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Scattering probability
P = delta(sigma) / A_beam
dsigma / domega I’d detector covers a small solid angle domega
Rutherford Scattering
Assume target does not recoil (heavy nuclei)
Using conservation of energy and momentum
Closest approach:
E_k = 0.5 * m * v_0
= (z * Z * e^2)/(4 * pi * epsilon_0 * D)
D = (zZe^2) / (2piepsilon_0mv_0^2)
= (zZe^2) / (4piepsilon_0*E_k)
Cross-Sectiion calculation
dsigma/domega = dsigma/domega * db/db * dtheta/dtheta = dsigma/db * db/theta * dtheta/domega
dsigma/domega = D^2 / (16*(sin(theta/2))^4)
1) Cyclotrons
2) Cyclotron frequency
3) Vertical beam stability
4) Cyclotron energy limit
1) 2 metal ‘D’s placed in constant magnetic field. Given high voltage supply oscillating at omega_0
Constant speed inside D but accelerates in gap as other D is negatively charged. Alternating current means this is always the case.
2) rotation frequency of a non-relativistic proton (f = 1/T) denoted as - omega_0 = qB / m
3) requires vertical B-field which decreases with radius because this converts any vertical movement into oscillations (B must be circular)
4) simple cylcotrons only work if gamma is approximately 1. Higher energies require B-field to increase with radius implying it requires a strong focussing. The most energetic means protons at 600MeV
1) Synchrotron
2) bending field limit
3) bremsstahlung loss limit
4) collider motivation
5) Luminosity
1) uses magnetic field varying with time to keep particles at a fixed radius. Injection happens at low field and momentum and particles accelerated to high momentum. Beam stability due to focussing magnets between dipole magnets
2) B = rho/qr usually limiting factor, up to 8.3T used
3) power loss per turn is U proportional to 1/r * E^4/m^4. Limit on energy
4) system invariant mass proportional to E^(1/2) on fixed target and E in collider
5) L = fn_1n_2 / (4pisigma_xsigmay)
N = sigma* integral of Ldt = sigma*Lagrange
Name, Role and Spin
Photon (gamma), electromagnetism, 1 Gluons (g), strong force, 1 (Acts on quarks) W & Z bosons, weak force, 1 (Acts on quarks and leptons) Higgs boson (H), giving mass, 0
Conserved properties
Energy, momentum, charge, colour, electron number, muon number, taon number, baryon number, quark flavour
Types of Quarks, their names and content
Meson: Pion^0 - u*ubar + d*dbar Pion^+ - u*dbar Kaon^+ - u*sbar Kaon^0 - d*sbar D^0 - c*ubar J/psi - c*cbar B^0 - bbar*d
Baryon:
Proton - uud
Neutron -udd
A^0 - uus (not actually A, it’s like an A with no line in the middle)
Relativity stuff
ct' = gamma(c*t-beta*x) x' = gamma(x-beta*c*t) E'/c = gamma(E/c - beta*P_x) P'_x = gamma(P_x - beta*E/c)
1) Statistical errors
2) systematic errors
3) Gaussian distribution
4) error propagation
5) error combination
1) from random outcome trials. For large number of trials becomes Gaussian
2) consistent mistakes due to equipment/experimental set up
3) all error Gaussian P proportional to e^(-1/2 * (x-mu)^2/sigma^2) with sigma being approximately mu^(1/2) if mu>10
4) y=f(x) x error unknown implies that sigma_y = dy/dx * sigma_x
5) independent Gaussians added in quadrature: sigma_tot^2 = sigma_1^2 + sigma_2^2
1) Ionization energy loss
2) MIP
3) Bragg peak
4) Particle Identification
1) charged particles lose energy by ionizing atoms they pass. Neutral ones do not.
2) Particles have a minimum energy loss rate at beta*gamma = rho/m of about 3. Depends on material but is at least 1MeVg^-1cm^2
3) Max energy loss rate experienced as a particle comes to rest
4) dE/dx rates give information on particle charge and velocity
Bethe-Bloch Formula
Predicts rate of dE/dx energy loss. Assumes:
1) only describes EM ionization loss. There are other sources
2) free stationary electrons (neglect recoil)
3) ignore energy transfer to nuclei
4) min radius - use de broglie wavelength (b_min=lambda_e=h/rho=2pih_bar/gammam_eV)
5) max radius use b_max = V*gamma/average v_e where v_e is orbital frequency
6) energy loss to individual electron is small
-dE/dx = nz^2/beta^2 * e^4/4piepsilon_0^2m_e^2c^2 * ln (m_ec^2beta^2gamma^2/2pih_bar*average v_e)
Discontinuous energy loss-
1) Delta rays
2) Bremsstrahlung
3) critical energy
4) pair creation
5) cherenkov radiation
6) hadronic interactions
1) an extreme case of ionization where electrons given sufficient energy to leave their own ionization
2) the emission of a photon from a moving particle when it is disturbed. E varies but can be very large. Effects electrons more than other particles
3) energy at which average losses to ionization and brem are equal
4) photons can be converted into e^+ and e^- pairs in presence of matter
5) charged particles move faster than local c in medium emit this. Minor energy loss easy to detect
6) these are infrequent but dominate interaction of hadrons. Nuclear fragments released accompanied by messons if energy is high enough.
Passage through thick materials -
1) Radiation Length
2) Bremsstrahlung loss
3) multiple scattering
4) pair creation length
1) property of material which characterises scattering, energy loss and photon conversion rate. Varies between m and mm
2) for high betagamma particle in matter mean energy decreases as E=E_0e^-(x/L) due to brem
3) a charged particle passing through thick material experiences Rutherford Scattering with many nuclei. Scattering approximated by Gaussian spread: sigma_theta=z13.6MeV/betacrho * (x/L_R)^(1/2) * (1+0.038ln (x/L_R)
4) mean distance a photon travels before converting into e^+ e^- pair L_pp = 9/7 * L_R
Tracking detectors -
1) Curvature in a field
2) position resolution
3) angle resolution
4) momentum resolution
1) r = rho/qB
2) the accuracy with which a point is measured
3) 2 detectors measure direction of particle at precision of
sigma_theta,meas = 2^(1/2) * sigma_x/L
Multiple scattering
4) min 3 detectors to find momentum. If identical spread L/2
Rho=0.3BL^2/8*x
