Radiation dose

I heard milk is radioactive. Should I worry?

Ionizing radiation is produced by radioactive materials and nuclear reaction. It is invisible, and thus worrying. How do we measure and calculate radiation dose, and what level should we worry about?


What is a Sievert?

On a sunny day the Sun shines down on us with an intensity of about 1 kW/m2. Lying on a beach sunbathing, the surface area we present to the Sun is about 1 m2. Depending on our clothing, we absorb about half of the incident sunlight, i.e. a total of about 500 W. A small person of mass 50 kg (chosen just to keep the numbers simple) would absorb 10 W/kg, or 10 J/s/kg.

Fortunately the Sun’s rays, although not totally benign, do not have enough energy to ionize atoms (i.e. to strip off electrons). If they did, we would last about half a second in the sunshine.   For ionizing radiation (X-rays, gamma-rays, electrons, neutrons etc.) the quantity of absorbed energy is called a “dose” and is measured in Sieverts (Sv), where a dose of 1 Sv means, for light or massless particles like X-rays, gamma-rays or electrons, one joule of radiation absorbed per kg of living tissue. For heavier particles like neutrons and alpha-particles that do more damage per joule absorbed, 1 Sv means some fraction of a joule (0.05-0.2) absorbed per kg. The fraction is chosen such that 1 Sv does the same amount of damage, regardless of particle type. Hence we can settle of one unit to talk about all ionizing radiation health effects. However, because a few Sv is a lethal dose to humans, we normally talk millisieverts, mSv (or micro, or nano…).

How many Sieverts is safe? Consider the following facts:

– Our natural background dose of ionizing radiation is typically a few mSv per year per.  This dose arises directly from cosmic rays and from naturally-occuring radioactive species (some of which pre-date the solar system like uranium and thorium and some are continually generated by cosmic rays like carbon-14). For most people in the industrialized world, this dose is augmented by medical diagnostics like dental x-rays.

– For workers in Canada’s nuclear industry (which is a well-regulated representative of the industry world-wide),  the Canadian Nuclear Safety Commission sets a limit of 50 mSv in a single year and 100 mSv over 5 years (i.e. an average of 20 mSv/y)[note] Health Canada rules on exposure to ionizing radiation, http://www.hc-sc.gc.ca/hl-vs/iyh-vsv/environ/expos-eng.php [2019-10-04].[/note].

– on 15 March 2011, the Japanese Health and Labour Ministry set a 250 mSv limit for its nuclear workers, in light of the situation at the Fukushima Nuclear Power Plant[note] The Fukushima Daiichi Nuclear Disaster, https://en.wikipedia.org/wiki/Fukushima_Daiichi_nuclear_disaster [2019-10-04].[/note].

– a dose of a few Sv (i.e. a few 1000 mSv) is likely fatal.

Simply put, a few mSv per year is probably completely benign. A few Sv over a short period of time is certainly not.

If the Sun did emit solely ionizing radiation, we would absorb in the sunshine about 10 Sv per second, and we wouldn’t last very long.

Calculation of dose: Activity?

To calculate dose, we need two things:

– the total amount of material ingested, or the rate at which we are exposed to ionizing particles or rays, and the time of exposure

– the energy of each particle or ray

The amount of radioactive material is normally measured by its activity, or rate of decay. This is measured in Becquerels (Bq), which is the mean number of decays per second. (Activity is not measured in Hertz (Hz) as this measure implies a fixed frequency, whereas radioactivity is a random process that has to be measured in terms of averages).

Let’s say we have ingested a nanogram (10-9 g) of iodine-131, a common fission product. This doesn’t sound a great amount, but 131I has a mean life of 11.6 days (1.00 million seconds)[note] Kenneth S. Krane, “Introductory nuclear physics”, (Wiley, New York, 1987), p.828. The half-life is 8.04 days but to obtain the activity we need the mean life, which is the half-life divided by 0.693. The mean life is the average time it takes an atom to decay from the time you start watching it. The half-life is the time is takes half of a large group of atoms to decay. [/note], so the initial activity can be calculated:

Mass of an 131I atom = (131)(1.66×10-27 kg) =  2.17×10-25 kg

Initial number of 131I atoms = (10-12 kg)/(2.17×10-25 kg) = 4.60×1012

Activity = number/mean-life = (4.60×1012)/(106 s) = 4.60×10Bq = 4.6 MBq

Nearly 5 million decays per second would send a Geiger-counter into orbit. A nanogram of 131I may not be much on a weigh-scale, but to a radiation detector (or to a living being, as we shall see), it is an enormous amount.

To calculate the dose, we need the mean energy deposition per decay. Iodine 131 decay into beta-particles (electrons, e), gamma-rays (γ) and neutrinos (ν):

131I → 131Xe + e + γ + ν

The electrons and gamma-rays can deposit energy in tissue, whereas the neutrinos fly off without further interaction. The total energy deposition is about 550 keV per decay. The mean range of the decay electrons is a few mm, i.e. all will be absorbed in the body. The decay product, 131Xe, is stable, so we do not need to consider its decays. The electrons and gamma-rays do not make other atoms radioactive. The iodine stays in the human body long enough for it all to decay5. (This is not true of some other radioactive species that can be flushed out before decaying).

1 keV is 1.6 ×10-16 J, so our 4.60×1012 of 131I atoms will release, eventually, (4.60×1012)(550 keV)(1.6 ×10-16 J/keV) = 0.41 J.

If this energy was deposited throughout a typical human mass of 70 kg, the specific energy deposition would be 0.0058 J/kg, i.e. for electrons and gamma-rays, a dose of 5.8 mSv. However, because it concentrates in the thyroid and the particle ranges are a few mm, the ionizing energy is deposited in the very small volume (in fact a mass of less than 200 g) and the dose to that tissue is very much higher. The Nuclear Data Sheet[note] Nuclear data safety sheet for iodine 131, http://hpschapters.org/northcarolina/NSDS/131IPDF.pdf [2019-10-04].[/note] lists the specific dose for 131I taken up by the thyroid to be 0.476 μSv/Bq. With an initial activity of 4.6 MBq, this means a dose to the thyroid of 2.2 Sv (2200 mSv), which is very dangerous.

Can 131I migrate to other parts of the world? Yes, with a mean life of 11 days, it can get a long way before decaying if lofted up into the jet stream. Species that last many years like cesium 137 can be spread around the globe. On the other hand, for worried North Americans, note that we can detect tiny fractions of one Bq, so while we may see 131I coming from Japan, that is many orders of magnitude away from it being a health concern. If you live a few km from Fukushima however, that is another matter entirely.

If the radiation field in a given area is measured in mSv per hour, or per day, one can easily calculate how long you may stay in a certain area, given whatever limits you are observing.

Putting it all in perspective: the case of radioactive milk

Milk contains enough radioactive potassium-40 to set a Geiger Counter humming. But it didn’t come from Fukushima, or any human activity.

The Human Health Fact Sheet from the Argonne National laboratory in Illinois[note] Human Health Fact Sheet for potassium-40 https://www.remm.nlm.gov/ANL_ContaminantFactSheets_All_070418.pdf [2019-10-04].[/note] has a page on potassium-40 (40K); it states “It is the predominant radioactive component in human tissues and in most food.  For example, milk [typically] contains about 2,000 pCi/L of natural potassium-40”.

What does 2,000 pCi/L mean? The curie (Ci) is the old measure of (radio)activity, and means 37 billion decays per second (3.7 x 1010 Bq).

2000 x 10-12 Ci/L =( 2000 x 10-12 Ci)(3.7 x 1010 Bq/Ci) = 74 Bq

In other words, in 1L of milk, 74 radioactive 40K nuclei decay each second. So how much 40K is in there to start with?

The half life of 40K is 1.28 billion years. In other words however much was made in the last round of supernova explosions that made the material for our solar system about 4.6 billion years ago, a little more than 8% (i.e. 2-4.6/1.28) of the original is left. In fact it makes up 0.012% of all potassium (that’s about one radioactive potassium atom per 8000 stable atoms), and potassium is a very common element. An average 70 kg human body contains about 140 g of potassium, i.e. about 17 mg of 40K.

The mean life of 40K (denoted by the Greek tau, τ) is:

τ = 1.28/0.693 = 1.85 billion years = (1.85 x 109 y)(3.15 x 107 s/y) = 5.82 x 1016 s.

The activity, A = N/τ, where N is the number of nuclei present. For 1L of milk:

N = A τ = (74 Bq)(5.82 x 1016 s) = 4.31 x 1018 nuclei.

The mass of this many 40K nuclei = (40)(1.66 x 10-27 kg)(4.31 x 1018) = 2.9 x 10-7 kg ≈ 0.3 mg (per L, or per kg).

This is almost the same as the mass of 40K in every kg of our bodies (0.2% K)(0.012% 40K/K) = 2.4 x 10-7 kg 40K per kg.

  Our total radiation dose from 40K in our own bodies is about 0.17 mSv/y[note] UNSCEAR 2008, http://www.unscear.org/docs/reports/2008/09-86753_Report_2008_Annex_B.pdf, p.236 [2019-01-04].[/note]. Our total natural dose is a few mSv/y.