Physics & Detection of AstroParticles A Introduction & Tools A.2...

Physics & Detection of AstroParticlesA Introduction & Tools

Andreas Zech, LUTH, Observatoire de Paris

A.2 The Standard Model of Elementary Particles

(in a Nutshell)

A.2 The Standard Model of Elementary Particles

(in a Nutshell)

A. Zech, Physics & Detection of AstroParticles, A2 Standard Model 2

Outline

A.2.1 Introduction: the discovery of subatomic particles

A.2.2 Fermions - constituents of matter

A.2.3 Bosons & fundamental interactions

A.2.4 Conservation laws

A.2.5 Very quick overview of nuclei, atoms, molecules

A.2.6 Interaction cross section

A.2.7 Particle accelerators & detectors

A.2.8 Beyond the standard model

A.2.9 Useful formulae

A.2.10 Bibliography


A.2.1Introduction: The discovery of

elementary particles

A.2.1Introduction: The discovery of



How elementary ?Elementary particles are pointlike in the sense that they have no known structure. Until the late 19th century, atoms were thought to be elementary (atomos = undivisible). The discovery of the first subatomic particles was the beginning of a new theory of elementary particles that led to a standard model of particles.

How pointlike is pointlike ?

This depends on the resolution of the "microscope". The smallest scales are probed with particle beams and their resolution depends on the de Broglie wavelength of the probing particle:

A particle beam of an energy of 10 GeV (101 0 eV) gives a spatial resolution of about 10- 1 6 m, about 10 times smaller than the radius of a proton.

The higher the energy of the probing particle beam, the better the resolution. Thus the need for every growing energies in particle accelerators.

(High energies are also needed for the creation of heavy elementary particles - like the Higgs boson...)

= h / p ( ~ 1 nm for an electron of 1 eV -> ~ 1000 times smaller than for a 1 eV photon)


A bit of history of sub-atomic and particle physics

● William Crookes experimented with vacuum tubes (the prototype of the TV) already in 1879. Applying high voltages between cathode and anode he observed what he called cathode rays. Objects in their way would cast a shadow on a fluorescent screen.

● J.J. Thomson applied electric and magnetic fields to the cathode rays and found that they had a mass ~1/1000 that of a hydrogen atom in 1897. This was the discovery of the first sub-atomic particle, the electron.

● Röntgen discovered in 1895 that wrapped unexposed photographic plates left close to a Crookes tube became darkened. He ascribed this effect to another new form of radiation more penetrating than the cathode rays – X-rays. Only in 1912 it was proved that these rays are a form of electromagnetic radiation, more energetic than UV light. (Laue observed their diffraction by crystals.)


Cosmic Rays and Particle Physics

● First evidence for a new form of radiation was found in the early 1900s when electroscopes were seen to be discharged, even in the absence of radioactive materials. In 1912 Victor Hess proved during several balloon flights that the radiation came from outside the atmosphere. This was the discovery of the cosmic rays.

● Only in 1929 and after the invention of the Geiger-Müller counter it was found that cosmic "rays" are in fact massive particles and not radiation.

● From the 1930s on, the highly energetic cosmic rays were observed in cloud chambers and bubble chambers. This led to the discovery of several new particles: positrons, muons, kaons, lambdas, Ξ and Σmesons.

● Another technique was developed and first used in the 1940s. With stacks of nuclear emulsions, pions were discovered in 1947.

● In the 1950s, the technology of particle accelerator facilities allowed to generate beams of artificial "cosmic rays" in the laboratory – with known energy and direction. Particle Physics and Cosmic Ray Physics became two separated fields of research.

© GNU Free Documentation License


The Particle "Zoo"

This is just a small excerpt.

For a complete listing, see:

http://pdglive.lbl.gov

A. Zech, Physics & Detection of AstroParticles, A2 Standard Model 8source: http://CPEPweb.org

The Standard Model of Particle Physics


A bit of historyThe standard model of particle physics, developed in the 60s, describes the elementary particles and fundamental interactions (except for gravitation) with the use of quantum field theory. It has been used to accurately predict the existence of many particles that have then been detected in accelerator experiments.

Important steps in the development of the model:

● quantum electrodynamics (QED) is developped by Feynman, Tomonaga, Schwinger and Dyson and accurately describes all electromagnetic interactions.

● in 1960, Glashow finds a way to combine electromagnetic and weak interactions into a single electroweak theory. Weinberg and Salam include the Higgs mechanism into the theory in 1967 to give it its modern form. The discovery of the W and Z bosons in the 70s and 80s, correctly predicted by the electroweak theory, gave confidence into the theory.

● quantum chromodynamics (QCD), the quantum field theory describing strong interactions, is completed in the early 70s and added to the standard model.

critical issues:

● the Higgs boson plays a central form in the current form of the standard model. Its discovery is still awaited.

● the standard model does not describe gravitational interactions. A unification of gravitation with the other forces has not yet been achieved.


A.2.2Fermions - constituents of matter

A.2.2Fermions - constituents of matter


Constituents of matterFermions are the elementary particles that make up matter.

Atoms are made of a nucleus of fermions (quarks that form protons and neutrons), surrounded by a "cloud" of a different type of fermions (electrons). Muons, pions, positrons and neutrinos are other fermions found frequently in nature.


Fermions in the standard model


Fermions and the Pauli PrincipleFermions are elementary particles with half-integral spin (1/2 ħ , 3/2 ħ , ...). "Spin" is a characteristic property of elementary and composite particles. While for composite particles, it can be interpreted as an angular momentum, for elementary particles, it is an intrinsic characteristic of the particle's quantum state. The spin of charged particles is associated with a magnetic dipole moment.

In Quantum Mechanics, particles or ensembles of particles are described by wavefunctions. The wavefunction of an ensemble of fermions is antisymmetric under the exchange of any pair of particles (spin-statistics theorem):

This implies that two identical fermions cannot be in exactly the same quantum state (i.e. have exactly the same characteristics). For example, two electrons in an atom have to differ in at least one quantum number (n, l, m

l or m

s ). This is the Pauli Exclusion Principle.

Implications:

- the characteristics of the elements in the periodic table depend on how the "shells" are being filled up by electrons with different quantum numbers.

- white dwarves: gravitational force is balanced by electron degeneracy pressure, i.e. electrons inside atoms "resist" being forced into lower "shells"

- neutron stars: protons and electrons have combined to form neutrons (i.e. also fermions). Neutron degeneracy pressure balances the gravitational force.

x1 , x2 , x3 , x4 ,... −x1 , x3 , x2 , x4 , ...


Fermi-Dirac statisticsFermi-Dirac statistics describe the distribution of identical fermions in thermal equilibrium. The average number of fermions in an energy state i is given by:

ni=g iei−/kT1

gi : degeneracy, i.e. number of states

with energy i (but with different quantum numbers)

µ : chemical potential, constrained by the total particle number

In the classical limit, i.e. for particle-separations much greater than the average de Broglie wavelength , the F.-D. statistic is approximated by the Maxwell-Boltzmann statistic:

ni=g iei−/ kT

The chemical potential measures the tendency of particles to diffuse and depends on the temperature.At T=0 the chemical potential is given by the Fermi energy.The Fermi energy is the energy of the highest occupied quantum state in a system of identical fermions at T=0.

One sees that forand also

= h / p

i ∞ : n i 0n i≤ g i -> Pauli principle


LeptonsLeptons (lepton - fine, small, thin) are the lightest fermions and come in 3 families (6 "flavours"). All leptons have integral charge (-e, 0, +e) and spin s=1/2, so they can have only two spin states m

s= + 1/2 or - 1/2 (spin "up" or "down").

electrons- mass = 0.511 MeV, charge = - e - stable particles, form the "shell" of atomsmuons- "heavy electron" with > 200 m

e , charge = - e

- mean lifetime Τ1 / 2

~ 2.2 10- 6 s , decay mostly as

tauons- very heavy (1.78 GeV) , found in acceleratorsand cosmic ray showers, charge = - e- mean lifetime Τ

1 / 2 ~ 2.9 10- 13 s , decays e.g.

and others...

− e− e

− e− e

− −

− −

neutrinos- 3 distinct flavours - neutral and massless in the standard model- it is now established that they do have (very small) masses- accompany their partner-lepton in decays (conservation of "lepton number")


QuarksQuarks are fermions with a spin of 1/2 and exist in 6 flavours like leptons. They carry fractional charges of -1/3 e and +2/3 e. Quarks do not exist as free particles, but only in combinations in which their charges add up to an integer charge (e.g. ddu ). This phenomenon is known as quark confinement. As a consequence, only "constituent" quark masses, which include binding effects, can be given.

up / down quarks:- the lightest quarks (3 / 6 MeV)- constituents of stable particles, e.g. proton (uud), neutron (ddu)

charm / strange / top / bottom quarks:- heavier than u and d, found only in accelerators and cosmic ray showers- combine to unstable particles that decay rapidly (~ 10 -13 s) into combinations of u and d. An additional characteristic is needed to describe the interactions of the quarks: each flavour of quark comes in three different "colours" (red, green, blue). These "colours" are just another name to distinguish e.g. three different types of up-quarks.

"Three quarks for Muster Mark!Sure he has not got much of a bark. And sure any he has it's all beside the mark."

James Joyce, Finnegan's Wake


antiparticlesAntiparticles are objects with the same mass and lifetime as the corresponding particles, but with opposite sign of charge and magnetic momentum. The first antiparticle to be found was the positron ("anti-electron") in 1932 in experiments with cosmic rays.

Antiparticles exist for fermions and for bosons. For fermions only, the difference in the number of fermions and antifermions is a constant, i.e. fermions and antifermions can only be created and destroyed in pairs.

Mathematically, antiparticles correspond to the negative solution of the energy-momentum conservation formula:

Classically, negative energies do not exist. In quantum mechanics, they have a meaning. Particles are presented by wavefunctions. The wavefunction of a free particle in 1 dimension is e.g.:

The propagation of the particle is described by a plane wave. Only the square amplitude of this wave ( Ψ2 ) has a real physical meaning: it is the probability of finding the particle in a certain position.

The wave function can present a particle with E>0 advancing in the direction of increasing x, but it can also represent particles of E<0 and momentum p<0 travelling in the negative x direction and backwards in time. For example one could imagine (negative charged) electrons flowing backwards in x and time. This would be equivalent to positive anti-electrons (positrons) flowing forward in x and time.

E2 = p2 c2m2 c4 E =± p2 c2m2 c4

= Ae−i / ℏ Et−px


hadrons: baryons & mesonsHadrons (hadros - stout, thick) are composite particles that are made up of quarks. The proton is the only hadron that seems to be stable (lifetime > 1032 yr). Neutrons decay with a half-life of about 15 minutes, unless they are bound in a nucleus. As part of atomic nuclei, the "nucleons" (i.e. p and n ) can be stable.

Two combinations of quarks exist and make up two different groups of hadrons: baryons and mesons. In each case, the fractional quark charges add up to integral charges in the hadron.

baryons (three quark states)

are fermions (fractional spin). They can only be created or annihilated in particle/antiparticle pairs.

mesons (quark-antiquark states)

are bosons (integral spin) -> see next chapter

The same combination of quark/antiquark pairs can make up different types of hadrons, distinguished by their total spin. Some mesons, like the neutral pion, are their own antiparticles.

0 = uu−d d /2


resonancesResonances are excited states of hadrons with a very short lifetime.

As an example, the Δ+(1232) resonance has the same quark content as a proton (uud), but a different spin. In the proton, one of the quarks is aligned opposite to the others, so its total spin is 1/2. The Δ+ has all three quarks aligned the same way and has a total spin of 3/2. Its mass is 1232 MeV, while that of the proton is only 938 MeV. Its lifetime is only 5.6 10 - 24 s.

Common production and decay path:

Resonances in nucleons lead to resonances in nuclei. In astrophysics, an important resonance is given by an excited state of the 12C nucleus. The element 12C is produced by helium fusion in red-giant stars:

p 1232 p0

nor

Be8

Be8 C ∗12

C ∗12 C12

The existence of carbon in the Universe depends on the existence of the 12C resonant state!

Heisenberg's uncertainty principle provides a relation between the duration of a resonance and the spread in energy of the decaying state: E t≈ ℏ


A.2.3Bosons & Fundamental interactions

A.2.3Bosons & Fundamental interactions


bosons in the standard modelBosons are elementary or composite particles with integral spin.

The four known elementary bosons ("gauge bosons") are the mediators of interactions ("forces") in the standard model. Interactions are described by the exchange of virtual gauge bosons between the fermions. Those are:the photons, the W + , W - and Z 0 bosons, and the gluons.

The existence of two additional gauge bosons, the gravitons and the Higgs bosons, is thought likely, but has not yet been proven.

Mesons are composite bosons. One distinguishes:

(pseudo-)scalar mesons with total spin 0 and even (odd) parity.(i.e. the quark spins are aligned in the opposite direction)

(pseudo-)vector mesons with total spin 1 and odd (even) parity.(i.e. the quark spins are aligned in the same direction)

spin

1

1

111


Bose-Einstein statisticsBose-Einstein statistics describe the distribution of bosons in thermal equilibrium. Contrary to fermions, the wavefunction of an ensemble of identical bosons is symmetric under the exchange of a pair of any two particles. An unlimited number of identical bosons can occupy the exact same quantum state, since they are not bound by the Pauli principle.

The expected number of bosons in an energy state with energy E

i is:

Note that n i can become very large.

For kT << ε i - µ , this reduces to the Maxwell-Boltzmann

distribution:

(this is true for systems at high enough temperature that are not too dense; in this case, quantum effects are negligible)

For kT >> ε i - µ , it reduces to the Rayleigh-Jeans distribution:

(this is true for small particle energies)

At very low temperatures (close to absolute 0), all bosons accumulate in the lowest energy state, forming a Bose-Einstein condensate.

ni=g iei−/ kT−1

ni=g iei−/ kT

ni=g i k Ti−

This is the basis for the Planck distribution that describes a photon gas.

hyperphysics.com


forces and force carriers

There are four types of interaction:

- gravitational force- weak force- electromagnetic force- strong force

Classically, interactions are described by potentials or fields spread out in the space around a charge or mass.

In quantum theory, interaction is described as the transfer of momentum by the exchange of a virtual gauge boson. These virtual particles are not directly observable.


Feynman diagrams (1)Feynman diagrams are a graphical way of describing interactions between particles. The different components of a Feynman diagram can be directly translated into a mathematical (quantum field theory) formulation of interactions. These formulations allow to calculate a decay rate or interaction cross-section (i.e. probability for the interaction to occur).

For example, electron-positron scattering is pictured in the following diagram:

electron

positron

virtual photon

vertex

rules for Feynman diagrams:- time is increasing from left to right (other conventions exist, e.g. bottom to top...)- solid straight lines represent fermions

- wavy or curly lines represent bosons

photons, W or Z bosons

gluons


Feynman diagrams (2)- anti-particles are indicated by an arrow that points backwards in time, e.g. anti-fermion

- fermion and boson lines meet at vertices, where charge, energy and momentum are conserved. The strength of the interaction is given by a coupling constant (e.g. the fine structure constant α for electromagnetic interactions). Single-vertex diagrams do not exist.

- lines that enter or leave the diagram represent real, detectable particles.

- lines that connect two vertices (e.g. the photon line in our example) represent virtual particles, which are not observable and do not follow the energy-momentum relation:

Translation of the above diagram into a formulaE2≠p2 c2m2 c4

p1

p2

p4

p3 ig

x

yig

e−i p1x

e−i p2 y

e i p3 x

e i p4 y

A =∫ d4 x∫ d 4 y ig 2 e−i p1 xe−i p2 y ei p3 xe i p 4 y∫ d4q i eiqx− y

24 q2

probability amplitude A for interaction (neglecting spin):

time-dependent wave functions of e- and e+

"Feynman propagator" of the virtual particle ( here a photon; for other gauge bosons q2 becomes q2 + m2 )

coupling constants at the vertices (prop. e2 for e- and e+ )


electromagnetic interactionsElectromagnetic interactions act on all charged particles. The interaction is mediated by the exchange of virtual photons (mass=0, charge=0, spin=1). The coupling constant specifies the strength of the interaction between the charged particle and the virtual photon. For e.m. interactions it is identified as the fine structure constant α:

Electromagnetic interactions are found in:

- interactions of charged particles with magnetic or electric fields in general- Coulomb interactions with atomic nuclei, photoeffect, photoabsorption, ionisation- synchrotron emission- bremsstrahlung, pair creation, pair annihilation- e- e- scattering, e+ e- ("Bhabha"-) scattering, Coulomb scattering- ...

Quantum Field Theory is the theory that describes particle interactions, including quantum and relativistic effects.Electromagnetic interactions are described by a very well tested branch of quantum field theory: quantum electro-dynamics (QED). Its predictions agree very well with experimental data from accelerator experiments.

=e2

4ℏc=

1137.0360...


weak interactionsWeak interactions take place between all quarks and leptons. They are mediated by the heavy W+ , W - and Z 0 bosons and are of short range.The weak coupling constant is much smaller than the e.m. coupling constant:

Weak interactions are usually not observable, because e.m. and strong interactions completely dominate them, except in two cases:

● interactions involving neutrinos (which do not interact via e.m. or strong interactions)

● interactions with quark flavour change (forbidden in strong interactions!)

Weak interactions by exchange of a W+ or W - are called "charged current" interactions, while those by exchange of a Z0 are called "neutral current" interactions.

Example of a weak decay:

Just as electricity and magnetism could be unified into one single theory,the electromagnetic and weak interactions have been unified into the electroweak theory by Weinberg, Salam and Glashow in 1960.

w

≈ 10−5

− e−e

W −

−

e

e−


strong interactions (1)Strong interactions act on the colour charges of quarks and gluons. The interaction is mediated by the exchange of virtual gluons (mass=0, charge=0, spin=1). The coupling constant for strong interactions is about a factor 100 larger than for e.m. interactions.

colour charge

The quantum field theory that describes the strong interactions (quantum chromodynamics - QCD) defines three types of strong charge ("colours" - red, green, blue). Each quark carries one of the three colours and its antiquark carries the corresponding anti-colour. The strength of the strong force is independent of the colour.

Gluons themselves carry colour charges as well. Each gluon carries a pair of colour-anticolour charges. This leads to 8 possible colour combinations (32 minus the "colourless" solution rr+bb+gg). The fact that gluons carry colour charges means that they interact with themselves (other than photons, which do not interact directly with other photons).

The combination of colours of different quarks leads to "colour-free" hadrons (e.g. rgb or rr ...).

s

≈ 100

Q Q'

r b

b r

G

rb

The Feynman diagram shows colour charge conservation in an interaction between two quarks: a red quark emits a rb gluon and turns into a blue quark (r -> r + b + b); a blue quark receives the rb gluon and turns into a red quark(r + b + b -> r).

time


strong interactions (2)confinement

The QCD potential can be described as

The second term (non-existant in the e.m. Coulomb potential ) results in a strong binding of the quarks with increasing distance and leads to the "confinement" of quarks inside the hadrons.

If one attempts to free a single quark from a hadron, this leads to the production of a quark- antiquark pair (= meson), which is energetically favoured.

residual strong force (nuclear force)

The strong force acts between quarks and gluons. A residual of this force leads to a (much weaker) nuclear force between hadrons. This residual force is responsible for the binding of nucleons into nuclei. It decreases with distance.

An analogue to this force can be seen in the van-der-Waals force between molecules, which is a residual of the e.m. force.

V s=−43 sr

k rV em=−

r


some examples of interactions


gravitational interactionsGravitational interactions act on masses. The strength of the gravitational force is given by the gravitational constant G. Compared to the electromagnetic coupling , the gravitational coupling is about a factor of 10- 3 8 weaker and thus in general negligible in particle physics.

Only for extremely heavy particles, with a mass equal to the "Planck mass" would the gravitational force be comparable to the e.m. force.

The graviton, a hypothetical gauge boson (mass=0, charge=0, spin=2), can be added to the standard model as the mediator of the gravitational force.

In the classical limit, General Relativity and the graviton interactions in quantum field theory arrive both at a correct description of gravity.

At very high energies close to the "Planck scale" (or at very short distances: the Planck length being given by ), quantum field theory no longer provides a valid description of gravity.

There is a fundamental problem in combining quantum mechanics and general relativity. Efforts to find a new theory that succeeds in unifying all interactions are ongoing (string theory, quantum gravity...). Unless one of these new theories proves successful, we have no means of physically describing the very early stages of the universe.

mP = ℏ c /GN 1 /2

= 1.22×1019GeV

e2/40GN m

2

lP = ℏG /c3 ≈ 1.62×10−35m


the Higgs bosonAn additional gauge boson is predicted by the Standard Model (Weinberg-Salam model) and is expected to have a mass < ~1 TeV and spin 0.

The mechanism of spontaneous symmetry breaking by the Higgs mechanism, which also produces the Higgs boson, gives mass to the W and Z bosons and also to fermions. At high enough temperatures (energies), the standard model predicts that all elementary particles are massless. This symmetry would be broken at a critical temperature and different masses would arise.

Mass is not understood as an intrinsic property of elementary particles, but as arising from an interaction with the Higgs field.

Ongoing searches, at Fermilab and at the LHC, are trying to find a signature of the Higgs boson in very energetic particle collisions (high mass !).

source: wikipedia


Summary: particle interactions

source: wikipedia


A.2.4Conservation laws

A.2.4Conservation laws


charge conservationElectric charge is a conserved quantity in all interactions.

Tight upper limits have been set by experiments on a potential violation of this rule. The decay of a neutron into a proton is known to produce also an electron, thus charge is conserved. One can search for neutron decays that do not produce electrons and would thus not conserve charges. It has been found that:

Charge conservation implies that only pairs of positive and negative charges can be created or annihilated.

Charge conservation is assumed to be exact and a direct consequence of the gauge invariance of the electromagnetic field. Gauge invariance in electrostatics means that one can choose the zero-point of the electric potential arbitrarily.

If charge was not conserved, one could create a charge q at a potential p. This would require an amount of work (energy) w. w is independent of p, since we assume that p can be chosen arbitrarily and that physical laws depend only on differences in potential Δp !One could then move the charge to a higher potential p', which would require the additional work q (p' - p). Then one could destroy the charge by liberating the same amount of work as for its creation. The total energy balance reads: ΔE = W - W + q (p' - p) > 0 . This would violate energy conservation !

Event though the electromagnetic potential is described differently in QED, charge conservation is still associated to gauge invariance.

n p e en p e− e

9×10−24


C,P,T invariance (1)● charge conjugation C

Charge conjugation changes the sign of charge and magnetic moment of a particle. Note that this turns a particle into its antiparticle! Strong and e.m. interactions are invariant under charge conjugation, while this is not the case for weak interactions. For example in the strong interaction the rates of positive and negative mesons are identical.

● parity P

Particle wavefunctions behave differently under the spatial inversion of coordinates x,y,z -> -x,-y,-z. They are said to be of even parity P=1 for Ψ(-r) = Ψ(r) and of odd parity P=-1 for Ψ(-r) = -Ψ(r) . Parity is conserved by strong and e.m. interactions, but not by weak interactions.Parity of a system of particles with orbital angular momentum L is given by P = (-1) L . For a single particle, an intrinsic parity is defined, just like any other quantum number.

One usually describes particle states by their total angular momentum J (a combination of orbital angular momentum L and spin S) and their parity P. A single pion has for example J P = 0 - .

● time reversal T

Strong and e.m. (not weak) interactions are also invariant under time reversal t -> -t .

pp −...


C,P,T invariance (1)

● CP and CPT invariance

Strong and e.m. interactions are invariant under a simultaneous operation of charge conjugation and parity change (CP), while CP invariance can be violated by weak interactions.

All interactions are seen to be invariant to a combined CPT change. (CPT theorem) This invariance implies that particles and antiparticles have the same mass and lifetime, and equal electric charges and magnetic moments (with opposite sign). Measurements on CPT invariance violation seem to confirm the CPT theorem.


baryon & lepton number conservationInteractions in the standard model have been found (experimentally) to conserve quantities called the "lepton number" and "baryon number".

In interactions including leptons, a lepton number is ascribed to each flavour: Le, L

µ , L

τ .

The value of the lepton (flavour) number is +1 for a lepton and -1 for an anti-lepton of the corresponding flavour.

ee

L −1 0 0 −1Le 0 −1 1 0

left: allowed process; lepton numbers are conservedright: forbidden process!

eLe 0 −1 0L −1 0 0

The recent discovery of flavour-change in neutrinos ("neutrino oscillations") shows that lepton number conservation does break down at long enough time-scales.

The total baryon number is equally conserved in most interactions.

Strong interactions conserve the flavour quantum number of strange (S), charme (C), bottom (B) and top (T) quarks. For example in a collision between hadrons made up of u and d quarks only, strange quarks can only be produced in strange-antistrange pairs in order to conserve "strangeness".

Weak interactions conserve total baryon number, but not the flavour quantum number.


isospin symmetry (1)It has been found that strong interactions between the nucleons are the same for pp, nn, or pn interactions. This has led to the introduction of another quantum number, the isospin. Protons and neutrons are described as two different states of the same particle - the nucleon.

The nucleon has isospin I=1/2. Its two substates, with Iz = +1/2 and I

z = -1/2 are the proton and

neutron, respectively. Their charge is given by Q/e = 1/2 + Iz .

Isospin is a conserved quantum number in strong interactions (not in electro-weak ones). Isospin symmetry (i.e. conservation) applies to all baryons and mesons which transform into each other by interchange of u and d quarks. For example the pion isospin triplet looks like this:

= u d I z =10 = 1/2 d d−uu I z= 0− = d u I z =−1

An example for isospin conservation:

i p p d

I 121

201

ii pn d0

I 0 / 1 01

(i) p has isospin 1/2 with Iz=+1/2. Total isospin for two protons is 1. A deuteron has I=0 (derived from the Pauli principle by considering all quantum numbers). The π+ has I=1 => isospin is conserved(ii) proton and neutron can have I=0 (singlet state) or I=1 (triplet state):

p ,n = 1/2 [ ] p ,n = 1/2 [ − ]

I=1 The reaction can only proceed via this channel !I=0


isospin symmetry (2)baryon decuplet: the 10 baryon states of lowest mass and of spin-parity J P = 3/2 +

Other multiplets have been defined to organize baryons and mesons into similar schemes.

strangeness

third componentof isospin

I = 3/2

I = 1

I = 1/2

I = 0

constant charge

The members of each isospin multiplet have approximately the same mass.


summary: conserved quantities

conserved quantity interaction

electromagnetic weak strong

energy-momentum p conserved conserved conserved

charge Q conserved conserved conserved

baryon number conserved conserved conserved

lepton number conserved conserved conserved

parity (P) conserved not conserved conserved

charge conjugation parity (C)

conserved not conserved conserved

CP (or T) conserved 10- 3 violation conserved

CPT conserved conserved conserved

isospin (I) not conserved not conserved conserved


A.2.5very quick overview of

nuclei, atoms, molecules

A.2.5very quick overview of

nuclei, atoms, molecules


liquid drop model for nucleiThe residual strong force ("nuclear force") binds nucleons (protons and neutrons) into nuclei.

Complete quantum-mechanically correct descriptions of nuclei or atoms are very difficult to achieve. In the largely simplified liquid drop model, the nucleus is described as a drop of incompressible fluid, made up of nucleons.

The number of nucleons is A , the number of protons is Z. The radius of the nucleus is approximated in a classical way: R = r

0 A 1 / 3 where r

0 ~ 1.25 10 - 15 m is an empirical constant.

The mass of the nucleus is given by:

The binding energy E B

, given by the Weizsäcker formula (not treated here), takes into account several contributions due to the Coulomb forces between protons, the Pauli exclusion principle and the combination of interactions between several nucleons in a sphere.

This simple approach arrives a good understanding of the stability of nuclei with certain Z/A ratios.

m = Z m p A−Z mn−EBc2


nuclear shell model (1)The nuclear shell model is a simple approximative way to describe nuclei in terms of energy levels, very similar to the atomic shell model. Energy levels are determined and occupied with nucleons according to the Pauli exclusion principle. The simplest form of the shell model assumes the potential of a three-dimensional quantum harmonic oscillator plus spin-orbit coupling.

Since neutrons and protons are not identical particles, there are two different sets of shells, filled with neutrons and protons respectively.

The Schrödinger equation of a spherically symmetric 3D harmonic oscillator leads to discrete energy levels:

n = 2k + l with the integer k >= 0. At each level n, the angular momentum quantum number l can thus take the values 0,2,...n for even n and 1,3,..,nfor odd n. (different from the atomic shell model !) The magnetic quantumnumber m

l is an integer with -l <= m

l <= l and ms is the orientation of

the nucleon spin (+- 1/2). This leads in a first approximation to the table below:

V r = 1/2m2 r2

En = 3/2n ℏ

www.lbl.gov

n l ml ms # of states

0 0 0 -1/2,1/2 21 1 -1,0,1 -1/2,1/2 62 0;2 0;-2,-1,0,1,2 -1/2,1/2 123 1;3 -1,0,1;-3,-2,-1,0,1,2,3 -1/2,1/2 20

degenerate states, i.e. different states of the same energy

(only n = 0 to 3 is listed)


nuclear shell model (2)To be more realistic, one has to include spin-orbit interactions. This interaction lifts the degeneracy of states of the same level n, but with different total angular momentum j. There is a difference in the energy of a state with l and s aligned (j = l + 1/2) and a state with l and s anti-aligned (j = l - 1/2). The set of quantum numbers l, ml, ms is replaced by j, mj, parity (-1) l One arrives at the following picture, closer to reality:

n j mj parity # of states

0 1/2 -1/2, 1/2 even 21 1/2;

3/2-1/2,1/2;-3/2,-1/2,1/2,3/2

odd 6

2 1/2;3/2;5/2

-1/2,1/2;-3/2,-1/2,1/2,3/2;-5/2,-3/2,-1/2,1/2,3/2,5/2

even 12

3 1/2;3/2;5/2;7/2

-1/2,1/2;-3/2,-1/2,1/2,3/2;-5/2,-3/2,-1/2,1/2,3/2,5/2;-7/2,...7/2

odd 20

energy levels without and with spin-orbit interactions (boxes: "magic numbers")wikipedia


The Discovery of Radioactivity● After the discovery of X-rays, Becquerel

experimented with phosphorescent substances to find which ones would be able to darken wrapped photographic plates. In these experiments he discovered the natural radioactivity of Uranium in 1896.

● P. and M. Curie isolated Polonium and Radium in 1898.

● In 1898 Rutherford showed that there are at least two different components of radioactivity: α-radiation (easily absorbed) and β-radiation (more penetrating). The latter was soon identified as electrons. It took 10 years to find out that α-radiation consists of Helium-nuclei.

● γ-radiation was discovered in 1900 by Villard and shown to be electromagnetic radiation more energetic even than X-rays.


radioactive decays & stabilityDifferent elements are distinguished by the charge number Z (=number of protons in the nucleus). For each element, different isotopes exist (same Z, different number of neutrons).

Only very few isotopes of an element are stable. Too few or too many neutrons in a nucleus will render it unstable and lead to radioactive isotopes.

The most stable nuclei have nucleon numbers that are "magic numbers" (2,8,20,28,...). These correspond to the number of states in a complete "nuclear shell". These nuclei have a higher binding energy per nucleon than nuclei with partially filled "nuclear shells".

wikipedia


Discovery of the Neutrinoβ-decay was first described as the reaction: n0 -> p+ + e-

In 1911, Lise Meitner and Otto Hahn saw in an experiment that the electrons produced in β-decay had a continuous energy spectrum. β-decay was found to violate energy and momentum conservation.

In 1930 Wolfgang Pauli postulated the existence of a third, very light and neutral particle (later called neutrino by Fermi) to take up the missing energy and momentum: n0 -> p+ + e- + ν

In the 1950s and 1960s, first the electron neutrino and then the muon neutrino were discovered. The tau neutrino was the last particle to be added to the Standard Model of Particle Physics in 2000. In the Standard Model, the neutrinos are massless particles.

Today several experiments have proven that neutrinos do have a very small mass and that the Standard Model needs to be extended to allow massive neutrinos.

left: E of alpha in alpha-decay

right: E of beta in beta-decay


atomsAtoms are formed of nuclei and electrons. The electrons are bound to the nuclei by e.m. interactions (Coulomb force).

Notation:

Since electrons are fermions, they are subject to the Pauli principle and occupy discrete states that are distinguished by at least one quantum number. A shell model can be constructed, similar to the nuclear shell model.

O816

nucleons protons element

Atoms have a size that is determined by the electron "shells" and is much larger than the size of the nucleus. Positive proton and negative electron charges cancel exactly and lead to electrically neutral atoms (net charge = 0).

In astrophysics, neutral atoms are indicated with a "I" (H I, Fe I). Ionized atoms, i.e. atoms with missing electrons are indicated as H II (ionized hydrogen), Fe III (doubly ionized iron atom), etc.


atomic shell model (1)The distribution of electrons over different energy states inside the Coulomb potential of the nucleus ("electron configuration") is often described in a shell model. The origin of this description was Bohr's semi-classical model of an atom surrounded by electrons on orbits of discrete energies.An electron shell is the set of allowed quantum states for electrons with the same principal quantum number n (energy level). For historical reasons, the shells are named by capital letters K (n=1), L (n=2), M (n=3), N (n=4), ... Each shell contains subshells for different azimuthal quantum numbers l (angular momentum) with 0 <= l < n. For historical reasons, the subshells are named by letters s (l=0), p (l=1), d (l=2), f (l=3) , ... A further distinction of states of a given n and l is made by the magnetic quantum number ml (projection of l) with ml = -l, -l +1, ..,l -1, l. And finally, two electrons with the same n, l , ml can still be distinguished by the spin projection quantum number ms (- 1/2, + 1/2).

n l ml ms # of states

names of states

1 0 0 -1/2,1/2 2 1s2 0;1 0;-1,0,1 -1/2,1/2 8 2s;2p3 0;1;2 0;-1,0,1;-2,-

1,0,1,2-1/2,1/2 18 3s;3p;3f

4 0;1;2;3 0;-1,0,1;-2,-1,0,1,2;-3,-2,-1,0,1,2,3

-1/2,1/2 32 4s;4p;4f;4d

wikimedia


atomic shell model (2)As in the nuclear shell model, one has to take into account spin-orbit interactions to arrive at a more realistic result. Again the correct set of quantum numbers to use changes from n,l,ml,ms to n,j,mj,parity. Here j = l +/- s, thus j=1/2,3/2,...,n-1/2. The projection mj=ml+ms can take values of -j,-j+1,...,j-1,j. And parity is defined as +1 for states that come from an even l and -1 for states coming from an odd l.

Spin-orbit interactions, relativistic effects and other corrections break the degeneracy of the electron energy levels ("fine structure"). States with the same n and l, but different j have different energies.

A further splitting of energy levels ("hyperfine structure") is due to interactions between the electron and nucleus spins. (In the figure, J isthe total angular momentum of the electrons,S is their spin, I is the nuclear spin and F the total angular momentum made up of J and I.)

In atoms with several electrons, interactionsbetween the electrons have to be taken intoaccount . Approximative methods are used to calculate the levels numerically.

Electron configurations in multi-electron systems are described with the term symbol:

(hyper)fine structure in hydrogen (wikimedia)LJ2S1


periodic table of the elements


atomic orbitals

wikimedia

V r =−1

40

Z e2

r

n l m = Rn l r Y l m ,

The electron distribution of different states in hydrogen atoms and their energy can be calculated by solving the Schrödinger equation for a Coulomb potential:

The resulting wavefunctions provide the atomic orbitals, i.e. the probability distribution for the location of the electrons: ( R: Laguerre polynomials, Y: spherical harmonic functions).

molecular orbitals

Two different notations exist to number atomicorbitals:

Er =−ℏ2

2m∇ 2r V rr with


Nuclear and Atomic Shell ModelAtomic Shell Model

The electrons in an atom occupy certain discrete energetic levels (shells or orbitals in the quantum-mechanics picture).

Absorption of photons with sufficient energy can promote electrons to a higher energy level (excitation) or lead to ionization.

Photons of energy E2-

E1 are emitted when an

electron «falls» back to shell 1 from shell 2 (de-excitation).

Nuclear Shell Model

The nucleons inside the nucleus occupy certain discrete energetic levels, similar to the electrons in the atom.

Nuclear excitation by particle collision, laser.

Excited nuclear states decay to more stable states by:- , and emission- Capture of K-shell electron and emission of X-rays.


moleculesEven though atoms have a net charge of 0, they can combine to form molecules by different types of bonding:

● covalent bond by increased density of common electrons between two nuclei

● ionic bond by exchange of electrons that forms ions

● van-der Waals force between atoms by polarisation of the electron "clouds"

● hydrogen bonds (e.g. between water molecules)

Like atoms, molecules have discrete energy levels andorbitals. Electrons can be excited/de-excited between those levels and absorb/emit radiation (mostly in UV).

In addition, molecules can also change between different states of rotation or vibration (depending on their geometry).

A transition between rotational levels leads to absorption/emission lines mainly in the microwave band. Vibrational transitions are visible mainly in the infrared.

www.images.iop.org


solidsAtoms, ions or molecules can combine to form solids. Concerning their structure, one distinguishes between crystalline solids (metals, ice, ceramics...) or amorphous solids (polysterene, glass,...). The atoms, ions or molecules in a solid can be held together by different bonding mechanisms: covalent bonding, ionic bonding, van der Waals force, metallic bonding.

The discrete energy levels in the atoms and molecules that form the solid superpose and form several continuous energy-bands (allowed states) that can be separated by gaps (forbidden states). The size of the band gap determines if a solid is a metal, insulator or semiconductor.

The quantized modes of lattice vibration in solids are called phonons and can be treated like particles.

http://www4.nau.edu wikipedia


A.2.6Beyond the standard model

A.2.6Beyond the standard model


super-symmetrySuper-symmetric (SUSY) models are an attempt to arrive at a more coherent description of fundamental particles and interactions. These theories postulate a fermion-boson symmetry that gives new fermion partners to the known fundamental bosons and new boson partners to the known fundamental fermions.

Super-symmetry is broken in the sense that the boson-fermions partners do not have the same mass, but they would have the same couplings. This would help solve a basic problem of the standard model: the hierarchy problem that does not allow masses of particles to be much above the W and Z boson masses.

particle spin sparticle spinquark 1/2 squark 0lepton 1/2 slepton 0photon 1 photino 1/2gluon 1 gluino 1/2W +- 1 wino 1/2Z 1 zino 1/2

The Minimal Super-symmetric Standard Model (MSSM) predicts a large number of new particles (5 Higgs particles, several charginos and neutralinos) that should be within reach of the current particle accelerators (LHC).

particles and their SUSY partners


grand unification

http://hyperphysics.phy-astr.gsu.edu

The so-called "grand unified theories" (GUT), none of which is currently accepted, try to unify the strong and electroweak interactions similar to the way the e.m. and weak interactions had been unified into the electroweak theory.

These theories suppose that the coupling strengths of all three interactions are equal at very high energy scales (close to the Planck scale).

GUT could account for the equality of proton and electron charge and explain the fractional quark charges.They predict a finite proton lifetime and in some cases the existence of magnetic monopoles. No generally accepted GUT exists at the time.

An even farther goal is the unification with gravity in a "Theory of Everything".


string theoryAnother approach to find a "theory of everything", i.e. to combine electroweak and strong forces with the gravitational force, is string theory. This theory replaces the pointlike particles by strings of finite length. The different elementary particles are represented as closed strings or loops with different modes of oscillation. These particles include a massless graviton. This approach requires at least 10 dimensions, out of which all but the four known space-time dimensions would be compactified into very small sizes (of the order of the Planck scale) and would be undetectable. Superstring theory is the super-symmetric version of string theory.

String theory is a quantum gravity theory (another being loop quantum gravity), i.e. a theory that tries to unify quantum mechanics and general relativity.

The problem with string theory (and other "theories of everything") is that it is very difficult if not impossible to test:

● one would need to produce many orders of magnitude larger energies than what is achievable in accelerators

● at the time, a huge number of equally acceptable solutions exist in string theory. The number of free parameters that cannot be predicted, but have to be measured, does not indicate a clear progress with respect to the current standard model.


A.2.7Interaction cross section

A.2.7Interaction cross section


= n1 v

P = n2 x

dNdt

= n2 x = n1 n2 v x

W =

(#part./m2/sec)

interaction cross-section (1)A cross-section is an effective area for particle interactions. A cross-section for a classical spherical target would be σ = π r 2. In particle physics, the cross-section is a more general parameter that determines the probability of a certain interaction to occur. It is measured in barn (1 barn = 10- 2 8 m2).

Interaction rate for a beam hitting a target

The flux of particles from the beam is:

Probability that any particle will hit a target particle is:

=> Number of interactions per unit time and unit area:

Reaction rate per target particle:


Interaction cross-section (2)Typical cross-sections for the different types of interactions are:

– 10 - 33 m2 (electromagnetic)– 10 - 39 m2 (weak)– 10 - 30 m2 (strong)

Non-relativistic perturbation theory allows to derive the reaction rate W from the matrix element between initial and final states (Fermi's Second Golden Rule):

M i f

is giving the overlap between the initial-state and final-state wavefunctions, caused by the interaction potential U:

The matrix elements for a certain interaction can be calculated with the help of Feynman diagrams.

is the energy density of final states.

W =2ℏ

∣M if∣2 f

∫ f∗Ui dV

f


dd

= Z2 e4

K M 2 v4⋅

1sin4 /2

The differential cross-section is defined as:

One determines the total cross-section from the differential cross-section by integrating over the solid angle:

d= sin d d

Rutherford determined the differential cross-section of the scattering of an α-particle off a nucleus (with charge Z) in his famous experiment:

α

differential cross-section


A.2.8Particle accelerators & detectors

A.2.8Particle accelerators & detectors


Particle Accelerators 1 : Linear Accelerators (LINAC)

● Acceleration of electrons, protons, ions for medical or research purpose. X-ray generation.

● e.g. Stanford Linear Accelerator Center (SLAC), 3km long. Several particles were discovered at SLAC (e.g. τ). It is now used as the first hard X-ray laser, using synchrotron radiation (LCLS).

● Initial particle source (i.e. cathode for e- )

● A long hollow, straight vacuum chamber contains the particle beam.

● Electrically isolated cylindrical electrodes are powered by an AC radio frequency power source. The length of the cylinders increases along the path of the particle.

● Within the cylinder, the particle is unaffected by the electric field (Faraday cage). Acceleration takes place in the gap between the cylinders.

● At the end of the pipe, the particle hits a fixed target. (or connected to a collider ring...)

+ : higher particle velocities than in synchrotron (no magnetic field required, no synchrotron loss) - : great length of the device, pulsed beam

taken from http://library.thinkquest.org


Particle Accelerators 2 : Cyclotrons Particle source in the center. Used for ion acceleration, electrons would suffer large radiative losses.● Ions move in a vacuum chamber.

● Two D-shaped electrodes are connected to radio frequency AC power.

● Within the "Dees", the ions are not affected by the electric field. Acceleration occurs in a small gap between the "Dees".

● A strong magnetic field keeps the ions on a spiral track. Radius increases with increasing velocity.

● At the end of their path, the ions hit a fixed target.

+ : compact and cost-effective, continuous beam

- : acceleration only up to a few % of c. Relativistic ions would get out of synch with constant frequency.

● Use in nuclear physics research at many universities.● Medical use (ion beam for radiation therapy, Positron Emission Therapy imaging)

taken from http://library.thinkquest.org


Particle Accelerators 3 : Synchrotrons● Injected particles (from a LINAC) are held on a circular path by a magnetic field and an electric field that are both synchronized to the particles' velocity.

● Particles move in a vacuum tube of torus-shape. Several (superconducting) magnets keep them on the circular path. Acceleration takes place in several (superconducting) accelerating cavities.

● The maximum velocity of ions is limited by the maximum strength of the magnetic field and the radius of the accelerator. The maximum velocity of electrons is limited by synchrotron losses.

● Two counter-rotating beams are used in particle colliders (e.g. Tevatron, LHC). This leads to an energy of interaction much larger than in a fixed target interaction, which is available for particle creation.

Particle physics research and generation of synchrotron radiation.

The largest accelerator of this type is the LHC, accelerating protons up to a few TeV and heavy ions to energies of ~1 PeV .

ECM = 2 E collider ECM ∝ E target


Particle Detectors in Accelerators (e.g. CMS)


Particle Detectors in Accelerators (e.g. CMS)

This sort of technology has inspired many of the High Energy Astrophysics detectors.


A.2.9Useful Formulae

A.2.9Useful Formulae


Definitions of Units, Constants etc. ● energy:

1 eV ≈ 1.6022 E-19 J ≈ 1.6022 E-12 erg

● orders of magnitude:a (Atto) = 1E-18f (Femto) = 1E-15p (Piko) = 1E-12n (Nano) = 1E-9

(Mikro) = 1E-6m (Milli) = 1E-3k (Kilo) = 1 E3M (Mega) = 1 E6G (Giga) = 1 E9T (Tera) = 1 E12P (Peta) = 1 E15E (Exa) = 1 E18Z (Zetta) = 1 E21

● elementary charge: e ≈ 1.6022 E-19 C● vacuum speed of light:

c = 2.99792458 E8 m/s = (ε0 µ

0)-1/2

● cross-section:1 barn = 1 b = 1 E-28 m2

● Planck's constant:h ≈ 6.626069 E-34 J s ≈ 4.13567 eV/GHz

● Boltzmann constant:k ≈ 1.380658E-23 J/K

● electric field constant:ε

0 = 8.85418781762 E-12 As/(Vm)

● electron mass: me ≈ 0.511 MeV / c2

≈ 9.11 E-31 kg

● proton mass: mp ≈ 938.272 MeV / c2

● neutron mass: mn ≈ 939.566 MeV / c2

ℏ = h / 2



wikipedia


A.2.10Bibliography

A.2.10Bibliography


BibliographyParticle Physics:

● Donald H. Perkins: Introduction to High Energy Physics , Cambridge University Press 2005

● Donald H. Perkins: Particle Astrophysics , Oxford Master Series

● Particle Data Group: http://pdg.lbl.gov/2006/reviews

Introduction to Quantum Field Theory:(largely beyond the scope of this course !)

● L.H.Ryder: Quantum Field Theory,Cambridge University Press

● Cheng & Lie: Gauge Theory of Elementary Particle Physics

Particle Detectors

● Dan Green: The Physics of Particle Detectors , Cambridge University Press 2000

● Particle Detector BriefBook: http://rkb.home.cern.ch/rkb/titleD.html

● Particle Data Group: http://pdg.lbl.gov/2006/reviews

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