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Half-value thickness

Radiation and the atom · level 7, lesson 2 of 6

A question to start

A gamma source in a lab needs a lead wall that lets through only a thousandth of its rays. How thick must the wall be?

The idea

Each layer of the same thickness lets through the same share of a gamma beam. The thickness that lets through half is the half-value thickness, \(x_{1/2}\). After a thickness \(x\), the share left is

\[\frac{I}{I_0} = \left(\tfrac12\right)^{x / x_{1/2}}.\]

The same law can be written with the attenuation coefficient \(\mu\): \(I = I_0 e^{-\mu x}\), where \(\mu = 0.693 / x_{1/2}\) (0.693 is \(\ln 2\)).

NIST lists the mass attenuation coefficient \(\mu/\rho\) for every material. Multiply by the density \(\rho\) to get \(\mu\). At 1 MeV:

Material \(\mu/\rho\) (cm²/g) Density (g/cm³) \(\mu\) (per cm) Half-value thickness
Lead 0.0710 11.35 0.806 0.86 cm
Concrete 0.0650 2.3 0.149 4.6 cm
Water 0.0707 1.00 0.0707 9.8 cm
11/21/40.86 cm1.72 cmlead thickness

Worked examples

1. Lead's half-value thickness. \(\mu = 0.0710 \times 11.35 = 0.806\) per cm. \(x_{1/2} = 0.693 / 0.806 = 0.86\) cm.

2. A thousandth. \(2^{10} = 1{,}024\), so 10 half-value thicknesses: \(10 \times 0.86 = 8.6\) cm of lead, or \(10 \times 4.6 = 46\) cm of concrete.

3. Five centimetres of lead. \(5 / 0.86 = 5.8\) half-value thicknesses. \((\tfrac12)^{5.8} = 0.018\): about 1.8% gets through.

Try it

  1. What is the half-value thickness?
  2. What share of 1 MeV gamma rays gets through 29.4 cm of water?
  3. Find the half-value thickness of a material with \(\mu = 0.35\) per cm.
  4. How thick a concrete wall cuts 1 MeV gamma rays to a millionth? (\(2^{20} \approx 10^6\).)
Answers
  1. The thickness that lets through half the radiation.
  2. \(29.4 / 9.8 = 3\) halvings: \(\tfrac18\).
  3. \(0.693 / 0.35 \approx 2.0\) cm.
  4. \(20 \times 4.6 = 92\) cm.

Common mistake

"Two half-value thicknesses stop it all." Half plus half is not the whole: each layer halves what is left. Two layers let through a quarter.

Where it leads