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Department of Physics and
Astronomy
Option 212: UNIT 2
Elementary Particles
SCHEDULE
7-Feb-08 1.00pm
Dr Matt Burleigh Intro lecture
11-Feb-08 9.00am
Prof Peter Maksym Problem solving
(15-Feb-08 1.00pm
18-Feb-08 9.00am
Problem Workshop)
Dr Matt Burleigh Follow-up
UNIT 2: OUTLINE SYLLABUS:
1st Lecture Introduction
Hadrons and Leptons
Spin & Anti-Particles
The conservation laws: Lepton Number
Baryon number
Strangeness
2nd Lecture Problem solving
Check a decay for violation of conservation laws
Quarks
Properties of a particle given quark combination
3rd Lecture Follow-up
Fundamental forces and field particles
The standard model
Recommended Books
 Chapter 41, PA Tipler
 Quarks Leptons and The
Big Bang, J Allday
 The Cosmic Onion, F Close
Web Sites
 Brief introduction to Particle Physics
http://superstringtheory.com/experm/index.html
 Introductions to Particle Physics
http://www.physics.ox.ac.uk/documents/WebGuide/default.html
 CERN web site
http://public.web.cern.ch/Public/
 212 Option - Lecture notes in MS Powerpoint
http://www.star.le.ac.uk/~mbu/
INTRODUCTION
to
Elementary Particle Physics
Fundamental building blocks of
which all matter is composed:
Elementary Particles
* Pre-1930s it was thought there
were just four elementary particles
electron
proton
neutron
photon
Cosmic Rays
*
1932 positron or anti-electron discovered, followed
by many other particles (muon, pion etc)
We will discover that the electron and photon are
indeed fundamental, elementary particles, but
protons and neutrons are made of even smaller
elementary particles called quarks
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are needed to see this picture.
CLASSIFICATON OF PARTICLES
An elementary particle is a point particle without structure
that is not constructed from more elementary entities
With the advent of particle accelerator
in the 1950’s many new elementary
particles were discovered.
The question arose whether
perhaps there were too
many to all be elementary.
This has led to the need
for classification of
particles.
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FUNDAMENTAL INTERACTIONS AND THE
CLASSIFICATION OF PARTICLES
Fundamental interactions
o gravitation
o electromagnetic
o strong nuclear force
o weak nuclear force
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Participating particles
• all particles with mass
• those carrying charge
• Hadrons (and quarks)
• Leptons (and quarks)
HADRONS
Hadrons interact through strong forces.
There are two classes, mesons and
baryons.
Mesons have zero or integral spin (0
or 1) with masses that lie between the
electron and the proton.
Baryons have half integral spin (1/2 or
3/2) and have masses that are always
greater than or equal to that of the
proton.
Hadrons are not elementary particles.
As we will see later, they are made of
quarks
LEPTONS
Leptons interact through weak interactions, but not via the strong force.
All leptons have spin of 1/2. There are
six kinds of lepton: electron e-, muon
m-, and tau t -, and 3 neutrinos ne, nm, nt
Note that each distinct neutrino is
associated with one of the other
leptons
Leptons were originally
named because they were
“Light-particles”, but we now
know the Tau is twice as
heavy as a proton
Neutrinos were originally
thought to be massless, but
they probably have a small
mass
Read more in Tipler p. 1336
Beta Decay and the discovery of the neutrino (Tipler p.1314)
-×
He
+
e
2
3
3
-
1H
-+ n
He
+
e
2
e√
3
3
1H
Energy Distribution
1.2
Relative
intensity
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
20
Energy (kev)
• In Beta decay a neutron decays into a proton plus an electron
• If decay energy shared by proton and emitted electron, energy of
electron would be unique
• But observed electrons have a range of energies – must be a third
particle involved: the neutrino
• Third particle must have no charge or mass, as they are accounted for
by the He nucleus and electron.
Spin
A particle has an intrinsic spin angular momentum
Spin ½ particles:
Electrons, protons, neutrons and neutrinos all have an
intrinsic spin characterised by the quantum number s = 1/2
Particles with half-integer spin (1/2, 3/2, 5/2, …) are called
Fermions
They obey the Pauli exclusion principle
Particles with integer spin (s = 0, 1, 2, …. ),
e.g. mesons, are called Bosons
They do not need to obey the Pauli exclusion
principle, and any number can occupy the same
quantum state
Matter & Antimatter
Every particle has an antiparticle partner
Read Tipler P.1339 to find out how Dirac predicted the existence of anti-particles in 1927
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Here are some examples
e- - electron
p - proton
e+ - positron
p - antiproton
n - neutron
n - antineutron
n - neutrino
n - antineutrino
Antimatter
For each particle there is
an associated
antiparticle
Anti-particles always created
in particle-anti particle pairs
s
Electron Pair Production
s
g -> e- + e+
Eg  2 x 511 keV
e-
e+
Antimatter
* Antiparticle has the same mass
and magnitude of spin as the
particle
Electron Pair Annihilation
* Antiparticle has the opposite
charge to the particle
* The positron is stable but has a
short-term existence because our
Universe has a large supply of
electrons
* The fate of a positron is
annihilation
s
ss
e- + e+ ->2g
e- s Each photon gets
eg = mec2
pg = mec
moc2
s = 1/2
e+
moc2
s = 1/2
Some Fundamental Particles
Particle
Mass less
boson
photon
Symbol
g
Rest energy MeV
0
Charge
Spin
Antiparticle
0
1
g
Leptons
Neutrino
Electron
Muon
n
em-
0
0.511
105.7
0
-1
-1
1/2
1/2
1/2
n
e
m
Meson
Pion

o
140
135
+1
0
0
0
o
Baryons
Proton
neutron
p+
no
938.3
939.6
+1
0
1/2
1/2
p-
-
n
The Conservation Laws
Can a conceivable reaction or decay occur?
• Conservation of energy
The total rest mass of the decay products must be less
than the initial rest mass of the particle before decay
• Conservation of linear momentum
When an electron and a positron at rest annihilate, two
photons must be emitted
• Angular momentum must be conserved in a decay or
reaction
• Net electric charge before must equal net charge
after a decay or reaction
The Conservation Laws
Can a conceivable reaction or decay occur?
• Conservation of Baryon number
We assign Baryon Number B=+1 to all Baryons, B=-1 to
all anti-Baryons, and B=0 to all other particles
Baryon number must be conserved in a reaction
• Conservation of Lepton number
Lepton number must be conserved in a reaction
BUT…..
The Conservation Laws
Can a conceivable reaction or decay occur?
• Conservation of Lepton number contd:
…..because the neutrino associated with an electron is
different to a neutrino associated with a muon, we assign
separate Lepton numbers Le, Lm and Lt to the particles
e.g. for e and ne, Le=+1, for their anti-particles Le=-1,
and for all other leptons and other particles Le=0
• Conservation of Strangeness
There are other conservation laws which are
not universal, e.g. strange particles have a
property called strangeness which must be
conserved in a decay or reaction
Some Fundamental Particles
Category
Particle
Symbol
Photon
photon
g
Leptons
Neutrino
Electron
Muon
Tau
n
emt-
Pion

o
K+
Ko
Hadrons
Mesons
Kaon
Baryons
Rest energy MeV
0
0
0.511
105.7
1784
140
135
493.7
497.7
938.3
p+
939.6
no
1115.6
L
1189.4

1192.5

1197.3
See also Tipler Table 41-1 Page 1337
For strangeness, examine Figure 41-3 Page 1344
Proton
Neutron
Lambda
Sigma
B Le Lm Lt S
Antiparticle
0
0
g




  
  
  
  




n
e
m
t












 
 
 
 
o
K
_
Ko




















-
-
-
-
p_n_

L
_
_ 
_ 
-
0 0
0
Department of Physics and
Astronomy
Option 212: UNIT 2
Elementary Particles
SCHEDULE
7-Feb-08 1.00pm
Dr Matt Burleigh Intro lecture
11-Feb-08 9.00am
Prof Peter Maksym Problem solving
(15-Feb-08 1.00pm
18-Feb-08 9.00am
Problem Workshop)
Dr Matt Burleigh Follow-up
UNIT 2: OUTLINE SYLLABUS:
1st Lecture Introduction
Hadrons and Leptons
Spin & Anti-Particles
The conservation laws: Lepton Number
Baryon number
Strangeness
2nd Lecture Problem solving
Check a decay for violation of conservation laws
Quarks
Properties of a particle given quark combination
3rd Lecture Follow-up
Fundamental forces and field particles
The standard model
Recommended Books
 Chapter 41, PA Tipler
 Quarks Leptons and The
Big Bang, J Allday
 The Cosmic Onion, F Close
Web Sites
 Brief introduction to Particle Physics
http://superstringtheory.com/experm/index.html
 Introductions to Particle Physics
http://www.physics.ox.ac.uk/documents/WebGuide/default.html
 CERN web site
http://public.web.cern.ch/Public/
 212 Option - Lecture notes in MS Powerpoint
http://www.star.le.ac.uk/~mbu/
Checking Baryon Numbers
_
a) p+ + n
b) p+ + n
2p+ +_p +_n
p+ + p + p
Answer: a) B = 1+1 on left hand side
B = 2 on right hand side too!
Allowed reaction!
b) B = 2 on left hand side
B = -1 on right hand side
Forbidden reaction
Checking Lepton Numbers
µ-
a)
b) π+
_
e- + ne + nm
µ+ + nm + ne
Answer: a) Before decay Le = 0 and Lm = +1
After decay Le = 0 and Lm = +1
Allowed reaction!
b) Before decay Lm = 0 and Le = 0
After decay Lm = 0 and Le = 1
Forbidden reaction!
Is Strangeness Conserved?
a) π+ + n
b) π- + p
K+ +  -+ 
Answer: a) Initial state has S = 0
Final state has S = +1 - 1 = 0
Allowed reaction!
b) Initial state has S = 0
Final state has S = -1
Forbidden reaction!
Conservation Laws
• Test the following decays for violation of the conservation
of electric charge, baryon number and lepton number.
• (a) n ->   -  m  m• (b)  - e+ + e- + g
Conservation Laws
Solution
• Method: Use Table 41-1 and the conservation laws for
Baryon number and Lepton number
• (a) n ->   -  m  m– Total charge on both sides = 0 : conserved
– Baryon number changes from +1 to 0: violated
– Lm = 0 on both sides : conserved
– Process not allowed
• (b)  - e+ + e- + g
– Total charge on both sides = 0 : conserved
– Baryon number on both sides = 0 : conserved
– Le = 0 on both sides: conserved
– Process is allowed
Quarks - The Smallest Building Blocks of Matter
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
Gell–Mann & Zweig 1963
Three Different Types of QUARKS
There are three elementary quarks (flavors)
That make up the fundamental particles:
Up
Down
Strange
Name
Up
Down
Strange
u
d
s
u
d
s
Spin Charge
1/2
+2/3
1/2
-1/3
1/2
-1/3
π+
u
d
p
Baryon
Baryon Strangeness
1/3
0
1/3
0
1/3
-1
Anti-quarks maintain spin, but change sign of S and B!
Meson
u
u
d
Different types of quarks contd.
• Mesons – quark + anti-quark ( q q )
• Baryons – three quarks ( q q q )
• Anti-baryons – three anti-quarks ( q q q)
By 1967 it was realised that new kinds of quarks were required
to explain discrepancies between the model and experiment
Charm (c)
Bottom (b) – discovered 1977
Top (t) – discovered 1995
Quark combinations
• Find the baryon number, charge & strangeness of the
following quark combinations and identify the hadron:
• (a) uud
• (b) udd
• (c) uus
• (d) dds
Quark combinations
Solution
Method: for each quark combination determine the baryon number B, the
charge q and the strangeness S; then use Tipler Table 41-2 to find a
match.
• (a) uud
–
–
–
–
B = 1/3 + 1/3 + 1/3 = 1
q = 2/3 + 2/3 – 1/3 = 1
S=0
It is a proton
• (b) udd
–
–
–
–
B = 1/3 + 1/3 + 1/3 = 1
q = 2/3 – 1 /3 – 1/ 3 = 0
S=0
It is a neutron
• (c) uus
– Ditto, B=1, q=1, S= -1 and it is a +
• (d) dds
– Ditto, B=1, q=-1, S= -1 and it is a -
Quark spin
• The angular momentum vector of a spin ½ quark
can have one of two settings up or down
• So a meson can have its two quark spins parallel
with each other or anti-parallel:
Spin 1
Spin 0
Quark spin contd.
• Baryons e.g. uud:
Spin 3/2
Spin 1/2
The spin ½ particle is a proton, spin 3/2 particle is a D
Note that
is also spin ½ (parallel, parallel, anti-parallel)
EIGHT FOLD WAY PATTERNS
(ddu)
(uud)
n
L
-



-
The Baryon
Octet Eight Spin 1/2
Baryons
S=0
p
S = -1
S = -2
Q = +1
Q = -1
Q=0
Department of Physics and
Astronomy
Option 212: UNIT 2
Elementary Particles
SCHEDULE
7-Feb-08 1.00pm
Dr Matt Burleigh Intro lecture
11-Feb-08 9.00am
Prof Peter Maksym Problem solving
(15-Feb-08 1.00pm
18-Feb-08 9.00am
Problem Workshop)
Dr Matt Burleigh Follow-up
UNIT 2: OUTLINE SYLLABUS:
1st Lecture Introduction
Hadrons and Leptons
Spin & Anti-Particles
The conservation laws: Lepton Number
Baryon number
Strangeness
2nd Lecture Problem solving
Check a decay for violation of conservation laws
Quarks
Properties of a particle given quark combination
3rd Lecture Follow-up
Fundamental forces and field particles
The standard model
Recommended Books
 Chapter 41, PA Tipler
 Quarks Leptons and The
Big Bang, J Allday
 The Cosmic Onion, F Close
Web Sites
 Brief introduction to Particle Physics
http://superstringtheory.com/experm/index.html
 Introductions to Particle Physics
http://www.physics.ox.ac.uk/documents/WebGuide/default.html
 CERN web site
http://public.web.cern.ch/Public/
 212 Option - Lecture notes in MS Powerpoint
http://www.star.le.ac.uk/~mbu/
• State which of the following decays or reactions violates
one or more of the conservation laws, and give the law(s)
violated in each case:
• (a) p -> n + e+ + ne
• (b) n -> p + • (c) e+ + e- -> g
• (d) p + p -> g  g
• (e) ne + p -> n + e+
(a) mp < mn : energy conservation is violated. Also Le=0 on lhs, but Le=-2 on rhs
(b) mn < mp + m : energy conservation is violated
(c) Momentum conservation is violated: in pair annihilation, two photons (g rays)
must be emitted to conserve momentum
(d) Allowed
(e) Le=-1 on both sides, but mp < mn so energy conservation violated
•





•
Consider the following decay chain
 - L  
L - p   - g  g
- - m-  nm
m- - e- + ne + nm
(a) write the overall decay reaction for  to the final decay
products
• (b) are the final decay products stable?
• (c) Check the overall decay reaction for the conservation of
electric charge, baryon number, and lepton number
• (d) Check the overall decay reaction for conservation of
strangeness. Is the reaction possible via the weak or strong
interactions?
(a)  - p  2g  nm  e- + ne + nm
• (b) Use Table 41-1. The proton is stable for 1031
years. In contrast, the neutron is only stable for
930secs. Answer: yes, stable.
• (c) Charge conservation: 0 -> p + e- = 0:
conserved. Baryon number 1 -> 1: conserved.
Lepton number Le: 0 -> e- + ne = 1 + (-1) = 0:
conserved. Lm: 0 -> -1 + 1 = 0.
• (d) See Tipler p.1344. Strangeness must be
conserved if reaction occurs via strong interaction.
Here S=-2 on lhs and S=0 on rhs. But if DS=+/-1,
then can occur via weak interaction. In first two
parts of reaction, DS=1 (L0 has S=-1) so is allowed
via weak interaction.
True or false?
• (a) Leptons consist of three quarks
• (b) Mesons consist of a quark and an antiquark
• (c) The six flavors of quark are up, down,
charmed, strange, left and right
• (d) Neutrons have no charm
(a) False: leptons are fundamental particles e.g e(b) True
(c) False: there is no left and right quark, but there are top and
bottom quarks
(d) True: neutrons are made of udd quarks
Quark confinement
• No isolated quark has ever been observed
• Believed impossible to obtain an isolated quark
• If the PE between quarks increases with separation
distance, an infinite amount of energy may be required to
separate them
• When a large amount of energy is added to a quark system,
like a nucleon, a quark-antiquark pair is created
– Original quarks remain confined in the original system
• Because quarks always confined, their mass cannot be
accurately known
Quark color
•
•
•
Consider the W- particle, which consists of three strange quarks
Remember that quarks have spin ½
The W- has spin 3/2, so its three strange quarks must be arranged thus:
•
But Pauli exclusion principle forbids these identical (same flavor, same mag of
spin, same direction of spin) quarks occupying identical quantum states
The only way for this to work is if each quark possesses a further property,
color:
•
•
•
Quarks in a baryon always have these three colours, such that when combined
they are “color-less” ( qr , qy , qb )
In a meson, a red quark and its “anti-red” quark attract to form the particle
Field Particles
• In addition to the six fundamental leptons (e-, m-, t-, ne, nm,
nt) and six quarks, there are field particles associated with the
fundamental forces (weak, strong, gravity and electromagnetic)
• For example, the photon mediates the electro-magnetic
interaction, in which particles are given the property “charge”
– The theory governing electro-magnetic interactions at the quantum
level is called Quantum Electrodynamics (QED)
• Similarly, gravity is mediated by the graviton
– The “charge” in gravity is mass
– The graviton has not been observed
Field Particles
• The weak force, which is experienced by quarks and
leptons, is carried by the W+, W-, and Z0 particles
– These have been observed and are massive (~100 GeV/c2)
– The “charge” they mediate is flavor
• The strong force, which is experienced by quarks and
hadrons, is carried by a particle called a gluon
– The gluon has not been observed
– The “charge” is color
– The field theory for strong interactions (analagous to QED) is
called Quantum Chromodynamics (QCD)
Electroweak theory
• The electromagnetic and weak interactions are considered
to be two manifestations of a more fundamental
electroweak interaction
• At very high energies, >100GeV the electroweak
interaction would be mediated (or carried) by four
particles: W+, W-, W0, and B0
• The W0 and B0 cannot be observed directly
• But at ordinary energies they combine to form either the Z0
or the massless photon
• In order to work, electroweak theory requires the existence
of a particle called the Higgs Boson
– The Higgs Boson is expected have a rest mass > 1TeV/c2
– Head-on collisions between protons at energies ~20TeV are
required to produce a Higgs Boson (if they exist)
– Such energies will only be achieved by the next generation of
particle accelerators (eg Large Hadron Collider at CERN)
The Standard Model
• The combination of the quark model, electroweak theory and QCD is
called the Standard Model
• In this model, the fundamental particles are the leptons, the quarks and
the force carriers (photon, W+, W-, Z0, and gluons)
• All matter is made up of leptons or quarks
– Leptons can only exist as isolated particles
– Hadrons (baryons and mesons) are composite particles made of quarks
• For every particle there is an anti-particle
• Leptons and Baryons obey conservation laws
• Every force in nature is due to one of four basic interactions:
– Stong, electromagnetic, weak and gravitational
• A particle experiences one of these basic interactions if it carries a
charge associated with that interaction
Properties of the basic interactions
Gravity
Weak
Electro- Strong
magnetic
Acts on
Mass
Flavor
Electric
charge
Particles
participating
All
Quarks,
leptons
Mediating
particle
Graviton W+, W-,
Z0
Electrical Quarks,
ly
Hadrons
charged
Photon
Gluon
Color
Grand Unified Theories (GUTs)
• In a GUT, leptons and quarks are considered to be two
aspects of a single class of particle
– Under certain conditions a quark could change into a lepton and
vice-versa
– Particle quantum numbers are not conserved
• These conditions are thought to have existed in the very
early Universe
– A fraction of a second after the Big Bang
– In this period a slight excess of quarks over anti-quarks existed,
which is why there is more matter than anti-matter in out Universe
today
• One of the predictions of GUTs is that the proton will
decay after 1031 years
– In order to observe one decay, a large number of protons must be
observed
– Such experiments are being attempted
Crib sheet
(or what you need to know to pass the exam)
• The zoo of particles and their properties
–
–
–
–
Leptons (e-, m-, -, ne , nm, n)
Hadrons (baryons and mesons)
Their anti-particles
The conservation laws and how to apply them (energy, momentum,
baryon number, lepton numbers, strangeness)
• Quarks and their properties
–
–
–
–
Flavors: up, down, strange, charm, top ,bottom
How to combine quarks to form baryons and mesons
Quark spin and color
The eight-fold way patterns
• Fundamental forces and field particles
• The standard model
• And from special relativity, its important to understand the concepts of
rest mass and energy, and the equations of conservation of relativistic
energy and momentum
QuickTime™ and a
GI Fdec ompressor
are needed t o see this picture.
Circular Accelerator
LEP: Large Electron Positron Collider
Requires
 large magnetic
field; B
 large radius; R
 charged particles;
q
Example 1
Maximum Energy
A cyclotron has magnet poles with radius 0.50 m and a magnetic
field with magnitude 1.50 T. Find the maximum particle energy
if this cyclotron is used to accelerate protons.
For protons q = 1.6x10-19 C and mp = 1.67x10-27 kg.
SOLUTION
2
2
2
(q) (B) (R)
K max 
2m p
 27MeV
Forces acting on charged
particles moving in a circular
accelerator
Cyclotron
1. Particles moving in constant
magnetic field:
F = qvxB
2. Particles moving in a circular
orbit:
F = mv2/r
3. Particles have an angular
frequency :
qB
; classical
m
qB
 
1- v 2 / c 2 ; relativistic
m
 
QuickTime™ and a
GIF decompressor
are needed to see this picture.
Wilson Cloud Chamber - 1912
DETECTORS
Cloud Chamber
QuickTime™ and a
GIF decompressor
are needed to see this picture.
Charged particles move through a super
saturated gas; a mixture of evaporated
liquid and non-condensing gas.
DELPHI Drift Chamber - CERN
A constant magnetic field is applied
perpendicular to the path of the
charged particles
Condensation starts around ions formed
by passing charged particles and the
resulting droplets are photographed
Quick Time™ an d a
GIF d eco mp res sor
ar e n eed ed to s ee this pic tur e.
PIONS
Example 3
Determine the magnitude of the
momentum and the speed of a proton
at a point where the observed radius
of curvature of the path in a cloud
chamber is r=2.67 m and the
magnitude of the magnetic field is
B=0.14T.
Solution
p = mv = qrB = 5.98x10-20 kgm/s
v = p/m = 3.58 x 107 m/s
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PROBLEMS
1. For each of the following decay reactions give all possible
electric charges of each particle and the energy Q released in
the decay.
a) π
µ+n
b) π
gg
c) µ
e- + n  n
2 Which of the following processes are absolutely forbidden and
why?
a) p
e+ + g
b) n
p + e- + n e
c) π + π
n+p
d) n
p + e+ + ne
e) πo + n
π+p
f) n + n πo+ π+ + πg) π+ + n π- + p
PROBLEMS
3. Two body decay: A ---> B + C
Show that in a frame of reference in which A is at rest the
kinetic energy of particle B is given by:
TB 
(
(M A
2
)
- M B ) - MC2 c 2
2M A
 In the case of scattering of pions by protons calculate the
momentum pπ and the kinetic energy Tπ of the pion in the
laboratory frame as a function of Etotal the total energy of the
pion-proton system in the centre of mass frame.
5. Calculate the threshold for the production of an e-+e+ pair
production by a photon in the presence of an initially
stationary e-.
SOLUTIONS
1. a) nµ and nµ have zero charge.
Q = Mπc2 - Mµc2 = 33.9 MeV; assuming that neutrinos have
have zero rest energy
For initial stationary pion
pn = -pµ = p
Eµ + En = Mπc2
c(p2+Mµc2)1/2 + cp = Mπc2
p = c(Mπ2 - Mµ2)/(2Mπ)
Eµ=c2(M2π+Mµ2)/(2Mπ)
Tµ=Eµ-Mµc2
= c2(Mπ-Mµ)2/(2Mπ)
= 4.12 MeV
1. b) g has no charge
Q = 135 MeV
SOLUTIONS
1. c) µ+ ---> e+ + ne + nµ
µ- ---> e- + ne + nµ
Neutrinos have charge zero.
Q = 105.2 MeV
2. a) Forbidden. It does not conserve the baryon number.
b) Allowed
c) Forbidden. It does not conserve charge.
d) Forbidden. It does not conserve charge.
e) Forbidden. It does not conserve the baryon number.
f) Allowed.
g) Forbidden. Does not conserve charge.
SOLUTIONS
3. Energy is conserved ===> EA,total = EB,total + EC,total
Momentum is conserved ===> pA = 0
pB = -pC
Total energy E for a particle is: E = ((pc)2 + mc2)1/2 (1)
EC = EA - EB (2)
(pBc)2 = (pCc)2; EB2 -MB2c4 = EC2 -MC2c4
Eq. 2 gives E2B - M2Bc4 = E2A -2EAEB+E2B - M2Cc4
EB = ((MAc2)2 + (MBc2)2 - (MCc2)2 )/(2MAc2)
TB = EB -MBc2 =
(
2
)
(M A - M B ) - M C 2 c 2
2M A
SOLUTIONS
4. ECM = Wc2 where W is the invariant mass of the pion-proton
system.
E2CM = c2(p2+m 2c2) where ECM is the total energy and p the momentum in an arbitrary reference frame.
In the laboratory reference frame:
EL = mπc2 + mpc2 + Tπ; where Tπ is the kinetic energy of the pion.
p = pπ = ( (Tπ + mπc2)2 - mπ2c4)1/2 / c
=[ Tπ (Tπ +2mπc2) ]1/2/c
Tπ = [m2 -(mp+mπ)2 ]c2/(2mp)
= (m+mp+mπ)(m-(mp+mπ))c2/(2mp)
pπ = [{m2 -(mp + mπ)2 }{m2 - (mp + mπ)2}]1/2
4. continuing
mπc2 = 0.1396 GeV
mpc2 = 0.9383 GeV
Therefore,
Tπ = (E+1.0779)(E-1.0779)/1.8766 [GeV]
Pπ = [Tπ(t + 0.2792)]1/2
For E = 0
For E = 2.64 GeV
Pπ = 0.74 GeV and Tπ = 0.61 GeV
Pπ = 3.26 GeV and Tπ = 3.12 GeV
SOLUTIONS
5. In the centre of mass system:
Mass is invariant
Net momentum = 0
pc = 0
Pair production at threshold leads to a final state which consists of
two electrons and one positron at rest with total energy
Ecentre = 3mcc2
In laboratory frame:
Elab = hn + mec2
plab = hn/c
Therefore, for the invariable mass we get : E2total - (cp)2
SOLUTIONS
5. continuing
(3mec2)2 = (hn+mec2)2 - (plabc)2
(3mec2)2 = hn2  2hn mec2 + me2c4 - hn2
(3mec2)2 = me2c4 + 2hn mec2
and
hn = 4 mec2
COLLIDERS & Centre of Mass Frame
CM: No net momentum
The invariant mass: W
1)
W2 c 4
2)
Wc2
=
E2
total
-
p2
L
c2
W2c4 = (EL + mtc2)2-(pLc)2
= mb2c4 + mt2c4 + 2mtc2EL
= ECM
mb = mbombard.
Laboratory Frame
EL = [(mb
Et = mtc2
c2 ) 2
+ pL
2c2]1/2
mt = mtarget
Available Energy
ECM

  mb c2

( )  (mt c )
2
2 2
1/2

 2mt c 2 E L 

Example 2
Available Energy
Threshold energy for pion, π, production: A proton with Kinetic
Energy T collides with a proton at rest, producing a pion (πo, rest
Energy = 135 Mev). What minimum value of T is recorded?
Solution
Total available energy must be at least
ECM = (2mp + mπ)c2
For the energy at CM this gives
ECM2 = (2mpc2)2 + (mπc2)2 + 2 mπc2(2mpc2) = 4.044x106 (MeV)2
ECM2 = 2(mpc2)2 + 2Etotal,pmpc2
Etotal, p = mpc2 + mπc2 (2+mπ/(2mp)) = (938 + 280) MeV
Incident protons Tk ≥ 280 Mev
Beta Decay
1H
3
He
+
e
√
2
Energy Distribution
1.2
1
Relative
intensity
3
3 He
2
3 H
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
Energy (kev)
WHAT IS THE PROBLEM HERE?
16
18
20
+ e- + n
ATOMIC SUBSTRUCTURE
10-18 m
Example 8, Chap.41, page 1333 in Tipler
State if one of the following decays or processes
violate one or more of the conservation laws and
if so identify the law/laws violated.
a)
b)
c)
d)
)
p+
n
e+ + ep + pn-e + p
n + e+ + v-e
p+ + g
g+g
n + e+
Answer: a) Violating energy conservation
b) Violating energy conservation
c) Violating momentum conservation
d) Allowed
e) Allowed