Formula for calculating dose rates from gamma emitting radioactive materials
Published: Sep 29, 2021
There are a number of formulae available for calculating gamma dose rates from radioactive material. Since the internet now has so much data available online, using a formula is probably consigned to academic interest. As long as you know the data is reliable (and seek Radiation Protection Adviser advice if required), then you will probably find using specific gamma ray constants of more use. Just try googling 'gamma ray constant Cs-137' and you will see what we mean! That said, you need to be really careful what you choose to use, you will sometimes find quite different values if you search for them.
Doing such a search for Cs-137 revealed the following results (we have converted all to SI units and scaled so they are directly comparable).
- 1156 micro Sv/h per GBq at 30 cm (search result 1)
- 848 micro Sv/h per GBq at 30 cm (search result 2, online calculator)
- 1146 micro Sv/h per GBq at 30 cm (search result 3)
Why the difference? Part of the difference is down to the derivation of the gamma ray constant and the dose unit it is quoted in. For example, do you have the exposure in air (e.g. micro Gy/hour) or exposure in tissue (micro Sv/h). Moreover, do you actually know the unit quoted (and is it correct)?
There is a temptation (and we did this above to make the point) to convert units without thinking too hard what they actually mean.
In the first example above. according to the web page we landed on, our first result should actually read 1156 micro Gy/ h (this being derived from the data which was actually reported as 115.6 mR/hour (the R being the roentgen!). Without more thought (actually this was deliberate) we just multiplied mR by 10 and settled on the units of micro Sv/h. Is is this wrong? Does it matter? The answer is 'is depends'. If you wish to consider this further then check out the 'Additional thoughts on this rule of thumb' at the bottom of the page'.
A simple rule of thumb
Here is the rule of thumb formula, it is something Mark Ramsay came across at the beginning of his career in radiation protection. It was in the book 'Introduction to Radiation Protection' by Alan Martin and was probably read about 1986.
In this expression
D = Dose rate in (micro Sv/h)
M = Activity in MBq
E = Gamma energy in MeV
r = distance from the source in m
The deceptively simple factor '6' actually ties up a load of variables converting activity (Bq) to disintegrations per second, eV to joules, joules per kg to Gy, Gy to Sv, seconds to hours, and accounting for an average (μen/ρ)air which is the mass energy absorption coefficient for air, and so on. The end result is an equation that works for the energy range of about 0.1 MeV up to 2 MeV (in most cases, but see below). The best way to think of D (micro Sv/h) is what you might expect to measure from an unshielded source using a typical workplace radiation monitor.
Using this rule of thumb with Cs-137
Let us use the equation with Cs-137 since its gamma emission (from the decay product Ba-137m) is simple (one gamma ray line of interest).
D = Dose rate in (micro Sv/h) - this is what we want
M = Activity in MBq (1000 MBq ,since we want 1GBq to match the search results we obtained earlier)
E = Gamma energy in MeV (0.662 MeV, obtained from any data book)
r = distance from the source in m (0.3 m to match the search results we obtained above).
We see that D = 1226 micro Sv/h. A little higher than the above search results, but pretty close. (If you are not satisfied with 'pretty close' then do please read to the bottom of this page).
Before concluding we should also consider a case where the radioactive material of choice emits more than one specific gamma ray energy. In fact, we need to know the energy lines and their emission probability.
Using this rule of thumb with Fe-59
Consider using the same expression with Fe-59. Looking in data tables we will find the following gamma energies with their emission probability:
1.292 MeV (43.2%)
1.099 MeV (56.5%)
0.192 MeV (3.1%)
0.143 MeV (1.0%)
We could probably leave out the bottom lower energy gamma as its emission probability is also low. However we will use them all so you can see how this works. Considering E in the above equation, we now have the following:
E=(1.292 * 0.432)+(1.099*0.565)+(0.192*0.031)+(0.143*0.01) =1.19
Using exactly the same expression as above, but replacing (E) 0.662 (Cs-137) with 1.19 (Fe-59) we find that D is 2204 micro Sv/h.
Remember that the only difference is emission energy probability, the activity (M) is still 1 GBq (1000 MBq) and distance (r) is still 30 cm (0.3 m).
Now that we know how the above expression works, and it appears to provide reasonable values for Cs-137 in line with the literature, intuitively our value for Fe-59 (2204 micro Sv/h) should be about right.
- 1710 micro Sv/h per GBq at 30 cm (search result 1, online calculator)
- 1985 micro Sv/h GBq at 30 cm (search result 2)
Our value is a little higher than data obtained from a quick search online, but its still the right order of magnitude.
As with all our rules of thumb resource, use carefully and contact a Radiation Protection Adviser (RPA) if unsure.
Additional thoughts on this rule of thumb
If you have made it down to here you may well just be curious, or perhaps you are not so happy with the phrases such as 'pretty close' or 'right order of magnitude'?
Will will explore some of the finer technical details in a future blog post (will post the link here when completed). However, here are some things to think about.
Its exposure in air. The origin of the above expression is based on exposure rate (X) in air and has the units of R/h (roentgen/hour). Whilst many will consider 100 R = 100 RAD = 1 Gy = 1 Sv etc, this is a simplification and a loose use of radiation dosimetry units. In actual fact the R (now replaced with C/kg) is based on ionisation in air such that 1 R = 0.877 RAD (old non SI unit of absorbed dose). Since 100 RAD = 1 Gy, it follows that 100 R = 0.87 Gy (for x-rays and gamma rays in air under these specific conditions). Therefore, if the above rule of thumb actually outputs as 'exposure in air', then to reflect Gy (and Sv) correctly we would need to adjust the answer in 'micro Sv/h' by multiplying by 0.877. Trying this with the Cs-137 dose rate we calculated above [1226 'micro Sv/h' * 0.877] = 1074 micro Sv/h. This is now much closer to some of the reported data we listed. If we try with the Fe-59 result we obtained above, [2204 'micro Sv/h' * 0.877] = 1933 micro Sv/h, we are again much nearer the other reported data.
Use of energy probability. This is a lesson in not making assumptions. You will note above we explained how to deal with the energy (E) and emission probabilities for Fe-59. Well for Cs-137 we just made an assumption that its single 0.662 MeV gamma ray (actually from Ba-137m), was emitted with 100% probability. This is wrong, its actually 85.1% (Ref: NNDC). If we apply this emission probability to the now adjusted 1075 micro Sv/h for the Cs-137 (i.e. 0.662 MeV * 0.851) we find the dose rate reported is now 914 micro Sv/h using the activity and distance as before. We have no way of knowing, but it might be that others have also incorrectly set the emission probability for Cs-137.
Use of a generic single average (μen/ρ)air (mass energy absorption coefficient). By 'use' we mean built into the above expression (tied up in the '6'). We have deconstructed the rule of thumb and now know that the value of (μen/ρ)air used is 0.03 cm2/g (air). We know that a more appropriate value is 0.0293 cm2/g for a photon energy of 0.662 MeV. If this value is used then our result for Cs-137 is 845 micro Sv/h and is then practically identical to 'search result 2, online calculator' reported at the top of this page (848 micro Sv/h). We happen to think that this final value for Cs-137 is probably what you would actually measure under controlled conditions (and real measurement are always best).
The effect of using average 0.03 cm2/g (air) is much more significant when using the rule of thumb with radioactive materials with lower emission energies. We will cover most of this in a future blog article, but in summary we tried the above rule of thumb with I-125. If you look at the energy and emission probability for I-125 you will find the follow:
0.035 MeV (6.5%)
0.027 MeV (112.5%)
0.031 MeV (25.4%)
Using the above rule of thumb formula, treating 'E' as before for multiple photon energies (see method used above for Fe-59), with an activity of 1000 MB and a distance of 30 cm, the dose rate calculated directly as before for I-125 is 25 micro Sv/h. A quick look online will indicate values much higher (e.g. 480-546 micro Sv/h). Something clearly does not look right here!! All becomes clear when you look at the cm2/g (air) values for the I-125 energies. Compare the standard value of 0.03 cm2/g (air) with what you actually should use in the calculation (in red below).
0.035 MeV (6.5%) (0.11)
0.027 MeV (112.5%) (0.26)
00.031 MeV (25.4%) (0.153)
This makes a significant difference to the calculation result and demonstrates that the rule of thumb would significantly underestimate dose rate for I-125 and other similar low energy gamma emitters. In an up and coming blog article we will show how the above values of ueu/p in cm2/g can be incorporated into the above rule of thumb, but by the time you have seen this you will probably want to use a reliable data book on online radiation safety calculator.