Redistribution of radionuclides in wall material and its effects on the room dose rate

Here we investigate the annual effective dose rate obtained from gamma radiation emitted from radionuclides in construction materials in a model room with fixed dimensions. The dose rate is calculated on the whole room area at half the room height. We focus our analyses on a comparison of the annual effective dose rate between the room centre and the room average at half the room height and provide wall-wise quadratic index equations for both. We find that the annual effective dose rate based on the room average is larger than for the room centre due to increased annual effective dose rates for positions in the room closer to the walls. Furthermore, we evaluate the annual effective dose rate under a non-equal distribution of radionuclides in the three wall types (floor and ceiling, long walls, short walls). When considering the room average of the annual effective dose rate, our analysis indicates that it appears advantageous to use construction materials with a higher radionuclide activity concentration for floor and ceiling and the material with a lower radionuclide content for long and short walls, if there is a choice in the construction process.


Introduction
Depending on their radionuclide concentration, building materials can emit a remarkable amount of gamma radiation and thus can contribute significant dose rates to the radiation balance of the population. A first evaluation for the radiation exposure in dwelling rooms was obtained via computer codes describing the emission of gamma rays from building materials. Those modelling approaches were conducted to evaluate the received annual dose rate due to the existence of radionuclides of the 238 U and 232 Th decay chains and 40 K in building materials. Those radionuclides represent the main sources for gamma radiation (e.g. Koblinger 1978, Stranden 1979, Mustonen 1984, Markkanen 1995. Many, more recent gamma radiation modelling studies for dwelling rooms are still based on these early publications and focus on the sensitivity of the dose rate with respect to several room and building parameters such as density of the wall material or wall thickness and room dimensions (e.g. Risica et al 2001, Nuccetelli et al 2015, Croymans et al 2018, while using more updated gamma ray emission probabilities and photon energies as well as constants for concrete build-up factors and attenuation coefficients.
Comparison of model results with measurements are rare due to the lack of gamma ray measurements in rooms but would help to verify those calculations. Only a relatively small number of studies measured gamma radiation dose rates in rooms and compared them to model results. Those studies mostly found a good agreement between model results and measurements (e.g. Risica et al 2001, Anjos et al 2011, Deng et al 2014. Room models for the evaluation of the gamma radiation exposure are also the standard tool for the dose rate assessment of emitted gamma radiation from various kinds of construction products with individual radionuclide activity concentrations (European Commission 1999, CEN/TR 17113 2017. Those models lay the foundation for index values (Markkanen 1995) and non-linear fitting equations (Nuccetelli et al 2015, CEN/TR 17113 2017 to rapidly estimate an upper limit of the radiation exposure expected in a dwelling without the need to apply a room model.
In most studies the radionuclides were assumed to be equally distributed in long and short walls as well as in floor and ceiling (e.g. Mustonen 1984, Risica et al 2001, Righi et al 2016, Croymans et al 2018. However, some studies even started to evaluate the annual effective dose rate in the centre of a room with respect to different radionuclide concentration in individual (pairs of) walls (Nuccetelli et al 2012). In the present study we also focus on a redistribution of radionuclides as it is often possible in building processes to switch between concretes or building materials with a different radionuclide composition for walls as well as for floor and ceiling. Especially as by-products of NORM-processing industries like metallurgical slags, fly and bottom ashes, phosphogypsum or red mud are investigated to use as complementary materials in cement matrices (Nuccetelli et al 2015, Snellings 2016, Adesanya et al 2020 to obtain a more closed recycling economy, the ability to account for different activity concentrations to predict and optimise the resulting dose rate becomes essential. However, applying different activity concentrations for individual (pairs of) walls or floor and ceiling will result in a change of the annual effective dose rate. In addition the annual effective dose rate with respect to various locations within the room will behave differently compared to the often analysed dose rate at the room centre point. Here, we investigate the dose rates for each point of the room area at the half room height and compare them with the dose rates received at the room centre for homogeneous but wall-individual radionuclide activity concentrations. After the description of the room model and calculation strategy (section 2) we present a thorough model sensitivity analysis as well as an evaluation of the contribution of individual walls at different points in the room to the annual effective dose rate (section 3). In the last section (section 4) we evaluate, if the annual effective dose rate calculated for the room center is a good estimate for the received dose rate elsewhere within the room. Finally we also discuss the effects and consequences of wall materials consisting of different radionuclide concentrations.

Methods
For our study we follow the approach of Mustonen (1984), which itself is a modification of the model used by Stranden (1979) in a self written python-code. We evaluate the dose rate obtained from building materials within the EN 16 516 model room (length × width × height: 4 × 3 × 2.5 m 3 ). For this study, doors and windows are not accounted for. In our approach we focus on the dose rate at all points on a plane at half the room height. The absorbed rate in air, R [Gy/h], at any point, P, on that plane due to the emission of gamma rays over all gamma lines i emitted by 232 Th, 226 Ra (as its parent nuclides are not significant gamma emitters) and their progeny (assuming activity equilibrium within the decay chains) as well as 40 K can be described by: where the emitted gamma lines are defined by their energies, E i , intensities, I i , absorption in air, ( µ en,i ρ ), and their attenuation coefficient, µ i , in the wall material. The latter three parameters depend on the energies of the gamma lines i. The wall material characteristics are described by its density, ρ W , and activity concentration, C W . The distances l and s are defined in equations (3) and (4). The factor f has its origin in the conversion of units with f = 10 −6 cm 3 m −3 * 1000 g kg −1 * 1.6022e −13 J MeV −1 * 3600 s h −1 = 5.77e −13 cm 3 g J s m 3 kg MeV h (CEN/TR 17113 2017). For our calculations we use a build-up factor, B i , according to the Berger-model (Berger 1957): with the two energy dependent constants C i and D i . For practical assessments sufficient accuracy is provided when using the emission probability weighted average values for all the individual gamma lines of the 226 Ra and 232 Th progenies (Markkanen 1995, Croymans et al 2018. This approach also allows a short calculation time compared to an approach where several hundred emission lines are accounted for. As the 2.615 MeV gamma line of the 232 Th decay chain contributes about 40% this line is used separately as well as the 1.46 MeV 40 K line (table 1). The integral in equation (1) is over the volume, V, of one wall, which is characterized by its dimensions (height, length and thickness-a, b, d, see figure 1), density, ρ W , and activity concentration, C W . The distance l between a point Q (x, y, z) in the wall, where the radiation is emitted, and the radiation receiving point P(x P , y P , z P ) in the room is described by:   (x, y, z) within one wall, where the distance l refer to a point P(xP, yP, zP) within the room. The wall dimensions, height a, length b and thickness d as well as the distance s, which is the length, the radiation will penetrate through the wall are indicated. All points in the wall and within the room are given with respect to the centre of the wall to the side facing the room (a/2, b/2, 0). The grey shaded parallelogram highlights the area at half the room height.
The distance s is this part of l, which is within the wall (figure 1): The absorbed rate in air is calculated per individual wall and summed up afterwards. This strategy allows the wall characteristics to be prescribed in an individual way and to distinguish the contribution to the total dose rate per wall. Technically, we use a Monte-Carlo (MC) approach to solve the integral of equation (1). The MC-approach picks randomly a large number N of points, S(x, y, z) (figure 1), per standard volume (1 m 3 ) of wall material for calculating the integral. It turns out to be computationally faster than solving the volume integral within its given limits in a numerical way. For N = 100 000 points m −3 this strategy is still more than three orders of magnitude faster than solving the volume integral numerically while providing an excellent accuracy and precision (see section 3). We show and discuss the data with respect to the annual effective dose rate, D, which relates to the absorbed rate in air by multiplication with the dose rate conversion factor of 0.7 Sv Gy −1 (UNSCEAR 1993) and a time of 7000 h per year spent in the dwelling (e.g. Markkanen 1995, CEN/TR 17113 2017. As R P (equation (1)) and D are linear with respect to the radionuclide activity concentration, C W , we prefer to refer to all results relative to the unit activity concentration of 1 Bq kg −1 for 226 Ra, 232 Th and 40 K where not explicitly stated otherwise.

Results
Gamma emissions from building material in dwellings and their effect on the annual effective dose rate in the room centre are already well investigated and many aspects are well known. The calculated dose rate at the room centre calculated with our approach is identical to the approach used in Markkanen (1995) and CEN/TR 17113 (2017) as we use the same values for all parameters as in those two studies. Here we focus only on aspects which are less frequently analysed, such as the annual effective dose rate contribution of individual points in the wall, which is possible due to the MC approach chosen to solve the volume integral of equation (1) (section 3.1). In a next step we test how many randomly chosen points per wall are necessary to obtain accurate and precise results for the dose rate. We do this exercise for the annual effective dose rate at the midpoint and for the room average (section 3.2). We also analyse the potential differences with respect to the stability of the results within individual locations in the room, i.e. room centre vs locations closer to the walls. Afterwards results of the model will be shown with respect to the annual effective dose rate distribution within the room (section 3.3) for the three different pairs of the wall (floor and ceiling, long walls, short walls). This also includes a comparison of the annual effective dose rate at the room centre and the room average. In a final step we provide a wall-wise quadratic index equation for the room average and room centre (section 3.4), which allows the handling of building materials for which individual radionuclide activity concentrations are specified.

Dose rate distribution for individual walls
In the 5 × 4 × 2.5 m 3 model room with 20 cm thick walls and a density of 2350 kg m −3 , the highest annual effective dose rate with respect to the room centre is obtained from the floor and ceiling (figure 2). The shape of the dose rate contribution of the three kinds of walls is the same, but as the long and short walls have a larger distance to the room centre than floor and ceiling their contribution to the annual effective dose rate is smaller. The small walls contribute the smallest amount to the dose rate as they have the largest distance to the room centre. The upper end of the spanned area reflects the annual effective dose rate obtained from points at the wall-room surface, following a 1/l 2 law. For those points no shielding or build-up occurs by wall material.
The values on the left in figure 2 reflect the centre of the respective wall ('centre of wall' labelled double-headed arrow; figure 2) at different depth within the wall. The shielding effect including the build-up effect is reflected by the various values of the annual effective dose rate along this arrow. The lowest values of each wall reflect a position at the opposite side of the wall with respect to the considered room ('back of wall' labelled double-headed arrow in figure 2). In contrast, the values with the highest annual effective dose rate along different distances from the room centre refer to the dose rate contributed by those parts of the wall, which are facing the room ('front of wall' labelled double-headed arrow, figure 2). Due to the largest distance to the room center, the corners of the room contribute only little to the annual effective dose rate ('corners of wall' labelled double-headed arrow, figure 2).
While the centre parts of the floor and ceiling are responsible for highest dose rates, parts more distant from the centre contribute only little radiation, even less than those of the long and short walls at the same distance. As floor and ceiling have the largest dimensions in the considered model room, radiation emitted from the distant parts potentially has to travel the longest distance through wall material and thus suffer from the strongest attenuation.

Sensitivity analyses
Meaningful interpretation of the annual effective dose rates in a room are only possible if stability of the results is ensured. However, as we are using an MC-type approach to solve the problem, we have to investigate the stability by varying the number of points in a wall. As all three types of a wall (floor and ceiling, long walls and short walls) have a different volume, it is not suitable to distribute the same number of points in each wall. Such a strategy would favour a larger uncertainty for larger walls. Hence, we consider here the more appropriate approach and analyse the results with respect to the number of points per standard volume (1 m 3 ) of wall material. This guarantees an equal treatment for all types of walls. For a first test we analyse the stability of the annual effective dose rate with respect to the number of points m −3 . For this purpose we produce 100 data sets, each with [100,200,500,1000,2000,5000, 10 000, 20 000, 50 000, 100 000] randomly chosen data points m −3 and analyse them with respect to the room centre and room average (figure 3). We only show data for gamma radiation due to 226 Ra. The main results for 232 Th and 40 K are similar.
For a small number of points m −3 individual calculations show a large spread in the results. If only 100 points are used for the calculations, the results for the room centre varies with a the relative standard deviation of about 3% around the mean value of of 4.394 µSv a −1 per Bq kg −1 (figure 3(b)). Some realisations even obtain values as low as 4.02 µSv a −1 per Bq kg −1 and as high as 4.71 µSv a −1 per Bq kg −1 . The more points m −3 are chosen to calculate the annual effective dose rate the smaller the data spread and relative standard deviation. For 100 000 points m −3 the standard deviation is below 0.1% with a mean value of 4.39 µSv a −1 per Bq kg −1 .
The observed general trend is similar for the room centre and average, however, the results for the room average are more stable than the results for the room centre (compare figures 3(a) and (b)). The relative standard deviation of the experiments with 100 points m −3 is 0.79% and smaller than 0.03% for the experiment with 100 000 random points m −3 . The average value for the annual effective dose rate of the room is 4.473 µSv a −1 per Bq kg −1 and thus somewhat higher than the dose rate at the room centre.
As we are dealing with dose rates at individual places in a room (at a height of 1.25 m), it is of interest to learn also about the stability of the annual effective dose rate with respect to the distance from a wall and how it relates to the stability of the annual effective dose rate at the room centre ( figure 3(b)), which is the point in the room with the largest distance to all walls. We analysed five different places with respect to the distance from the short walls relative to the short wall centre (figure 4). Again, we varied the number of points m −3 of wall material and repeated each experiment 100 times to evaluate the expected precision of individual experiments.
The sensitivity analysis reveals that the closer the point of interest to the walls the larger is the standard deviation. This is valid for any chosen number of points m −3 . However, the standard deviation is smaller when using a higher number of randomly chosen points m −3 . At 100 000 points the standard deviation is despite its increase towards the walls still in the ‰-range for all points in the room. We also analysed the  annual effective dose rate at the wall surface, but received very high standard deviations in relation to those further away from the wall and hence, do not show them in figure 4. Based on that, we do not include calculated annual effective dose rates at the wall for all following analyses.
For the analysis of the distribution of the gamma radiation exposure in the room (section 3.3) it is necessary to calculate the annual effective dose rate on a grid. Thus, in a final step in this sensitivity analysis we investigated the impact of the chosen grid density in the room on the average annual effective dose rate within the room. We applied a grid density starting from a low resolution (0.8 m × 0.6 m, 16 grid points) successively increasing the number of points towards a high grid point density (0.02 m × 0.015 m, 39 601 grid points) and calculate the average dose rate in the room with respect to 226 Ra only (figure 5). For each run, we also extracted the dose rate at the room centre. At those resolutions, where the grid does not have a point at the exact room centre, we average the four grid points nearest to the centre.
We found an increasing annual effective dose rate with increasing grid density within the room (figure 5, triangles). While the average annual effective dose rate on all grid points in the room is only somewhat higher than the annual effective dose rate at the room centre (figure 5, crosses), the annual effective dose rate with the highest grid point density is about 0.1 µSv a −1 per Bq kg −1 higher than the annual effective dose rate at the centre point and seems to run against an upper limit. The annual effective dose rate at the room centre is relatively constant and does not vary above the expected range (see figure 3).
The step wise behaviour of the average room dose rate for smaller values is an artefact due to the distribution of the grid points. Especially for those experiments with a relatively low resolution it is important how the grid is placed within the room. For example there is no exact centre point for the first and fourth experiment, leading to a disturbance of the general trend of an increase in the average annual effective dose rate of the room.

Dose rate distribution in the room
The annual effective dose rate at various locations in the model room is an additive feature with respect to the three pairs of the opposite walls. We visualise this for the model room and with a grid resolution of 0.2 m × 0.15 m (figure 6). This becomes especially important, if walls have different individual radionuclide activity concentrations.
The annual effective dose rate is symmetrical to the room centre for each considered pair of walls and for when all walls are considered to be subject of increased radionuclide activity concentration. The effective annual dose rate obtained from the pairs of floor and ceiling, long walls and short walls (figures 6(a)-(c)) show very smooth shapes in the model room with relatively large gradients. At the room centre the contribution from the floor and ceiling are largest, while the contribution from the short walls is smallest. This is related to the distance of the room centre to the according wall pairs. The closer the distance to a radionuclide loaded wall the higher the dose rate. However, the dose rate obtained from the floor and ceiling is decreased for a position located closer to the walls.
When all walls as well as floor and ceiling contain radionuclides, the total annual effective dose rate in the room is the sum of the individual contributions. This is valid for each position in the room ( figure 6(d)). The gradients due to the gamma emissions of the individual pairs of walls nearly cancel each other, when added. The increasing trend towards the walls under a scenario, where all walls (including floor and ceiling) have the identical radionuclide activity concentration, is much smaller compared to a scenario when considering individual pairs of opposite walls only. The range of the annual effective dose rate at various places in the model room is relatively small, which is the reason that the annual effective dose rate distribution in the room does not look fully smooth and symmetrical. However, when considering the variations in the results for individual points due to the applied MC-approach (section 3.2, figures 3 and 4) this is an expected feature. The annual effective dose rate at the midpoint is smallest within the room when all six walls have the same radionuclide activity concentration (figure 6(d)). Consequently, the room average of the annual effective dose rate must be higher. Here, we investigate the difference between the annual effective dose rate at the room centre vs. the average within the room ( figure 7(a)). For this analysis we use the concept of 'mass per unit area' , where the density and wall thickness are multiplicatively coupled (e.g. Nuccetelli et al 2015, CEN/TR 17113 2017). This coupled parameter pair lays the background for the 'quadratic index formula' (Nuccetelli et al 2015, CEN/TR 17113 2017 and was previously provided for a typical wall thickness of 20 cm. For other wall thicknesses the annual effective dose rates vary slightly (figure 7(b)) and are largest in the range between ρd = 200−500 kg m −2 .
As observed elsewhere the dose rate contribution per Bq kg −1 of 40 K is smallest, followed by 226 Ra and 232 Th for all masses per unit area. The annual effective dose rate is increasing with increasing mass per unit area. The increase for lower masses per unit area (<200 kg m −2 ) might be approximated in a linear way, while for a higher mass per unit area the annual effective dose rate approaches a limit (figure 7(a)). As seen in figure 5 the annual effective dose rate at the room centre is smaller than that for the room average. This can be especially well observed for values <500 kg m −2 . For masses per unit area higher than this value both, centre and average are approximately the same. The same characteristics are observed for various wall thicknesses ( figure 7(b)). Furthermore, thinner walls appear to emit more gamma radiation under a constant mass per unit area, i.e a larger wall material density. This should be kept in mind, when considering any 'quadratic index formula' with respect to the product of density and wall thickness.

Wall-wise modular index formula
Based on the results of the sensitivity analysis (section 3.2) and on the considerations for the distribution of the annual effective dose rate in a room (section 3.3), it is now possible to provide an index formula, which enables an easy estimation of the annual effective dose rate under consideration of different radionuclide activity concentrations in individual pairs of walls. As the annual effective dose rate of the room average is generally somewhat higher than that at the room centre, we provide an individual index formula for both aspects. For consistency reasons with respect to previous studies we focus on establishing the index formulas in a quadratic form (Nuccetelli et al 2015, CEN/TR 17113 2017. We fitted the results for masses per unit area between 50 and 500 kg m −2 using a wall thickness of 20 cm for the three pairs of walls ( figure 8).
Between 50 and 500 kg m −2 quadratic equations approximate the calculated annual effective dose rates well (figure 8). For larger masses per unit area those type of fitting curves are not valid any more. A hyperbolic tangent will fit better at higher masses per unit area and should be considered for application under those circumstances. But for usual dwellings 500 kg m −2 is at the upper end for construction materials and wall thickness and thus in most cases sufficient.
The construction of dwellings can rely on different building materials with different radionuclide activity concentration, which are used for the walls or floor and ceiling. Thus, we provide here an quadratic index equation for pair-wise walls. The sum of those pair-wise wall contributions provides an estimate about the total effective annual dose rate in a room according to the following equations: (−0.0046(ρd) 2 + 4.6bρd + 55c) * C Ra (−0.0053(ρd) 2 + 5.3ρd + 60) * C Th (−0.000 34(ρd) 2 + 0.37ρd + 3.7) * C K   short walls * 10 −6 +   (−0.0065(ρd) 2 + 6.5ρd + 100) * C Ra (−0.0072(ρd) 2 + 7.4ρd + 108) * C Th (−0.000 49(ρd) 2 + 0.52ρd + 6.8) * C K   long walls * 10 −6 +   (−0.0061(ρd) 2 + 6.1bρd + 53c) * C Ra (−0.0067(ρd) 2 + 7ρd + 58) * C Th (−0.000 45(ρd) 2 + 0.48ρd + 3.5) * C K   floor + ceilling * 10 −6 − D bg (6) where D average and D centre are provided in mSv a −1 if the activity concentrations, C, of the building materials are given in Bq kg −1 . D bg is the value for the local natural background dose rate. If not known, D bg might be set to 0.29 mSv a −1 , which is the population weighted average for Europe (UNSCEAR 1993). Equations (5) and (6) assume that the local background is completely shielded out by the wall material. For the model room, this is a valid assumption, but if openings, such as windows and doors, must be considered, this would not be the case. Note, that the obtained values for the fitted parameters depend on the chosen range of the mass per unit area and the number as well as distribution of points used for the fit. Here, we report parameters, derived with calculated annual effective dose rates in the interval between approximately 63 and 505 kg m −2 with a constant 63 kg m −2 spacing (figure 8).

Sensitivity
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 m 3 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.

Index formula
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.

Dose rate distribution in the room
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)).
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 226 Ra, 232 Th and 40 Ka 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)).
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.

Conclusion
We have established a room model for calculation of the annual effective dose rate due to gamma emissions of radionuclides in building materials at any point on an area at half the room height. All analyses were performed for the EN 16 516 model room (CEN/TR 17113 2017). However for user specific questions it is possible to easily change some parameters within the python scripted code like the dimensions of the room, pair-wise wall-specific wall material density, wall thickness and activity concentrations (Fohlmeister and Hoffman 2023). After a sensitivity analysis, where technical model parameters were investigated, we compared the annual effective dose rate obtained at the room centre with other places in the room and the room average.
Closer to the walls, the annual effective dose rate is higher than in the room centre and thus the average obtained dose rate in the room is also higher than at the room centre. This might have consequences for wall materials with dose rates estimated for the room centre, which are close to the 1 mSv a −1 threshold as the room average might exceed this value. For an easy estimation of the annual effective dose rate at the room centre or for the room average, we provided wall specific quadratic index equations, where radionuclide activity concentrations, wall thickness and wall material density can be adjusted wall-wise.
The basic idea of the shown model experiments was to guide to possibilities to minimise and optimise the radiation exposure in cases for which some wall materials (must) have a high radionuclide activity concentration. Finally, we discussed the effect of a potential non-equal distribution of radionuclide concentrations between individual pairs of walls. In this thought experiment, we focussed on the case in which long and short walls were kept radionuclide free, while floor and ceiling had to accommodate an accordingly higher activity concentration. Such a redistribution would lead to an increase in the annual effective dose rate at the room centre but to a lower one for the room average and places close to the wall.
Especially in the light of an often used second layer of building materials for floors in dwellings (screed, tile), this is an interesting aspect. If the second layer could be mostly made free of radionuclides it would contribute some shielding and further reduce the radiation from radionuclides in the floor, which would effectively reduce the annual effective dose rate for both the room centre and the room average. However, at present, the code is not able to handle a second building material layer. But it appears interesting to further investigate the strategy for redistributing the radionuclide activity concentration in different walls from the radiation protection point of view once the second layer is implemented.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https:// figshare.com/s/b1be23ca22072649c1b9.