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This is probably a very simple question, but I wonder how the amount of radioactive substance is related to the radiation it emits and therefore the harm it does to the body.

If I held 0.1g of barium-137 in my left hand and 10g of it in my right hand, for the same amount of time, would my right hand have been damaged 100 times as much as my left hand?

(I used barium-137 as an example because it's a gamma emitter. Alpha and beta particles would be absorbed internally more by the 10g mass because it's bigger.)

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    $\begingroup$ The rate of decay (number of atoms undergoing fission per unit time) is proportional to the number of atoms (i.e. the mass). However I am not sure about the relation between the radiation absorb and biological damage. How would you quantify biological damage in the first place? $\endgroup$ Jun 19 '18 at 12:01
  • $\begingroup$ In practice, one experiment would not be a reliable indication of damage. teachnuclear.ca/all-things-nuclear/radiation/… and following the links from who.int/ionizing_radiation/about/what_is_ir/en/index2.html especially this page who.int/ionizing_radiation/pub_meet/en may illustrate how biological damage is assessed in general terms. $\endgroup$
    – user198207
    Jun 19 '18 at 12:13
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    $\begingroup$ @ArchismanPanigrahi - Well, you go from rad (radiation absorbed dose) to rem (radiation effect man) (or your other favorite set of units) using a factor that depends on the type and energy of the radiation, based on radiation health physics studies. $\endgroup$
    – Jon Custer
    Jun 19 '18 at 13:02
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The search term to use in order to learn about this is "linear no-threshold," or LNT. This is the hypothesis that the biological harm is proportional to the radiation dose. It's hard to get reliable data, but LNT appears to be a poor approximation at low doses.

At low doses, there may even be an effect called radiation hormesis. A variety of experiments seem to show cases in which low levels of radiation activate cellular damage control mechanisms, increasing the health of the organism. For example, there is evidence that exposure to radiation up to a certain level makes mice grow faster; makes guinea pigs' immune systems function better against diphtheria; increases fertility in female humans, trout, and mice; improves fetal mice's resistance to disease; reduces genetic abnormalities in humans; increases the life-spans of flour beetles and mice; and reduces mortality from cancer in mice and humans.

I've had a hard time evaluating the evidence about radiation hormesis and LNT. A couple of authors who have worked on this sort of thing are Tubiana and Little, and they reach different conclusions.

There are obviously important public health and public policy implications for all this, so it would be nice to have a definitive answer, but I just don't think we're ever likely to get one. Ideally you'd raise a gazillion lab rats to adulthood with different amounts of ionizing radiation, and compare their health. In reality, the kinds of population doses we talk about in cases like Hiroshima, Chernobyl, and Fukushima are so low that if LNT were true, the number of lab rats required in order to do a statistically conclusive controlled study would be prohibitively expensive. (Even in the cases of Hiroshima and Nagasaki, most of the surviving population was exposed to relatively low doses. In studies of survivors that look for cancer, for example, it's hard to disentangle the effects of nuclear radiation from the effects of things like burning buildings, which emit nasty carcinogenic smoke.)

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There are different aspects to this, as you may have gathered from the other answers.

One aspect is the amount of ionization that is produced in your tissue. This is a simple linear relationship. Each unit of a particular kind of radiation produces the same amount of ionization. It produces the same amount of energy deposited in affected tissue. It produces the same amount of radicals, dislocation of molecules, etc. Twice as much radiation produces twice the result, etc.

The other aspect is, how much harm is produced by different amounts of radiation. On this aspect, there are some subtle points, and some controversy.

First, it is complicated by the fact that there is a statistical component. Some of the effects show up as probabilities of certain harm. Thus, radiation effects are often quoted in terms of probability of producing excess cancers, and similar statements.

The next complication is the idea of a minimum value of radiation that might not be harmful, or might even be beneficial. This is controversial. The buzz word is "hormesis."

Above some threshold, it seems fairly clear that twice as much radiation is twice as bad. This holds up to some other threshold where death is likely to occur. Above that it does not make sense to talk about additional harm. You are dead.

So, that much isn't controversial. There is some discussion about fixing those limits, and the exact degree of harm from a given unit of radiation.

What is controversial is that lower threshold. There is some evidence that, below some minimum, living things can recover from small exposure to radiation. That it won't produce adverse effects. There is even some evidence, controversial evidence, that small doses of radiation give resistance to larger doses. It's kind of like you get a base tan before going to the tropical beach.

The problem here is, it is very difficult to be accurate enough to be confident of any such claims.

The competing idea to this threshold is the no-threshold idea. That idea says that radiation is bad down to arbitrarily small dose. So if dose produces 1 additional cancer in a population of 1000 people so exposed, then the no-threshold idea says a dose of half will, on average, produce 1 additional cancer in a population of 2000 people so exposed. And so on to smaller and smaller doses.

Consider what would be required. If, for example, you wanted to test a dose that would be expected to produce 1 extra cancer in 10,000 people. And you want to do this in a way that lets you have a standard deviation in your answer of 1%. You would need to expose many 100's of thousands of people and observe the excess cancers. But look. You would then have given, possibly, 100 people cancer. This is generally considered monstrous, and so medical researchers are not allowed.

But we do have x-ray data. And some other exposures. And researchers spend a lot of time arguing about how to interpret these. We also have exposures of lab rats and such. And again, there is a lot of controversy over these.

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  • $\begingroup$ You would need to expose many 100's of thousands of people and observe the excess cancers. [...] This is generally considered monstrous, and so medical researchers are not allowed. The number would be $10^8$ people ($10^4/0.01^2$), but you could do the study with mice. It's unlikely that it would matter whether you use one mammal or another. So the issue isn't ethical, it's money. It's too expensive to raise 0.1 billion mice. The other big issue is that even if you did such a study, nobody would accept it as conclusive. Cf. the endless studies of cell-phone radiation causing cancer. $\endgroup$
    – user4552
    Jun 19 '18 at 18:59
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Mass (kg) is proportional to activity (Bq) which in turn is proportional to dose (Sv). @AndreiGeanta mentioned effective dose which is statistically proportional to the adverse effects of radiation for the whole body averaged.

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  • $\begingroup$ I don't think this is very accurate. Essentially the OP is asking whether LNT is valid, and this answer is asserting without qualification that LNT is valid. Actually there doesn't seem to be much controversy about the fact that LNT is wrong in general. The controversy is about to what extent it's wrong. $\endgroup$
    – user4552
    Jun 19 '18 at 14:53
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If I held 0.1 g of barium-137 in my left hand and 10 g of it in my right hand, for the same amount of time, would my right hand have been damaged 100 times as much as my left hand?

There are a few problems with this example.

First, Ba-137 is not radioactive; this is a stable nuclide. Since you are talking about “a gamma emitter”, you probably mean Ba-137m. However, you cannot hold $0.1\ \mathrm g$ or $10\ \mathrm g$ of Ba-137m in your hand.

The half-life of Ba-137m is only $2.552\ \mathrm{min}$. Accordingly, its specific activity is very high, about $2.0\times10^{19}\ \mathrm{Bq\ g^{-1}}$. Thus, a $0.1\ \mathrm g$ sample of Ba-137m would have a tremendous activity of about $2\times10^{18}\ \mathrm{Bq}$. By way of comparison, this would be more than the Ba-137m activity in the equilibrium core of the largest operating nuclear reactor, which is about $5.6\times10^{17}\ \mathrm{Bq}$. $10\ \mathrm g$ of Ba-137m would be more than the total inventory of the reactor cores of all operating nuclear power plants in the world.

The next problem is the high isotopic power that is generated in the samples. A $0.1\ \mathrm g$ sample of Ba-137m would produce about $200\ \mathrm{kW}$; a $10\ \mathrm g$ sample of Ba-137m would produce about $20\ \mathrm{MW}$. Roughly one tenth of that power would be absorbed within the samples. Thus, the samples would evaporate in about $2\ \mathrm{ms}$ and then burn in air.

The short half-life of $2.552\ \mathrm{min}$ would make working with Ba-137m difficult anyway. There are a few experiments that actually use isolated Ba-137m (such as the usual caesium/barium separation experiment that is done in radiochemistry beginners courses); however, most applications don’t use pure Ba-137m. Usually, Ba-137m is used in secular equilibrium with its mother Cs-137. Cs-137 has a longer half-life of $30.1671\ \mathrm a$, which makes the handling of samples much easier.

Therefore, we might want to adjust your example to:

If I held 0.1 g of caesium-137 in my left hand and 10 g of it in my right hand, for the same amount of time, would my right hand have been damaged 100 times as much as my left hand?

A $0.1\ \mathrm g$ sample of Cs-137 would have a Ba-137m activity of about $3\times10^{11}\ \mathrm{Bq}$; A $10\ \mathrm g$ sample of Cs-137 would have a Ba-137m activity of about $3\times10^{13}\ \mathrm{Bq}$. This change moves the example in a reasonable direction; nevertheless, these activities are still very high.

A real Cs-137 sample would usually not be carrier free; i.e., the sample would contain some stable Cs-133. Furthermore, it would consist of a caesium salt and not metallic caesium. A sealed source would also include some packing material. Thus, the total mass of the source would be higher than the net mass of the Cs-137 and Ba-137m alone. Even for gamma radiation, there would be some self-absorption of radiation within the sample itself.

For the sake of simplicity, we might want to assume that the geometry, mass, and chemical composition of the samples is identical so that they only differ in activity concentrations of Cs-137 and Ba-137m. Thus, the fraction of self-absorbed radiation would be the same for both samples. For example, we might want to dissolve each sample in water so that we get two aqueous solutions, each with a volume of $1\ \mathrm l$. Then, the shielding would be mainly determined by the water so that it would be approximately the same for both samples.

The next problem is that you want hold one sample in your left hand and the other sample in your right hand. Since the human body is not symmetrical, the effective dose is different for radiation that is coming from the left and radiation that is coming from the right. The effective dose per fluence for the gamma radiation of Ba-137m $(E=662\ \mathrm{keV})$ is $2.12\ \mathrm{pSv\ cm^2}$ for the left lateral geometry and $1.97\ \mathrm{pSv\ cm^2}$ for the right lateral geometry. Since you are asking about the radiation damage to your hands, we can ignore this difference.

Nevertheless, we cannot ignore the effective dose to the whole body yet. Assuming that you hold the samples in a distance of about $0.5\ \mathrm m$ from your body, the effective dose rate is about $50\ \mathrm{mSv\ h^{-1}}$ for the $0.1\ \mathrm g$ sample and about $5\ \mathrm{Sv\ h^{-1}}$ for the $10\ \mathrm g$ sample. If you stand like this for about $1\ \mathrm h$, you would probably get a lethal dose. You would probably die in $30{-}60\ \mathrm d$ after exposure due to damage to the bone marrow, i.e. haemopoietic failure, resulting primarily from a lack of progenitor cells that produce functional short-lived granulocytes, as well as from haemorrhages without the replacement of red cells.

This intermediate result shows that we should limit the exposure period for our example and that we are not talking about low doses. The assumption of a linear-non-threshold (LNT) model for the induction of cancer and heritable effects (according to which an increment in dose induces a proportional increment in risk even at low doses) that has been discussed in other answers is a actually not an issue in this case since the dose is so high.

Since the considered doses are so high, we are now concerned with deterministic effects. These tissue reactions are caused by damage to populations of cells. They are characterised by a threshold dose and an increase in the severity of the reaction as the dose is increased further. At doses below the dose threshold value, the deterministic effects do not occur.

For example, we might want to limit the exposure period to less than about $2\ \mathrm{min}$ in order to avoid a lethal whole-body dose (see above). The resulting dose to the body would be about $0.1{-}0.2\ \mathrm{Sv}$. Severe reactions do not occur in most body tissues of adults at such a dose. This dose is well below the threhold for developing the first radiation sickness symptoms (nausea, loss of appetite, and fatigue).

The dose to your left hand (holding the $0.1\ \mathrm g$ sample) would be about $0.3\ \mathrm{Sv}$. This would not cause any deterministic effects, not even reddening of the skin (which has a threshold of about $\lt3{-}6\ \mathrm{Sv}$).

The dose to your right hand (holding the $10\ \mathrm g$ sample) would be about $30\ \mathrm{Sv}$. This would cause skin reddening of a large area (maybe a week after exposure) and skin burns (maybe two weeks after exposure). The skin basal layer cells die and the blood vessels are damaged, which results in poor oxygen supply to the tissue of the hand. This is difficult to treat; amputation of the hand may become necessary.

You can see, since we are talking about deterministic effects with a threshold dose, there is generally no linear dose–response relationship. Your right hand would not simply be damaged 100 times as much as your left hand.

By way of comparison, a linear-non-threshold (LNT) model is used for stochastic effects. This dose-response model is based on the assumption that, in the low dose range, radiation doses greater than zero will increase the risk of excess cancer and/or heritable disease in a simple proportionate manner. We didn't consider such stochastic effects yet, but looking at the resulting whole-body dose we may estimate the additional risk of cancer at about $1\ \%$.

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