Subatomic And Cosmology Course Notes

8/13/2019 Subatomic And Cosmology Course Notes

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4 CONTENTS

3.3.4 Scintillation Detectors . . . . . . . . . . . . . . . . . . . 38

3.3.5 Cherenkov Detectors . . . . . . . . . . . . . . . . . . . . 383.3.6 Transition Radiation Detectors . . . . . . . . . . . . . . . 39

3.3.7 Semiconductor Detectors . . . . . . . . . . . . . . . . . . 40

3.3.8 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Detectors at the LHC . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 ATLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 ALICE . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.4 LHC-b . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 The Particle Zoo 474.1 Mesons and Isospin . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Baryons, Strangeness, and strong Hypercharge . . . . . . . . . . 50

4.3 The eightfold way . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Symmetry 53

5.1 Invariance and symmetry groups in quantum physics . . . . . . . 53

5.2 Symmetry Groups. . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2.1 Groups and group representations . . . . . . . . . . . . . 54

5.2.2 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.3 The Lie groups of the standard model . . . . . . . . . . . 59

5.3 Building representations out of the fundamental representation . . 63


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Chapter 1

Nuclear Radiation

1.1 -decay

See Mathematica notebook on nuclear physics.

1.2 -decay

In -decay the number A of nucleons in an atomic nucleus doesnt change, butthe number of protons does. Generally one hasA = 0 and|Z| = 1 for thechange ofZandA.

-decay: AXZA YZ+1 + e+ e.One neutron is converted into a proton; an electron and an (electron-) anti-

neutrino are emitted. This usually happens if a nucleus has an abundance of

neutrons.

+-decay: AXZA YZ1 + e+ + e.One proton is converted into a neutron; a positron and an (electron-) neu-

trino are emitted. This usually happens if a nucleus has an abundance ofprotons.

Electron capture: AXZ + eA YZ1 + e.An electron from one of the inner shells (K shell) of the atom is captured and

one proton is converted into a neutron; an (electron-) neutrino is emitted.

Usually also X-rays are emitted because electrons from higher atomic shells

quickly fill the hole in the inner shell. This usually happens if a nucleus

has an abundance of protons.


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8 Nuclear Radiation

direct observations of neutrinos and their interaction in the context of the standard

model of particle physics.The neutrino must be a spin 1/2 particle because of angular momentum con-

servation. Thespinof a particle is an intrinsic angular momentum of a particle,

and spin 1/2 means that the z-component (or x, or y) of this intrinsic angular mo-

mentum can take on the values 12. If we have two spin 1/2 particles then we

have to add up their angular momentum. So the total spin of two particles must

be an integer number (-1, 0, 1). Generally, an even number of spin 1/2 particles

must have integer total spin and an odd number of spin 1/2 particles must have

half-integer total spin. If you look at decay from angular momentum conser-vation you haveSX = Sy +Se+S. Now X and Y have the same number of

nucleons and the electron is a spin 1/2 particle. To make sure that both sides haveeither both integer or both half-interg spin the neutrino has to have spin 1/2.

The neutrino is a very special particle because it can only beleft-handed. This

means that its spin is always pointing into the opposite direction of its momentum.

For a right-handed particle, spin and momentum point in the same direction. The

anti-neutrino can only be right-handed, therefore neutrino and anto-neutrino are

distinct.

The left-handedness of the neutrino has an amazing consequence: nature dis-

tinguishes between a physical situation and its mirror image! In more technical

terms, left-handedness implies violation of parity. Parity is a spatial symmetry

operation in which time is not changed but all space-points are inverted, x x.All interactions but the weak interaction (which is behind -decay) are invari-ant under a parity transformation. To see why left-handedness violates parity we

have to consider the transformation of angular momentum vectors. Under a parity

transformation we have

p= mdx

dt m

dx

dt = p , (1.5)

and thereforeL= x p (x) (p) =L . (1.6)

Spin is also an angular momentum, so spin points in the same direction but the

momentum is inverted under parity transformation. Hence left-handed particles

should turn into right-handed particles, but since there are no right-handed neutri-

nos parity is violated.

1.2.3 Baryons and Leptons

All nuclear radiation processes, and in fact all fundamental interactions, do also

conserve the number of BaryonsB and LeptonsLin the universe. For now it suf-fices to say that proton and neutron both have Baryon number +1. Anti-Baryons


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1.3 -decay and M obauer effect 9

haveB = 1, and photons and electrons and neutrinos have B = 0. The baryon

number is best understood in the context of the quark model. The name Baryoncomes from heavy and thus is associated with more massive particles.

The Lepton number of Baryons is zero. Leptons are the electron and its neu-

trino. There are more massive versions of the electron, namely the muon andthetauon. They both have their respective muon- and tauon-neutrino and all ofthese particles haveL = 1. Their anti-particles haveL = 1. It is then an easyexcercise to see that-decay conservesL.

From the point of view of Baryons and Leptons -decay, or more generallythe weak interaction, can be summarized as

n p+ e+ e (1.7)p n+ e+ + e (1.8)

p+ e n+ e (1.9)

along with their more exotic variations

n p+ + (1.10)

p n+ + + (1.11)

p+ n+ (1.12)

and

n p+ + (1.13)

p n+ + + (1.14)

p+ n+ . (1.15)

It appears that tauon capture has not been observed yet. For muon capture see

http://arxiv.org/abs/0704.2072.

A free proton is lighter than a free neutron and therefore cannot decay. The

+-decay of the proton therefore only happens inside an atomic nucleus. On theother hand, free neutrons are indeed unstable particles. Their half-lifetime for

-decay is 613 s.

1.3 -decay and Mobauer effect

Nuclearradiation is essentially the same phenomenon as spontaneous emissionof light from an excited atom. After an ordecay the daughter nucleus is oftenin an excited state. This means that one or more of the nucleons are occupying an

energy state that is higher than the Fermi energy. This state is unstable and conse-

quently the nucleon will fall back to the ground state by emitting electromagnetic
http://arxiv.org/abs/0704.2072http://arxiv.org/abs/0704.2072http://arxiv.org/abs/0704.2072


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10 Nuclear Radiation

Figure 1.2: decay of 228Th to excited states of 224Ra, followed bydecay tolower levels.

radiation.decay is therefore a purely electromagnetic effect and has nothing todo with the nuclear force or the weak interaction.

There are some notable differences between atomic and nuclear spontaneous

emission. First and foremost the energy of the emitted photon, which for atomic

processes is in the order of 1 eV but for nuclear processes is in the order of 100keV because of the larger energy spacing between the energy eigenstates in the

nucleus (see Fig.1.2).

The other difference is the large momentum kick that a nucleus gets when it

emits aphoton of energyE= , where is the photons angular frequency.If the nucleus is first at rest, its momentum after decay ispnucl = p = E/c(recall thatE= |p|cfor massless particles). To understand the consequences letscompare spontaneous emission with the decay of a classical damped harmonic

oscillator (this analogy is really well working in general),

x+ x+ 20x= 0, (1.16)

with the decay rateof the oscillator. For 0the oscillator has a decay timeof1/ and oscillates at a frequency of roughly 0. Fourier transformation thenshows that the spectrum of the oscillator has a width of and is centered around0.

Fordecay the decay rate, and hence the frequency width, is in the order of = 108 s1. On the other hand, because the nucleus is moving afterdecay, theDoppler effect shifts the frequency of the photon by an amount of

= v

c =

pMc

=E2

Mc2 (1.17)


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12 Nuclear Radiation


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Chapter 2

Applications of nuclear physics

2.1 Nuclear fission

This is the most famous application as it is used for nuclear power plants and

nuclear bombs. It is based on the observation that low-energy neutrons can trig-

ger the fragmentation of certain heavy nuclei into two medium-sized nuclei, for

instance235U+n 148 La+87 Br+n. (2.1)

The binding energy for 235

U is only7.5 MeV per nucleon whereas it is about8.4 MeV for the two products. Consequently one can gain about 0.9 MeV ofenergy per nucleon in a fission process. For 235U this amounts to about 200 MeV

in a single fission.

In particular, on average 2.5 neutrons are produced in 235U processes similar

to that in Eq. (2.1). These neutrons can trigger other fission processes so that a

chain reactionmay be created. One defines the ratio

k =# of neutrons produced in the (n+1) stage of nuclear fission

# of neutrons produced in the n stage of nuclear fission . (2.2)

This ratio depends nt only on the number of produced neutrons but also on how

likely they are to trigger another fission process. If the nuclei are far apart, for

instance, this likelihood is very low. Ifk < 1 the reaction is calledsub-criticaland the reaction eventually stops. For nuclear power plants one operates at the

criticalthresholdk = 1, and nuclear bombs operate atk >1 (super-critical).

In nuclear bombs one achieves k > 1 by bringing the 235U nuclei close to-gether. Basically one has several small pieces of Uranium that each are sub-critical

inside a conventional bomb. When the conventional bomb explodes these pieces

are pressed together and become super-critical.


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14 Applications of nuclear physics

Figure 2.1: Binding energy per nucleon as a function of the number of nucleons.

In a nuclear power plant one can control kby inserting retractable control rodsmade of Cadmium into the Uranium material. Cadmium has a high absorption

cross section for neutrons; by varying the amount of control rod material between

the uranium one therefore can control the number of neutrons available for fission.

The energy generated by the fission process produces heat that is used to produce

steam. This steam then turns a generator in which electric current is produced

through induction. 1g of 235U contains about 3 1021 atoms, so that in can beused to produce about1011 J of energy. This is enough to provide the entire energyfor an average North American household (11209 kWh per year, which is more

than twice what Europeans are using) for 2.5 years. The problem is that nuclear

power plants produce waste that remains highly radioactive for billions of years.

Waste management is therefore not at all easy in this case.

2.2 Nuclear fusion

One cannot only release nuclear binding energy by fragmenting heavy nuclei into

larger ones but also by combining very light nuclei into heavier ones. The reason

is that the binding energy per nucleon has a maximum around iron (see Fig.2.1)

Nuclear fusion provides the power for the sun and stars. To combine two

light nuclei they have to brought very close together so that the nuclear force can

provide the binding. This means the nuclei have to overcome the Coulomb energy

barrier between the protons, which is about 4 MeV high. The nuclei therefore have

to have a kinetic energy of this order. In a thermal gas this implies kbT 4MeV,


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2.3 Radioactive decay chains and radioactive dating 15

which corresponds to a huge temperature ofT 1010 K. In stars the gravitational

attraction can provide enough pressure to keep such a hot gas together (fusiononly happens in the inner regions of a star).

For technical applications the need to contain a very hot gas is a formidable

challenge. So far this has been achieved in the hydrogen bomb for military ap-

plications, and in fusion reactors for energy generation. Fusion reactor research

has been going on for 40-50 years and while some reactors were able to create

energy of short durations (in the order of seconds) the big breakthrough is still

to come. In fusion reactors the hot plasma gas (atoms are completely ionized at

these temperatures) is often contained by a magnetic field so that the plasma does

not touch any walls.

The prinicipal fusion processes in the sun are theproton-proton cycle,

1H+1 H 2 H+ e+ + e+0.42 MeV (2.3)1H+2 H 3 He+ + 5.49 MeV (2.4)

3He+3 He 4 He+ 2(1H) +12.86 MeV (2.5)

The doubly magic 4He nucleus is very stable and is the exhaust of this reaction

chain. However, it can produce carbon nuclei through the reaction

3(4He) 12 C+7.27 MeV (2.6)

These carbon nuclei provide the fuel for theCNO cyclein the sun,12C+1 H 13 N+ + 1.95 MeV (2.7)

13N 13 C+ e+ + e+1.20 MeV (2.8)13C+1 H 14 N+ + 7.55 MeV (2.9)14N+1 H 15 O+ + 7.34 MeV (2.10)

15O 15 N+ e+ + e+1.68 MeV (2.11)15N+1 H 12 C+4 He+4.96 MeV (2.12)

The complete cycle therefore can be summarized as

12C+ 4(1H) 12 C+4 He+ 2e+ + 2e+ 3+ 24.68 MeV (2.13)

2.3 Radioactive decay chains and radioactive dating

2.3.1 Exponential decay of individual isotopes

Radioactive decay is best quantified through the decay rate of a particular pro-cess. Assume we haveNnuclei of a radiactive isotope. During a timet each


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Figure 2.2: Visualization of the CNO cycle (http://www.zgapa.pl/zgapedia/Cykl

w%C4%99glowo-azotowo-tlenowy.html )

nucleus decays with a certain probability. The total change Nin the number of

nuclei during that time must be proportional toN(the more you have the morecan decay). The decay rate is then defined by the relation N = Ntfor sufficiently short timest. In the limitt 0 this becomes a differentialequation,

dN

dt = N . (2.14)

This is a linear differential equation for N(t) with constant coefficients. Suchequations can be often solved by making an expontial ansatz N(t) = exp(t)N(0).1 Inserting this in Eq. (2.14) yields= so that

N(t) =etN(0). (2.15)

Thehalf lifet1/2 of an isotope is defined as the time when half of the initialnumber of nuclei has decayed. HenceN(t1/2) =N(0) exp(t1/2) =N(0)/2ort1/2 = ln(2)

1. One also often finds the (mean)lifetime1 which is the timeduring which the initial number has been reduced by a factor ofe1. This alsocorresponds to the average time it takes a nucleus to decay.

1In certain situations this ansatz needs to be modified to N(t) = tn exp(t). This happenswhen there are certain algebraic relations between the constant coefficients. An example is the

overdamped harmonic oscillator: Eq.(1.16)with = 20.
http://www.zgapa.pl/zgapedia/Cykl_w%C4%99glowo-azotowo-tlenowy.htmlhttp://www.zgapa.pl/zgapedia/Cykl_w%C4%99glowo-azotowo-tlenowy.htmlhttp://www.zgapa.pl/zgapedia/Cykl_w%C4%99glowo-azotowo-tlenowy.htmlhttp://www.zgapa.pl/zgapedia/Cykl_w%C4%99glowo-azotowo-tlenowy.html


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the right-hand side. Likewise fore2t and constant terms. Consequently we get

1a= 1N0 2a (2.20)

c= 0, (2.21)

so thatN2(t) = 1/(2 1)e1t + be2t. Assume that initially we dont have

anyN2isotopes. ThenN2(0) = 0so that

N2(t) = N012 1

(e1t e2t) (2.22)

A similar treatment forN3gives

N3(t) =N0+ N012 1

e1t

1+

e2t

2

. (2.23)

This demonstrate how one can solve decay chains in general by successively inte-

grating the equation from the top to the bottom.

This analysis can be tremendously simplified if the isotopes reached a secular

equilibrium, which means that their abundances hardly change, dNi/dt 0. Ifthis condition is fulfilled the abundances must be related byN11 N22 .Hence in secular equilibrium the ratio beteen the numbers of isotpoes is equal to

the ratio of their decay rates.

Let us refine the example above a bit: strictly speaking 99Mo does not directly

deceay to the ground state of99Tc but rather to an excited nuclear state99mTc (m

formetastable) that has a half life of 6 hours and then decays to the ground state99Tc. This fact is of utmost importance for medicine (see below). Mathematically

it means that we have to consider a chain of four isotopes, which can be solved in

exactly the same way as above. The result of this calculation is shown in Fig. 2.3.

2.3.3 Nuclear dating, 14Cmethod

The fact that one knows the entire time evolution of a decay chain if one knows

the initial abundances of the isotopes in that chain can be used to determine the

age of objects. The most famous dating method is the 14C dating to determine

the age of organic material that has been produced during the last few thousand

years.

The trick is that living organisms are in constant exchange with the atmo-

sphere (e.g., breathing), but this exchange is stopped when the organism dies. The

atmosphere contains in particular nitrogen and CO2 molecules, and the latter are

overwhelmingly often formed using the stable 12C isotope of Carbon. However,


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2.4 Nuclear physics in medicine 19

Figure 2.3: Abundances of 99Mo (black), 99mTc (red) and 99Tc (blue) for 1000

initial Molybdenum nuclei.

the atmosphere is constantly bombarded by cosmic rays which produce slow neu-

trons. These neutrons led to the generation of14C through

14N7 + n 14 C6 +p . (2.24)

Hence a small fraction of the CO2 molecules in the atmosphere are formed with14C isotopes. Now 14C is unstable anddecays through

14

C6

14

N7

+ e+ e (2.25)

with a half life of 14C is 5730 years. In a living organism the amount of 14C

is determined by the atmosphere; in a living plant the corresponding acitivity is

12 disintegrations per minute per gram. If the plant dies it stops to take on 14C

isotopes. Those in the plant decay so that, after a very long time 5730 yearsonly 12C will be found. But on time scales in the order of a few thousand years

the remaining activity in the plant can tell us how old it is.

2.4 Nuclear physics in medicine

There are many different ways in which nuclear physics can be used in medicine,

but all of them are related to the way in which radiation is absorbed by organic

tissues. The most important aspect is how deeply radiation can penetrate through

biological tissues. One often uses lead as a standard to compare this because lead

strongly absorbs radiation. We have the following typical data:

particles (4He): absorbed by a 0.01 mm lead foil.

particles (electrons): can travel through 0.1 mm of lead.


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radiation (high energy photons): can pass through 100 mm of lead.

The corresponding passage lengths in biological tissue are of course much greater.

radiation can easily pass through the body andparticles typically travel a fewmm to cm. Consequently,rays are best suited for medical imaging.

Generally, when nuclear radiation interacts with tissues its effects are mostly

generated through ionization of atoms. To ionize an atom or molecule one needs

between 1 eV and 35 eV of energy. Because nuclear radiation particles have typi-

cally energies in the order of keV to MeV a single particle can ionize many atoms.

The produced ions or radicals are highly reactive and the subsequent chemical re-

actions interfere with the normal operation of cells or may damage the DNA. An

example is when radiation hits a water molecule and breaks it apart, H2O H+ HO. These radicals can break the sugar phosphate backbone of the DNA.

2.4.1 Radioactive isotopes used in diagnostics

The most common diagnostic tool that uses radiation is based on X-rays. As X-

rays are generated in atomic processes (when inner shell electrons are removed)

they will not be discussed here.

Nuclear magnetic resonance (NMR)

We will only very briefly discuss NMR because it is not really based on nuclear

processes. This technique makes use of the fact that atomic nuclei usually possess

a spin (the sum of the spins of all nucleons). In a magnetic field the spin-up

state and the spin-down state have different energies, they so to speak form an

excited state and a ground state. One can trigger transitions between these states

using radiofrequency waves. Because the energy difference between the two states

depends strongly on the specific isotope one can observe the abundance of specific

atoms by observing how much the radiofrequency waves are absorbed. Because

most particles possess a spin this technique is not really relying on nuclear physics.

The only reason why one uses the nuclear spin is that it is well shielded from the

effect of other atoms. The spin of electrons is much more disturbed because the

electrons sit in the atomic shell so that their spin state is strongly affected by the

presence of other atoms.

Radioactive tracing

The most important diagnostic technique that is based on real nuclear physics uses

theradionuclide 99mTc, an excited (metastable) state of the isotope 99Tc. It has a

half life of 6 hours, which makes it very suitable for diagnostics, and decays to the


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ground state by emitting a particle. Most importantly, Technetium can be build

into a large variety of chemical compounds. Some compounds have the tendencyto concentrate in a specific organ or region of the anatomy; it is therefore possible

to transport 99Tc to a region of interest. By observing the emitted radiation onecan gain information about the state of the organ. High or low radioactivity may

represent an over- or under-activity of an organ, or the presence of a lesion or

tumor.

Another advantage of99mTc is the energy of its emittedradiation (140 keV).This energy is in the order of that used in X-ray diagnosis, the photons aretherefore easy and cost-effective to detect. Currently a big disadvantage is its

production. There are only a few reactors in the world that produce 99Mo which

decays into 99mTc with a half live of 2.7 days (convenient for transport!). Two ofthese reactors (the National Research Universal (NRU) reactor located in Chalk

River, Ontario, and a Dutch reactor) provide about 65% of the worlds supply of99mTc, but both are old and face frequent disruptions of service, resulting in a

crisis in medical diagnosis.99mTc is not the only isotope that is used to trace the activity and effects

of chemical compounds in the body. There are many other examples, for in-

stance 51Cr which is used to label red blood cells. A list of isotpoes that are

used in medicine can be found at http://www.radiochemistry.org/nuclearmedicine/

radioisotopes/ex iso medicine.htm.

Positron emission tomography (PET)

PET is a beautiful example of how even basic physics can be used to improve

technology. PET utilizes the annihilation between matter and anti-matter: when

an electron and a positron collide the can be completely destroyed by emitting two

photons. Energy and momentum conservation require that always two photonshave to be created. Their energy is rather large because the minimum energy of

an electron + positron system is 2mec2 1 MeV. If electron and positron are

initially at rest (or very slow) the total momentum must be approximately zero.

Hence the two photons have to fly away in almost opposite directions and both

have an energy of 511 keV. In a PET this consequence of momentum and energy

conservation is used for medical imaging.

The positrons are generated by using radioactive tracer that perform a+ de-cay such as 11C, 13N, 15O, or 18F. All of these isotopes have short half times in

the order of tens of minutes. The positron emitted in this decay has initially a lot

of kinetic energy which it looses when it ionizes atoms or is scattered by atoms.

After a few mm the positron is slow enough to form a positronium, which is like

a hydrogen atom with the proton replaced by the positron. Positronium is unstable

and decays within1010 s by electron-positron annihilation, see Fig.2.4(a).
http://www.radiochemistry.org/nuclearmedicine/radioisotopes/ex_iso_medicine.htmhttp://www.radiochemistry.org/nuclearmedicine/radioisotopes/ex_iso_medicine.htmhttp://www.radiochemistry.org/nuclearmedicine/radioisotopes/ex_iso_medicine.htmhttp://www.radiochemistry.org/nuclearmedicine/radioisotopes/ex_iso_medicine.htmhttp://www.radiochemistry.org/nuclearmedicine/radioisotopes/ex_iso_medicine.htm


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(a) (b)

Figure 2.4: (a) Electron-positron annihilation after a + decay in a PET.(b) Principle of a PET scan. Sources: (a): http://japan.gehealthcare.

com/cwcjapan/static/rad/nm/etraining/images/annihilations.gif, (b): http://en.

wikipedia.org/wiki/Positron emission tomography

The PET device directly utilizes the fact that the two emitted photons fly away

in opposite directions (actually the fly away at 180 0.25). The patient is sur-rounded by a ring ofdetectors, see Fig.2.4(b). Only coincidence events, whentwo opposite detectors are triggered, are taken into account. When such a coinci-

dence event takes place the Coincidence processing unit measure the time delay

between the two detections and in such a way can determine where the annihila-

tion took place with a resolution of 3-5 mm.

2.4.2 Radioactive isotopes used for therapy

Most therapeutic applications of nuclear physics make use of the damage that

radiation can do to biological tissues. As discussed above, ions and free radicals

produced through radiation can disrupt the normal chemical processes in a cell. A

very effective way to use this effect is similar to the tracer method in diagnostics:

the isotope is build into a compound that is know to accumulate in cancer cells. In

such a way one can use rather low doses of radioactive material because it targets

exactly the desired location in the body. An example is 131I which is used for

cancer in the thyroid gland (an organ at the base of the throat). This isotope is also

a hazard and was released in great quantities after the Chernobyl reactor accident

in 1986. Iodine tends to accumulate in the thyroid gland.

Sometimes radionuclides are also used for pain relief. An example is 186Re

whoseemission has enough energy (1.07 MeV) and a convenient half life (90hours) for destroying metastases of bone cancer. 186Re also emits radiation at
http://japan.gehealthcare.com/cwcjapan/static/rad/nm/etraining/images/annihilations.gifhttp://japan.gehealthcare.com/cwcjapan/static/rad/nm/etraining/images/annihilations.gifhttp://en.wikipedia.org/wiki/Positron_emission_tomographyhttp://en.wikipedia.org/wiki/Positron_emission_tomographyhttp://en.wikipedia.org/wiki/Positron_emission_tomographyhttp://en.wikipedia.org/wiki/Positron_emission_tomographyhttp://japan.gehealthcare.com/cwcjapan/static/rad/nm/etraining/images/annihilations.gifhttp://japan.gehealthcare.com/cwcjapan/static/rad/nm/etraining/images/annihilations.gif


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137 keV which is suitable for imaging. Again the trick is that one only needs little

radioactivity when the isotope is build into a compound that accumulates in themetastases.


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Chapter 3

Accelerators and Particle Detectors

To gain a more detailed picture of the nucleus one needs to produce projectile

particles of higher energy. One reason is that to break up a nucleus the (e.g.,)particle needs to tunnel through the Coulomb barrier which is only possible for

higher kinetic energies. A more fundamental reason is Heisenbergs uncertainty

principle,xp . To achieve a smaller spatial resolutionx one needs toincreasep which usually goes in hand with increasing the overall momentum.For instance, if the projectile particle is a proton and the desired resolution is 0.1

fm we need a momentum uncertainty pof1018 kg m/s, which corresponds toa relativistic velocity ofv = 0.9c (which results from settingp m

pv) and

a kinetic energy ofp2c2 + m2pc4 mpc2 = 1.24GeV. The acceleration neces-sary to achieve this over a length of, say, 100 m would be about 1014g. Modernacceleration rings can even achieve1020g, so lets see how that works.

3.1 Linear Accelerators

Projectile particles are usually charged (electrons, protons, ions) so using electric

fields is convenient. However, to achieve an acceleration ofa = 1014g requiresa huge electric field. Setting mpa = qEandE= V /Lcorresponds to a electric

potential difference ofV = 1.26GV over a lengthL = 100m.

3.1.1 Cockcroft-Walton Machines

For smaller energies the concept of just a simple potential difference is still used.

It was invented in 1932 by Cockroft and Walton and is rather simple, see Fig. 3.1.

The projectile particles are in some way created (in the figure by shooting elec-

trons at hydrogen atoms to ionize them) and these particles are then accelerated by

an electric potential difference. The kinetic energy gain can be easily calculated


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26 Accelerators and Particle Detectors

a) b)

Figure 3.1: a) Principle of a Cockcroft-Walton Machine (from http://library.

thinkquest.org/19662/low/eng/accelerators.html ) b) Cockcroft-Walton Machine

as a pre-accelerator used at CERN during the 1980s (from http://linac2.home.

cern.ch/linac2/default.htm)

from the potential energy difference of the particles V. For V = 150000Vand protons we find a kinetic energy ofT = eV = 150000 eV = 0.15 MeV.This also explains why eV are the preferred unit of energy in particle physics.

3.1.2 Linear Accelerators (Linac)

Linear accelerator work on similar principle as Cockcroft-Walton Machines. To

increase the gain in kinetic energy the potential difference is chopped up in smaller

pieces, see Fig.3.2a).

Again the potential difference between two subsequent tubular electrodes (drift

tubes) is used to accelerate the particles. To repeat the process a radio frequency

ocillator (RF) inverts the potential difference while the particles move through adrift tube so that, when they leave the tube, they are accelerated again towards the

next drift tube.

For heavy particles such as protons the geometry of and distance between the

drift tubes must take into account that the velocity of the particles changes. There-

fore the initial drift tubes are shorter than the later drift tubes. For electrons the

geometry must be different because they reach velocities close toc very quickly.Furthermore, electrons radiate strongly when they get accelerated (something you

may learn about in Phys 422) so that one needs relatively much energy to increase

their kinetic energy. That energy is provided by the microwave fields that travel in

step with the electrons to provide the potential difference.

Linacs can be quite large, the most famous perhaps being SLAC at Stan-
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3.2 Circular Accelerators 27

a) b)

Figure 3.2: a) Linear accelerator of Alvarez type. b) Stanford Linear Acelerator

(SLAC) (from http://www.fnal.gov/pub/ferminews/ferminews03-12-01/p3.html)

ford (see Fig. 3.2 b) which has a length of 3 km and can accelerate electrons

to an energy of 50 GeV. In particle physics they are now most often used as pre-accelerators for collider rings.

3.2 Circular Accelerators

3.2.1 Cyclotrons

Because of the limitations of linacs people quickly started to examine the com-

bination of electric and magnetic fields to accelerate particles. The basic idea is

to still use electric fields to accelerate the particles but to also use magnetic fields

to keep the particles on a circular or spiroidal trajectory. This is useful to storeparticle (storage rings) and to save space.

A particle in a magnetic field experiences a force

F =qv B (3.1)

that is perpendicular to the velocity. Hence the speed does not change but the

direction of the velocity does so that the particle follows a circular trajectory ifBis orthogonal to the initial velocity v. In circular motion the radial acceleration isgiven byar =v

2/rso that Newtons 2nd law yields

mv2

r =qvB. (3.2)

orv = qBr/m. For non-relativistic motion we therefore find

T =1

2mv2 =

1

2m(qBr)2 . (3.3)

The cyclotron is the simplest way to exploit this magnetic confinement. A sim-

ple cyclotron (see Fig.3.3a) consists of a magnetic field (Electromagnet in the
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a) b)

Figure 3.3: a) Sketch of a cyclotron. b) The worlds largest cyclotron at TRIUMF

(University of British Columbia) accelerates H ions to 500 MeV (from http://www.physics.ubc.ca/research/particle.php).

figure) and to semi-circular electrodes which provide an electric potential differ-

ence to accelerate the particles. Similarly to linacs the potential difference mustoscillate for sustained acceleration and is created by a radio frequency ocillator

(RF). The frequency of this RF field must match the frequency of the circulatingparticles, thecyclotron resonance frequency

=

2 =

1

2

v

r =

1

2

qB

m .

In a cyclotron the particles start at a small radius r which is then increased

while the particles are accelerating. Consequently changes and the RF frequencyis adapted until the particles achieve the maximum radius and are ejected.

The above considerations were all based on non-relativistic physics. When

the particles achieve a speed that is close to the speed of light these results do not

hold anymore. Consequently, cyclotrons cannot be used to accelerate particles

to extremely high energies. The speed of a particle becomes relativistic when its

kinetic energy is comparable to (or larger than) its rest energy mc2. TRIUMF(see Fig.3.3b) is therefore as good as it gets because the 500 MeV energy of the

hydrogen ions is already quite close to the proton rest energy of about 1 GeV.
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3.2.2 Synchrotrons

Synchrotrons and synchrocyclotrons operate on the same principles as a cyclotron

but take into account relativistic corrections. The relativistic momentum of a par-

ticle is given bymv so that Newtons second law becomes

md

dtv=qv B . (3.4)

Again the force is perpendicular to the velocity so that the speed (and thus ) doesnot change. Hence, with a= dv/dt, we get

ma= qv B . (3.5)

If B is perpendicular to v we can identify the acceleration with the radial ac-celeration of circular motion, ar = v

2/r, to find the modified angular cyclotronfrequency

=v

r =

qB

m . (3.6)

When v is close to c, the factor becomes very large so that, for fixed B, thecyclotron frequency actually decreases with growing v. Machines where Bis heldconstant but the frequency is varied are called synchrocyclotrons, and machines

where the magnetic field is changed, and the frequency may be changed or not,

are called synchrotrons.Eq. (3.6) can be used to estimate the maximum energy of particles in a syn-

chrotron. Withp= mvwe can rewrite it as

p= qBr . (3.7)

Recall that for very large momenta the energy-momentum relation E=

p2c2 + m2c4

can be approximated byE pc, so that

E qBrc . (3.8)

For superconducting electromagnetic one can achieve B 10 T so that for ele-mentary particles with |q| =ewe haveE 3000rGeV, whereris in km. For theradius of the Large Hadron Collider (LHC), 4.3 km, this corresponds to a theoret-

ical maximum energy of 13 TeV, which is comparable to the actual performance

goal of 7 TeV for protons. A way to further increase this energy without increas-

ingr is to increase the charge qby using heavy ions. For the ALICE (A LargeIon Collider Experiment, see http://aliceinfo.cern.ch/Public/Welcome.html) com-

pletely ionized Pb+82 lead ions will be used to (hopefully) generate a so-called

quark-gluon plasma. Each lead nucleus will gain an energy of 574 TeV, or 2.76

TeV per nucleon.
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a) b)

Figure 3.4: a) Sketch of a synchrotron. b) The Large Hadron Collider at

CERN near Geneva (fromhttp://scienceblogs.com/startswithabang/2009/05/the

lhc black holes and you.php).

In synchrotrons the homogeneous magnetic field of the cyclotron is replaced

by periodically spaced (superconducting) electromagnets long the circular path

of the particles, see Fig. 3.4 a). An arrangement of pre-accelerators is used to

accelerate the particles to a speed that is consistent with the dimensions of the

synchrotron. An RF field in the collider ring is then used to further acceleratethese particles.

3.2.3 Some notable colliders

Lets start at home in Canada, where a few very notable accelerators do exist that

are used for special purposes.

TRI-University Meson Facility (TRIUMF)

Located at UBC, TRIUMFs 500 MeV cyclotron for H ions is used for a va-riety of purposes. Collisions between the protons ofH create a large numberof muons. Muons are used to test the electroweak interaction, and muon spin

resonance can be used to probe the small scale magnetic structure of materi-

als in solid state or chemistry. Heavy ion beams are used to produce and study

short-lived isotopes. This is also of interest for medical applications. Home page:

http://www.triumf.ca/
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a) b)

Figure 3.5: a) Sketch of CLS. b) The CLS.

Canadian Light Source (CLS)

The Canadian Light Source (http://www.lightsource.ca/) is located at the Univer-

sity of Saskatchewan in Saskatoon. It features a 2.9 GeV synchrotron for electrons

whose primary purpose is to produce an extremely bright (a million times brighter

than the sun) source of light and other electromagnetic radiation. Thissynchrotron

lightis a consequence of the Bremsstrahlung effect: an accelerated or decelerated

electron emits electromagnetic radiation. In a synchrotron this acceleration isachieved by bending the electron beam using magnets. The relativistic speed of

the electrons leads to very strong focusing of the radiation in the forward direction

(Remember that vectors in a fast moving inertial frame are tilted in the forward or

backward direction; this is basically ehat happens to the light). At the CLS, the

spectrum of radiation ranges from thar far infrared to hard X-rays and is used for

a large variety of purposes, including chemistry, living tissues, surface science,

and biochemistry. Bremsstrahlung is an unwanted effect in high energy physics

experiments because it reduce the particle energy, but at CLS it is the central tool

for research. The CLS is one of the largest science projects in Canadian history.

Large Hadron Collider (LHC)

The LHC is the largest machine ever built. It consists of a 27 km long synchrotron

whose proton or lead ion beams run 50-150 m beneath Swiss and French soil, see

Fig.3.6. As discussed above the proton will reach 7 TeV (which corresponds to

= 0.999999991 or= 7453) of energy while the lead ions can reach 574 TeV.The LHC contains in fact two parallel beam pipes that cross at four points where

collisions can be studied. With this arrangement it is possible to accelerate two

proton beams in opposite directions so that the center-of-mass collision energy
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a)

b)

Figure 3.6: a) Sketch of LHC accelerators and detectors. b) Lay-

out of the LHC.http://lhc-machine-outreach.web.cern.ch/lhc-machine-outreach/

images/complex/Cern-complex.gif
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3.3 Fundamental Types of Detectors 33

doubles to 14 TeV. Several thousand magnets keep the beams on a circular trajec-

tory, most of them superconducting and cooled to about 2K using liquid Helium.Liquid Helium is not only expensive but also difficult to work with. In fact, LHC

was shut down after its initial launch in 2008 because of problems with the He-

lium. The magnetic fields created by the superconducting magnets can be varied

from 0.54 to 8.3 T.

The main purpose of the LHC is fundamental research on elementary particle

physics and cosmology. This includes in particular the search for the Higgs Boson.

This is the last unobserved particle that occurs in the standard model and plays a

central role: it generates the masses of the W and Z bosons (see below). Without

the Higgs Boson the standard model would become inconsistent. LHC should

provide enough energy to observe it, so either it will be found or the standardmodel needs to be changed on a very fundamental level.

Other research topics will include the search for supersymmetry (a symmetry

between bosonic and fermionic particles), grand unified theories (all fundamental

forces apart from gravity may result from a single symmetry group obeyed by

nature instead of three separate symmetry groups as in the standard model), and

quark-gluon plasmas which mimic the physics of the very early universe.

3.3 Fundamental Types of Detectors

Similarly to medical applications of nuclear physics, elementary particles make

themselves visible mostly through ionization processes, i.e., its interaction with

matter.

3.3.1 Cloud Chamber

This is the simplest detector for elementary particles and only of historical interest

(however, you can build one yourself, see http://njsas.org/projects/atoms/cloud

chamber/cache/cloud.html). A cloud chamber uses a super-cooled vapour of water

or alcohol. When a particle ionizes a molecule in the vapour this ion acts as a

condensation nucleus: in super-cooled vapours even a small perturbation like an

ion can create small liquid droplets. In a cloud chamber these droplets are visible

as particle trajectories.

3.3.2 Ionization detectors

Ionization detectors follow a similar principle as the cloud chamber. They consist

of a chamber filled with an easily ionizable gas. However, the ions and elec-

trons are not used as condensation nuclei. Rather, the chamber also possesses two
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Figure 3.7: Cloud chamber image (from http://en.wikipedia.org/wiki/Cloud

chamber). Thicker, curly trajectories represent particles, longer, straight trasjec-tories representparticles. An animated version of this picture can be found athttp://en.wikipedia.org/wiki/File:Cloud chamber ani bionerd.gif.

electrodes with a large electric potential difference. The respective electric field

prevents the recombination of ions and electrons and accelerates the electrons to

the positive electrode and the ions to the negative electrode. This generates an

electric current that can be measured.

How exactly the current is related to the ionization process depends strongly

on the potential difference that is applied. Fig.3.9show the dependence of the

ionization current on the applied voltage. Several regions of this diagram are used

to build detectors for different purposes.

Ionization Counters

Ionization counters operate at low voltage, e.g., in the recombination region of

Fig.3.9. In this region a good portion of the ions and electrons recombine so that

the output signal, albeit proportional to the number of ionized particles, does not

contain all particles that have been ionized and is therefore quite small.

Ionization counters have the advantage that the gas quickly recovers from an

event and can therefore be used in situations with a lot of radiation. They also

have a very good energy resolution of the events that they record. One way to

measure the energy is by monitoring the signal current while decreasing the gas

pressure. As long as the gas is dense enough to completely stop the particle the

current will remain constant. When the density is too low the particle will not use
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Figure 3.8: Ionization chamber (from http://en.wikipedia.org/wiki/Ionizationchamber).

up all its energy to produce ions so that the signal current drops. Since the number

of ions produced is proportional to the initial energy of the particle the latter can

be found by monitoring at which pressure the current starts to drop.

Proportional Counters

Proportional counters operate at electric fields in the order of104 V/cm which

corresponds to the proportional region of Fig. 3.9. In this case the electric fieldsprevents recombination by accelerating ion and electon away from each other.

This happens in the ionization region, too, but in the proportional region the elec-

tric field provides the electrons with enough energy to ionize other atoms. As a

consequence, the initial signal is amplified in an avalanche kind of scheme. The

signal current is then roughly proportional to the number of initial ions but much

stronger than in ionization counters (amplification factor105).Proportional counters are very good for detecting small signals, but their en-

ergy resolution is limited. The energy of a particle can be determined from the

pulse height of the signal current pulse, but because that height is quite sensitive

to the applies voltage the resolution is not perfect.

Usually the positive electrode in a proportional counter consists of a very thin

(10 50m) wire. The electric field is particularly strong in the vicinity of thewire so that in this region most of the ionizations do occur.

Geiger-Muller Counters

At even higher voltages a single ionization by radiation can lead to a complete

discharge of the gas, i.e., virtually all atoms are ionized through an avalanche

process. This is advantageous for measuring very small initial signals, particularly
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Figure 3.9: Dependence of the ionization current on the applied voltage in ion-

ization detectors.

for detecting nuclear radiation. On the other hand, the total discharge erases any

information about the initial energy of the particle. Furthermore, the recovery

time for the medium to get de-ionized is quite long (1 ms dead time). They are

therefore not suitable for measuring large amounts of radiation.

3.3.3 Multiwire Proportional Chambers and Drift Chambers

Multiwire proportional chamber are actually a variant of proportional counters.

The basic idea is very simple: the chamber does not only contain a single but many

electrode wires. Because the gas is predominantly ionized in the vicinity of the

wires we can gain information about where the particle has triggered an ionization.

This principle is heavily used in modern high-energy particle detectors so that this

idea and its subsequent refinement has earned Georges Charpak the 1992 Nobel

Prize in Physcis.

Through an arrangement of crossed positive and negative electrode wires (seeFig.3.10b) one can increase the spatial resolution of a multiwire proportional

chamber. Basically each positive electrode wire acts as an independent propor-

tional chamber and carries a signal current if an ionization took place in its vicin-

ity. In a drift chamber, additional wires are used to make the electric field as

homogeneous as possible between the electrodes (see Fig. 3.10a).

Often the multiwire proportional chamber also contains a region with a mag-

netic field in which a particles momentum can be measured by measuring the

angle by which its velocity has changed.


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a)

b)

Figure 3.10: a) Electric field inside a multiwire chamber. b) Wire arrangement to

maximize the spatial resolution of a multiwire chamber.


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3.3.4 Scintillation Detectors

Scintillation detectors do not directly observe the ionization of a medium when a

charged particle passes through, but the excitation of atoms or molecules to higher

energy levels that is created by the ions. When the atoms de-excite they emit light

that can be detected. Materials that emit visible light when a charged particle

passes through it are called scintillators; they can be made of plastic (organic) or

crystals (inorganic). Organic scintillators tend to be faster than inorganic scintil-

lators (108s spontaneous decay time vs. 106s) and are therefore better to detectlarge amounts of radiation.

Often scintillators are combined with other devices to increase the signal strength.

In particular, photomultiplier tubes are used to convert a photon into an electric

current. A photon hits a material from which it can release an electron. That elec-

tron is accelerated in a voltage difference towards another material in which it can

release secondary electrons. If this process is repeated one gets an avalanche of

electrons and thus a strong electric signal for each detected photon.

Scintillators combined with photomultipliers can also be used to detect neu-

trons or high-energy photons. The photons can either produce electrons through

the photo-electric effect by releasing one electron from the medium, or through

pair production (creation of an electron-positron pair). The latter possibility of

course only exists for photons with an energy of 1 MeV or more.

Because of their fast response times, scintillation detectors are very suitable

for time-of-flight measurements (TOF) by tracking the time differences on thetrajectory of the particle. This is an important tool for measuring the velocity of a

particle and thus helps to distinguish particles that have a similar momentum but

different masses. Momentum can be found from energy measurements, but the

velocity can only be found through TOF if the particles mass is not known. Of

course this only works for relatively slow particles because for energies that are

considerably larger than the rest mass all particles move at velocities that are close

to the speed of light.

3.3.5 Cherenkov Detectors

These detectors exploit the Cherenkov effect: when a charged particle travels

through a medium at a speedv = c that exceeds the speed of light c/n in thatmedium (withn the refractive index), it emits a cone of blue and UV light. Theopening anglecof the light cone is given by

cos c= 1/(n). (3.9)

This effect is similar to the sound emitted by an airplane that travels faster than

the speed of sound, see Fig.3.11a).


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a) b)

Figure 3.11: a) Sound cone emitted by a jet that moves at a speed greater than the

speed of sound. b) Sketch of Cherenkov radiation emitted by a muon travelling

through the proposed IceCube neutrino observatory. (from http://www.astro.wisc.

edu/heroux/)

Cherenkov detectors exploit this effect to distinguish different particles at low

energies which have about the same energy but different masses and thus different

speed. For n < 1 Eq. (3.9) has no solution and so the particle cannot emitradiation. One particle may be so slow that n < 1 and consequently does notemit radiation. The other particle may be fast enough so that it can be clearly

distinguished. Clearly this is only possible when at least one of the particles has

an energy that is moderately low so thatcan be significantly different from one.Another possibility to distinguish different particles is through the opening

anglec. This requires a higher detector sensitivity. Cherenkov detectors that cando this are called differential counters.

Another variant is known as ring-imagining Cherenkov counters (or RICH

counters). This type exploits the UV photons in Cherenkov radiation, which have

sufficient energy to ionize certain media. The electrons produced in this photo-

ionization have a ring-like distribution (hence the name) and can be detected in

multiwire proportional chambers.

3.3.6 Transition Radiation Detectors

When a relativistic charged particle passes through an interface between media

of different dilectric constants it emits transition radiation. The details are more

complicated, but the main reason is that the electric field in different dielectrics is

different. Remember that D = free =charge density of the particle and thatD = 0 E in a linear dielectric, so D remains the same but Echanges whenchanges. The sudden change in the electric field is then carried away as radiation
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(which is an electromagnetic field as well).

The amount of energy that is converted into radiation depends on the Lorentz

factorof the particle. Transition radiation is particularly usefulr to discriminatebetween electrons and hadrons with energies between 1 and 100 GeV. In that

range thefactor of hadrons is moderate but that of the electrons very large. Toincrease the amount of radiation produced one uses many layers of different media

of specific thicknesses and refractive indices.

3.3.7 Semiconductor Detectors

Radiation in a semiconductor (silicon, germanium) can trigger the creation of

electron-hole pairs (i.e., the excitation of one electron from the valence band to theconduction band) at an energy cost of only 3 eV. This makes semiconductors very

suitable as ionization chambers with large signals for low energy radiation. Thin

wafers (few hundredm) of such a material are sufficient to generate sufficientlystrong signals. An array of wafers can be used to measure the velovity or kinetic

energy of a particle.

3.3.8 Calorimeters

A calorimeter measures the energy of a particle by completely absorbing the ki-

netic energy of it. For hadrons this usually happens through collisions with thenuclei of the detector material. These collisions may generate other particles that

ultimately ionize the medium and thus deliver an electric signal that is propor-

tional to the absorbed energy. Because themean free path, that is the length that a

hadron covers on the average between two collisions, is relatively large hadronic

calorimeterstend to be rather thick.

Electromagnetic calorimeters measure the energy of high energy photons.

These photons mainly deposit their energy through pair creation: an electron and

a positron is created which then ionize the detector material. They also generate

Bremsstrahlung which then can do pair creation again etc.; thus, an electromag-

neticshowerof low-energy photons and particle pairs is generated. Because somehadronic particles can decay into electromagnetic radiation the energy resolution

of electromagnetic calorimeters is usually better than that of hadronic calorime-

ters.

3.4 Detectors at the LHC

Detectors at the LHC (and other colliders) are very large machines that combine

many of the techniques that we have discussed. They have a radial structure in


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a)

b)

Figure 3.12: a) The ATLAS detector (fromhttp://atlas.ch/photos/index.html). b)

The toroidal magnetic field of ATLAS.
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3.4 Detectors at the LHC 43

Figure 3.13: The CMS detector (from http://www-hep.colorado.edu/

experimental/cms/)

ultra-high energy cosmic rays. These are very rare (about 15 observations so

far) events of cosmic ray particles with energies exceeding 1020 eV. Fundamentalconsiderations about supernovae and the propagation of particles through the in-

terstellar space suggest that such particles should be impossible. Understanding

their origin would solve one of the big riddles of astrophysics.

3.4.2 CMS

The Compact Muon Solenoid detector (http://cms.web.cern.ch/cms/index.html )

has this name because its magnetic field differs from the ATLAS design. it is par-

allel to the beam in the inner part and anti-parallel in the outer part. The remain-

ing components are quite similar to ATLAS but different in design. A very good

description of its elements can be found at http://cms.web.cern.ch/cms/Detector/

index.html.
http://www-hep.colorado.edu/experimental/cms/http://www-hep.colorado.edu/experimental/cms/http://cms.web.cern.ch/cms/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/Detector/index.htmlhttp://cms.web.cern.ch/cms/index.htmlhttp://www-hep.colorado.edu/experimental/cms/http://www-hep.colorado.edu/experimental/cms/


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CMS also hunts for the Higgs boson. The purpose of having two similar but

different experiments for the same purpose is to make sure that one doesnt fallfor design flaws in a single experiment. This attitude is actually touching the heart

of physics research in general: physics deals with the description of repeatable

experiments. Repeatable is very important here; if we wouldnt insist on this

then the occasional lucky guess of a fortune teller could be considered science as

well. Since LHC is the only accelerator available that produces 7 TeV protons, the

only chance to repeat the ATLAS/CMS experiment is to have a second detector at

LHC.

At the same collision point at which CMS is placed, another, smaller exper-

iment is taking place, theTotal Cross Section, Elastic Scattering and Diffraction

Dissociation (TOTEM) (see http://totem-experiment.web.cern.ch/totem-experiment/).Its purpose is to measure the total scattering cross section using detectors in the

forward direction of the beam. The detectors are much smaller (3m long) and are

positioned a few hundred meters away from the collision point.

3.4.3 ALICE

This is A Large Ion Collider Experiment in which the collisions between Pb+82

ions are observed to study the formation of a quark-gluon plasma. This is an

important state of matter that would allow to better understand the strong force.

Detector components from innermos to outermost:

Inner tracking system (ITS) consisting of six layers of silicon semiconduc-tor detectors.

Time Projection Chamber (TPC): multi-wire proportional chambers as mainparticle tracking device.

Transition radiation detector (TRD) to discriminate electrons and positronsfrom hadrons.

Time of Flight (TOF) measurement to distinguish heavy from light particles.

Electromagnetic calorimeter (EMCAL) to measure the energy of electronsand photons.

A Photon Spectrometer (PHOS) which is basically a scintillation counter.

A RICH detector (HMPID) to determine the speed of particles.

Muon spectrometer: to measure the momentum of muons.
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3.4 Detectors at the LHC 45

Figure 3.14: The ALICE detector (from http://aliceinfo.cern.ch/Public/en/

Chapter2/Chap2Experiment-en.html)

3.4.4 LHC-bThe purpose of the Large Hadron Collider beauty experiment is have a closer

look at the physics of the b-quark. The b-quark, or bottom quark, was originally

also known as the beauty quark, hence the name. As we will see below the b-

quark is not a constituent of neutrons or protons but is part of other hadron such

as the B mesons. A particularly intersting aspect of b-physics is the violation

ofCP symmetry through weak interaction. CP stands for the combined effect of

charge conjugation (matter anti-matter) and parity (space inversion). This isone of the most fundamental symmetries in nature: because of thePCT theorem

CP combined with a time inversion (T) must be a symmetry of any relativistic the-

ory. In addition, there are speculations that CP violation is behind the differencein the abundance of matter and anti-matter that we observe in the universe.

LHC-b aims at observing the effects of B mesons which are predominantly

found in the forward cone of the scattering cross section. This is also behind the

unvonventional design of the detector: instead of trying to cover as much solid

angle as possible, LHC-b is optimized to detect particles that leave the collision in

a direction between 15 and 300 mrad (0.86 to17) along the forward direction,see Fig.3.15.

Detector components from front to back (left to right in Fig. 3.15):
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Figure 3.15: The LHCb detector (from http://www.mpi-hd.mpg.de/lhcb/index.

php?id=exp)

Vertex locator: silicon detectors to precisely determine the position of thesecondary vertex. During the initial collision B mesons are created. They

have a life time tB of only 1012 s, but because they are fast ( 0.99, 7) they can actually travel over a distance of 1 mm or so. The reasonis the relativistic time dilation. In their rest frame they have a life timetB ,but in the lab fram they live for a time tB and can travel a distancectBbefore they decay. A high spatial resolution therefore allows to see exactly

in which parts the B mesons decay.

Two RICH detectors to distinguish between pions, kaons, and protons.

Tracking (drift) chambers (TT, T1, T2, T3) to observe the trajectories ofcharged particles.

Electromagnetic calorimeter (ECAL) to measure the energy of electrons andphotons.

Hadronic calorimeter (HCAL): to measure the energy of heavy particles.

Muon chambers (M1-M5) to measure the momentum of muons.
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Chapter 4

The Particle Zoo

What kind of processes are observed in particle detectors? Let us start by recapit-

ulating the findings of nuclear physics where we found elementary particles:

Baryons: proton, neutron

Leptons: electron, muon, tauon, and their respective neutrinos

Photon

These particles are involved in different kinds of processes:,, anddecay that

respect certain conservation laws: electric charge, lepton number, baryon number,energy, momentum, and angular momentum are all conserved. The three differ-

ent kinds of decays were associated with different kinds of fundamental forces:

the strong, weak, and electromagnetic force. Baryons are involved in all kinds of

interactions while leptons are not strongly interacting and photons only appear in

electromagnetic interactions.

4.1 Mesons and Isospin

In 1935 Yukawa suggested the existence of a particle that mediates the strong

interaction. It was dubbedmeson and experimentally discovered in 1947 in the

form of the pion. Although this point of view is now obsolete Yukawas theory

still works well to describe some basic features of the strong interaction.

Pions are generated in proton proton collisions, for instance (see Fig. 4.2). In

the figure, blue tracks correspond to negatively charged pions (charge = -e)while most of the red track correspond to + mesons.


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48 The Particle Zoo

Figure 4.1: Bubble chamber picture of proton proton collisions (fromhttp://www.

mathstat.uottawa.ca/rossmann/posters files/creation(caption).htm)

p+p p+p+ 0

p+p p+ n+ +

n+p p+p+ (4.1)

There is also a neutral pion0 and the following processes were observed, and allthree particles have very similar masses, m = 139 MeV and m0 = 135 MeV.The neutral pion is its own anti-particle andis the anti particle of+. The spinof poins is zero.

Pions decay in different ways. Charged pions have a relatively long mean

lifetime of2.6108s while neutral pions decay within1016s. The reason is thatcharged pions can only decay through the weak interaction while neutral pions can

decay elecromagnetically:

+ + + ore+ + e (4.2)

+ ore+ e (4.3)

0 2 or+ e+ + e (4.4)

This indicates that pions interact via all four fundamental forces.

What kind of quantum numbers do pions have? We know already spin and

charge. Their decay channels indicate that their lepton number is zero. An-

other argument supports that their baryon number must be zero, too: the pion

is much lighter than the proton but we do not observe a proton decay of the form

p e+ + 0, for instance. The proton is (luckily and quite literally) rock-stableso that this process must be forbidden by some condervation law. If pions (and

in fact all mesons) have baryon number zero, this process would violate baryon

conservation, and this is indeed the case.
http://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htmhttp://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htmhttp://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htmhttp://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htmhttp://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htmhttp://www.mathstat.uottawa.ca/~rossmann/posters_files/creation(caption).htm


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4.1 Mesons and Isospin 49

The small mass difference between neutral and charged pions indicates that

with respect to the strong force all pions are all similar. We have seen that beforefor neutrons and protons which seem to differ mostly because of their different

electromagnetic properties. We can phenomenologically describe this similarity

by introducing a new quantum number,isospinI. Within this framework, nucle-ons and pions are assigned the following quantum numbers:

particle I I3p 1

212

n 12 1

2

1 -1

0

1 0+ 1 1

For the anti-particles, I3 inverts its value while I remains the same. Isospinis conserved under the nuclear force but not under electromagnetic (e.g., 0 0 +) and weak interaction (e.g., n p+ e + e) . The proton protoncollision process(4.1) shows that the strong interaction preserves isospin while

the decay channels of the pion also show that isospin is not conserved under weak

and electromagnetic interaction. The introduction of isospin will make a lot of

sense once we get to the quark model becaus pions, like all mesons, are composed

out of a quark and an anti-quark.

Mathematically, isospin works in the same way as ordinary spin, although

physically there is no connection between the two. Isospin is not an angular mo-

mentum, it just happens to have the same quantum numbers: a particle of total

isospinIcan haveI3values ranging from Ito I. This should be familiar to youfrom orbital angular momentuml and magnetic quantum numberm of electronsin an atom. One may also say that isospin operators obey the same commutation

relations as angular momentum operators, but we havent discussed this yet.

Also in 1947, another neutral particle later dubbedkaonwas found in cosmic

ray showers which decays into pions,

K+

+

+ 0

. (4.5)

The kaon is like a heavy (495 MeV) pion, except that the antiparticle K0 of theneutral kaon and the neutral kaon K0 itself are different particles. There weresome strange facts about the kaon, e.g., it is produced in strong interactions but

decays ith a mean lifetime that is comparable to that of weak interactions, but this

will be the subject of the next section.

Soon more mesons were found and nowadays they also include the , , D, B , , ,and J/ meson. Some of them occur in different variations, e.g., B, Bs, Bc, B

, Bs , Bc .

Things get quickly complicated.


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50 The Particle Zoo

Figure 4.2: 1964 bubble chamber picture of proton proton collisions in which the

particle was first observed. vertical lines corresponds to the proton beam. (fromhttp://www.pd.infn.it/%7Edorigo/omega-dia-w.gif)

4.2 Baryons, Strangeness, and strong Hypercharge

Baryons are composed of three quarks (see below) and therefore are usually heav-

ier than mesons. The nucleonNwith its two isospin states (proton and neutron)was of course the first known baryon. More were found in cosmic ray showers.

The and particles had something to do with the kaon because the kaon wasalways produced together with it,

+p K0 + 0 (4.6)

+p K+ + + 0 (4.7)

+ +p K+ + + (4.8)

+ +p K+ + + (4.9)

A number of different processes were theoretically possible, e.g. +p K++ + 0, but the0 was never produced together withK, only withK+

(or the neutral kaon). Similarly it was observed that the+ particle was alwaysproduced together withK+ but not withK.

To explain this behaviour a new quantum number, strangeness, was intro-

duced. Strangeness is conserved under strong and electromagnetic interaction but

not under weak interaction. Most known baryons were given strangeness S= 0,butSK0 =SK+ = 1, with their anti-particles getting the opposite value S= 1.Further particles with non-zero strangeness were found, like the cascade particle
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4.2 Baryons, Strangeness, and strong Hypercharge 51

Figure 4.3: List of some baryons and mesons and their quantum numbers.

and the particle. Nowadays we know hundredths of baryons, including vari-ations of, , , and new ones like the . The latter particle is actually a very(1023s) short lived resonance that corresponds to an excited pion-nucleon state.Fig.4.3gives an overview over some particles and their quantum numbers.

It is also useful to introduce the strong hyperchargeY =B + S. Gell-Mannand Nishijima found that for all hadrons there is a relation between Yand isospinin the form

Q= I3+Y

2 . (4.10)

When analyzing the violation of quantum number conservation one has

to keep in mind that most of the strong quantum numbers (isospin, strangeness,

strong hypercharge) are not defined for leptons or photons. Hence, to discuss the

violation of strong quantum numbers one can only take into account the intitial

and final hadrons that take part. For instance, inn p+ e+ e the leptonnumber L is conserved. To test isospin we ignore the leptons and find I3 : 1/2 1/2so isospin is not conserved in this weak process.


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52 The Particle Zoo

a) b)

c)

Figure 4.4: a) Meson octet, b) baryon octet, and c) baryon decuplet of the eight-

fold way. (fromhttp://en.wikipedia.org/wiki/Eightfold Way (physics))

4.3 The eightfold way

By the early 1960ies the situation in particle physics was pretty confusing. Hun-

dredths of particles and resonances were found, and finding a structure in this

particle zoo was paramount. Murray Gell-Mann and Yuval Neeman indepen-

dently found such a structure using the methods of group theory. The observed

that particles could be arranged in such a way that their quantum numbers formed

symmetric patterns, see Fig.4.4. In principle these figures are only 2D arrange-

ments of quantum numbers. However, they can be interpreted as the eigenvalues

of generators of the group SU(3), and this observation paved the way to the dis-

covery of quarks. What this means will become clear in the next chapter.
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Chapter 5

Symmetry

Symmetries are one of the most important topics in physics, particularly in quan-

tum mechanics. Because the beauty of the standard model can only be understood

with some insight in symmetry groups we will discuss this here.

5.1 Invariance and symmetry groups in quantum physics

Symmetry generally means that a system doesnt change if a certain operation is

performed. A good example is the hydrogen atom, where an electron moves in the

Coulomb potentialV(r) of the nucleus. This potential is spherically symmetricbecause it only depends on the distance between electron and nucleus. Conse-

quently, if we rotate the system we still have the same potential.

For instance, lets consider the Schrodinger equation in 3D in spherical coor-

dinates,

E(r,,) = 2

2m

+ V(r)(r,,) (5.1)=

2

2m

1

r2rr

2r+ 1

r2 sin sin +

2

(r,,) +V(r)(r,,)

(5.2)

The wavefunction(r,,)is a solution to this equation. If we rotate it by a fixedangle0 around the z-axis it should still be a solution because the differentialequation remains unchanged under a change of variable =+0. So (r,,+0)is also a solution.


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54 Symmetry

5.2 Symmetry Groups

It is clear that0can take any value between 0 and2 and all these values corre-spond to a symmetry operation, i.e., they leave the Hamiltonian invariant. In fact

an arbitrary rotation around any axis leaves it invariant but thats beyond the scope

of this course.

Let us denote the operation rotate the wavefunction by an angle 0 as anoperator,

U(0) (r,,) :=(r,,+ 0). (5.3)

Clearly we can perform two rotations to obtain another rotation:

U(2)U(1) (r,,) =(r,,+ 1+ 2) = U(1+ 2) (r,,). (5.4)

Furthermore, we have an identity operation, U(0) (r,,) = (r,,), whichwe will denote by1. For each rotation we have an inverse operation U1(0) :=U(0)such that U

1(0)U(0) =1.In Mathematics, the three properties

U(2)U(1) = U(1+ 2) (5.5)

U(0) =1 (5.6)

U1(0)U(

0) =1 (5.7)

are the defining properties of a group. In other words, the set of rotations around

the z-axis forms a group.

5.2.1 Groups and group representations

Generally, a groupGis formed by a set of elementsg and some group muliplica-tion (or addition) law such that the above properties are fulfilled, i.e.,

g1 g2=g3 G (5.8)

g1 g= 1 (5.9)

g 1= 1 (5.10)

In other words, the group must contain a neutral element 1 and for each group

element gits inverse g1 must also be an element of the group. Groups can have aninfinite number of elements like the rotations around the z-axis about an arbitrary

angle, or they can have a finite number of elements (see the groups Cnbelow). Thegroup itself is entirely determined by the specific multiplication laws that define

the group (again see the groupsCnbelow).


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5.2 Symmetry Groups 55

The definition of a group is an abstract one and it is important to keep in mind

that groups are abstract mathematical constructs. Group structures can be realizedin nature, but these realizations are not the group itself, they rather form a repre-

sentation of a group. The distinction between a group and its representations is

a central aspect of understanding symmetries in Physics.

Lets consider an example: three protons form a regular triangle and are not

moving. The charge distribution of theprotons is then invariant under rotations of

120and240. One can consider these operations as the representation of a groupthat has three elements: 1(rotation by0),g1 (rotation by120), andg

11 which

correspond to the240rotation. This group is called C3and is a special case of thegroupsCn that are formed by repeated discrete rotations by an angle of360

/n.

A special property of theC3group is thatg1 g1 = g11 .Let us try to find amatrix representationof the group C3. We start by writing

the position vectors of the protons as

rn= cos(n120)x+ sin(n120)y , n= 0, 1, 2. (5.11)

Let us now find a matrix representationM1 of the elementg1 that transforms thecomponents of vector rninto that of vector rn+1, e.g.

M1,xx M1,xyM1,yx M1,yy

cos(120)sin(120)

=

cos(240)sin(240)

(5.12)

It is not hard to verify that this is accomplished by the matrix

M1 =

1

2

32

32 1

2

(5.13)

Together with the2 2 unit matrix and M11 this forms a matrix representationof the groupC3. There are also representations of groups that are not based onmatrices. They can, for instance, involve differential operators (see below).

However, this is not the only matrix represenation ofC3. Any matrix that

fulfillsM1 M1 = M1

1 would also generate a representation. This is because thegroupC3 is defined by the relations g1 g1 = g

11 so thatg

31 = 1. An example

would be

M1 =

14

34

34

34

34

14 3

4

34

34 3

414

34

34

34

34

14

(5.14)which obviously does act on a four-dimensional vector space. The physical or

geometrical nature of this space is quite unclear in our example. However, in real


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56 Symmetry

physical applications one very often has to find matrix representations for spaces

of a particular nature. For instance, in molecular physics that four-dimensionalspace could correspond to the Hilbert space spaned by the three electronic ground

states (one bound to each proton) and a symmetric superposition of electronic

excited states. There is a rather involved theory that allows to classify representa-

tions of finite groups in terms of their conjugacy classes, but this is beyond the

scope of this course. If you are interested, there is a classic paper by Robert S.

Mulliken on this topic (Physical Review 43, p. 279 (1933)).

The representations of finite symmetry groups that are composed of discrete

rotations, inversions, mirror reflections, and discrete shifts along some axis is of

tremenduous importance in molecular physics, solid state physics, and crystallog-

raphy. For particle physics, a very specific kind of groups is most important. Wewill deal with it in the next subsection.

5.2.2 Lie groups

Lie groups are roughly speaking groups for which the group element can be writ-

ten as exponentials of so-called generators. In this course we will discuss their

features in the briefest possible way, just sufficient to understand the structure

of the standard model. If you like to learn more about Lie groups and algebras

you can download Robert Cahns book for free from the authors web page at

http://phyweb.lbl.gov/rncahn/www/liealgebras/book.html.Lets consider the rotations around the z-axis again, U(0)() =( + 0).

We can express the right-hand side of this equation as a Taylor expansion,

( + 0) =() + 1

1!(1)() 0+

1

2!(2)() 20+ (5.15)

=n=0

1

n!

n

nn0 . (5.16)

We can interpret the factor n/n as the n-fold application of the differential

operatoron the wave function(). Hence,

( + 0) =

n=0

1

n!n n0

() (5.17)

=e0() (5.18)

The exponential (or any other function) of an operator may look somewhat strange

at first, but all that is behind it is that you do a Taylor series and then repeatedly

apply the operator on the wavefunction. The only difference between a function
http://phyweb.lbl.gov/~rncahn/www/liealgebras/book.htmlhttp://phyweb.lbl.gov/~rncahn/www/liealgebras/book.htmlhttp://phyweb.lbl.gov/~rncahn/ww

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