The Units of Radiation

As discussed in the previous section, the activity $({\cal A})$ has units of decays-per-second. This tells us the number of decays and, indirectly, the intensity of the radiation. Unfortunately, the intensity of radiation does not express the effect of radiation. The effects of radiation depend on the energy of the particle as well as the particle being emitted. Alpha particles are large and heavy. When they impact, they transfer a lot of energy through a few collisions. When they hit, they hit hard; but they do not penetrate deeply. You can shield yourself from alpha particles with a barrier as thin as paper or a few inches of air; however, without protection (such as in your stomach or lung) the tissue will absorb lots of energy. Beta particles can penetrate deeper than the alpha particles, but not significantly. A few millimeters of aluminum will shield you from beta particles; but, again, if ingested, they will deposit a lot of energy into the tissue. Alphas and betas are referred to as internal hazards. Gamma radiation, on the other hand, is an external hazard since it needs a few inches of lead to shield against it. Given the half-life and the amount of material, the activity (and intensity) can be predicted. Given the mass values and the decay type, the energy-per-decay can be predicted. It would be convenient if the energy output of the radiation told us the damage caused by the radiation. Unfortunately, the energy absorbed relates the damage inflicted, not the energy emitted. The absorption will vary between the three types of radiation as well as between the same type of radiation with different energy. In order to compare the ``strength'' of different types of radiation, a few standards have been introduced, such as the amount of ionization in air. (The ionization of the air is a measure of the absorption by the air of the energy passing through the air.) However, these standards are specific to the material used and only secondarily reflect the damage caused to a different material such as biological tissue. The absorption of radiation in one (standard) material only approximately reflects the absorption of the same radiation by different material. In other words, in order to know how much damage some radiation will cause, somebody must have at some point irradiated that material and tested the absorption rate. This is not to say that the activity is useless information. The amount of radiation absorbable is still related to the amount of incident radiation. As we have already seen, the SI⁴ unit of Baquerels (Bq) is defined by

$$1 \,{\rm Bq} = 1 \mbox{$\phantom{1\!}^{{\rm decay}\!\!}/_{\!\!{\rm sec}}$ }$$

and the Curie (Ci) which was originally introduced as the amount of radiation given off by $1\,{\rm gram}$ of radium, but is currently defined by

$$1 \,{\rm Ci} = 3.7\times 10^{10} \,{\rm Bq}$$

The amount of radiation given off by $1\,{\rm gram}$ of radium is not quite $1 \,{\rm Ci}$.
Dose
In order to compare radiation strength, a dose expresses the amount of energy absorbed by some amount (by some mass) of material. Since different material absorbs radiation uniquely, the explicit material which absorbs the radiation must be expressed with the dose. The SI unit the Gray (Gy) is defined as

$$1\,{\rm Gy} = 1 \mbox{$\phantom{1\!}^{{\rm J}\!\!}/_{\!\!{\rm kg}}$ }$$

Another convenient unit which expresses a dose is the rad ( radiation absorbed dose) defined as

$$1\,{\rm rad} = 0.01\,{\rm Gy}$$

Both units are applicable to any type of radiation and both need to have the absorbing material specified (kilograms of such-and-such).
Exposure
In order to compare a variety of radioactive material, some standard is necessary. X-rays and $\gamma$-rays with photon energies less than $3\,{\rm MeV}$ ionize the air reasonably well. It takes about $88\times 10^{-7}\,{\rm J}$ to ionize $1\,{\rm g}$ of air. The unit defined by this standard is the Roentgen (R) defined by

$$1\,{\rm R} = .88 \,{\rm rad\ of\ air}$$

(Notice that this is a dose in air.) Because the energy is explicitly measured via the amount of ionization and each ion has a charge, the roentgen is also expressed in terms of electrical charge produced (each ion has $1.602\times 10^{-19}\,{\rm Coulombs}$ of charge) per kilogram of air

$$1\,{\rm R} = 2.58\times 10^{-4}\mbox{$\phantom{1\!}^{{\rm C}\!\!}/_{\!\!{\rm kg}}$ }$$

Further, since air has a density⁵ of $\rho = 1.29\mbox{$\phantom{1\!}^{{\rm kg}\!\!}/_{\!\!{\rm m^3}}$ }$, The Roentgen can also be expressed in terms of energy-per-volume

$$1\,{\rm R} = 1.13\times 10^{-2}\mbox{$\phantom{1\!}^{{\rm J}\!\!}/_{\!\!{\rm m^3}}$ }$$

The Roentgen does in a sense measure an energy-density, but we need to bear in mind that an exposure explicitly measures absorbed radiation by some amount of air. We are not accounting for the energy which does not get absorbed. There is a complicated relation to the incident radiation which is easier to measure experimentally than to predict mathematically. Having said this, the possibility of estimating the radiation absorbed by tissue is not wholly unrelated to the radiation absorbed by the air. In fact, a rough estimate can be made from considering the electron density of a molecule. Since there are as many electrons as protons, we can use the ratio of proton number to nucleon number (the atomic number to the mass number): Z/A. Air has an electron density of 1/2. Water has an electron density of 10/18. Mostly water, tissue has an electron density of about 14/25. Further, since one Roentgen is about $.88\,{\rm rad\ in\ air}$, we can find the equivalent dose in water

$$\begin{eqnarray*}1 \,{\rm R} & \approx & (0.88\,{\rm rad_{air}}) \times\left(\fr... ...}\,{\rm air}}\right) \\ & \approx & 0.978\,{\rm rad_{water}} \end{eqnarray*}$$

or in tissue

$$\begin{eqnarray*}1 \,{\rm R} & \approx & (0.88\,{\rm rad_{air}}) \times\left(\fr... ...\,{\rm air}}\right) \\ & \approx & 0.986\,{\rm rad_{tissue}} \end{eqnarray*}$$

The $\,{\rm rad}$ was originally developed with the intention that $1\,{\rm R}$ of radiation would be essentially equivalent to $1 \,{\rm rad_{tissue}}$. This is almost the case.
Exposure Rate
The amount of exposure should be, and is, related to the activity of the sample, among other things. However, the activity (expressing the amount of radiation emitted) is a rate of the decay process. The relationship between the activity (amount of energy emitted during some time) and the exposure (amount of energy absorbed) is via the exposure rate. (Recall that a rate is how something changes during some time.) Explicitly,

$$\frac{\Delta X}{\Delta t} = \Gamma \frac{{\cal A}}{d^2}$$

d is the distance of the absorber from the material, ${\cal A}$ is the activity, and $\Gamma$ is a decay-specific constant. The exposure rate decreases as the activity decreases as well as decreasing as the distance increases. $\Gamma$ ranges anywhere from the very small ( $\Gamma = 0.059$ for Co⁵⁷) up through almost 2 ( $\Gamma = 1.84$ for Na²⁴). Since the exposure is explicitly a $\gamma$-decay and X-ray phenomenon, the exposure rate (and this equation) is also only discussable for $\gamma$-decay. The $\Gamma$ value is related to the energy and intensity of the given radiation and will change accordingly as these values change. The exposure rate is not the same as the intensity of the radiation.
Relative Biological Effectiveness
In the discussion of dose, the radiation absorbed by biological tissue was measured in $\,{\rm rad_{tissue}}$. We found that $1\,{\rm R}$ (a dose of $0.88\,{\rm rad_{air}}$) will produce about $0.986\,{\rm rad_{tissue}}$. However, the exposure, the $1\,{\rm R}$ is explicitly a dose in air, not a dose in tissue. Since we are generally concerned with the dose in tissue, we need a useful method of finding the effect in tissue. This method is a multiplicative factor called the Relative Biological Effectiveness, or RBE for short. The RBE is defined as the following ratio:

$$\frac{\left(\mbox{dose in tissue of a specific radiation}\rig... ...ys}\\ \mbox{which produces the same effect}\end{array}\right)}$$

This factor ranges anywhere from 1 to about 20. Unfortunately, the denominator is difficult to figure out. So, this fraction is approximated by the Quality Factor.
Quality Factor
Since the Relative Biological Effectiveness (RBE) is difficult to figure out, the Quality Factor (QF) is used as an approximation. The QF is related to the energy deposited over some path-length. Since alpha-decay dumps all of its energy without going very deep, the QF for alpha-decay is close to 20. Very energetic neutrons spread their energy over a slightly deeper region and have a QF between five and ten. Less energetic neutrons spread their energy over a slightly deeper region and have a QF between two and five. X-rays and $\gamma$-rays have a QF of one. They go quite deep, spreading their energy throughout the penetration depth.
Dose Equivalent
This finally relates the damage done to biological tissue by accounting for the difference between the different types of radiation. The dose (in $\,{\rm rad_{tissue}}$) refers to the amount of energy absorbed and the QF relates how that energy is spread through the tissue. The dose equivalent is measured in units of $\,{\rm rem}$, roentgen equivalent man. The expression is

$$(\mbox{dose equivalent}) = (\mbox{dose})({\rm QF})$$

The QF converts a $\,{\rm rad_{tissue}}$ to a $\,{\rm rem}$. Notice that $1 \,{\rm rad_{tissue}}$ of X-rays is only $1 \,{\rm rem}$ but that $1 \,{\rm rad_{tissue}}$ of alpha-rays can be as much as $20 \,{\rm rem}$!