Cambridge Notes, Past Papers, Revision Questions

Production and Use of X‑rays

1. Production of X‑rays in an X‑ray Tube

1.1 Basic principle

Electrons are emitted from a heated cathode and accelerated through a potential difference V (typically 10–150 kV).

The high‑energy electrons strike a metal target (the anode) and are suddenly decelerated.

The loss of kinetic energy is emitted as X‑ray photons.

1.2 Minimum (cut‑off) wavelength

The highest‑energy photon that can be produced corresponds to the full kinetic energy of one electron:

E_{\max}=eV\qquad\Longrightarrow\qquad

\lambda_{\min}= \frac{hc}{eV}

h = 6.626 × 10⁻³⁴ J s (Planck constant)

c = 3.00 × 10⁸ m s⁻¹ (speed of light)

e = 1.60 × 10⁻¹⁹ C (elementary charge)

Worked examples

Tube voltage V	λ_min (nm)	Comment
50 kV	0.025 nm	Typical for a dental X‑ray tube.
120 kV	0.010 nm	Common in medical diagnostic radiography; photons are ~2.5 times more energetic.

1.3 Types of X‑ray radiation

Bremsstrahlung (braking radiation) – a continuous spectrum produced when electrons are decelerated in the Coulomb field of the target nuclei.

Characteristic X‑rays – discrete lines that appear when an incident electron ejects an inner‑shell electron (usually K or L). An outer‑shell electron fills the vacancy and a photon with energy equal to the difference between the two atomic energy levels is emitted:
\[
E{\gamma}=E{i}-E_{f}
\]

Common K‑α and K‑β energies (target material)

Target (Z)	K‑α (keV)	K‑β (keV)
Cu (Z = 29)	8.04	8.90
Mo (Z = 42)	17.48	19.61
W (Z = 74)	59.3	67.2

These energies are often used for calibration and for producing specific X‑ray lines in laboratory experiments.

1.4 Factors that affect the X‑ray spectrum

Parameter	Effect on spectrum	Quantitative relationship (Cambridge AO2)
Tube voltage V	Higher V → shorter λ_min (more energetic photons) and more intense bremsstrahlung.	λ_min ∝ 1/V ; bremsstrahlung intensity ∝ V
Target atomic number Z	Higher Z → stronger bremsstrahlung and higher‑energy characteristic lines.	Bremsstrahlung intensity ∝ Z² ; characteristic line energy ∝ Z² (approx.)
Tube current I₀	More electrons per second → higher overall X‑ray output.	X‑ray intensity I ∝ I₀ (linear relationship)

1.5 Attenuation of X‑rays in matter

The transmitted intensity after a thickness x of material is given by the exponential attenuation law:

I = I_{0}\,e^{-\mu x}

I₀ – incident intensity.

I – intensity after the material.

μ – linear attenuation coefficient (cm⁻¹), depends on photon energy and absorber composition.

Example 1 – Aluminium (30 keV X‑rays)

\mu_{\text{Al}}\approx0.15\;\text{cm}^{-1}\quad\Rightarrow\quad

I = I{0}e^{-0.15\times1}=0.86\,I{0}

(≈ 14 % of the photons are absorbed in 1 cm of Al.)

Example 2 – Lead (120 keV X‑rays)

\mu_{\text{Pb}}\approx1.2\;\text{cm}^{-1}\quad\Rightarrow\quad

I = I{0}e^{-1.2\times0.1}=0.89\,I{0}

(Only 0.1 cm of lead is needed to attenuate ≈ 90 % of 120 keV photons.)

1.6 Safety and shielding

Safety box – protecting people from X‑rays

Shielding materials – dense, high‑Z substances such as lead (Pb) or concrete. For diagnostic energies (≤ 150 keV) a lead equivalent of ≥ 1 mm is usually sufficient; higher energies require proportionally thicker shields.

Radiation dose concepts
- Absorbed dose (D) – energy deposited per unit mass (Gy = J kg⁻¹).
- Equivalent dose (H) – absorbed dose weighted by radiation type (Sv).

ALARA principle – keep radiation “As Low As Reasonably Achievable” by:
- Using the minimum voltage and current needed for the task.
- Collimating the beam to limit stray exposure.
- Employing interlocks, warning lights and audible alarms.
- Providing personal dosimeters for staff in high‑dose areas.

2. Use of X‑rays in Astronomy and Cosmology

2.1 Why observe the Universe in X‑rays?

Temperatures of 10⁶–10⁸ K (e.g., hot plasma in galaxy clusters) emit most of their radiation as X‑rays (thermal bremsstrahlung).

Strong gravitational fields near black holes and neutron stars accelerate particles to relativistic speeds, producing high‑energy photons.

Non‑thermal processes such as synchrotron radiation and inverse‑Compton scattering also generate X‑rays.

2.2 Major X‑ray astronomy missions (selected)

Einstein Observatory (1978–1981)

ROSAT (1990–1999)

Chandra X‑ray Observatory (1999–present)

XMM‑Newton (1999–present)

NuSTAR (2012–present)

eROSITA (survey mission, 2019–present)

2.3 Typical X‑ray sources

Source type	Typical X‑ray luminosity (L_X)	Energy range (keV)	Dominant physical process
Active Galactic Nuclei (AGN)	10⁴²–10⁴⁶ erg s⁻¹	0.1–100	Accretion‑disc corona, relativistic jets
Neutron stars / Pulsars	10³²–10³⁸ erg s⁻¹	0.1–10	Magnetospheric emission, hot surface spots
Supernova remnants (SNR)	10³⁴–10³⁶ erg s⁻¹	0.1–10	Shock‑heated plasma, synchrotron
Galaxy clusters	10⁴⁴–10⁴⁵ erg s⁻¹	0.1–10	Intra‑cluster medium (thermal bremsstrahlung)
Cosmic X‑ray background (CXB)	≈ 10⁻¹² erg cm⁻² s⁻¹ sr⁻¹	0.1–10	Integrated emission from distant AGN & hot gas

2.4 X‑ray detection techniques

Scintillation detectors – X‑ray photon excites a scintillator; the emitted visible light is amplified by a photomultiplier. Typical energy resolution ≈ 10 %.

Proportional counters – Photon ionises a gas; the resulting electron avalanche is proportional to the photon energy. Resolution ≈ 15 %.

Charge‑Coupled Devices (CCDs) – Photon creates electron‑hole pairs in silicon; charge is transferred pixel‑by‑pixel and read out. Resolution 2–3 %.

Transition‑Edge Sensors (TES) / microcalorimeters – Operated at ≈ 50 mK; a tiny temperature rise changes the resistance, giving very high energy resolution. Resolution ≈ 1 % (ΔE/E ≈ 0.01).

2.5 Applications in cosmology

Cluster mass determination – The temperature T of the intra‑cluster medium (ICM) is linked to the gravitational potential via the virial theorem:
\[
\frac{3}{2}kT \approx \frac{GM{\text{cluster}}\mu m{p}}{2R}
\]
Measuring T from the X‑ray spectrum yields the total mass M_cluster.

Large‑scale‑structure mapping – Wide‑field X‑ray surveys trace hot gas in filaments and superclusters, providing a complementary view to optical galaxy surveys.

Dark‑energy constraints – The redshift evolution of the cluster number density N(z) depends sensitively on Ω_m and Ω_Λ. X‑ray‑selected cluster samples give robust constraints on cosmological parameters.

Growth of supermassive black holes – The spectrum and intensity of the cosmic X‑ray background encode the integrated accretion history of AGN over cosmic time.

2.6 Future directions

High‑resolution X‑ray spectroscopy with microcalorimeter missions (e.g., XRISM, Athena).

All‑sky surveys (eROSITA) to detect millions of AGN and thousands of new galaxy clusters.

Time‑domain X‑ray astronomy – rapid response to transients such as tidal‑disruption events and neutron‑star mergers.

Multi‑messenger campaigns – coordinated X‑ray, gravitational‑wave and neutrino observations to probe the most energetic phenomena.

Suggested diagram: Schematic of a grazing‑incidence X‑ray telescope showing nested mirrors that reflect X‑rays at shallow angles onto a focal‑plane CCD detector.

Astronomy and cosmology