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Flashcards in Neutrino Physics Deck (68):

Energy/Momentum relation

p = Eβ


What is the invariant mass?

S = (ΣE)^2 - (Σp)^2


What does p(a)•p(b) equal?



Out of the strong, electromagnetic, weak neutral current and weak charged current, which bosons elicit a change in flavour?

Strong: No flavour change
EM: No flavour change
Weak CC: Always flavour change
Weak NC: No flavour change


What quantities are conserved in reactions?

Lepton number
Baryon number
Quark flavour numbers (Strong, EM & Weak NC {Z})


CM energy is > or = to what?

Sum of masses of products


What does an α-decay spectrum look like?

Discrete vertical lines. Kinetic energy of α-particle on x-axis


What does a β-decay spectrum look like? How can one be produced?

Continuous curve like Boltzmann distribution. Produced via either deflection in a magnetic field or via the ionisation they inflict.


What was expected from the Ellis & Wooster experiment, and what was discovered and then hypothesised?

It was expected that every decay process would leave the electron with the same energy - on the contrary what was found was a continuous spectrum of electron energies.
They hypothesised 1 of 2 things:
- Either, different nuclei of the same substance emit electrons with different energies.
- Or, electrons from β-decay lose energy via secondary interactions with the source material, leading to variation in the detected energy.


What assumption can you make when calculation ping α-decay energies?

Mass of initial and final nuclei are about the same, and much greater than the mass of α-particle.


What's the difference between α & β ray energies?

α-rays all emitted with same energy from given substance, whereas β-rays have a wide range of energies for same substance.


What were the problems with β-decay theory in the 1920s that led to the proposition of a neutrino particle?

• Angular momentum didn't seem to be conserved (n --> p+e , all with 1/2 spin)

• Continuous (not discrete) energy spectrum


How do the energy spectra of 2 & 3 body decays differ?

2 body decay spectrum is discrete, 3 body decay is continuous.


How are neutrinos produced in nuclear fission?

In fission, multiple neutrons are converted to protons in the nucleus, this happens via multiple beta decays, with each producing a single neutrino.


Describe the Feynman diagram for β-decay.

Neutron goes to a proton and a W(-) gauge boson, leading to an electron and an anti-electron neutrino.


Explain the relationship between Z and N in the nuclide chart.

As the proton number (Z) increases, the N/Z ration increases due to the increasing Coulomb repulsion between protons.


How were anti-neutrinos, produced nuclear fission reactions, detected?

The anti-neutrinos move into water and undergo inverse beta decay with protons, producing e(+) + n. The positron then annihilates almost immediately with an electron in the water, producing 2 photons, the neutron on the other hand travels around a while before undergoing neutron capture with the proton of a water molecule (forming deuteron and releasing a photon). Therefore there is a prompt, followed by a delayed signal.


How are photons detected in a material?

Photons interact with electron via Compton scattering, electrons become ionising radiation and move into scintillation detector, detector absorbs ionising radiation and emits light.


What is a typical nuclear reactor neutrino's energy?

~ a few MeV


What was the Cowan-Reines experiment?

- Gave first conclusive section of neutrinos

- Anti-neutrinos interact with water + Cadmium protons (in 2 tanks) via inverse beta decay
- Adding Cadmium to water reduced the time between prompt &a delayed signal, reducing uncertainty.
- Same result in both tanks.


How do you distinguish e & μ in a spark chamber?

μ is ~200x heavier, hence smaller energy loss due to Bremsstrahlung - so muons travel further and leave straight tracks when compared to electrons!


What is the optimum β-emitter for neutrino mass measurements and why?

- Second lowest endpoint
- Relativelty short half-life & low atomic mass -> high specific activity
- Only require small amounts -> less electron scattering
- Simple electronic configuration -> precise electron spectrum calculation


Which point on the β-energy spectrum is most sensitive to neutrino mass?

The maximum.


What are the ideal properties of a β-emitter in order to measure neutrino mass?

- Low energy release -> more data at endpoint energy
- Low half-life


State the 3 body decay energy equation for b.

m(b) <= E(b) <= [m(a)^2 + m(b)^2 - (m(c) + m(d))^2] / 2m(a)

(Upper energy limit same as 2 body decay but with [m(c) + m(d)])


How can we use a β-decay energy spectrum to determine electron neutrino mass?

The difference in endpoint energy of spectrum with massive neutrino and with massless neutrino is equal to minus the mass of the neutrino.
Therefore, find precise endpoint experimentally and deduct the theoretical endpoint with a 0 neutrino mass.


What do ε, p, ω and F(Z,ω) equal in a Fermi-Kurie plot?

• ε = E - E(0) = Electron kinetic energy - Endpoint energy
• p = electron momentum
• ω = electron total energy
• F(Z,ω) = Fermi function = probability of electron energy


What are primary and secondary cosmic rays?

Primary: High energy protons and 4He from astrophysical sources.
Secondary: Light particles produced in inelastic interactions with the air (N and O nuclei), called air showers.


What are he main sources of atmospheric neutrinos?

Decays of secondary pions and kaons.
π(+) -> μ(+) + ν(μ)


What is the formula for the mean free path of secondary cosmic ray particles?

λ = γβcτ = (E/m)βcτ ~ (E/m)cτ


State the ratio of muon neutrino to electron neutrino flux at low pion / muon energies (< 1 GeV) and at high energies.

~ 2 for < 1 GeV
> 2 at higher energies


State the properties of atmospheric neutrinos

- Neutrino flux ~ 10 cm^-2 s^-1
- Roughly isotopic directions
- Up/down spin symmetric
- Most common energy = 1 GeV


What method gives the most precise absolute neutrino mass?

Endpoint energy difference of Tritium β-decay, E(ν) < 2eV. Absolute mass unknown.


Give an estimate for the cross-section of a weak elastic scattering of an anti-electron neutrino and an electron. Does this have neutrino energy dependence?

σ ~ 10^(-43)•[E(ν)/ MeV] cm^2

Cross-section has linear energy-dependence.


Typical rate of neutrino product in a nuclear reactor?

~ 10^20 s^-1


How can we estimate neutrino interaction rates using cross-section and flux?

Interaction rate ~ σφ


How can we find the neutrino flux of a nuclear reactor?

φ = Rate of production / Area they pass through = 10^20 / (4πL^2)


Describe a possible neutrino detector.

Multi-kilotonne water Cherenkov detector.
- 40m in length
- Charged leptons take ~100ns to traverse it, emitting ~10^5 Cherenkov photons.
- Photons detected by photomultiplier tubes on walls (1ns time resolution).
- Speed and direction of charged lepton are measured.


What's the difference between the Cherenkov images of muons and electrons?
Why are τ leptons not observed?

- Muons leave sharply defined ring edges, since they don't interact much with water.
- Electrons leave a more diffuse ring edge, because being lighter they scatter off water molecules and shift Cherenkov cone.
- τ not observed because of very short lifetime.


Define the zenith angle, θz.

If neutrino was produced in atmosphere directly above detector, zenith angle = 0, if produced directly below (on other side of world) then angle = 180.
- θz is between 0 and 180
- cos(θz) between -1 and +1
- Distance is between 10km and 13000km


What anomaly did the Cherenkv detector discover?

Muon/electron neutrino flux ratio < 2 (atmospheric neutrino anomaly)
- Implies either a deficit in muon neutrinos or excess of electron neutrinos in atmosphere.


What relationship did the Cherenkov detector find between electron and muon neutrino flux and the zenith angle?

- Electron flux as expected, no excess.
- Muon flux asymmetric about zenith angle, muon deficit from below detector (through earth). Due to longer distance that muons below detector must travel - neutrinos can change flavour over long distances.


State the neutrino mixing hypothesis.

During weak interactions ν's assume flavour eigenstates (3 possible: e,μ,τ), during propagation ν's assume mass eigenstates (3 possible: 1,2,3).
- One set is a linear superposition of the other.

- Flavour eigenstates have well-defined weak interactions
- Mass eigenstates have well-defined masses


Give the formula for 2-flavour neutrino mixing and define the fundamental parameter.

|ν(α,β)> = Rotation matrix*|ν(1,2)>
θ is mixing angle


Give the formula form neutrino wave propagation.

|ν1(t)> = |ν1(0)>*e^(-iE(1)t)


Give the appearance probably eqn. in natural units and state the factor required to change to more strand are units, giving the final eqn.

P(L) = sin^2(2θ)·sin^2(Δm^2·L/4E)
- hbar·c factor
-> P(L) = sin^2(2θ)·sin^2(1.27Δm^2·L/E)


Define all the terms in appearance probability.

• sin^2(2θ) = probability amplitude
• θ = mixing angle
• L = distance travelled from source, in Km
• Ε = neutrino energy, in GeV
• Δm^2 = difference of square of masses, in eV^2


What is survival probability?

P[αα](L) = 1 - P[αβ](L)


What must be true for neutrino oscillations to occur?

- Δm^2 must not = 0, so masses can be the same
- At least one of the masses must be non-zero


Define the oscillation length, L[0]

- πL[0] is length required for one full neutrino oscillation.
- L[0] = E/(1.27*Δm^2)

- Hence, P[αβ](L) = sin^2(2θ)·sin^2(L/L[0])


What was the initial form of the appearance probability equation? What approximation is made to get to the final form?

P[αβ](L) = sin^2(2θ)·sin^2([E2-E1]t/2)

Highly relativistic, so assume p~E, hence using Taylor expansion, E2-E1 = Δm^2/2E


What is the special case for U in 3-flavour decays?

If U is identity matrix there is no mixing.


What values for Δm^2 and mixing angle have experiments found for ν(μ) -> ν(τ)?

Δm^2 = 2.4*10^-3 eV^2
θ ~ 45° (near maximum mixing)


Describe the normal and inverted hierarchy of neutrino masses

Normal hierarchy: m3 > m2 > m1
Inverted hierarchy: m2 > m1 > m3


How do you find the average survival probability?

Average it over L/E


How do you find the average survival probability of neutrinos?

Average the sin^2(1.27*Δm^2L/E) part of SP, getting SP = 1 - sin^2(2θ)·(1/2), which is greater than or equal to 1/2.


What are the two possible equations for inverse beta decay?

1) ν(e) + n -> e(-) + p
2) (anti)ν(e) + p -> e(+) +n


Describe the mechanisms by which water Cherenkov detectors can measure solar (Boron) neutrinos.

Electron neutrino undergoes elastic scattering with an electron in the water, this electron then whizzes through water at greater than light speed and releases Cherenkov radiation which is detected by PMTs.


Why do solar neutrino Cherenkov detectors have alight detection threshold energy?

To suppress natural β-decays.


Describe the Homestake experiment and how it differs from the Gallium experiments.

- Purpose of Homestake experiment was to measure solar neutrino flux. It did this by using the inverse β-decay of chlorine to Argon, extracting the argon (by flushing tank with He) and detecting its β-decay back into chlorine (via K-capture).
- The threshold energy for the chlorine decay was too great to detect pp neutrinos, but it could detect Boron neutrinos.
- Gallium experiments detected the pp neutrinos whereas Homestake detected Boron neutrinos.


What did the Gallium & Homestake solar neutrino experiments conclude?

There was an ~40% solar neutrino deficit when compared to expected solar neutrino flux (with both pp and boron neutrinos).


How did they know that solar neutrino were not being lost in atmospheric (inverse β-decay) reactions?

Most solar neutrinos have energies < 1MeV so they have insufficient energy for the IBD reaction (GeV).


What are the dominant neutrino producing processes in the sun?

pp neutrinos (99.8%)

Boron neutrinos (most energetic, easiest to detect)


If each full fusion process produces 2ν and 26MeV, use the power of the sun (2.4*10^39MeV) and its distance from Earth (1.5*10^8km) to calculate the solar neutrino flux on Earth.

One ν produced per 13MeV, so rate of ν production = 2.4*10^39MeV /13 s^-1
φ(Earth) = Ans/4πL^2 cm^-2 s^-1 (with L in cm not km)


Why is electron scattering of solar neutrinos in the Cherenkov detector dominated by electron neutrinos?

The electron neutrino can scatter via either CC or NC electron scattering, with the threshold energy being zero for both.
The other solar neutrinos can only undergo NC elastic scattering because the threshold energy for their CC interactions is far greater than the MeV energies of the solar neutrinos.


Why are neutrino detectors often placed underground?

To shield against cosmic ray muons.


What was the SNO experiment and what mechanisms did it measure?

SNO was a heavy water (D2O) Cherenkov detector used to measure the total neutrino flux - it found flux to be as predicted by Standard Solar Model (SSM)
There were 3 mechanisms that it measured:
• Elastic scattering, measured using the Cherenkov radiation of the resulting electron (this is dominated by ν(e)):
- Charged current ES (only possible for ν(e))
- Neutral current ES (possible for all ν flavours)
• Charged current, (deuteron changes to 2 protons) measured using the Cherenkov radiation of the resulting electron (only possible for ν(e))
• Neutral current, (deuteron separates into p+n) measured using neutron capture of resulting neutron, possible for all ν flavours -> leads to total solar neutrino flux measurement.


What change was made to the SNO experiment to increase the energy released by each neutron capture?

Salt was added to the D2O water, meaning that neutrons were captured by Cl instead of deuteron so more energy was released by each capture, and neutrino detection efficiency increased.