One advantage of our approach to solve the volume integral over a wall (equation (1)) is the easy visualisation of the annual effective dose rate contribution at a certain point in a room due to individual parts of one wall (figure 2). It becomes immediately clear, that for the room centre the floor and ceiling contribute most to the annual effective dose rate and that most of the gamma radiation stems from their central parts. Only a small fraction of the annual effective dose rate is obtained from the outer parts of the wall. The same is valid for the short and long walls. Thus, from a radiation protection point of view it is most effective to put doors and windows in the central parts of the walls to reduce the radiation exposure. A large reduction of the radiation at the midpoint can be obtained by such a strategy. This is in agreement with results presented in a previous study using a similar model (Croymans et al 2018).

As our approach to solve the volume integral for the walls is based on an MC algorithm, it is necessary to run our model with enough iterations (randomly chosen points in the wall) to ensure a precise and accurate solution. As a measure for precision and accuracy we prefer not to use the total number of points per wall as all wall pairs have different sizes. Instead we decided to use the number of points per m3 of wall volume. This guarantees a similar treatment of each wall, independent of its size in this study or when performing calculations for rooms with other dimensions. The results for the room centre suggest to use at least 1000 points m−3 as this ensures a result within the 1% standard deviation from the true value (figure 3). About 100 000 iterations are necessary to obtain a result, which is within a 1‰ standard deviation interval.

The annual effective dose rate for each of the individual points in the room is calculated with a new, independently determined set of points within the wall material. Thus, when focussing on the precision and accuracy of the annual effective dose rate for the room average a lower number of points per wall volume is necessary than for the room centre, to obtain a similar 1% or 1‰ standard deviation range (compare figures 3(a) and (b)) for both measures. In particular, to obtain a similar standard deviation for the room centre and the room average, the number of points m−3 of wall material can be about two orders of magnitude smaller for the room average than for the room centre, when the annual effective dose rate for the room average relates to about 100 grid points.

Nevertheless, one should keep in mind that the standard deviation for individual points in the room increases with a decreasing distance from the long or short walls (figure 4). Thus, too low a number of points m−3 of wall material should be avoided. A similar statement can be drawn from the analysis of the room average with respect to the number of points in the room. When the number of points in the room is too low (less than 100 points) the annual effective dose rate can be slightly underestimated (by 1–2 %, figure 5). With a higher number of points in the room (greater than approximately 1000) the average dose rate becomes virtually constant. Hence, we suggest to run the code with about 1000 points in the room and about 1000 points m−3 of wall material.

We provide a quadratic index formula for individual pairs of walls. This is especially convenient for an estimation of the annual effective dose rate in a room (average or centre), where the construction material and its radionuclide activity concentration for one pair of walls (e.g. floor and ceiling) is different compared to that of other pairs of walls (e.g. long and short walls). Albeit its somewhat lengthy form, the equation is similar to the well known quadratic index formula, where all material has the same radionuclide activity concentration (CEN/TR 17113 2017). With the form presented here it is possible to easily describe the radiation exposure to inhabitants in a more precise way in case pairs of opposite walls consist of material with different activity concentrations.

It is a widely applied practice to use an index formula for the centre point (Markkanen 1995, Deng et al 2014, Nuccetelli et al 2015, CEN/TR 17113 2017) to estimate the annual effective dose rate. Such a strategy provides a good approximation of the true value and is very convenient to use. However, we showed that the annual effective dose rate at the room centre is somewhat underestimating the average annual effective dose rate for the whole room for typical wall thicknesses and concrete densities (figures 4–8). Therefore, we also provide a quadratic index formula for the average annual effective dose rate obtained in a room (equation (6)). We showed that thinner walls will result in a somewhat higher radiation exposure, than calculated with the quadratic index formula, which is derived for a wall thickness of 20 cm (figure 7). Thus, if a room has thinner walls than 20 cm another small underestimation of the radiation exposure can be expected. However, as the model room has no windows and doors the true annual effective dose rate would still be smaller than that estimated for the room average and the room centre. Thus, the index formula for the room centre can still be considered as conservative enough and is still helpful to provide an estimate if material can be used for dwelling constructions from a radiation protection point of view.

The question is, if it is still necessary to apply the index formula? Recent codes for the calculation of the gamma radiation exposure are easy to use and written in popular coding languages such as python (this study). Their application is easy as it is often not necessary to change parameters within the code but simply offer individualised parameters for the room (such as room dimension, wall thickness or concrete density) in well-arranged external files. This becomes especially important, when the room is significantly different in its dimensions compared to the model room (Risica et al 2001, Croymans et al 2018). Under those cases the results from the application of the index formula, which was derived for model room dimensions, is questionable and it is suggested to preferably use such codes instead. Currently, these codes allow individual room dimensions to be used as as long as the rooms have a rectangular shape. But future versions might also be able to calculate the effective dose rate for more irregular shapes and thus could be implemented in building information modelling systems.

Quite often it is possible to rely on different building materials with material specific radionuclide activity concentration for the construction of dwellings. In those situations it is helpful to determine the annual effective dose rate from those types of building material, used for the different types of walls. First, we acknowledge this by providing quadratic index equations for the annual effective dose rate for the room centre and room average, which account for individual activity concentrations, thickness and density of the three pairs of opposite walls. In a situation, where for example floor and ceiling have another radionuclide concentration than the long and short walls, it is an easy task to use equation (5) or (6) to calculate the annual effective dose rate at the room centre or the room average.

Also the distribution of the annual effective dose rate in the room provides interesting insights. If all walls have the same radionuclide activity concentration, the room centre obtains the smallest amount of radiation (figure 6(d)). The places closer to the walls receive higher dose rates. However, the range of the annual effective dose rate in the room is relatively small. Nevertheless, quite often inhabitants of dwellings are often spend their time closer to the walls than in the room centre. Often there is a usual habit to place e.g. couches and beds close to walls. On both pieces of furniture a considerable time is spent. So it would be helpful to minimise the radiation from the long and short walls.

There are two ways to achieve this. The first way includes an increase of the density of the long and short wall material. A hint on that is provided by the observation of a continuous alignment of the dose rates obtained at the room centre and in the room average values with increasing mass per unit area (figure 7). As the wall thickness is fixed in this figure, this means that the annual effective dose rate becomes more similar with a denser wall material. The reason for this alignment is caused by an advancing range minimisation of the annual effective dose rate throughout the room (compare figures 6(d) and 9(a), (b)).

Figure 9. Distribution of annual effective dose rates within the model room under a wall density of 1175 kg m−3 (a) and 3525 kg m−3 (b) for a 226Ra contribution only. Please note the larger annual effective dose rate range from the centre towards the long and short walls for the low-density scenario compared to the high-density scenario. (The difference between the minimum and maximum value is the same in both sub-figures.) A similar distribution is observed for 232Th and 40K. Further parameters for both plots: 20 cm thick walls, 100 000 points m−3; 0.2 m × 0.15 m resolution.

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In the three experiments with the spatial distribution of the annual effective dose rate in the room we varied solely the density from 1175 kg m−3 (figure 9(a)) over 2350 kg m−3 (figure 6(d)) to 3525 kg m−3 (figure 9(b)). As expected the annual effective dose rate in the room centre (3.14 4.39 4.92 µSv a−1 per Bq kg−1) and the room average (3.34 4.49 4.93 µSv a−1 per Bq kg−1) are increasing with increasing density, while the difference between the room average and room centre decreases. The spatial annual effective dose rate distribution in the room suggests that this is due to a more homogenised annual effective dose rate within the room with a more elevated wall density. For the examples above with ρ = 1175, 2235, 3525 kg m−3 the annual effective dose rate difference between a position close to central part of a long wall and the room centre decreases from 0.47 over 0.24 to 0.11 µSv a−1 per Bq kg−1.

For the second option, to minimise the annual effective dose rate close to the walls more strongly than at the room centre, we perform a small thought experiment. We consider how the annual effective dose rate in the room centre, close to the walls and the room average changes under a redistribution of the radionuclide activity concentration within pairs of opposite walls. This redistribution of the radionuclide occurs under the assumption that the sum of all radionuclides within the material in the six walls surrounding the model room remains constant. For example, if the long and short walls are thought to be radionuclide free, the floor and ceiling additionally have to accommodate this amount of radionuclides.

When we assume a radionuclide vector for 226Ra, 232Th and 40Ka of [80, 80, 800] Bq kg−1 in a scenario where all walls contain the same activity concentration for those radionuclides, a scenario, where the long and short walls would be thought to be radionuclide free the activity concentration of floor and ceiling is 2.4583*[80, 80, 800] Bq kg−1. The factor 2.455 83 is the ratio of the wall material volume between all six walls surrounding the room and that of floor and ceiling only. Under those redistributed conditions, the spatial annual effective dose rate distribution in the room is inverted compared to the scenario, where all walls have the same radionuclide activity concentration (compare figures 10(b) and 6(d)).

Figure 10. Distribution of annual effective dose rates within the model room under a wall density of 1175 kg m−3 (a) and 2350 kg m−3 (b) for a 226Ra contribution only. In both plots all radionuclides are in floor and ceiling only and the factor of 2.4583 is already accounted for (see text). Please note the smaller annual effective dose rate range from the centre towards the corners of the room walls for the low-density scenario compared to the high-density scenario. (The difference between the minimum and maximum value is the same in both sub-figures.) A similar distribution is observed for 232Th and 40K. Further parameters for both plots: 20 cm thick walls, 100 000 points m−3; 0.2 m × 0.15 m resolution.

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The room centre obtains the largest radiation dose rate, while the corners of the room receive the lowest amount of radiation. Under the scenario of an equal radionuclide distribution, the dose rate at the room centre was 4.39 µSv a−1 per Bq kg−1 and the dose rate for the room average was 4.49 µSv a−1 per Bq kg−1. For the scenario, where only the floor and ceiling contain radionuclides (but in sum the same total amount of radionuclides as in the first scenario), the dose rate at the room centre increased to 4.97 µSv a−1 per Bq kg−1 while it reduced for the room average to 3.9 µSv a−1 per Bq kg−1. Thus, from a point of view where the room average is a measure for radiation evaluation, it would be advisable to prefer the redistribution of radionuclides into the floor and ceiling.

Especially, as the places in the room closer to the walls, where beds and couches are often located, receive less radiation, the redistributed radionuclide scenario appears safer from a radiation protection point of view. For office buildings, where work places and desks are often placed closer to the room centre this statement should be treated with care. However, it is possible to minimise the radiation close to the room centre by reducing the density of the construction material (compare figures 10(a) and (b)). This reduction at the room centre is even more effective than for locations close to the walls and within the room edges. In the example (figures 10(a) and (b)), where densities of 1175 and 2350 kg m−3 are used, the dose rate reduction close to the long walls is about 1.07 µSv a−1 per Bq kg−1, while it is about 1.35 µSv a−1 per Bq kg−1 for the room centre. Thus, the increase of the annual effective dose rate at the room centre under the redistribution of radionuclides towards floor and ceiling could be even more effectively countered than for locations closer to the walls. Under the current trend towards a more lightweight concrete based construction of dwellings this appears to be a favourable characteristic.

Of course, this thought experiment cannot be translated into a real world application as it is practically impossible to have a radionuclide-free material, but it exemplifies how a dose rate reduction can be obtained by an advantageous use of different building materials. According to the thought experiment it is appropriate to use the material with the lower radionuclide activity concentration for the long and short walls and preferentially use that material with a higher activity concentration for floor and ceiling. As floors in many buildings are or will be equipped with a second floor layer, e.g. screed, material for floor heating, tiles, which provide some additional shielding, the annual effective dose rate from the floor will be even more reduced.

However, the present model version does not incorporate this second layer. So a more detailed elaboration on this issue is not possible at the moment. Equally, with the present model version it is not yet possible to calculate the annual effective dose rate distribution in all three dimensions. Currently, the area, where the annual effective dose rate can be calculated is fixed at the half room height. Thus, there is some potential to improve the model. Especially as with the redistribution of the radionuclides towards the floor and ceiling a higher annual effective dose rate can be expected closer to the floor. The magnitude of this increase should be investigated further, ideally in connection with the application of a second radionuclide depleted shielding layer, before a final statement about the appropriateness of a radionuclide redistribution in the wall material can be given.