4.5 - Nuclear and Particle Physics A2

A nucleus is described by its nucleon number $A$ and proton number $Z$. The nucleon number, also called mass number, counts all protons and neutrons. The proton number, also called atomic number, counts protons and fixes the element's identity.

For $^{121}_{51}\mathrm{Sb}$, the nucleus contains 51 protons and $121-51=70$ neutrons. An ion of this nuclide still has the same $A$ and $Z$ because gaining or losing electrons does not alter its nucleus.

Rutherford scattering tested the diffuse positive-charge ('plum pudding') model by directing alpha particles at thin metal foil. The observations did not fit diffuse charge, so the atomic model changed to one with a tiny central nucleus.

A large deflection requires a large force acting for a short time. This occurs only when an alpha particle passes very close to a concentrated positive charge. The rarity of these events shows that the nuclear region is tiny compared with the atom.

Heating a metal cathode gives some conduction electrons enough energy to escape its surface: this is thermionic emission. A positively biased anode then attracts the emitted electrons, so the electric field transfers energy and increases their speed.

Across potential difference $V$, an electron gains kinetic energy of magnitude $eV$ when other energy changes are negligible. Electric force is opposite to electric-field direction because the electron is negatively charged.

A magnetic field exerts a force perpendicular to the electron's velocity and the field. It changes the direction of velocity, producing acceleration and a curved path, but it does no work and cannot by itself increase the electron's speed.

In a cyclotron, speed and momentum increase at every gap crossing, so $r=p/(B|Q|)$ increases and the path spirals outward. The magnetic force supplies centripetal acceleration without adding energy.

In a linac, drift-tube lengths increase as a non-relativistic particle speeds up, allowing it to reach each next gap when the electric field again points in the accelerating direction.

For motion perpendicular to a uniform magnetic field, magnetic force supplies centripetal force. Using $B|Q|v=pv/r$ gives $r=p/(B|Q|)$. At non-relativistic speed, $p=mv$, so $r=mv/(B|Q|)$.

An electron with $p=1.50\times10^{-24}\,\mathrm{kg\,m\,s^{-1}}$ in $B=48\,\mu\mathrm{T}$ has $r=p/(Be)=0.195\,\mathrm{m}$. Convert microtesla to tesla before substitution.

At an isolated particle interaction, total electric charge, total energy and total momentum before the vertex equal their totals after it. Momentum is a vector, so balance components rather than adding track momenta as unsigned numbers.

Choose two perpendicular axes through the vertex. Resolve every known momentum, apply $\sum p_x$ and $\sum p_y$ conservation separately, then reconstruct the missing magnitude and direction. Check charge and total energy independently.

To investigate a nucleon's internal structure, the probe must interact on a distance scale smaller than the nucleon. Increasing particle momentum shortens its de Broglie wavelength, so higher-energy beams can resolve finer structure in scattering patterns.

High collision energy can also produce new massive particles through mass-energy conversion. Colliding two equal, opposite beams gives nearly zero total momentum, so less of the available energy must remain as kinetic energy of the products than when a beam hits a stationary target.

Deviations from scattering expected for a structureless target reveal constituent behaviour. The beam energy is therefore selected to make the wavelength small enough and, where relevant, to exceed the energy needed to create detectable products.

Mass-energy equivalence connects a change in system mass with an energy change: $\Delta E=c^2\Delta m$. In particle creation, supplied energy becomes rest energy and usually kinetic energy; in annihilation, particle and antiparticle rest energy becomes radiation and kinetic energy of other products.

Compare the complete initial and final systems. The minimum energy to create particles is their total rest energy, but momentum conservation may require additional kinetic energy. For annihilation of an electron and positron initially at rest, two photons can carry equal and opposite momentum.

Each electron has rest energy $mc^2\approx0.511\,\mathrm{MeV}$. Electron-positron annihilation from rest releases $1.022\,\mathrm{MeV}$ in total, shared by two photons of $0.511\,\mathrm{MeV}$ moving in opposite directions.

Convert energy by multiplying the number of electronvolts by $1.602\times10^{-19}\,\mathrm{J/eV}$. Thus $2.0\,\mathrm{GeV}=2.0\times10^9\times1.602\times10^{-19}=3.2\times10^{-10}\,\mathrm{J}$.

To convert $938\,\mathrm{MeV}/c^2$ to kilograms, first convert $938\,\mathrm{MeV}$ to joules, then divide by $c^2$, giving about $1.67\times10^{-27}\,\mathrm{kg}$. The notation $/c^2$ belongs to a mass unit, not an energy unit.

An unstable particle moving at a speed close to $c$ has a longer measured lifetime in the laboratory frame than its proper lifetime measured in its own rest frame. The effect becomes significant only at relativistic speeds.

Muons formed high in the atmosphere have a proper mean lifetime of about $2.2\,\mu\mathrm{s}$. Even at nearly $c$, a non-relativistic estimate gives only about $660\,\mathrm{m}$ travelled in that time, yet many are detected after travelling several kilometres. Their increased laboratory lifetime explains the observation.

Compare the journey time or observed travel distance with what the proper lifetime would allow. If the speed is a substantial fraction of $c$ and the observed survival is much greater, relativistic lifetime increase cannot be neglected.

Baryons and mesons are hadrons because they are made of quarks. Quarks and leptons are treated as fundamental in this model. A proton or neutron is therefore not fundamental even though it is a subatomic particle.

The pattern of quarks in the model contains paired members. Completing that symmetry led physicists to predict a top quark before it was observed; later experimental detection supported the classification.

Every particle has a corresponding antiparticle with the same rest mass but opposite electric charge and opposite additive quantum numbers such as baryon number and lepton number. The antiparticle is written with a bar or a distinct symbol, for example $e^+$ for the electron's antiparticle.

To deduce antiparticle properties, retain rest mass and reverse charge, baryon number and lepton number. A particle-antiparticle pair can annihilate, provided the products conserve total energy, momentum and all required quantum numbers.

An allowed particle interaction must conserve electric charge $Q$, baryon number $B$ and lepton number $L$. Assign each incoming particle's values, total each column, and compare with the totals for all outgoing particles.

For beta-minus decay $n\rightarrow p+e^-+\bar\nu_e$: charge gives $0=(+1)+(-1)+0$; baryon number gives $1=1+0+0$; lepton number gives $0=0+(+1)+(-1)$. All three ledgers balance.

A particle equation lists every incoming particle to the left of an arrow and every product to the right. Read each supplied symbol first, then use conservation laws to identify a missing symbol or test the completed equation.

The decay $\pi^+\rightarrow\mu^++\nu_\mu$ has charge $+1=(+1)+0$, baryon number $0=0+0$, and lepton number $0=(-1)+(+1)$. The antimuon is an antilepton, while the muon neutrino is a lepton.