understand that annihilation occurs when a particle interacts with its antiparticle and that mass–energy and momentum are conserved in the process

Published by Patrick Mutisya · 8 days ago

Production and Use of X‑rays – A‑Level Physics 9702

Production and Use of X‑rays

Learning Objective

Understand that annihilation occurs when a particle interacts with its antiparticle and that mass–energy and momentum are conserved in the process.

Key Concepts

  • Bremsstrahlung (braking radiation)

  • Characteristic X‑ray emission

  • Positron–electron annihilation

  • Conservation of energy and momentum

  • Applications of X‑rays

1. Bremsstrahlung

When high‑energy electrons are decelerated in the electric field of a nucleus, they lose kinetic energy in the form of photons. The spectrum is continuous up to a maximum photon energy equal to the incident electron kinetic energy \$K_e\$.

Maximum photon energy:

\$E{\max}=Ke = eV\$

where \$V\$ is the accelerating potential and \$e\$ the elementary charge.

2. Characteristic X‑rays

If an incident electron ejects an inner‑shell electron from an atom, an outer‑shell electron can fill the vacancy, emitting a photon with energy equal to the difference between the two binding energies.

TransitionNotationEnergy \$E\$ (keV)
K‑shell to L‑shellK\$_\alpha\$≈ 8.0 (for Cu)
L‑shell to M‑shellL\$_\alpha\$≈ 0.9 (for Cu)

3. Positron–Electron Annihilation

A positron (\$e^+\$) is the antiparticle of the electron (\$e^-\$). When they meet, they can annihilate, producing photons.

  1. The total rest mass energy of the pair is \$2m_ec^2\$.

  2. To conserve both energy and momentum, at least two photons are produced.

  3. In the centre‑of‑mass frame the photons have equal energy \$E\gamma = mec^2 = 511\ \text{keV}\$ and travel in opposite directions.

Energy conservation:

\$2mec^2 + K{e^+}+K{e^-}= \sumi E_{\gamma i}\$

Momentum conservation (vector form):

\$\mathbf{p}{e^+}+\mathbf{p}{e^-}= \sumi \mathbf{p}{\gamma i}\$

For the simplest case (both particles at rest):

\$\mathbf{p}{\gamma 1} = -\mathbf{p}{\gamma 2},\qquad E{\gamma 1}=E{\gamma 2}=511\ \text{keV}\$

4. Conservation Laws in Annihilation

Consider a positron moving with kinetic energy \$K_{e^+}\$ colliding with a stationary electron. The total initial four‑momentum is

\$P{\text{initial}} = \left(mec^2+K{e^+}+mec^2,\; \mathbf{p}_{e^+}\right).\$

If two photons are emitted, their four‑momenta \$P{\gamma 1}\$ and \$P{\gamma 2}\$ must satisfy

\$P{\gamma 1}+P{\gamma 2}=P_{\text{initial}}.\$

Solving the equations gives the photon energies and emission angles. The result shows that one photon can have energy greater than \$511\ \text{keV}\$ while the other has less, but the sum always equals \$2mec^2+K{e^+}\$.

5. Applications of X‑rays

  • Medical imaging (radiography, CT scans)

  • Crystallography – determination of crystal structures

  • Material analysis – X‑ray fluorescence (XRF)

  • Positron Emission Tomography (PET) – relies on \$511\ \text{keV}\$ annihilation photons

Suggested diagram: Energy–momentum vector diagram for electron‑positron annihilation showing two 511 ke \cdot photons emitted in opposite directions.

Summary

Production of X‑rays can occur via bremsstrahlung, characteristic transitions, or particle–antiparticle annihilation. In annihilation, the total rest mass energy of the electron–positron pair is converted into photon energy, and the laws of conservation of mass–energy and momentum dictate that at least two photons are emitted, each carrying \$511\ \text{keV}\$ in the centre‑of‑mass frame. These principles underpin many practical uses of X‑rays, from medical diagnostics to materials science.