance scales
• What is the energy needed to probe 0.1–fm?
– From de Broglie Wavelength, we obtain
P
Wednesday, Oct. 25, 2006

PHYS 3446, Fall 
2006
Jae Yu
c 197fm  MeV 2000MeV / c


c
0.1fm c
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Forces and Their Relative Strengths
• Classical forces:
– Gravitational: every particle is subject to this force, including
massless ones
• How do you know?
– Electromagnetic: only those with electrical charges
– What are the ranges of these forces?
• Infinite!!
– What does this tell you?
• Their force carriers are massless!!
– What are the force carriers of these forces?
• Gravity: graviton (not seen but just a concept)
• Electromagnetism: Photons
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Forces and Their Relative Strengths
• What other forces?
– Strong force
• Where did we learn this force?
– From nuclear phenomena
– The interactions are far stronger and extremely short ranged
– Weak force
• How did we learn about this force?
– From nuclear beta decay
– What are their ranges?
• Very short
– What does this tell you?
• Their force carriers are massive!
• Not really for strong forces
• All four forces can act at the same time!!!
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
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Forces’ Relative Strengths
• The strengths can be obtained through potential, considering
two protons separated by a distance r.
• Magnitude of Coulomb
and gravitational potential are
2
2
e
e
Fourier x-form
VEM  r  
VEM  r   2
r
q
2
2
G
m
GN m Fourier x-form
Vg  r   N 2
Vg  r  
q
r
– q: magnitude of the momentum transfer
• What do you observe?
– The absolute values of the potential E decreases quadratically with
increasing momentum transfer
– The relative strength is, though independent of momentum transfer
 e2  1
e2
VEM
c5  1  1 1039 GeV 2
36



~
10


2
2 3446, Fall 2006
Wednesday,
Oct.
2006
PHYS
20
2
G25,
m
Vg
c
GN  137  1GeV 2
6.7
N
  mc


Jae Yu
Forces’ Relative Strengths
• Using Yukawa potential form, the magnitudes of strong and
weak potential can be written as
2
VS  r  
g S2
m c 2r

e c
Fourier x-form
r
mW c 2r
2
gW  c
VW  r  
e
r
Fourier x-form
VS  r  
VW  r  
gS
q 2  m2 c 2
gW2
q 2  mW2 c 2
– gW and gs: coupling constants or effective charges
– mW and m: masses of force mediators
• The values of the coupling constants can be estimated from
2
2
experiments
g
g
S
c
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 15
PHYS 3446, Fall 2006
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W
c
 0.004
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Forces’ Relative Strengths
• We could think of  as the strong force mediator w/
m  140MeV / c 2
2
m

80
GeV
/
c
• From observations of beta decays, W
• However there still is an explicit dependence on
momentum transfer
– Since we are considering two protons, we can replace the
momentum transfer, q, with the mass of protons
2 2
qc
Wednesday, Oct. 25, 2006
2 4
 mp c
 1GeV
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Forces’ Relative Strengths
• The relative strength between EM and strong
potentials is
g S2
2
VS
c
q


2
2
2 2
VEM
c e q  m c
2
gS
c
2 4
m pc
c
2
2 4
2 2
e m p c  m c
 15  137  1  2  103
• And that between weak and EM potentials is
VEM e c q 

c gW2
q
VW
2
2
mW2 c 2
2
2 4
2 2
m
c

m
e c p
Wc

c gW2
m 2p c 4
2
1
1


 802  1.2  104
137 0.004
Wednesday, Oct. 25, 2006
PHYS 3446, Fall 2006
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Interaction Time
• The ranges of forces also affect interaction time
– Typical time for Strong interaction ~10-24sec
• What is this?
• A time that takes light to traverse the size of a proton (~1 fm)
– Typical time for EM force ~10-20 – 10-16 sec
– Typical time for Weak force ~10-13 – 10-6 sec
• In GeV ranges, the four forces are different
• These are used to classify elementary particles
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Elementary Particles
• Before the quark concepts, all known elementary
particles were grouped in four depending on the
nature of their interactions
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Elementary Particles
• How do these particles interact??
– All particles, including photons and neutrinos, participate in
gravitational interactions
– Photons can interact electromagnetically with any particles
with electric charge
– All charged leptons participate in both EM and weak
interactions
– Neutral leptons do not have EM couplings
– All hadrons (Mesons and baryons) responds to the strong
force and appears to participate in all the interactions
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Elementary Particles: Bosons and Fermions
• All particles can be classified as bosons or fermions
– Bosons follow Bose-Einstein statistics
• Quantum mechanical wave function is symmetric under exchange
of any pair of bosons
 B  x1, x2 , x3 ,...xi ...xn    B  x2 , x1, x3 ,...xi ...xn 
• xi: space-time coordinates and internal quantum numbers of
particle i
– Fermions obey Fermi-Dirac statistics
• Quantum mechanical wave function is anti-symmetric under
exchange of any pair of Fermions
 F  x1 , x2 , x3 ,...xi ...xn    F  x2 , x1, x3 ,...xi ...xn 
• Pauli exclusion principle is built into the wave function
– For xi=xj,
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 F   F
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Bosons, Fermions, Particles and Antiparticles
• Bosons
– All have integer spin angular momentum
– All mesons are bosons
• Fermions
– All have half integer spin angular momentum
– All leptons and baryons are fermions
• All particles have anti-particles
– What are anti-particles?
• Particles that has same mass as particles but with opposite quantum numbers
– What is the anti-particle of
•
•
•
•
A 0 ?
A neutron?
A K0?
A Neutrinos?
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Quantum Numbers
• When can an interaction occur?
– If it is kinematically allowed
– If it does not violate any recognized conservation laws
• Eg. A reaction that violates charge conservation will not occur
– In order to deduce conservation laws, a full theoretical
understanding of forces are necessary
• Since we do not have full theory for all the forces
– Many of general conservation rules for particles are based on
experiments
• One of the clearest conservation is the lepton number
conservation
– While photon and meson numbers are not conserved
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Baryon Numbers
• Can the decay p  e   0 occur?
– Kinematically??
• Yes, proton mass is a lot larger than the sum of the two masses
– Electrical charge?
• Yes, it is conserved
• But this decay does not occur (<10-40/sec)
– Why?
• Must be a conservation law that prohibits this decay
– What could it be?
•
•
•
•
An additive and conserved quantum number, Baryon number (B)
All baryons have B=1
Anti-baryons? (B=-1)
Photons, leptons and mesons have B=0
• Since proton is the lightest baryon, it does not decay.
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Assignments
1. Carry out Fourier transformation and derive
equations 9.3 and 9.5
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