← Back to Part III: Relativistic Mechanics

Chapter 5: Relativistic Collisions

Analyzing collisions at relativistic speeds requires four-momentum conservation. Energy and momentum are conserved together as a single four-vector, enabling us to calculate threshold energies and predict particle creation.

Four-Momentum Conservation

\( \sum_{\text{initial}} p^\mu_i = \sum_{\text{final}} p^\mu_f \)

Four equations: one for energy, three for momentum

The power of four-momentum: use invariants! \( (\sum p^\mu)(\sum p_\mu) \) is the same in all frames.

Threshold Energy

To create new particles, the collision must have enough energy. The minimum (threshold) energy is calculated using invariants.

Example: Creating a Pion

p + p → p + p + π⁰ requires E_threshold = 280 MeV kinetic energy (much more than the pion's 135 MeV rest mass, because momentum must also be conserved).

Compton Scattering

Photon scatters off electron, losing energy (wavelength increases):

\( \Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta) \)

λ_C = h/m_ec = 2.43 pm (Compton wavelength)

Interactive Simulations

Relativistic Collision Simulator

Python

script.py60 lines

import numpy as np

print("=" * 70)
print("RELATIVISTIC COLLISION SIMULATOR")
print("=" * 70)

# Collision 1: Head-on proton-proton (like LHC)
print("\n--- PROTON-PROTON HEAD-ON COLLISION (LHC) ---")
m_p = 0.938  # GeV/c^2
E_beam = 7000.0  # GeV per beam

gamma = E_beam / m_p
beta = np.sqrt(1.0 - 1.0/gamma**2)
p = gamma * m_p * beta

# Center of mass energy
E_cm = 2.0 * E_beam  # head-on collision
print(f"Each beam: E = {E_beam:.0f} GeV, gamma = {gamma:.0f}")
print(f"Center-of-mass energy: sqrt(s) = {E_cm:.0f} GeV = {E_cm/1000:.0f} TeV")

# Fixed target equivalent
E_fixed = E_cm**2 / (2.0 * m_p)
print(f"Fixed target equivalent energy: {E_fixed:.0f} GeV")
print(f"Collider advantage factor: {E_fixed/E_beam:.0f}x")

# Collision 2: Threshold for particle creation
print("\n--- THRESHOLD ENERGY FOR PARTICLE CREATION ---")
particles = [
    ("Pion (pi0)", 0.135),
    ("Kaon (K+)", 0.494),
    ("Proton-antiproton pair", 2 * 0.938),
    ("Higgs boson", 125.0),
    ("Top quark pair", 2 * 173.0),
]

print(f"\nMinimum sqrt(s) needed to create particle (+ proton pair):")
print(f"{'Particle':>25} {'Mass (GeV)':>12} {'Threshold sqrt(s)':>20} {'E_fixed_tgt':>14}")
print("-" * 74)

for name, m_new in particles:
    # Threshold: sqrt(s) = 2*m_p + m_new (minimum, all at rest in CM)
    sqrt_s_min = 2 * m_p + m_new
    # Fixed target: E_lab = (s - 2*m_p^2) / (2*m_p), where s = sqrt_s^2
    E_lab = (sqrt_s_min**2 - 2*m_p**2) / (2*m_p)
    print(f"{name:>25} {m_new:12.3f} {sqrt_s_min:18.3f} GeV {E_lab:12.1f} GeV")

# Collision 3: Invariant mass reconstruction
print("\n--- INVARIANT MASS RECONSTRUCTION ---")
print("Two photons detected. Is it a pi0 or Higgs?")
print(f"{'E1 (GeV)':>10} {'E2 (GeV)':>10} {'theta (deg)':>12} {'M_inv (GeV)':>12} {'Particle':>10}")
print("-" * 58)

cases = [(0.1, 0.08, 30), (62.5, 62.5, 5), (0.5, 0.4, 10), (60, 70, 3)]
for E1, E2, theta_deg in cases:
    theta = np.radians(theta_deg)
    M_inv = np.sqrt(2 * E1 * E2 * (1 - np.cos(theta)))
    particle = "pi0" if abs(M_inv - 0.135) < 0.05 else "Higgs" if abs(M_inv - 125) < 10 else "other"
    print(f"{E1:10.1f} {E2:10.1f} {theta_deg:12d} {M_inv:12.3f} {particle:>10}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Relativistic Two-Body Collision Kinematics

Fortran

program.f9076 lines

program collisions
  implicit none
  double precision :: m1, m2, E1, p1, beta1, gamma1
  double precision :: s, sqrt_s, E_cm1, E_cm2, p_cm
  double precision :: m_p, E_lab
  integer :: i

m_p = 0.938d0  ! proton mass (GeV/c^2)

print '(A)', repeat('=', 65)
  print '(A)', 'RELATIVISTIC COLLISION KINEMATICS'
  print '(A)', repeat('=', 65)

! Fixed target: proton hits stationary proton
  print '(A)', 'Proton beam on stationary proton target'
  print '(A12, A12, A14, A14)', &
    'E_lab(GeV)', 'p_lab', 'sqrt(s)(GeV)', 'gamma_CM'
  print '(A)', repeat('-', 55)

block
    double precision :: E_labs(8), p_lab, beta_cm, gamma_cm
    E_labs = (/1.0d0, 2.0d0, 5.0d0, 10.0d0, &
               100.0d0, 1000.0d0, 7000.0d0, 100000.0d0/)

do i = 1, 8
      E_lab = E_labs(i)
      p_lab = sqrt(E_lab**2 - m_p**2)

! Mandelstam s = (E1 + E2)^2 - (p1 + p2)^2
      ! For fixed target: s = 2*m_p*E_lab + 2*m_p^2
      s = 2.0d0 * m_p * E_lab + 2.0d0 * m_p**2
      sqrt_s = sqrt(s)

! CM frame velocity
      beta_cm = p_lab / (E_lab + m_p)
      gamma_cm = 1.0d0 / sqrt(1.0d0 - beta_cm**2)

print '(F12.1, F12.3, F14.3, F14.3)', &
        E_lab, p_lab, sqrt_s, gamma_cm
    end do
  end block

! Threshold calculations
  print '(/,A)', repeat('=', 65)
  print '(A)', 'THRESHOLD ENERGIES FOR PARTICLE PRODUCTION'
  print '(A)', 'p + p -> p + p + X (fixed target)'
  print '(A)', repeat('=', 65)

block
    double precision :: m_x, s_thresh, E_thresh
    double precision :: masses(5)
    character(len=20) :: names(5)

names = (/'pi0 (135 MeV)       ', 'K+ (494 MeV)        ', &
              'p-pbar pair         ', 'J/psi (3.1 GeV)     ', &
              'Z boson (91.2 GeV)  '/)
    masses = (/0.135d0, 0.494d0, 1.876d0, 3.097d0, 91.19d0/)

print '(A20, A12, A16, A16)', &
      'Particle', 'm_X (GeV)', 'sqrt(s)_min', 'E_beam_min'
    print '(A)', repeat('-', 65)

do i = 1, 5
      m_x = masses(i)
      sqrt_s = 2.0d0 * m_p + m_x
      s_thresh = sqrt_s**2
      E_thresh = (s_thresh - 2.0d0 * m_p**2) / (2.0d0 * m_p)

print '(A20, F12.3, F14.3, A2, F14.1, A4)', &
        names(i), m_x, sqrt_s, '  ', E_thresh, ' GeV'
    end do
  end block

end program collisions

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Share:X Reddit LinkedIn

← Mass-Energy Equivalence Next: Photons and Massless Particles →