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Mass per area

Radiation and the atom · level 7, lesson 3 of 6

A question to start

Lead halves a 1 MeV gamma beam in 0.86 cm; water takes 9.8 cm. So is lead eleven times better? Or is something else going on?

The idea

Look at the column \(\mu/\rho\) in the last lesson: about 0.07 cm²/g for lead, concrete and water alike. For 1 MeV gamma rays, a gram of lead stops about as much as a gram of water. Lead is thinner only because it packs more grams into each centimetre.

So shielders measure a shield by its mass per area: the grams of material over each square centimetre,

\[\text{mass per area (g/cm}^2) = \text{density (g/cm}^3) \times \text{thickness (cm)}.\]

At 1 MeV, every one of these materials halves the beam in about 10 g/cm². For cosmic rays it is not a simple halving, but mass per area is still the measure that counts, and it is what a rocket must launch.

The air is a shield too. The pressure at the ground is the weight of the air above each square metre:

Air above you Mars Earth
Mass over each m² about 171 kg about 10,300 kg
Mass per area about 17 g/cm² about 1,030 g/cm²
The same as a layer of water 17 cm deep about 10 m deep
lead 0.9 cmconcrete 4.3 cmwater 10 cmeach one: 10 g over every cm²

Worked examples

1. Mars's air. The surface pressure is 636 Pa and gravity 3.73 m/s². Mass over each m² \(= 636 / 3.73 \approx 171\) kg. That is \(171{,}000\) g over \(10{,}000\) cm²: 17 g/cm².

2. Earth's air. \(101{,}400 / 9.82 \approx 10{,}300\) kg over each m², or 1,030 g/cm². Water is 1 g/cm³, so that is the same as 1,030 cm, about 10 m, of water.

3. Earth's air against a lead wall. 1,030 g/cm² of lead is \(1{,}030 / 11.35 \approx 91\) cm thick. Everyone on Earth lives under as much mass as nearly a metre of lead, spread out as air.

Try it

  1. Find the mass per area of a 20 cm concrete wall (2.3 g/cm³).
  2. How thick a layer of water has the same mass per area as Mars's air?
  3. Why does a gram of lead stop about as much 1 MeV gamma radiation as a gram of water?
  4. Earth's air gives about how many times the mass per area of Mars's air?
Answers
  1. \(2.3 \times 20 = 46\) g/cm².
  2. About 17 cm.
  3. The coefficient \(\mu/\rho\) is nearly the same, about 0.07 cm²/g, for both.
  4. \(1{,}030 / 17 \approx 60\) times.

Common mistake

Comparing shields by thickness alone. A thin lead sheet and a thick water tank can give the same protection. Compare the grams over each square centimetre.

Where it leads