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3rd Year Undergraduate Physics > Nuclear and Particle Physics > Flashcards

Nuclear and Particle Physics Flashcards

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1
Q

How is relativistic energy, E, related to momentum, p, and mass, m?

A

E^2 = p^2c^2 + m^2c^4

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2
Q

How is a fermion defined?

A

It is a particle with spin 1/2. They include quarks and leptons.

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3
Q

By which force do neutrinos interact?

A

Only by the weak force, via the exchange of W and Z bosons.

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4
Q

By which force to quarks interact?

A

By either the EM force (via the exchange of photons), the weak force (via exchange of W and Z bosons), but most likely by the strong force (via the exchange of massless gluons).

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5
Q

In natural units, what are the units for energy, momentum, mass and length?

A
  • Energy: GeV (converted from J)
  • Momentum: GeV (i.e GeV/c, from E = pc)
  • Mass: GeV (i.e. GeV/c^2, from E=mc^2)
  • Length: GeV^-1 (i.e hbar*c / GeV, from dimensional analysis: hbar * c / GeV = Js * ms^-1 / J = m)
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6
Q

What is the de Broglie equation?

A

Wavelength lamda = h / p

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7
Q

What are the rules for particle decay?

A
  • A particle must decay to 2 or more particles, in order to conserve momentum and energy.
  • Electric charge must be conserved.
  • Lepton number must be conserved.
  • The mass of the products must be equal to or less than the mass of the decaying particle.
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8
Q

Which way to particles and antiparticles move in a Feynman diagram?

A

Particles move left to right, while antiparticles move right to left, to signify moving backwards in time.

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9
Q

What is the energy-momentum 4-vector, P? What is its inner product (i.e. P^2)?

A
  • P = (E, cp) = (E, cp_x, cp_y, cp_z)
  • P^2 = P.P = E^2 - p^2*c^2

This inner product is invariant under Lorentz transformations.

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10
Q

What is the momentum of a particle in its rest frame? What therefore is its 4-vector?

A

In its rest frame, p = 0.

Therefore its momentum 4-vector simplifies to P^2 = E^2 = m^2*c^4.

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11
Q

What is the formula for the invariant mass, W, of N particles?

A

W^2 * c^4 = (Sum[E_N])^2 - (c*Sum[p_N])^2

This is an invariant quantity under Lorentz transformations.

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12
Q

What is the formula for the Yukawa potential?

A

V(r) = - g^2/4pi * e^(r/R)/r

Where g is the coupling constant, and R is the range:

R = hbar / (M_x * c),

where M_x is the mass of the interaction particle.

For 2 particles:

W = sqrt( 2 * E_1 * E_2 * (1 - cos(theta))

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13
Q

What is the formula for the scattering amplitude, M?

In terms of M, what is the probability of a particle to be scattered?

A

M = (g^2 * hbar^2 * c^2) / (p^2 - M_x^2 * c^4)

Where g is the coupling constant and M_x is the mass of the interaction particle.

The probability of a particle to be scattered is then given by its square modulus, |M|^2.

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14
Q

What is the Pauli Exclusion Principle?

A

2 identical fermions cannot occupy the same quantum state simultaneously.

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15
Q

How is Lepton number defined?

A

Negatively charged leptons and their corresponding neutrinos (i.e. e-, t- mu-, v_e, v_t, v_mu) carry a lepton number of 1. Their antiparticles carry a lepton number of -1.

Lepton number, L, as well as individual lepton numbers L_e, L_t and L_mu, are conserved across all interactions, similar to electric charge.

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16
Q

What is the branching fraction?

A

The branching fraction for a decay is the fraction of particles which to decay to an individual decay mode w.r.t. the total number of particles which decay.

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17
Q

What is the probability for neutrino oscillation, P(v_a -> v_b)?

A

P(v_a -> v_b) = sin^2 (2*theta) sin [1.27 * Δm^2 * c^4 * L / E]

Where theta = mixing angle, L = length in km, and E = energy in GeV.

Know how to derive from neutrino wavefunctions.

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18
Q

What is Colour Confinement?

A

Quarks carry a colour charge; red, green or blue. When forming hadrons or mesons, colour charge always cancels to form colourless particles.

As quarks are forced apart, the potential energy between them, due to the Strong Force, increases, and at a certain point it becomes energetically favorable to create a new quark-antiquark pair which couples to the original particle, creating colourless mesons and keeping colour confinement.

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19
Q

What is the Baryon Number?

A
  • The number of quarks, N, is conserved (with antiquarks having negative quark number).
  • A baryon is formed of 3 quarks, so baryon number B = N/3
  • Both numbers are conserved in all interactions
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20
Q

Why are protons stable?

A

They are the lightest possible baryon, and so cannot decay into anything lighter without violating the baryon number.

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21
Q

In what interactions can quarks change flavour?

A

Weak interactions.

E.g. a anti-charm quark can split into an anti-strange quark and a W- boson, without violating quark number.

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22
Q

How does luminosity L relate to the rate of a certain process?

A

dN/dt = L*sigma

Where sigma is the cross section of the process.

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23
Q

What is the formula for the luminosity of a fixed target experiment?

A

L = n_b * v_i * N

Where n_b = number density of particles in a beam, v_i = the velocity of the beam, and N = number of illuminated particles in the target.

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24
Q

What are the processes by which photons can lose energy in a material?

A
  • Photoelectric Effect: A photon hits an electron in the orbit around a nucleus hard enough to liberate it from its shell, while the photon gets absorbed in the process.
  • Compton Scattering: A photon scatters off an electron, gets deflected, and transfers some of its energy.
  • Pair production: Once a photon has high enough energy, it creates an e+ e- pair. This process is dominant at high energies.
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25
Q

What formula describes photon intensity, I, while travelling through a material?

A

I = I_o * e^( -mu * x)

Where I_o is initial intensity, and mu is the attenuation coefficient.

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26
Q

What formula describes energy loss of photons in terms of radiation length, X_o?

A

E = E_o * e^( x/X_o)

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27
Q

What is an electromagnetic shower?

A

When a high energetic photon enters material an electromagnetic shower develops through continuous repetition of bremsstrahlung and pair creation, until it reaches a critical energy at which energy loss contributions from bremsstrahlung and ionisation (i.e. Bleth-Bloch) are equal.

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28
Q

In a particle detector, what are the properties of interest?

A
  • The position of particle trajectory
  • Particle momentum
  • Particle energy
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29
Q

How does a scintillator work?

A

Scintillators indicate the passing of a charged particle by emitting light. To obtain an electronic readout, they need to be coupled to a light sensitive detector, usually a photomultiplier tube.

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30
Q

How do calorimeters work?

A

Calorimeters stop particles and measure their energy.

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31
Q

What properties do gluons have?

A
  • Spin-1
  • Massless
  • Electrically neutral
  • Bosons
  • Self-interacting
  • Only couple to particles with colour charge (i.e. quarks).
32
Q

What is deep inelastic scattering?

A

The scattering of high-energy leptons off nucleons, in order to see the the nucleon structure at very small distance scales.

33
Q

What is asymptopic freedom?

A

The strong interaction gets weaker at higher energies (and therefore short distances).

34
Q

How is parity defined?

A

Parity is a conserved quantity (except for in weak interactions), defined as symmetry under space-inversion.

I.e. the physical laws of a system do not change is spatial coordinates are inverted; x -> -x, y -> -y and z -> -z.

35
Q

If Feynman diagrams showing interactions via the stong force, EM force, and the weak force are all allowed, which one dominates?

A
  • The strong force force is dominant, if possible
  • EM force has a smaller coupling
  • Weak force is usually negligible, unless it is the only process allowed
36
Q

Around what masses are the W and Z bosons?

A

M_w ~= 80GeV

M_z ~= 90GeV

37
Q

How does the Higgs mechanism work?

A

A field known as the Higgs field exists at all points in the universe. Interactions with this field give particles mass, in such a way that the theory describing the interactions remains gauge-invariant.

Excitations in this field come in the form of Higgs Bosons (similar to how photons are excitations in the EM field). The Higgs Boson couples to all particles with mass, with a coupling that increases with the mass of the particle.

38
Q

What are the processes that Higgs Bosons can be formed?

A
  • The dominant process is gluon-gluon fusion, which occurs via a quark loop.
  • The next most likely process is weak boson fusion, where either a W+W- or ZZ pair fuse to form a Higgs.
  • Then the next most likely process is the W+- or Z boson associated production, where quark-antiquark annihilation produce a W+- or Z boson, which radiates a Higgs.

See diagrams in notes.

39
Q

What are the decay methods for Higgs bosons?

A

As a Higgs Boson coupling depends on the mass of the particle, it is most likely to decay to the heaviest particles that it is kinematically allowed to decay to.

  • The dominant decay process is for a Higgs to decay to a b(anti)b pair. This is because a b(anti)b pair is the heaviest pair of particles that are still lighter than the mass of a Higgs Boson.
  • The next most likely is a W+W- pair. The mass of W+W- is actually larger than the Higgs, but it is still possible if one of the W bosons is produced with a mass less than usual, which is allowed for virtual particles.
  • Other important decays is a ZZ or t+t- pair.
  • One of the most important channels is into a yy (photons) pair, which occurs via a particle loop.
40
Q

In nuclear nomenclature, what are Z, N and A?

A

Z - the atomic number, i.e. the number of protons.

N - the neutron number, i.e the number of neutrons.

A - the mass number, i.e. the number of nucleons (A = Z + N)

41
Q

What is binding energy?

A

The energy required to split up a nucleus into into its constituent nucleons.

This energy translates into a mass deficit in the nucleus, given by E = mc^2.

42
Q

In the semi-empirical mass formula, what to coefficients a_v, a_s, a_c, a_a, and a_p represent?

A

a_v: Volume term, due to each nucleon binding to nearest neighbors, proportional to total number of nucleons, A.

a_s: Surface term, due to nucleons on the surface having less neighbours to bind to.

a_c: Coulomb term, due to protons electrostatically repelling each other.

a_s: Asymmetry term, due to energy energy penalty for having excess of protons or neutrons. Comes from Pauli exclusion principle.

a_p: Pairing term, due to nucleons with opposite spins binding mor heavily to one another.

43
Q

What are the formulae for beta-decay, B- and B+?

A

B-: n -> p + e- + (anti)v_e

Mass of the atom generally decreases.

B+: p -> n + e+ + v_e

Mass of atom also generally decreases.

44
Q

How does spontaneous fission occur?

A

For large atoms (A > 100), stretching of an atom increases surface energy (activation energy). Energy then decreases as the nucleus splits fully, resulting in 2 daughter that repel each other due to to Coulomb forces.

This fission becomes more and more energetically favourable as nucleus size increases.

45
Q

In nuclear decay, what is the formula for the number of original atoms, N, in terms of mean lifetime, T?

A

N(t) = N_o * e^(-t/T)

46
Q

Why is iron the most stable element?

A

Iron has the highest binding energy per nucleon. Heavier or lighter elements have less binding energy per nucleon, and so are able to release energy by either undergoing fission or fusion.

47
Q

What are the ideas behind the Fermi gas model for nuclei?

A
  • Assumes a 3D potential well potential (different for protons and neutrons).
  • Nucleons can move freely within the nucleus. This agrees with the idea that they experience an overall effective potential created by the sum of the other nucleons.
48
Q

What is the formula for the highest occupied state in a nucleus (i.e. Fermi energy, E_f)?

A

E_f = p_f^2 / 2M

Where p_f is the Fermi momentum.

49
Q

What are the ideas behind the Shell Model for nuclei? What are the quantum numbers?

A
  • Protons and neutrons occupy energy states, similar to electrons orbiting a nucleus:
  • Atomic energy n, n = 1,2,3…
  • For every n, there is orbital angular momentum number l, l = 0,1,2…,(n-1)
  • For every l, there are (2l + 1) sub-states for magnetic quantum number m_l, m_l = -l, -l +1, … -1,0,1,…l-1, l.
  • As each nucleon then possesses up or down spin, there can be 2 nucleons in each state with quantum number m_s, m_s = +1/2 or -1/2.

(See notes for Shell Model diagram)

50
Q

How is total angular momentum j defined?

A

j = s+l

E.g for l=3, s = +1/2 or -1/2, so j = 7/2 or 5/2

Note, nuclei with larger j have lower energy E, different than electrons.

51
Q

In the Shell Model, what is the Pairing Hypothesis?

A

For ground state nuclei, pairs of n and p in a given sub-shell always couple to give a total angular momentum of 0, even when the sub-shell is not totally filled.

Therefore the last unpaired neutron/proton determines the net nuclear spin.

52
Q

What is the net nuclear spin in nuclei where A is odd, and when A is even but Z or N is odd?

A
  • In odd-A there is only one unpaired nucleon, so net spin can be determined precisely. It is given by the angular momentum j of the unpaired nucleon.
  • In even-A odd-Z/odd-N nuclei there is an unpaired p and an unpaired n. Hence the nuclear spin will lie in the range |j_p-j_n| to (j_p+j_n).
53
Q

In the shell model, how is the parity of a nucleus, P, predicted?

A

Parity is given by P = (-1)^l, where l is the total angular momentum of the nucleus.

But due to the pairing hypothesis, only the angular momentum of the unpaired proton/neutron is uncancelled. So l is given by the product of l (last proton) x l (last neutron).

54
Q

What are the basic principles behind nuclear fission and fusion?

A

Fission: Splitting heavy nuclei into 2 lighter nuclei results in a larger total binding energy (which is negative). This change in binding energy is released as kinetic energy.

Fusion: Fusing 2 light nuclei creates a larger nucleus with larger total binding energy. The change in binding energy is again released as kinetic energy.

This process can continue until you reach iron, the most stable element.

55
Q

How are chain reactions created in fission?

A

When a nucleus undergoes fission, it creates 2 daughter nuclei and some fast-moving neutrons. These neutrons go on to induce fission in other nuclei, releasing more neutrons, and so on, creating a chain reaction.

56
Q

What determines the criticality of a fission chain reaction?

A

If the probability of a neutron inducing a fission reaction is q, and the number of neutrons released per reaction in n, then number of neutrons present at time t is given by N(t):

N(t) = N(0) e^[(nq-1)t/t_p]

Where t_p is the average time before absorption of the neutron occurs.

  • If nq 1, N(t) increases exponentially, so the process is ‘supercritical’ and will eventually lead to a nuclear explosion.
57
Q

How is the criticality of a nuclear reaction kept constant? What role do delayed neutrons play?

A

Delayed neutrons are created around 13 seconds after a fission reaction, and contribute to nq (which should be kept at nq = 1). These delayed neutrons ensure that a chain reaction does not become uncontrollably supercritical, as it slows down the exponential growth of the chain reaction enough that it can be controlled via control rods, which can be inserted to reduce the reaction.

58
Q

What are the most common decay methods for muons and tau particles?

A
  • Muon: mu -> v_mu + e- + (anti)v_e
  • Tau: t -> pion- + pion0 + v_t

See Feynman diagrams for details.

59
Q

What evidence is there that the number of colour charges Nc = 3?

A

The ratio for the cross-section for e+e- -> Hadrons to e+e- -> mu+mu- shows that an e+e- pair is around 3 times more likely to produce quarks (and therefore hadrons) than muons than models would predict without colour charge.

Without the charge the ratio of cross sections should be around 11/9, but the experimental data shows the ratio is actually around 33/9.

When discovered, the experimental data suggested there were 5 quark flavours with 3 colour charges, as the energy used at the time was insufficient to produce the very heavy top quarks.

60
Q

For which coefficients in the semi-empirical mass formula do the Liquid Drop Model, Fermi Gas Model and the Shell Model account for?

A

Liquid Drop Model: Volume term, Surface term, Coulomb Term

Fermi Gas Model: Asymmetry term

Shell Model: Pairing term

61
Q

Why do odd-A nuclei have a single parabola along a Mass/A graph, while even-A have 2?

A

In odd-A nuclei, either Z,N = even,odd, or Z,N = odd,even, so the pairing term vanishes for the top nucleon, so there is one parabola.

In even-A nuclei, either Z,N = even,even or Z,N = odd,odd, so the pairing term either exists or doesn’t for the top nucleons, leading to 2 seperate parabolas.

62
Q

Even though the dominant decay channel for a Higgs Boson is b(anti)b, why has it still not been discovered in physical evidence?

A

The b(anti)b channel suffers from an extremely large background, with QCD production of b(anti)b states being 10 million times more likely than Higgs -> b(anti)b.

The jets coming from bottom quarks also have very poor resolution, making any anomalies very difficult to resolve.

63
Q

What is the quark composition for neutrons and protons?

A
  • Neutrons: ddu

- Protons: uud

64
Q

What is the formula for the fraction of an incoming proton’s momentum carried by a struck parton, x, in deep inelastic scattering?

A

x = -q^2 / (2p.q)

Where q = k’-k. k is the 4-momentum of the of the incoming electron, and k’ is the 4-momentum of the outgoing electron.

p is the 4-momentum of the incoming proton.

65
Q

What are the two main methods of confinement in fusion reactions?

A
  • Magnetic confinement fusion is an approach to generating fusion power that uses magnetic fields to confine the hot fusion fuel in the form of a plasma.
  • Inertial confinement fusion (ICF) attempts to initiate nuclear fusion reactions by heating and compressing a fuel target, typically in the form of a pellet that most often contains a mixture of deuterium and tritium, using high energy laser light.
66
Q

What is electron capture decay?

A

Electron capture is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron. This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino.

p + e- -> n + v_e

67
Q

How is helicity defined?

A

Helicity is the projection of the angular momentum onto the direction of momentum. I.e. the combination of spin and linear velocity.

68
Q

In what interactions is parity violated?

A

Parity is most famously violated in beta-decay.

Electrons emitted in beta-decay most likely have spins that oppose that of the nuclear spin, where under parity conservation they should have spins that are equally likely to be aligned or anti-aligned to the spin.

69
Q

What is C-Parity?

A

The C-parity of an interaction is conserved if the interaction is symmetric under particle anti-particle exchange.

The weak interaction is not invariant under a charge conjugation transformation (changing particles to anti-particles) while strong and EM interactions both are.

70
Q

What is the heaviest quark?

A

The top quark.

It is as massive as a gold atom.

71
Q

What are some examples of stable particles (I.E. particles that cannot decay)?

A
  • Electrons and positrons
  • The lightest neutrino
  • Photons
  • Protons
72
Q

What is Lepton Universality?

A

The coupling of a W boson to any Lepton Doublet (a charged lepton and a neutrino) are the same for all 3 generations (e, mu or tau).

73
Q

What is the dirac notation for the wavefunctions of neutrinos, |v_a> and |v_b>?

What factor does time evolution |v,t>|v_a> introduce?

A 
|v_a> = |v_1>cos(theta) + |v_2>sin(theta)
|v_b> = -|v_1>sin(theta) + |v_2>cos(theta)

|v,t>|v_a> = |v_a>cos(theta) * e^(-iE_1 t/hbar) + |v_2>sin(theta) * e^(-iE_2 t/hbar)

This is useful in deriving the neutrino mixing formula.

74
Q

What evidence is there for neutrino mixing?

A

The Sun produces neutrinos, but showed a deficit with respect to theoretical calculations for neutrinos received on Earth.

Neutrino mixing fixed this problem, and allows us to calculate neutrino mixing angle theta.

75
Q

What is a quark jet?

A

When a e+e- annihilate, they can create a q(anti)q pair. Due to conservation of momentum, these quarks will start travelling back-to-back.

The strong force between them therefore increases, and at some point it is energetically favourable to create a new quark-antiquark pair between them. This process is repeated over and over, creating two quark jets, travelling back to to back in the directions of the original quark and antiquark.

76
Q

How is the rate of a process, dN/dt, related to flux, Phi?

A

dN/dt = Phi * N * sigma

Where N = number of target particles, and sigma = interaction cross-section.

77
Q

What is a Bragg Peak?

A

A particles loses most of its energy when it is slowest while travelling through a material. As a particle is stopped, it will deposit most of its energy at the end of the path (giving a peak in energy loss, the Bragg Peak).

3rd Year Undergraduate Physics (3 decks)

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