Introduction to Nuclear and Particle Physics Flashcards
ħ and c size approximation
ħc ≈ 200 MeV fm
DeBroglie relation
λ = h/p
What units should we be using
fm, MeV, fine structure constant α, factors of ħ and c
Heisenberg uncertainty principle
∆x∆pₓ ≥ ħ/2
Getting from momentum to energy
p = h/λ = 2πħc/λc = 2π 200 MeV fm / λc
Basically try to get ħc factor
Momentum in MeV/c corresponds to energy in MeV
Schrödinger equation
– ħ²/2M d²ψ(x)/dx² + V(x)ψ(x) = Eψ(x)
Time dependance of a wavefunction
Ψ(x,t) = ψ(x)exp(iEt/ħ)
Probability density of a wavefunction
P(x) = Ψ∗(x, t)Ψ(x, t)
which respects unitarity
∫±∞ P(x)dx = ∫±∞ Ψ∗(x, t)Ψ(x, t) dx = 1
Total energy of a particle
E² = p²c² + m₀²c⁴
Angular momentum operators for each component
L = r × p
L̂ₓ = –iħ (y ∂/∂z – z ∂/∂y)
L̂ᵧ = –iħ (z ∂/∂x – x ∂/∂z)
L̂₂ = –iħ (x ∂/∂y – y ∂/∂x)
Angular momentum commutations
[L̂ₓ, L̂ᵧ] = iħL̂₂
[L̂ᵧ, L̂₂] = iħL̂ₓ
[L̂₂, L̂ₓ] = iħL̂ᵧ
[L̂², L̂ᵢ] = 0
Atom notation
ᴬzX
Z = atomic number = number of protons
A = Z + N = mass number = number of protons + neutrons
X = Chemical symbol
What is binding energy
The energy that holds nucleons together in a nucleus, E_B
Equation for binding energy
Difference in mass from constituents
M(Z,A) = ZM(¹₁H) + NMₙ – E_B/c²
E_B/c² = ZM(¹₁H) + NMₙ – M(Z,A)
where M(Z,A) is atomic mass of isotope ᴬzX and M(¹₁H) is the mass of a hydrogen-1 atom
Relation between atomic and nuclear mass
Mₐₜ(Z,A) = Mₙᵤ꜀(Z,A) + Zmₑ
Mass excess
NOT actually related to any physics
∆(ᴬzX) ≡ M(Z,A) – A u
where M(Z,A) is atomic mass of isotope ᴬzX in atomic mass units, u
Direct method of measuring nuclear mass
Only able to be used for long-lived, stable particles
Ion moving in EM field, forces will be described by the Lorentz force
F = q(E + v × B)
If no E-field and velocity is perpendicular to B-field, ion exhibits circular motion,
F = mv²/r r̂
Equating forces
m = q|B|r/|v|
To select velocity, can include E field to only take ions passing through a slit. Could also just measure the speed.
Could also use a Penning trap, which uses an electric quadropole field. Motion of nuclei confined here can be measured and related to charge-mass ratio.
Transmutation equation notation
Inelastic collision of projectile a onto target nucleus, A, to produce B and outgoing projectile b
A + a → B + b
Can write as
A(a, b)B
For example:
¹⁴₇N(α, p)¹⁷₈O
Q-value of a reaction
Difference between nuclear rest masses of initial and final states
Q = Σᵢₙᵢₜᵢₐₗ mᵢc² – Σբᵢₙₐₗ mբc²
+ve Q values, energy is released in reaction, seen as kinetic energy of products. In general, larger Q, faster reaction, more energetically favourable it is.
-ve Q values, implies energy must be supplied for reaction to occur.
Indirect method of measuring mass
Use Q value, if we know masses of three constituents and look at final velocities, we can work out the fourth constituents mass.
Graph of binding energy per nucleon
Sharp rise, peak at Fe, slow decrease.
See fig 1.4, pg 14
Rate of nuclear decay
Activity
A(t) = dN/dt = –λN
where N is number of nuclei present and λ is decay constant.
A(t) has units of becquerels (Bq, decays/second)
Can solve differential to find number of nuclei present at a certain time, N(t), given how many there were at the start, N₀:
N(t) = N₀exp(–λt)
A(t) = λN₀exp(–λt)
Half life
t₁ᵢ₂
average time needed for number of nuclei of that isotope to halve.
N(t₁ᵢ₂) = N₀/2
t₀ = ln(2/λ)
Gamow’s saturation hypothesis and nuclear radius
As A increases, volume per nucleon remains constant.
Spherical nuclei,
4/3 πR³ = V₁A
where V₁ is related to volume of a single nucleon.
Implies
R ∝ A¹’³
where R is nuclear radius
Using scattering for charge dist. of a nucleus
Electron scattering. Process is identical to diffraction; beam incident on a small obstacle produces same diffraction pattern as a slit of equal area.
Rough estimate, assuming nucleus is a 2D disk, thus relating diameter to the first minima by
D = 1.22 λ/sinθ
where λ is de Broglie wavelength for the incoming electron.
See fig 1.7, pg 17
Cross section of a reaction
General scattering experiment, beam incident onto a target, production rates of the final states are counted.
Rate of specific reaction, Wᵣ, is proportional to flux of the beam, J, (# of particles passing through surface at the target perp. to beam direction) and number of target nuclei illuminated by the beam, N
Wᵣ ∝ JN
Constant of proportionality is the cross-section, σᵣ, such that
σᵣ ≡ Wᵣ/JN
Barn unit
1 barn = 100 fm²
Rate of particles passing through an area of solid angle dΩ (= d(cosθ)dϕ)
dWᵣ ≡ JN dσᵣ(θ,ϕ)/dΩ dΩ
Calculating differential cross section for electron scattering
Differential cross-section is a probability, so must be related to quantum-mechanical amplitude of the reaction being studied, Aᵣ
σᵣ/dΩ ∝ |Aᵣ|²
Aᵣ ∝ ∫ ψբ∗ V(r) ψᵢ d³r
Assuming point-like nucleus, can use Coulomb potential
V꜀(r) = –Zαħc/r
where α is fine-structure constant.
End up finding
A꜀ ∝ Ze² 4π/k²sin²(θ/2)
which gives differential cross section
dσ/dΩ ∝ Z²e⁴/k⁴sin⁴(θ/2)
where k is magnitude of the momentum of the incoming electron and θ is angle from initial trajectory to final (fig 1.8)
Actual differential cross section
Need to account for nucleus having finite size, so include charge density for potential
Charge density
∫ ρ(r) d³r = Z
produces potential
V(r) = –αħc ∫ ρ(r’)/|r–r’| d³r’
Eventually end up with
d/dΩ ∝ Z²e⁴/q⁴ × F(q²)² × (rel. factor)
OG Rutherford term x finite size modifier x relativity factor
F(q²) is the form factor, depends on momentum transfer, q, between incoming and outgoing electrons.
Skin thickness
Skin thickness of a nucleus is linked to the fact that nuclei dont have sharp edges.
Quantified by density falling from 90% to 10% of its original value.
What can we learn from a graph of radial charge distributions
See fig. 1.9, page 20
- Density within core appears more or less the same regardless of A. Suggests nuclear force is saturated and nucleons experience repulsion at very short distances.
- Skin thickness, t, roughly the same for all nuclei. Attractive nuclear force must be very short ranged.
- Combining these, nucleon close to
the centre of ¹²₆C is as tightly bound as a nucleon close to the centre of ²⁰⁸₈₂Pb, and similarly for nucleons close to their respective surfaces. - Densities resemble Fermi-Dirac distribution, expected as we are considering fermions. Can approximate charge distribution as
ρ(r) = ρ₀ / 1+exp[(r–R)/a]
where ρ₀ can be read off graph and
a = t/(4 ln(3))
Nuclear radius from charge radius
Mean square charge radius
⟨r꜀²⟩ ≡ 1/Z ∫ r²ρ(r) d³r
Fitting this to data for A ≥ 40 gives
⟨r꜀²⟩¹’² = 0.95 A¹’³ fm
Approximating nucleus as a homogeneously charged sphere
R꜀ₕₐᵣ₉ₑ = √5/3’ ⟨r꜀²⟩¹’² = 1.23 A¹’³ fm
Nuclear density, distribution, radius from charge density, distribution, radius
Differ by a factor of A/Z, assuming protons and neutrons are distributed evenly throughout nuclei.
ρₙᵤ꜀ₗₑₐᵣ ≈ A/Z × ρ꜀ₕₐᵣ₉ₑ
= A/Z × Z / ⁴⁄₃πR³꜀ₕₐᵣ₉ₑ
= A / ⁴⁄₃πR³꜀ₕₐᵣ₉ₑ
Find nuclear radius
rₙᵤ꜀ₗₑₐᵣ ≈ 1.2 A¹’³ fm
Charge symmetry and independence of the nuclear force
Charge here means different nucleons.
Consider elastic nucleon scattering at low energies.
Like-nucleon scattering:
Two particles are identical, cannot distinguish between small and large angle scattering. After taking Coulomb interaction into account for p-p, experimental results show proton-proton and neutron-neutron interactions are identical and hence charge symmetric.
Vₚₚ ≈ Vₙₙ
Proton-neutron interactions: See peaks of similar magnitude at both low and high angles. Low peak is from glancing collisions, not surprising. High angles would be from almost head-on collisions, which aren’t as common as glanging blows.
dσ/dΩ ∝ 1/sin⁴(θ/2)
Nuclear force not strong enough to produce backscatters.
Instead can be explained by an exchange force causing the proton to turn into a neutron, and vice versa.
Virtual exchange particles:
Different pions have similar mass, so pn scenario is approx. equivalent to pp and nn. Nuclear force is charge independent.
Find exchange particle of the nuclear force
-
Spin
Both proton and neutron are spin-1/2 particles, quanta must have integer spin. -
Charge
Protons and neutrons differ in electric charge, must have charge. For consistency with like-nucleon, must also be a neutral case. - Energy and mass
for nn scattering
n₁ → n₁ + x
x + n₂ → n₂
Energy is ‘borrowed’ using the uncertainty principle
∆x∆pₓ ≥ ħ/2
∆E τ ≥ ħ
Energy must at least be rest mass
∆E ≥ mₓc²
and it can only exist for time τ
τ ~ ħ/∆E ≤ ħ/mₓc²
Cannot travel faster than c, range is limited to
R ≲ cτ = ħc/mₓc² = 200 MeV fm / mₓc²
Nuclear force saturated, can argue range is of nucleon size order ~1fm
Rest mass of x is of order 200 MeV
Meson!
Pion factfile
Lightest mesons, come in three varieties based on electric charge: π±, π₀ with electric charges ±e and 0 respectively.
Masses vary slightly depending on if they carry electric charge.
All have 0 spin.
Lowest order Feynman diagrams for pion exchange between a proton and neutron.
See figure 1.14, pg 27
Basic form of the SEMF
M(Z,A) = ZM(¹₁H) + (A–Z)Mₙ – E_B/c²
Binding energy is composed of 5 terms.
The five binding energy terms in SEMF
- Volume
- Surface
- Coulomb
- Asymmetry
- Pairing
Volume term
Nuclear force is short range, so a nucleon only experiences the presence of its nearest neighbours, independent of the size of the nucleus.
Nuclear radius is proportional to A¹’³, so volume term is proportional to A.
Term is attractive, so positive, increasing total binding energy.
aᵥ A
Surface term
Corrects for over-estimation of the volume term. Nucleons at the surface are not surrounded by other nucleons and so are less bound.
Negative term, decreasing the binding energy, proportional to nuclear radius squared.
–aₛ A²’³
Coulomb term
Accounts for electromagnetic repulsion between protons within the nucleus.
Electric potential experienced by each proton re-written in variables common to the rest of the terms:
V = 3/5 1/4πε₀ q²/r = a꜀ Z(Z–1)/A¹’³
where all constants have been absorbed into a꜀.
For large values of Z, can approximate this to
a꜀ Z(Z–1)/A¹’³ ≈ a꜀ Z²/A¹’³
–a꜀ Z(Z–1)/A¹’³
Asymmetry term
Purely quantum mechanical effect from nuclei tending to have equal numbers of protons and neutrons.
Originates from Pauli exclusion principle:
a large excess of either type of nucleon would have to occupy increasingly higher energy states. Would then be energetically favourable for excess nucleons to decay to the other type to balance the numbers of each.
Effect is less dominant for heavy nuclei (proton Coulomb potential raises all proton energy levels), accounted for by the factor of A in the denominator.
–aₐ (A–2Z)²/A
Pairing term
Quantum mechanical effect that is justified based on observations.
Nuclei with even numbers of both protons and neutrons are the most favourable, and nuclei with odd numbers of both are least favourable.
It is energetically favourable for nucleon spins to be paired.
Form is based on fitting to data.
+aₚA⁻¹’² for even-even
0 for odd-even or even-odd
–aₚA⁻¹’² for odd-odd
Sizes of SEMF contributions
Constants must be extracted from data. Approach doesnt work for light nuclei, results will depend on where we choose to start fitting from (e.g. A ≥ 20)
Terms are ordered from greatest to least contribution.
Rough plot of contributions to E_B/A in SEMF
See fig 2.4, pg 32
Valley of stability from SEMF
Minimising SEMF w.r.t. Z gives max binding energy, most stable isotope for a given mass number.
Zₘᵢₙ = A (a꜀A⁻¹’³ + 4aₐ + Mₙ – M(¹₁H)) / 8aₐ+2a꜀A²’³
Plotting this follows trend of line of stability, including trend for light nuclei, N=Z.
What is beta decay
Nuclear reaction whereby a nucleon can change type and is accompanied by the emission of an electron (or positron) and an electron anti-neutrino (or neutrino)
β decay reaction equations
n → p + e⁻ + ν̄ₑ
p → n + e⁺ + νₑ
Generic β⁻ decay
ᴬzX → ᴬz₊₁Y + e⁻ + ν̄ₑ
Q value of β⁻ decay
Q = (Mₙᵤ꜀(ᴬzX) – Mₙᵤ꜀(ᴬz₊₁Y) – mₑ – mᵥ)c²
where Mₙᵤ꜀ is nuclear mass.
Neutrinos are thought to have masses of a few eV at most, we approximate their mass to zero;
mᵥ ≈ 0
In terms of atomic masses,
Mₐₜ(Z,A) = Mₙᵤ꜀(Z,A) + Zmₑ
Q = (Mₐₜ(ᴬzX) – Zmₑ – (Mₐₜ(ᴬz₊₁Y) – (Z+1)mₑ) – mₑ)c²
= (Mₐₜ(ᴬzX) – Mₐₜ(ᴬz₊₁Y))c²
Electron capture and its equation
Proton rich nuclei, orbiting electron is absorbed by a proton, which then turns into a neutron emitting a neutrino.
p + e⁻ → n + νₑ
Isobar
Same A, differing Z
Beta decay and SEMF
Beta decay allows nuclei to balance asymmetry between proton and neutron fermi levels.
Odd nuclear mass isobars have min. mass excess accessible through beta decays. (fig 3.1 pg 36)
Even nuclear mass isobar graphs appear as two parabola, with two local minima accessible through beta decays. Double beta decay is possible to go from the local to true minimum, but this is highly unlikely.
(fig 3.2 pg 37)
Fusion vs fission using binding energy
For A < 56, binding energy per nucleon increases with A. Energy is released when light nuclei fuse.
Opposite happens for A > 56, energy is released when nuclei split.
What are s & r processes
Paths to reach higher A elements.
Slow and rapid neutron-capture processes.
Path taken depends on intensity of neutron flux.
If neutron flux is such that the nucleus is more likely to β-decay before it absorbs another neutron, s-process will be followed.
Example,
⁵⁶₂₆Fe → ⁵⁷₂₆Fe → ⁵⁸₂₆Fe all stable.
⁵⁹₂₆Fe has half-life of ~45 days.
If likelihood of absorbing another neutron is less than once per 45 days, this unstable isotope will β-decay to ⁵⁹₂₇Co.
Process could repeat to produce ⁶⁰₂₈Ni and beyond.
Exact path followed will vary, but in general will zigzag along the line of stability until ²⁰⁹₈₃Bi, where there are no more isotopes stable enough for the s-process to continue.
Rapid neutron-capture process, r-process, is where nuclei are bombarded with neutrons faster than they can β-decay. Such environments are thought to only occur in violent astrophysical processes such as supernovae and neutron star mergers.
Example,
Fe isotopes would continue until the isotope is so unstable that it does decay faster than it can absorb an additional neutron.
r-process path follows more or less straight horizontal and vertical lines along the neutron drip lines.
Alpha decay and its reaction equation
For A > 56, E_B/A decreases with A. Eventually energetically favourable for the nucleus to split into two lighter nuclei with larger total binding energy (smaller total mass).
Fission is an example of cluster decay. When the cluster is an α-particle, ⁴₂He, this is α decay.
⁴₂He is good because it is light, has a small number of protons, but is strongly bound. Probability of it happening involves quantum tunneling out of the nucleus.
α decay reactions have form
ᴬzX → ᴬ⁻⁴z₋₂Y + ⁴₂He
Energy released through α decay
Q = (Mₙᵤ꜀(ᴬzX) – Mₙᵤ꜀(ᴬ⁻⁴z₋₂Y) – Mₙᵤ꜀(⁴₂He))c²
In terms of atomic masses
Q = (Mₐₜ(ᴬzX) – Zmₑ – (Mₐₜ(ᴬ⁻⁴z₋₂Y) – (Z–2)mₑ) – (Mₐₜ(⁴₂He) – 2mₑ))c²
= (Mₐₜ(ᴬzX) – Mₐₜ(ᴬ⁻⁴z₋₂Y) – Mₐₜ(⁴₂He))c²
In terms of binding energies
Q/c² = ZMₐₜ(¹₁H) + (A–Z)Mₙ – E_B(Z,A)/c²
– [(Z–2)Mₐₜ(¹₁H) + (A – 4 – (Z–2))Mₙ – E_B(Z–2, A–4)/c²]
– [2Mₐₜ(¹₁H) + (4–2)Mₙ – E_B(2,4)/c²]
Mass terms cancel
Q = E_B(Z–2, A–4) + EB(2,4) – EB(Z,A)
Decay only allowed if Q greater than zero, only favourable if
E_B(2,4) > EB(Z, A) – EB(Z–2, A–4)
Alpha decay constant
λ = PfT
where P is the preformation factor (probability of finding four nucleons as an α particle in a given nucleus), f is the frequency the α hits the barrier (v_α/2R), and T is the transmission
factor (Gamow)
Spontaneous fission
Nuclear process where a heavy parent nucleus breaks apart into two daughter nuclei of similar mass without external action. Process is random and the products can vary. As heavy nuclei are neutron rich, the decay is often accompanied by release of free neutrons and neutron rich daughter nuclei that may β decay towards stability.
Variation of potential for spontaneous vs induced fission
Fig 3.6, pg 44
Nucleus shape and fission
If nucleus surface becomes prolate, surface term will increase whilst Coulomb term decreases. Physically the protons repel one another, which leads to a deformation, stretching the nucleus until ultimately it falls apart. SEMF estimates this process to be likely for A ≥ 270.
Activation energy in fission
Naturally occurring isotopes rarely undergo spontaneous fission. Instead, fission can be induced, typically by adding a neutron to an isotope such that it overcomes the activation energy to become fissile.
In reactors, a fuel susceptible to induced fission is chosen such that it emits more neutrons than needed to be absorbed. This leads to a chain reaction and is a very efficient source of energy.
SEMF failings and magic numbers
Difference in SEMF predicted and measured binding energes shows unexpected stability (higher binding energies) for certain Z and N numbers.
These magic numbers are experimentally found to be
N = 2, 8, 20, 28, 50, 82, 126
Z = 2, 8, 20, 28, 50, 82
3D harmonic oscillator equations
–ħ²/2M ∇²ψ(r) + 1/2 Mω²r²ψ(r) = Eψ(r)
Splitting into 1D eqs, get e-vals
Eₙₓ,ₙᵧ,ₙ₂ = (nₓ + nᵧ + n₂ + 3/2) ħω
Splitting instead into radial and angular, ψ(r) = Rₙₗ(r)Yₗₘ(θ,ϕ), get e-vals
Eₙₗ = (2n + l + 3/2) ħω
where n = 0, 1, 2, … and l is the angular momentum of the state
l, mₗ, mₛ possibilities
(2l + 1) states available for each l, corresponding to azimuthal quantum number, mₗ, which takes values from l to +l in integer steps.
For each mₗ there are two mₛ states corresponding to spin; mₛ = ±1/2.
Unlike atomic physics, no restriction imposed on l by n as we do not have a potential proportional to r⁻¹.
3D harmonic oscilator magic number predictions
Correctly predicts 2, 8, 20, through degeneracy of energy levels. Fails to predict 28.
See pg 47.
How does changing potential from SHO to infinite quare well adjust predicted magic numbers
Shifts energy levels down, predicts 2, 8, 20, shifts next down to 34 as opposed to 40 for SHO.
Woods-Saxon potential form
More similar to form of nuclear density. Based on Fermi dist.
Vᵥᵥₛ(r) = –V₀/1+exp((r–R)/a)
where a is the nuclear skin and can be extracted from data when finding the nuclear radius.
A typical value for a is 0.5 fm.
Still doesn’t accurately predict magic numbers.
Spin orbit potential
Vₜₒₜ = Vᵥᵥₛ(r) – E/ħ² L̂ · Ŝ
where E is a constant extracted from data.
L̂ and Ŝ are now coupled, so mₗ and mₛ are no longer useful quantum numbers. Instead must use eigenstates of total angular momentum operator, Ĵ = L̂ + Ŝ
Finding L̂ · Ŝ eigenvalues
(dropping hat notation for ease)
J² = L² + S² + 2L · S
L · S = 1/2 (J² – L² – S²)
⟨L · S⟩ = ħ²/2 [j(j+1) – l(l+1) – s(s+1)]
⟨L · S⟩ values for nucleons
Values for j depend on if the orbital and spin angular momenta are aligned or anti-aligned, j = |l ±s|. This, plus the fact that protons and neutrons are spin-1/2, gives
⟨L · S⟩ = ħ² l/2 for j = l + 1/2
and = ħ² –l+1/2 for j = l – 1/2
States previously degenerate now have energies that differ by ∆Eₛₒ.
Experimentally find higher j gives lower energy
¹⁶₈O nucleon shells
protons: (1s₁ₗ₂)²(1p₃ₗ₂)⁴(1p₁ₗ₂)²
neutrons: same
Doubly magic.
Subshell notation
nlⱼ
each sub-shell able to contain up to 2j + 1 nucleons.
Angular momenta of a nucleus
M_J = Σmⱼ
Sum of angular momenta of constituents
Parity equation
P = (–1)ˡ
Parity of a nucleus
Product of constituents
P = Πₗ (–1)ˡ
Nucleus angular momenta and parity notation
Jᴾ
eg all doubly magic nuclei are 0⁺
Valence nucleons
Valence nucleons (outside of filled shells) determine total angmom and parity of entire nucleus
What is a mirror pair
Two isotopes with the same A, but Z and N have been swapped. Exhibit similar characteristics due to nuclear force being charge symmetric
Nucleon pairing
SEMF, nucleus more tightly bound when pairs are formed of equal and opposite mⱼ.
Contrasts Hunds rule in atomic physics, attraction between nucleons as opposed to repulsion of electrons.
Angular momentum of odd-odd nuclei
Consider ¹⁸₉F, one unpaired proton and one unpaired neutron, each in a 1d₅ₗ₂ shell.
Different particles, so no Pauli restrictions.
Each can take any mⱼ value from –5/2 to +5/2.
Angular momentum could be any of
J = 0, 1, 2, 3, 4, or 5.
Can’t know which one with our basic single-particle shell model.
Can know parity though, in this case +
Energy to excite nucleons
Within same shell, typically of order 1 MeV
Promoting to next shell, needs roughly 10x as much energy. Can estimate using 3D harmonic oscillator energy spacing
∆E = ħω₀ ~ 40 A⁻¹’³
Similarities in mirror pairs and isobaric multiplets
Mirror pairs, A same, Z and N swapped. Find similar energy spacings.
Isobaric multiplets. Same A, mirror pair has N and Z differ by 2. Intermediate odd-odd nucleus of same A between them.
eg ¹⁸₈O, ¹⁸₉F, ¹⁸₁₀Ne
All share alot of similar energy level spacings (but odd-odd has alot more due to less Pauli restriction)
Shows charge symmetry and independence.
What is the island of stability
Currently can explain stable nuclei up to uranium. Current research trying to predict/find the ‘island of stability’, where the next nuclear shell is filled leading to more stable nuclei.
What is the collective model
For far-from-closed shells, valence electrons exhibit collective behaviour, appearing to rotate and/or vibrate in unison.
Approx energy for 32 valence nucleons rotating collectively
Assume all 32 valence nucleons rotate collectively with energy
E = L(L+1)ħ²/2I
where inertia
I ≈ 32 Mₙᵤ꜀ₗₑₒₙ R²
and R ≈ R⁰ A¹’³
Natural units
ħ = c = 1
Forces involved in nuclear decays
- α decay:
α particles are Helium atom cores, that must tunnel through an energy barrier. The α particle itself is tightly bound by the strong force, and tries to get out of a Coulomb potential - β decay:
Weak force changing quark flavour. Note: weak force is the only force of the standard model that changes flavours - γ decay:
Excited state of a nucleus de-excites by emitting a high-energy photon. This is an electromagnetic process.
Quark generations
Up, down
Charm, strange
Top, bottom
Lepton families
Electron, electron neutrino
Muon, muon neutrino
Tau, tau neutrino
What forces do quarks and leptons feel
Quarks feel all three:
Strong, weak, electromagnetic
Charged leptons feel:
Weak, electromagnetic
Neutrinos feel:
Weak
What are the force carriers
- Strong force: gluons
- Electromagnetic force: photons
- Weak force: W⁺, W⁻ and Z bosons
Notation used in Feynman diagrams
- Space vertical
- Time horizontal
- Arrows must ‘go on’
- Fermion: solid line, arrow forwards in time
- Anti-fermion: solid line, arrow backwards in time
- Photon: wiggly line
- Gluon: spring line
- W/Z/Higgs boson: dashed line
See fig 3, pg 7
Draw all possible
e⁺e⁻ → e⁺e⁻
Feynman diagrams
See figure 5, pg 8
Checks in Feynmann diagrams
- Check that at all times, conserved quantities are conserved.
- Check that all arrow are continuous.
- Check all vectors make sense in terms of force interactions.
What does ‘leading order’ mean
Minimum number of possible vertices in a Feynman diagram to get the desired process
Muon decay Feynman diagram
µ → eν̄ₑν_µ
See figure 12, pg 11
Lepton number conservation
Number of leptons of each flavour must be conserved between initial and final states.
Quark number conservation
Number of quarks, (N(q)), must be conserved, irrespective of flavour. Individual flavour number does not need to be conserved in weak interactions, but must be in EM and strong.
Individual quark number notations
- Nc = C charmness
- −Ns = S strangeness
- −Nb = B˜ bottom-ness
- Nt = T topness
Hadrons
- Mesons: qq̄ pairs. An example is the Pion, π⁺, which has quark content ud̄.
- Baryons: qqq (and anti-baryons q̄q̄q̄). Examples are the proton with quark content uud and the neutron with quark content udd.
Baryon number
B = N_q/3
= N(q)−N(q̄)/3
Baryon number is conserved in all known interactions.
Three discrete symmetries
- Parity
- Charge conjugation
- Time reversal
What is parity
Behaviour of a state under spatial reflection
r → –r
Eigenvalues of the parity operator
P = ±1
Parity of a multistate system
Parity is multiplicative,
P̂(ψ₁,ψ₂,…,ψᵢ) = P₁ · P₂ ·…· Pᵢ
Intrinsic parity
Parity for a particle at rest.
Getting parity calculation from parity operator
Wavefunction is the product of radial and angular parts
ψₗₘₙ = RₙₗYₗᵐ(θ, φ)
Using
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
Parity transformation:
r → r
θ → π − θ
φ → π + φ
Spherical harmonic becomes
Yₗᵐ(θ, φ) → Yₗᵐ(π−θ, π+φ)
= (−1)ˡ Yₗᵐ(θ, φ)
This means
P̂ψₗₘₙ(r) = P̂ψₗₘₙ(−r)
= P(−1)ˡ ψₗₘₙ(r)
P(−1)ˡ is the eigenvalue, and ψₗₘₙ(−r) is the eigenstate.
Which forces conserve parity
Electromagnetic and strong
Parity of a fermion-antifermion pair
Always –1
Convention is to assign +ve to fermion and -ve to antifermion.
What is charge conjugation
Operation of changing a particle into its antiparticle
C parity of neutral particles that are their own antiparticles
C = ±1
Charge parity for fermion-antifermion (or boson-antiboson) pairs with distinct antiparticles
C = (−1)ᴸ⁺ˢ
For which forces are charge parity conserved
Electromagnetic and strong
What is time reversal invariance
Invariance under the transformation t → −t
What forces are invariant under time reversal
Electromagnetic and strong
What is the CPT theorem
Any relativistic quantum field theory is invariant under the combined operation of CPT (charge, parity, time reversal) – where the order of the three operations does not matter
Consequences of CPT theorem
Last state has same energy as the original particle, and same momentum.
Leads to particles and antiparticles having identical masses and lifetimes.
What is conserved in EM and strong interactions
C, P, T, charge, lepton number, and baryon number (plus quark numbers C, S, B̃ and T)
What is conserved in weak interactions
Charge number, lepton number, and
baryon number are conserved.
C, P and T are not. Individual quark numbers, C, S, B̃ and T are not.
What is isospin
Symmetry between up and down quarks.
Analagous to ordinary spin.
Protons and neutrons are ‘up’ and ‘down’ components of a single nucleon N. N has isospin quantum number I = ½
Nucleon and quark isospin
I₃ = −½ for neutrons
I₃ = +½ for protons
up quark has
I₃ = +½
down quark has
I₃ = −½
All other quarks (and all leptons), have isospin I = 0. Only the up and down quarks carry non-zero isospin.
When is isospin conserved
In electromagnetic and stong interactions, but not weak.
Isospin multiplets
For a given isospin I, there should be I₃ = -I, … I states (in steps of 1)
2I + 1 states for each multiplet.
Example: pion multiplet
π⁺ : ud̄ : I₃ = +1
π⁻ : ūd : I₃ = -1
π⁰ : uū + dd̄ : I₃ = 0
These are three projections of I = 1 with a multiplicity of (2I + 1) = 3.
π⁺, π⁻, and π⁰ form an isospin triplet.
Total baryon wavefunction
ψ = ψբₗₐᵥₒᵤᵣ · ψₛₚₐ꜀ₑ · ψₛₚᵢₙ · ψ꜀ₒₗₒᵤᵣ
Hypercharge
Y = B + S + C + B̃ + T
Colour states
Colour quantum number can be red, green, or blue.
Properties of colour
- Any quark (u, d, · · ·) can have one of the possible three colour states r, g, b.
- Anti-quarks carry one anti-colour (r̄, ḡ, b̄).
- Only states with overall zero value for the colour charge are observable as free particles. These are called colour-singlets.
- E.g. A ”zero colour” state can be obtained by combining r, g, b altogether, or by combining a colour and its respective anti-colour: rr̄, gḡ, or bb̄.
When are colour zero states required?
When the state is bound. If initial state isn’t bound, doesnt need to be colour neutral.
Differences between strong and electromagnetic interaction.
Gluons carry colour.
Gluons couple to other gluons.
Gluon interaction vertices
2 quarks - gluon
2 gluons - gluon
2 gluons - 2 gluons
Which hadron decays are possible
Can only decay via the strong interaction if lighter states composed solely of hadrons exist.
Decay of π⁰ and π⁺ pions
Figures 21 and 22, pg 29
Photon factfile
Force carrier of the EM force.
Spin-1 boson.
Couples to electric charges.
Does not carry its own electric charge.
Coupling strength for photon EM coupling
αₑₘ = e²/4πε₀ħc ≈ 1/137 (where e is electron charge)
Scales as q² where q is fractional charge wrt electron charge.
What type of particle are the weak exchange particles
Spin-1 bosons
Range of weak force
Range R ≈ c∆t = ħ/Mc
where M is the mass of the force carrier.
Mass of the W boson is rather large, implying the weak force has a very short range. Shorter range, longer for a particle to decay via that force.
What can the W boson couple to
Leptons:
Can only couple to leptons within the same family.
Quarks:
Can couple within the same family, or between ‘up’ and ‘down’ varients of different families. See fig. 29 pg 33
Hadron notations
Λ꜀⁺ → Kₛ⁰ + p
Subscript shows quark involved, here charm and strange. From charge can work out quark constituents.
Λ꜀⁺ : c, u, d
Kₛ⁰ : s, d̄
What does the Z boson couple to
All known fermions. Cannot, however, change flavours.
d and s quark linear combinations when participating in the weak force
d’ = dcosθ꜀ + ssinθ꜀
s’ = −dsinθ꜀ + scosθ꜀
where θ꜀ is the Cabbibo angle, ~13°
vertex ud’ as a combination of other vertices
Vertex of ud’ can be viewed as a combination of two vertices, ud and us.
Weak coupling constant gᵥᵥ now has two components g(ud) = gᵥᵥ cosθ꜀ and gᵤₛ = gᵥᵥ sinθ꜀
Electroweak unification condition
gᵥᵥ sinθᵥᵥ = g₂ cosθᵥᵥ
gᵥᵥ is coupling constant of the W boson, g₂ is the coupling constant of the Z boson, and θᵥᵥ is the weak mixing angle (also called the Weinberg angle).
Weak mixing angle is given by
cos θᵥᵥ = mᵥᵥ/m₂
where mᵥᵥ and m₂ are the W and Z boson masses respectively.
All different electroweak boson couplings
(not including Higgs)
Fig 35 pg 37
Feynman diagrams for the process
µ⁺µ⁻ → µ⁺µ⁻
at leading order (ignoring Higgs)
Fig 36 pg 37
Can Z boson change flavour
NO!
What can a Z boson decay into
Any fermion-antifermion pair (of same particle)
Relative branching factor o hadronic to leptonic Z boson decays
Dont include top-antitop because too heavy.
5 quark combos, 6 lepton combos.
Each quark combo can exist in three flavours.
Hadronic : Leptonic
15:6
Overview of Wu experiment
Cold ⁶⁰Co in magnetic solenoid, spin aligns parallel to field direction.
Beta decays, expect electron emission in certain direction, opposite to cobalt spin. Changing parity is equivalent to changing Co spin, experiment redone in this way showed electrons emitted opposite to expected direction.
Shows weak parity violation.
If parity were conserved, would expect equal number of electrons in both directions regardless of Co spin, dont see that in practice.
How does chirality affect weak interaction
Weak force only couples to left-handed fermions (spin down in direction of travel) and right-handed anti-fermions (spin up in direction of travel)
Define helicity
The helicity of a particle is the spin projection along its direction of motion (divided by its maximal possible value)
Helicity and handedness only the same for massless particles.
Weak force charge and parity violation
Both proved by charge and parity operators causing right-handed neutrinos, which is not possible in nature.
Higgs mechanism
Spontaneous symmetry breaking of the Higgs field, causing drop from higher potential to radially symmetric minimum. Interaction with this field gives W and Z bosons their masses, and self-excitation gives the spin-0 Higgs boson.
7 possible Higgs vertices
See fig 46, pg 47
Higgs production and deca Feynman diagrams
2 gluons (or a quark-antiquark pair) from two protons, Higgs, 2 Z bosons, two pairs of charged leptons
See fig 47 pg 48
What is a parton
Part of a structure? In higgs stuff it references part of a proton, such as a quark, gluon, or virtual particle within caused by quantum fluctuation.
Probability a specific bit of a proton participates in an interaction
PDF: parton distribution function
Function of fraction of proton momentum.
Variables involved in Higgs detection
Initial states
* Probability specific part of a proton is involved in interactions: Parton distribution function
* Energy of the collisions
* Probability protons see eachother: linked to understanding of the accelerator itself
* Production cross section
* Luminosity?
Final states
* Branching fractions of different decays in the change leading ot the lepton-antilepton pairs we wish to study - gives expected number of Higgs events
* kinematic distributions, momenta, angles, etc. - requires Monte Carlo simulations
ATLAS structure
Multilayered, multipurpose detector, akin to an onion.
Has pixel, transition radiation, electromagnetic, hadronic, muon spectrometer parts
Irreducible physics background
Other physics processes that result in similar end products
Reducible detector-related background
Processes whos final products look similar, but may have been misidentified by the detector.
What does the standard model not explain
Dark matter, dark energy, matter-antimatter asymmetry, and how neutrinos acquire mass.
