A19920
ANY CALCULATOR
.
School of Physics and Astronomy
DEGREE OF B.Sc. & M.Sci. WITH HONOURS
FIRST-YEAR EXAMINATION
03 17484
LC INTRODUCTION TO PARTICLE PHYSICS AND COSMOLOGY
SUMMER EXAMINATIONS 2014
Time Allowed: 1 hour
Students should attempt two questions. If you answer more than two questions,
credit will only be given for the best two answers.
The approximate allocation of marks to each part
of a question is shown in brackets [ ].
Calculators may be used in this examination but must not be used to store text.
Calculators with the ability to store text should have their memories deleted prior to
the start of the examination.
Two tables of physical constants and units that may be required
will be found at the end of this question paper.
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1. (a) If a parsec is defined as the length of an adjacent side of a right triangle
whose apex angle is one arcsecond (1/3600 of a degree) and opposite
side is one astronomical unit, show that 1 parsec = 3.1 × 1016 m.
[2]
(b) An astronomer measures the distance to a galaxy cluster to be 100 million light years. Show that this distance is consistent with 31 megaparsecs
[2]
(Mpc).
(c) Spectral measurements of this galaxy cluster indicate that wavelengths appear shifted by fraction of 0.01. What value of the Hubble constant H (in km
s−1 Mpc−1 ) is inferred from these measurements?
[2]
(d) Estimate the age of the Universe using the value of H calculated in part (c).
[2]
(e) The Cosmic Microwave Background (CMB) temperature is approximately
2.725 K at the present moment. What was the temperature of the CMB at
[2]
the redshift of this observed cluster?
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2. (a) Consider the Friedmann equation that describes the evolution of the Universe
H2 =
kc2
8πG
ρ− 2 ,
3
a
where the symbols have their usual meaning.
i. Derive a power-law expression for the scale factor a(t) as a function of
the cosmic time t for a flat, radiation-dominated Universe.
[3]
ii. What is the functional form of a(t) at late times for an open Universe?
[2]
Describe the behavior of this Universe.
(b) Describe briefly how the following detector systems are typically arranged to
form a general-purpose detector at a particle collider, justifying your answer
in terms of how an electron, a muon, a charged hadron and a neutral hadron
are detected:
i. tracking chamber;
ii. vertex detector;
iii. muon chamber;
iv. solenoidal magnet;
v. hadronic calorimeter;
vi. electromagnetic calorimeter.
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[5]
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3. Consider a very simple model of electromagnetic shower development in which
a high energy particle of energy E0 travels a distance of 1 radiation length in
a medium, then interacts to produce two particles each having energy E0 /2.
This process of doubling the number of particles and equal sharing of energy
continues until all particles in the shower have energy less than the critical energy,
Ec , of the medium.
(a) What are the two physical processes involved in this shower model?
[2]
(b) Why does the increase in the number of particles in the shower stop when
the mean energy of the particles is less then Ec ?
[2]
(c) What is the mean number of particles in the shower after a depth of t radia[2]
tion lengths?
(d) Consider a point in the shower when all particles have energy E . Deduce
an expression for the depth t of the shower in terms of E and E0 .
[2]
(e) Deduce an expression for the number of shower particles at the maximum
shower density and hence show how the depth of the shower maximum,
tmax , (in radiation lengths) depends on the incident particle energy.
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[2]
Physical Constants and Units
Acceleration due to gravity
9.81 m s−2
g
G
Tice
NA
6.674 × 10−11 N m2 kg−2
Ice point
273.15 K
Avogadro constant
6.022 × 1023 mol−1
[N.B. 1 mole ≡ 1 gram-molecule]
Gas constant
R
8.314 J K−1 mol−1
Boltzmann constant
k, kB
1.381 × 10−23 J K−1 ≡ 8.62 × 10−5 eV K−1
Stefan constant
σ
5.670 × 10−8 W m−2 K−4
Rydberg constant
R∞
1.097 × 107 m−1
R∞ hc
13.606 eV
Planck constant
h
6.626 × 10−34 J s ≡ 4.136 × 10−15 eV s
h/2π
~
1.055 × 10−34 J s ≡ 6.582 × 10−16 eV s
Speed of light in vacuo
c
2.998 × 108 m s−1
~c
197.3 MeV fm
Charge of proton
e
1.602 × 10−19 C
Mass of electron
me
9.109 × 10−31 kg
Rest energy of electron
0.511 MeV
Mass of proton
mp
1.673 × 10−27 kg
Rest energy of proton
938.3 MeV
One atomic mass unit
u
1.66 × 10−27 kg
Atomic mass unit energy equivalent
931.5 MeV
Electric constant
0
8.854 × 10−12 F m−1
Magnetic constant
µ0
4π × 10−7 H m−1
Bohr magneton
µB
9.274 × 10−24 A m2 (J T−1 )
Nuclear magneton
µN
5.051 × 10−27 A m2 (J T−1 )
Fine-structure constant
α = e2 /4π0 ~c
7.297 × 10−3 = 1/137.0
Compton wavelength of electron
λc = h/mc
2.426 × 10−12 m
Bohr radius
a0
5.2918 × 10−11 m
angstrom
Å
10−10 m
barn
b
10−28 m2
torr (mm Hg at 0 ◦C)
torr
133.32 Pa (N m−2 )
Gravitational constant
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Astrophysical Constants and Units
Astronomical Unit
AU
Parsec
pc
1.50 × 1011 m
3.1 × 1016 m
365.242 mean solar days
Solar luminosity
L
3.84 × 1026 W
Absolute bolometric magnitude of the Sun
Mbol
+4m 75
Bolometric correction for the Sun
BC
−0m 08
Apparent visual magnitude of the Sun
mv ()
−26m 74
Solar constant
f
1.36 × 103 W m−2
Solar mass
M
1.989 × 1030 kg
Wien constant
b
2.898 × 10−3 m K
Hubble constant
Ho
70 km s−1 Mpc−1
Solar radius
R
6.95 × 108 m
Distance of the Sun from galactic centre
8.3 kpc
Earth mass
M⊕
5.972 × 1024 kg
Earth radius
R⊕
6371 km
Tropical year
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