Serpent 2 simulations of the historic Haigerloch B8 nuclear reactor of 1945

1.1 Historical background

The Italian Enrico Fermi tried to create new artificial (transuranic) elements by bombarding uranium with slow neutrons. When repeating the experiments in Berlin in December 1938, Otto Hahn and Fritz Strassmann discovered nuclear fission, while Lise Meitner, being Jewish, who had to flee from Nazi Germany from Berlin to Sweden in July 1938, provided the theoretical calculations and the explanation of the unstable uranium nucleus after the capture of an additional neutron. Already four months later, Paul Harteck, the director of the Institute of Physical Chemistry at the Universität Hamburg, and his assistant Wilhelm Groth wrote a letter dated April 24, 1939 to the Heereswaffenamt (German Army Ordnance Office). They alerted the military of a potential of a chain reaction and the principle possibility of an “explosive”. The Army Ordnance Office in the Reich Ministry of War was in the Third Reich responsible for the development of new weapons. Several publications on the potential practical application of the nuclear fission arrived there in the sequel. Another four months later in August 1939, Siegfried Flügge wrote a public newspaper article in the Deutsche Allgemeine Zeitung entitled with “The utilization of nuclear energy” (orig. “Die Ausnutzung der Atomenergie”). Directly afterwards, Dr. Kurt Diebner, physicist and the person in charge in the German Army Ordnance Office, invited to the first Uranium Conference in Berlin on September 16, 1939. The “uranium club” was founded.

The result of this first uranium conference was that all measures for clearly answering the question of the technical feasibility of nuclear energy production should be taken. Werner Heisenberg should first work out the theory of the nuclear reactor. During a follow-up conference in October 1939, Heisenberg was asked whether it would be possible with the discovery of Otto Hahn to produce nuclear weapons. Heisenberg basically answered that from the physics point of view, it would be conceivable, but the technical and economic problems he rated as serious enough that a production of the bomb would not be possible within the next few years. After the conferences, the experimental work aiming at the construction of a nuclear reactor (called uranium machine) was distributed over different locations, where different geometrical arrangements of the set-up were realized and tested.

At the Universität Leipzig, the sequence L1 to L4 of experiments was realized by Werner Heisenberg and Robert Döpel (Heisenberg and Wirtz 1947). They used a spherical arrangement of fuel and heavy water as a moderator. A positive neutron multiplicity could be achieved for the first time with the experiment L4 in June 1942, indicating the realization of a short-time chain reaction. A similar experiment was realized in parallel by Fermi in Chicago only a few weeks later. During the same month, a reactor exploded in Leipzig because of an unexpected chemical reaction of uranium metal with water. Kurt Diebner was the first to use an array of uranium cubes in the sequence G1 to G3 at the Army Research Center in Berlin-Gottow. This brought improved results as compared to the arrangement with uranium plates used before that (Heisenberg and Wirtz 1947).

In the Kaiser-Wilhelm-Institute of Physics (KWI) in Berlin-Dahlem, Werner Heisenberg, Karl Wirtz, Carl-Friedrich von Weizsäcker realized the series B1 to B7 of experiments (Heisenberg and Wirtz 1947). The KWI worked with uranium plates and heavy water as a moderator. Due to air strikes in Berlin, the research work was transferred to a small town in the South-West of Germany. The institute was moved to Hechingen, while an experimental research site was started in a small cave in the town of Haigerloch located a few kilometers away from Hechingen. A former beer cellar was rented by the Kaiser-Wilhelm-Gesellschaft and the experimental set-up started in late 1944. Together with Fritz Bopp and Emil Fischer, the last experiment B8 was realized in April 1945. After the transfer from Berlin, this group then also used an arrangement with uranium cubes as suggested by Diebner for the first time.

The transfer from Berlin to Hechingen and Haigerloch was initiated by Walter Gerlach, in charge of nuclear physics at the Reichsforschungsrat (German Research Council) acting under the Ministry of Armaments and War Production. He had been appointed successor of his Ph.D. advisor Friedrich Paschen in 1925 as a professor of physics at the Universität Tübingen (Hagmann 2023), about 40 km away from Haigerloch. During his time in Tübingen, Gerlach probably traveled the surroundings, thereby visiting the idylic town of Haigerloch situated in a narrow limestone valley, such that it could be safe from air attacks.

In this work, we re-address the historic Haigerloch nuclear reactor experiment B8 from the point of view of modern simulation tools of neutron dynamics. As is known, it never reached criticality, but the measured neutron multiplicity reached a maximum value of M = 6.7 of all experiments carried out during World War II in Nazi Germany. The B8 reactor was realized as a cylindric device, with an outer vessel placed in a concrete pit in the floor of an underground cave which had been used as a beer cellar in Haigerloch. The outer vessel contained a layer of graphite blocks used as a neutron reflector. Inside, an inner cylinder vessel was placed that contained heavy water as the moderator. The fuel was shaped in the form of cubes that hang from the lid of the closed vessel along aluminum wires. An external neutron source was inserted through a central chimney. In addition, radially placed chimneys were used to insert glass tubes as measurement probes. The major advancement compared to the previous experiments until B8 was the use of a regular arrangement of the uranium fuel in the form of cubes of natural uranium metal, suspended in the inner vessel.

1.2 Modern simulation techniques applied to the historic B8 reactor

Modern numerical simulation techniques, especially those based on the Monte Carlo method, have nowadays reached a high level of maturity and are routinely being applied in the design of modern nuclear reactors. Back in the 1940s of the last century, scientists could only dream of such tools. It is therefore quite appealing to analyze historical reactor concepts in view of present knowledge and understanding using such modern tools. The Haigerloch B8 reactor has been studied by numerical techniques before. Grasso et al. (2009) reported results for k_eff and M, its neutron energy-distribution spectrum, and its neutron-flux distribution in its central horizontal and vertical planes for both thermal and fast neutrons using the Monte Carlo neutron transport code MCNP5 (Sweezy et al. 2003). It is a general-purpose, continuous-energy, generalized-geometry, time-dependent, coupled neutron-photon-electron Monte Carlo transport code that was developed at Los Alamos National Laboratory. It is a widely used tool for designing and analyzing nuclear reactors and is an alternative to Serpent 2. Grasso et al. (2009) estimated k_eff = 0.86 based on a simplifying assumption that the uranium-fuel cubes and the aluminum-wire links are treated as a kind of homogenized material, in the sense that they could be modeled as effective cylinders of a phenomenological diameter. The cylinders were assumed to be filled with a uniform and homogeneous mixture of uranium fuel, aluminum, and heavy water, with the materials placed inside each cylinder.

Pešić (2018) used a more accurate modelling of the geometry and a follow-up version MCNP6.1 of the Monte Carlo neutron transport code with the latest ACE type neutron nuclear cross section data at that time. A careful description of essential geometrical details of the arrangement of the uranium cubes and the wire suspensions together with updated material parameters yielded a much larger effective neutron multiplication factor of k_eff = 0.953. In addition, Pešić (2018) could study many different detailed variants of the reactor configurations, taking into account the unknown geometric details or material configurations (alloys, purities, contaminations, etc.).

Park (2022) reported improved MCNP simulations. He used the precise form of the uranium cubes and their historically reported arrangement. In addition, he referred to recent mass spectroscopy measurements of the density of an original uranium cube at the Pacific NW National Lab (PNNL). The measured value amounts to 18.53 g/cm³, which has to be compared by the nominal magnitude of 19.05 g/cm³ of natural uranium as used in the present work and by Pešić (2018). Furthermore, Park (2022) used a historically determined purity of the heavy water obtained in a measurement in 1947 of the Haigerloch heavy water kept at NIST. By this, Park (2022) further refined the value to k_eff = 0.958 in his analysis of the Haigerloch reactor.

1.3 Outline of the present study

In this study we use the contemporary Monte Carlo tool Serpent 2 (Leppänen et al. 2015) to perform simulations of the B8 experiment and to calculate the criticality parameters. We emphasize especially on building a model in Serpent 2 that reconstructs the historical design and material composition of the B8 experiment as close as possible, using data from either historic reports, from recent measured data at the Atomkeller Museum Haigerloch, Germany, or from literature. We therefore present first our best estimate of the design of the Haigerloch B8 reactor as well as of the properties of the used materials. We then compare the results of the effective neutron multiplication factor from our Serpent 2 simulations with the historically reported value of Heisenberg and Wirtz (1947) as well as the results reported from previous work (see above). We find that the historic configuration realized in the B8 experiment is rather close to the optimum given the circumstances of limited resources and equipment. In addition, we have varied key reactor parameters in the simulations in order to understand which of the parameters have a major impact on criticality. We investigate the enrichment of uranium and also analysed the effect of the unknown degree of contamination of the heavy water with light water and with boron. Furthermore, we conducted a systematic parametric study to evaluate the impact of the fuel element spacing on the effective neutron multiplication factor k_eff. While keeping the total volume of heavy water constant, we adjusted the geometry of the core accordingly in order to maintain consistent moderation conditions. Finally, we answer the question by what decent modifications of the reactor set-up under the historic circumstances criticality would have been reachable.

2.1 Concrete cylinder

The reactor as a whole was embedded in a pit in the floor of the Atomkeller cave in Haigerloch. For this, a cylindrically shaped concrete wall and a concrete floor were designed in which the reactor was inserted and which eventually served as a (normal) water pool. The diameter of the original concrete construction was measured in Haigerloch to be 300.0 ± 0.2 cm (Würth WDM 30 laser rangefinder) and the height of the concrete cylinder has been measured as 254.1 ± 0.2 cm.

2.2 Uranium cubes and their suspension

The fuel consisted of cubes of natural uranium with a edge length of 5 × 5 × 5 cm. In agreement with the historic reports, the replica of the device also contained replica cubes (of aluminum) in the same size. In addition, one original uranium cube is still conserved up to present in the Haigerloch Atomkeller museum. From previous experiments, Heisenberg realized that an optimal cube edge length would be 6–7 cm (Heisenberg and Wirtz 1947). Since a larger amount of cubes with edge length of 5 cm were left over from the Gottow experiments, those cubes were used and complemented by further produced cubes of 5 cm edge length (Heisenberg and Wirtz 1947).

The uranium cubes themselves were not clad, but coated with a polystyrene emulsion to protect them from chemical reactions with air or water and to minimize radiation exposure for the operators (Grasso et al. 2009), but this was not taken into consideration in the simulation. 664 cubes were arranged vertically in a chain along aluminum wires such that the (end-to-end) distance between the cubes amounts to an average value in the replica of 6.85 cm. To obtain this value, we measured the distance between several randomly sampled cubes in the replica. This is consistent with the reported center-to-center distance of the cubes of 12 cm. In total, there are 40 chains carrying nine uranium cubes and 38 chains carrying eight cubes. The 78 chains are arranged in alternating order in four concentric rings (counted from the outside inwards with 32 chains in the outermost ring, and then 23 chains, 15 chains, and eight chains (innermost ring). The diameter of the aluminum suspension wire forming the chain was measured in the replica at four different random samples, yielding a mean value of 3.35 ± 0.05 mm (Caliper). In the historic report, Heisenberg and Wirtz (1947) note that neither the cube dimensions nor their distances were optimal, but had to be taken as given due to the general circumstances in early 1945.

2.3 Reactor core and inner magnesium vessel

The dimensions of the reactor core cylinder vessel have been measured at the replica. We have found an inner diameter of the magnesium cylinder of 115.6 ± 0.20 cm (Würth WDM 30 laser rangefinder) and have used this in the simulations. The number slightly deviates from the number reported by Heisenberg and Wirtz (1947) which is 124 cm. The total height of the magnesium cylinder is 162.0 ± 0.20 cm. This magnitude is in decent agreement with the historic one reported as 164 cm (Heisenberg and Wirtz 1947). We have measured a wall thickness of the replica of 0.368 ± 0.005 cm (Caliper) which also coincides with the magnitude of the wall thickness of 0.3 cm reported by Heisenberg and Wirtz (1947).

2.4 Graphite reflector

The thickness of the graphite layer as the reflector is well documented (Heisenberg and Wirtz 1947). We use a layer thickness of 40.0 cm at the cylinder walls and at the bottom of the cylinder. We note that the replica in the museum shows a graphite layer thickness of 30.6 cm, which does not correspond to the value published in the original historic report. The thickness of the graphite layer in the lid is in reality slightly larger, but for simplicity, we use the same value of 40.0 cm also for the lid (the replica in the museum is somewhat irregular due to construction and yields a value of 47.8 cm which we believe is not realistic). Furthermore, the graphite had been used in the form of rectangular blocks with the dimension 5 × 10 × 50 cm (Heisenberg and Wirtz 1947) and arranged in the cylinder layer. Due to these circumstances, small gaps filled with air remained, but are considered as irrelevant for the present analysis.

2.5 Aluminum outer vessel

The wall thickness of the aluminum tank could be measured directly at the original device which is still conserved at the Haigerloch museum as the only original piece. It has been destroyed by explosives in May 1945 and left back by the military ALSOS mission. Despite being strongly deformed, its wall thickness can still be measured. We have measured the wall thickness of the outer aluminum vessel as 0.783 ± 0.005 cm (Caliper). This value slightly deviates from the number of 0.5 cm reported by Heisenberg and Wirtz (1947). We believe that this difference is of minor relevance.

Due to the disagreement of the thickness of the graphite layer between the published value and the value measured at the replica, we believe that the value of the diameter of the outer aluminum tank of the replica in the museum is not reliable. Instead, we have calculated the inner diameter of the aluminum tank according to twice the wall thickness plus twice the graphite layer thickness plus the diameter of the magnesium core cylinder. We find 199.1 cm for the diameter of the outer aluminum tank. This value is also slightly in disagreement with the number of 210 cm for the diameter reported by Heisenberg and Wirtz (1947).

The height of the aluminum outer tank is taken from the literature as 210 cm, which is consistent with the sum of the height of the magnesium cylinder and twice the thickness of the graphite layer (one in the bottom and one in the lid) which amounts to 212 cm.

2.6 Neglected components and simplifications

In order to simplify the Serpent 2 simulations, we neglect the metal enforcement construction inside the top part of the reactor which carries the suspended chains with the uranium cubes. The enforcement construction needed to be used in order to carry the total weight of the suspended uranium cubes of about 1.5 tons. Then, we use a planar geometry of the lid of the aluminum vessel and neglect the vault present in the original form designed for mechanical stability reasons. Furthermore, we did not take into account the fact that the uranium cubes were suspended in an intertwined pair of aluminum wires, but used only one wire in the simulations. In addition, we did not include the wooden bars at the bottom of the concrete pit on which the outer aluminum vessel was placed. Finally, we did not take into account the gaps in the inner vessel and the graphite reflector used in the original construction to insert six cylindrically shaped tubes of the neutron detecting probes. Likewise, the chimney in the center of the lid which was used to insert the initial neutron source and to gradually fill the heavy water into the inner vessel is not part of the arrangement used in the simulations. In agreement with Pešić (2018), we believe that these simplications are adequate and do not affect the overall result in a substantial manner. The geometrical parameters of the reactor components used in the simulations are summarized in Table 1 and can also be visually traced in the schematic representation shown in Figure 1.

3.1 Natural uranium

The fuel consisted of the form of natural metallic uranium (nUm) that has a typical density of 19.05 g/cm³ and contained impurities measured as the equivalent boron content (EBC). The EBC value varied between 1 ppm and 20 ppm, with 5 ppm EBC assumed as the average (Pešić 2018).

3.2 Heavy water D₂O

D₂O was used as a moderator. Values of 95 % D₂O and 5 % H₂O were used for the reference calculation of the nominal design. However, due to historical uncertainty, the purity of the heavy water is varied between 89 % and 99 % D₂O in the sensitivity calculations. The density and atomic density are determined at a temperature of 293.15 K (Pešić 2018).

3.3 Aluminum alloy 1100

Aluminum alloy 1100, also known as Al 1100, is a commercially pure aluminum alloy with an aluminum content of at least 99.0 %. A density of 2.71 g/cm³ is specified in this work (Alloys International Inc. 2024). It is characterised by excellent corrosion resistance, good formability and high thermal and electrical conductivity. This alloy is frequently used in heat exchangers, chemical equipment and food processing equipment. In the Haigerloch B8 reactor, it was used as the material for the outer aluminum vessel (Pešić 2018). The composition of Al 1100 includes (United Aluminum 2024): aluminum (Al): min. 99.0 %, iron (Fe): max. 0.95 %, silicon (Si): max. 0.95 % (combined with iron), copper (Cu): 0.05–0.20 %, zinc (Zn): max. 0.10 %, manganese (Mn): max. 0.05 %, other elements (each): max. 0.05 %, and other elements (total): max. 0.15 %.

3.4 Aluminum alloy 5025

For the suspension of the uranium cubes in the reactor core, wires made of aluminum alloy 5025 was used (Pešić 2018), which consists of 95 % aluminum and 5 % magnesium (Aluminum Association 2024). The density is given as 2.64 g/cm³. This alloy offers a good mix of mechanical strength and malleability.

3.5 Magnesium alloy AZ91

In the Haigerloch B8 reactor, AZ91 magnesium alloy was used for the inner magnesium vessel (Pešić 2018). This alloy is widely used and consists mainly of magnesium (approximately 90 %), aluminum (8.5–9.5 %) and zinc (0.5–1.0 %) (Magnesium Elektron 2024). A density of 1.83 g/cm³ is specified in the source. It is known for its excellent castability, high strength and good corrosion resistance. It is frequently used in the automotive industry, for electronic housings and in the aerospace industry.

3.6 Graphite

For the reflector material, a detailed analysis of the graphite used is difficult. A detailed discussion is presented by Pešić (2018). Different types of graphite were used in the reactor (Pešić 2018). Pure graphite has excellent neutron moderating properties, while graphite with 1 ppm EBC corresponds to nuclear-grade graphite, which has low impurities and is therefore ideal for nuclear applications (Pešić 2018). Natural graphite has an impurity density of 4.212 ppm EBC. This graphite contains natural impurities that can slightly affect its moderation properties (Pešić 2018). Eventually, we have used pure graphite (or, graphitic carbon) with a density of 1.7 g/cm³, as described by Heisenberg and Wirtz (1947). We did not explicitly account for certain air gaps between the prebuilt graphite blocks used. They have emerged due to the geometric arrangement of the blocks in the outer wall of the reactor.

6.1 Effect of fuel arrangement: variation of uranium cube size and distance

It is well known that the criticality of a reactor depends strongly on the geometrical arrangement of the fuel and the moderator, since this has, together with thermal effects, an important impact on moderation and capture/fission probabilities. In general, increasing the size of a fuel element increases the probabilities for fissions but also for parasitic captures by non-fissile material. Increasing the space between the fuel elements (filled with moderator) improves the thermalization of the neutrons and thus the probability of fissions, but also increases the risk of captures (e.g. due to impurities). Thus, there should exist optimum values or ranges for the fuel element dimensions and spacing. As mentioned above, already, Heisenberg and Wirtz (1947) noted that a larger size of the fuel cubes with an edge length between 6 and 7 cm should be optimal for criticality. However, due to practical reasons (5 cm cubes left over from the Gottow experiments), the edge length of the cubes of the historical B8 reactor was chosen as 5 cm. Also, their distance in both horizontal and vertical directions of 14 cm was rather determined by the available amount of heavy water than by optimality criteria. In modern days, the parameters of the geometrical arrangement of the core region can be easily varied in simulations in order to study the effect on the criticality. This was also done by Pešić (2018), though he used specifically for his parametric calculations on the fuel element size and the spacing a simplified model. This was comprised only of a representative fuel cube within a cube filled with heavy water (of varying purity) with an edge length corresponding to the quadratic pitch between elements. Selective boundary conditions were applied for this representative fuel cell, i.e. he determined the effective neutron multiplication factor k_inf while neglecting the escape of neutrons. In our study, we used the reactor model to perform parametric calculations with variations of the geometrical configuration in the core. Doing so, of course unrealistic configurations, e.g. when the edge length of the cube exceeds the distance between the cubes, or, the cube becomes too small in relation to the distance, as would be the case for an edge length of 1 cm and a distance of 20 cm, were omitted in our simulations. The results are given in Table 3 and shown in Figure 4.

Figure 4:

k_eff as a function of the edge length in cm of the cubes for three different horizontal distances between the cubes of 10 cm, 14 cm and 20 cm as indicated.

In addition, in Figure 4, k_eff is plotted as a function of the edge length of the cubes for three different horizontal distances between the cubes. The diagram shows that k_eff is largest for a distance of 10 cm for a cube edge length of ≤4 cm. Since the cubes in this configuration are small compared to the distance, this is an expected result, as the neutrons are less likely to reach the nearest uranium atom outside the cube at larger distances. For an edge length of 5–7 cm, the distance between the cubes of 14 cm is optimal. For an edge length of 8 cm, the distance of 20 cm yields the maximum effective neutron multiplication factor.

Our results obtained with the Serpent 2 code are largely consistent with those of Pešić (2018), who used the MCNP code and concluded, that the optimum edge length should be 6–7 cm. Figure 4 shows that in our calculations the maximum effective neutron mulitplication factor arises for edge lengths of the uranium cubes in the range of 5–8 cm. However, in Pešić’s results k_eff reached values up to 1.3 (with a lattice pitch of 20 cm and pure heavy water) using the simplified fuel cell model, while according to our results increasing the edge length and the pitch is not sufficient to make the reactor critical.

To summarize, there are two configurations with a significant maximum, namely with an edge length of 6 cm and a distance of 14 cm, and an edge length of 10 cm and a distance of 20 cm. The advantage when increasing the edge length of the cubes is due to the larger amount of fuel. However, larger cubes also require larger distances because a sufficient amount of moderator is needed to induce criticality. It should also be kept in mind that a larger distance must lead to an enlargement of the reactor, which consequently also requires more heavy water. However, a larger edge length also means a larger demand of uranium, which was difficult to obtain in 1945. Since an edge length of 5 cm at a distance of 14 cm produces a result similar as for the other edge lengths, an edge length of 5 cm at a distance of 14 cm is the optimal solution in view of the scarcity of fuel and moderator materials, if the cubic shape of the fuel is utilized. It is a remarkable result that Heisenberg hit the optimal reactor design in 1945 by analytical calculations, without access to nowadays simulation tools.

6.2 Effect of fuel geometry: uranium spheres and cylinders

In this section, we examine the question of whether a different form of the fuel element would have led to a large effective neutron multiplication factor under certain conditions. As an alternative to cubes, we examine spheres and cylinders. In order to be comparable with the previous arrangement, the mass (resp. the volume) of the fuel should be kept constant. Determining the diameter of a sphere that has the same volume as a cube with edge length of 5 cm yields a diameter of 6.2 cm. Using the spheres instead of the cubes while keeping all other parameters identical to the reference case (see Section 5), we find k_eff = 0.9511 ± 0.00043.

Next, we investigate the effect of the spatial arrangement of the fuel spheres by reducing the distances between the spheres. This has the effect that the “core region”, i.e. the region filled with dispersed fuel elements becomes smaller, i.e. the fuel is more concentrated in a smaller part of the reactor. This has, firstly, the effect that the fuel-to-moderator ratio increases and, secondly, that the lateral or vertical water colums acting as reflectors become thicker. The results for k_eff depending on different spacings between fuel elements are given in Table 4. We find that a reduction in the distances in the z- or x-direction between the uranium spheres leads to a smaller effective neutron multiplication factor.

A reduction in the size of the spheres and the associated reduction in spacing could lead to an improvement in the effective neutron multiplication factor, while the amount of uranium used should remain the same, meaning that more spheres would have to be suspended. However, the suspension of the spheres poses a challenge, as this is difficult to realize for smaller sphere diameters. In addition, the fuel-to-moderator ratio should be kept optimal and there should remain enough amount of moderator between the fuel spheres.

Another option would have been to arrange the fuel in the form of cylinders, similar to the fuel rods of nowadays light water reactors. Here, we investigate the effect of the shape of the fuel element in a simplified way and neglect the necessity of suspending the fuel e.g. by stacking uranium pellets inside a clad tube. To ensure that the same amount of uranium is used as in the reactor with uranium cubes, an equivalent diameter of the cylinders has to be determined. It is assumed that each column of cubes suspended by the same wire is replaced by one cylinder and that the uranium cylinders have the same height as the entire reactor core. Equating the total volume of the uranium cubes with the total volume of the cylinders and solving for the diameter gives approximately 3 cm. For simplicity, we keep the horizontal distance from one cylinder to the next at 14 cm. All other geometries of the original B8 reactor are kept unchanged. By this, we obtain k_eff = 0.9420 ± 0.00074.

To further investigate this configuration, we varied the cylinder diameter and the horizontal spacing. The parameter “inner circle” refers to the radial distance from the reactor center to the first ring of concentric fuel wires, as used in the original B8 setup. Reducing the inner circle from its original value of 11 cm leads to a higher concentration of fuel near the center and reduces the amount of moderator available in the core region. This has a similar effect to the reduction of the core region discussed for the spherical configuration in Table 4, where a smaller fuel zone increases the fuel-to-moderator ratio and thickens the outer reflector, but also reduces moderation inside the core. The results are given in Table 5.

We find that the lateral distance between the cylinders and their diameter have a significant influence on the effective neutron multiplication factor. An optimum is achieved with larger distances, such as for 14 cm and 15 cm, as shown in Table 5. This is due to the increasing amount of moderator present for increasing the distance between the cylinders. Furthermore, another optimum is found for a cylinder diameter of 5 cm and a distance of 14 cm, where k_eff = 0.9500 ± 0.00074 is achieved which is merely due to the increased amount of fuel. Compared to the calculated effective neutron multiplication factor of the reactor with uranium cubes of 5 cm edge length and a distance of 14 cm in all directions of k_eff = 0.9517 ± 0.00048, the magnitude for the cylinder configuration with a diameter of 3 cm and a distance of 14 cm yields k_eff = 0.9420 ± 0.00074, which shows that the choice of cubes, under fixed parameters, is better.

We find that the lateral distance between the cylinders has a significant influence on the effective neutron multiplication factor. An optimum is achieved with larger distances, such as for 14 cm and 15 cm, as shown in Table 5. This is due to the increasing amount of moderator present when increasing the distance between the cylinders. Compared to the calculated effective neutron multiplication factor of the reactor with uranium cubes of 5 cm edge length and a distance of 14 cm in all directions of k_eff = 0.9517 ± 0.00048, the configuration with cylindrical fuel elements of 3 cm diameter and the same 14 cm spacing results in k_eff = 0.9420 ± 0.00074, which shows that the choice of cubes, under fixed parameters, is slightly more favourable.

Yet, the spherical and especially cylindrical geometry of the fuel could have been an alternative option to obtain a critical reactor. However, this would have required extensive modifications to the historical reactor design.

6.3 Effect of fuel enrichment

The potential of enrichment of the isotope U-235 has been early recognized by the German scientists. However, the technical challenges of implementing the project on the required scale were considered too great and the focus was laid on reaching criticality with natural uranium and heavy water only. From a nowadays perspective is nevertheless interesting, which degree of enrichment would have been sufficient to make the B8 reactor in Haigerloch critical. For this purpose we carried out parametric calculations using enriched uranium oxide of different degrees of enrichment. The enrichment stages are based on the stages used by the NEA and are enrichment of UOX levels from 0.7 % to 4.55 %.

Figure 5 shows that the effective neutron mulitiplication factor strongly increases with enrichment, as expected. The reactor would have been supercritical already with an enrichment of 1 % leading to a k_eff = 1.0062 ± 0.00036.

Figure 5:

k_eff of the Haigerloch B8 reactor as a function of the enrichment of uranium. The left-most data point marks natural uranium which was used in the historical set-up.

However, in this configuration the reactivity ρ = (k_eff − 1)/k_eff, which measures the departure from criticality, has a value of 0.0062, while the fraction of delayed neutrons according to the Serpent 2 results is β_eff = 0.00712, i.e. the reactor would already have been dangerously close to prompt supercriticality. If the reactivity ρ exceeds the fraction of delayed neutrons β_eff, the neutron flux and therefore the power increases exponentially due to the prompt neutrons alone. The corresponding time constant is then determined by the average lifetime of the free neutrons, which is of the order of magnitude of 1 ms in a moderated reactor (instead of the time constant of a delayed critical reactor, which is in the order of magnitude of seconds). This extremely rapid increase leads to an uncontrolled power excursion, as it can no longer be influenced quickly enough by external technical means.

The designers of the B8 reactor were well aware of the risk of an uncontrolled power excursion and had foreseen respective countermeasures. Heisenberg’s concept realized this in the form of cadmium rods (Heisenberg and Wirtz 1947), which could have been introduced via the chimneys on the upper side of the reactor, as can be seen in Figure 2. The experiment in Haigerloch proceeded by gradually filling in the heavy water into the reactor core and constantly monitoring the neutron multiplicity. In this sense, prompt supercriticality would have been rather unlikely, since the experiment could have been stopped in the case of a significantly rising multiplicity. In this sense, the experiment B8 was a “critical experiment” in modern language since modern approaches to reactor control typically involve measuring k_eff in the subcritical state with a neutron source to safely approach criticality. Since the neutron multiplicity is proportional to 1/(1 − k_eff), the reciprocal of the neutron detector reading is plotted against the level of the heavy water inside the reactor core.

In passing, we note that the mentioned criticality can only be achieved with the historically original geometries and materials. The heavy water has a particularly large influence here. If the heavy water in the reactor core was to be replaced by light water, a much smaller magnitude of k_eff = 0.3439 ± 0.00072 would result, even with an enrichment of 1.1 %. In comparison, the CROCUS reactor at EPFL in Switzerland becomes critical at 1.806 % enrichment (Frajtag et al. 2018). For light water, the moderation length is much smaller, and therefore it is much better to distribute the same amount of uranium in many rods of small diameter (as in CROCUS) than in large cubes. This is due to the much larger scattering cross-section of H compared to D. In Table 6, we provide k_eff in dependence of different enrichment levels.

6.4 Effect of outer reflector thickness and material

In order to reduce neutron losses, the Haigerloch B8 reactor had a shell of width of 40 cm around the inner vessel made of graphite blocks. In this section, we investigate the influence of the width and of alternative materials of the shell on the effective neutron multiplication factor. The results are shown in Figure 6. As can be seen, k_eff grows when the shell thickness is increased. For graphite, the dependence saturates around a thickness of 70 cm, indicating that the optimal width of the reflector was not yet realized, although graphite was available in abundant quantities. Other materials could have been used as reflectors as well, such as beryllium, iron or natural uranium. Their reflection properties are shown in Figure 6 as well. Beryllium would have been slightly superior to graphite because of the (n, 2n) reaction in which beryllium absorbs one neutron and emits two neutrons, which significantly increases the neutron number inside the reactor. Yet, due to the historic circumstances, beryllium was difficult to obtain. The reflection properties of the other two materials iron and uranium are inferior to graphite or beryllium.

Figure 6:

k_eff of the Haigerloch B8 reactor as a function of the width of the reflector shell for different reflector materials graphite, beryllium, iron and uranium as indicated. The Haigerloch reactor experiment used graphite.

6.5 Effect of contamination of the heavy water

6.5.1 Light water contamination

In this section, the influence of the contamination of the heavy water as the moderator by light water is examined using historical geometry and materials. The contamination of the heavy water inventory (D₂O) with light water (H₂O) is highly probable, although the precise rate is unknown. The heavy water available in 1945 was produced already in 1941 by Norsk Hydro in Norway. Due to different usage until 1945 and due to potential degradation by absorbing light water present in air as moisture, a contamination by light water is likely. It is therefore important to analyze the influence of light water contamination on the effective neutron multiplication factor of the B8 experiment. In the simulations of the reference case as well as in the parametric calculations presented so far, we have assumed a contamination by 5 % light water. Park (2022) reported that the purity of samples of B8 heavy water kept at NIST was measured at NIST in 1947 to be 96.8 %. So, it is useful to vary the contamination rate and study the influence on criticality. The corresponding results are shown in Table 7 and in Figure 7.

Table 7:

Impact of light water contamination on k_eff with corresponding errors.

Light water
Contamination (%)	k eff
0 %	0.9749 ± 0.00089
2.5 %	0.9711 ± 0.00079
5 %	0.9517 ± 0.00048
7.5 %	0.9313 ± 0.00078
10 %	0.9092 ± 0.00065
15 %	0.8626 ± 0.00074
20 %	0.8184 ± 0.00066

Figure 7:

Comparison of the effect of boron (left) and light water (right) contamination in the moderator on the criticality. Even small amounts of boron lead to a more pronounced reduction in k_eff than the corresponding levels of light water, illustrating the high neutron absorption efficiency of boron impurities.

7.1 Scaling of the entire B8 geometry

In order to scale the reactor, we need to assume a realistic enlargement of the reactor. This means that an additional ring with aluminum cables and uranium cubes must be built. This also means that the amount of heavy water and the size of the magnesium vessel, the graphite reflector and the aluminum vessel have to be changed. For simplicity, we keep the height of the reactor as before. To determine the number of the suspending wires in the fifth ring of a structure of concentric cylinders, whose total diameter is 115.6 cm and each ring has an effective shell thickness of approximately 14 cm, we first analyze the existing rings. The first ring contains seven cables, the second ring has 15 cables, the third ring has 24 cables, and the fourth ring has 32 cables. These cables are evenly distributed around the circumference of each ring. To see the pattern in the increase in the number of cables, we consider the differences between the rings: from 7 to 15 (with eight more), from 15 to 24 (with nine more), and from 24 to 32 (with eight more). There is an average increase in the number of cables of about 8. Taking this symmetry and the average increase into account, an increase of eight can be assumed for the fifth ring. Therefore, the fifth ring would probably have about 40 cables, which are also arranged symmetrically. This means an increase of 272 uranium cubes with alternating hanging of eight and nine cubes. For each additional ring with an effective thickness of 14 cm, the diameter of the magnesium tank, the graphite reflector and the aluminum tank must increase by 28 cm. The angle of the cable position to an imaginary line through the reactor is adjusted in each case by dividing 360° by the number of cables.

By this we find that an extension by a fifth ring with an additional 272 cubes to a total of 936 cubes leads to an increased neutron multiplication factor of k_eff = 0.9792 ± 0.00096. This implies a total mass of natural uranium of 2,234.7 kg, which is an increase by 649.4 kg of the original approximately 1,585.3 kg. This corresponds to a multiplication of 1.41 of the uranium mass. In this configuration, the Mg vessel has a radius of 78 cm. The graphite reflector has a radius of 119 cm, the Al tank has a radius of 120 cm, and the overall outer dimension of the reactor is 170 cm. All dimensions are measured radially from the center of the reactor.

As one additional ring is still not sufficient to reach criticality, a sixth ring is next added. This increases the number of the uranium cubes by 408 to total 1,344. The effective neutron multiplication factor increases to k_eff = 0.9982 ± 0.00083. This implies a total mass of natural uranium of 3,208.8 kg, which is an increase by 1,623.5 kg compared to the original approximately 1,585.3 kg. This corresponds to a scaling factor of 2.02. In this configuration, the Mg vessel has a radius of 93 cm. The graphite reflector has a radius of 134 cm, the Al tank has a radius of 135 cm, and the overall outer dimension of the reactor is 185 cm. The reactor would have come very close to criticality in this set-up with six rings.

For completeness, we investigate how much a seventh ring would increase the criticality of the up-scaled B8 reactor further. The seventh ring increases the number of uranium cubes by 476 to total 1,820 cubes. Likewise, this means a total mass of natural uranium of 4,345.25 kg, which is an increase by 2,759.95 kg to the original approximately 1,585.3 kg. This corresponds to a scaling factor of 2.74. In this configuration, the Mg vessel has a radius of 98 cm. The graphite reflector has a radius of 139 cm, the Al tank has a radius of 140 cm, and the overall outer dimension of the reactor is 190 cm. For this configuration the Serpent 2 simulation gives an effective neutron multiplication factor of k_eff = 1.0108 ± 0.00090. This reactor would however be prompt supercritical and should never be realized due to safety reasons.

To summarize this part, it would have been required to augment the reactor with at least a sixth ring, corresponding to a scaling factor of 2.02 of the original amount of uranium and a total mass of uranium of 3,208.8 kg, with the same shape and spacing of the uranium cubes. This would have required a radius of the inner core vessel (Mg vessel) of 93 cm. Thus, an increase of the radius of the inner core vessel from 60 cm to 80 cm, as estimated by Heisenberg and Wirtz (1947) (which corresponds to the addition of a fifth ring) would not have been sufficient to reach criticality.

7.2 Variation of fuel element spacing

After exploring scaling scenarios of the historic B8 reactor setup, we now turn to optimization strategies that respect the historical constraint of a limited amount of heavy water. Instead of increasing the reactor size or the number of uranium cubes, this section investigates whether a more favorable spatial configuration of the existing fuel could have improved the effective neutron multiplication factor. Specifically, we analyze the impact of the distance between fuel cubes on k_eff, keeping the total heavy water volume constant.

To preserve the overall moderator content while varying the fuel element distance, the radius of the cylindrical core was adjusted accordingly. In order to maintain the geometrical structure of the suspended chains, the height of the core was also varied in proportion to the number of cubes per chain. This ensures that only the spacing between fuel cubes changes, while the overall fuel mass and moderator mass remain fixed. Furthermore, a 5 % contamination of the heavy water with light water was taken into account in all calculations to realistically reflect the historical boundary conditions.

The results, presented in Table 9 and Figure 8, show that k_eff reaches a maximum at a fuel element distance of approximately 14 cm. Although a larger spacing increases the moderator volume between the cubes and improves moderation, it also raises the probability of neutron absorption in the light water component. This trade-off leads to an optimum configuration at 14 cm, where the fuel-to-moderator ratio and reflector conditions are most favorable. These findings indicate that the original B8 design was already close to the optimal spacing with respect to the achievable effective neutron multiplication factor under the material constraints of the time.

Figure 8:

Variation of k_eff as a function of the fuel element distance. The amount of heavy water was kept constant by adjusting the reactor core radius and the core height according to the chain length, which depends on the fuel element spacing. A maximum k_eff was observed at a distance of approximately 14 cm.

Our investigations also show that, assuming completely pure heavy water, the original B8 configuration with a fuel element spacing of 14 cm and the corresponding cylindrical core dimensions would lead to a significantly higher value of k_eff = 0.98209 ± 0.00078. This underlines the strong influence of the moderator purity on the neutron economy, which is mainly due to the significantly higher absorption cross section of hydrogen compared to deuterium.

These results suggest that the historical B8 configuration was already very close to criticality under more favourable conditions. This raises the question of whether criticality could have been achieved by adjusting the reactor geometry, especially assuming that pure heavy water is available. In the following, we therefore investigate modified core geometries that retain the basic design principles but offer improved neutron moderation and reflection. This could also allow a reduction in the amount of fuel while still achieving the critical threshold.

7.3 Adjusting the geometry

Pešić (2018) showed that a modification of the geometry of the reactor core, which contains both uranium and heavy water, by increasing the diameter from 115.6 cm to 160 cm and simultaneously reducing the height from 169 cm to 160 cm can lead to an increase in the effective neutron multiplication factor. All other parameters, such as the materials used and the wall thickness of the Mg and Al vessels, remained unchanged. The wall thickness of the graphite reflector was set to 40 cm and was also not changed, taking into account the stacking of the graphite blocks in the reflector. This change in geometry increases the total moderator volume, thus allowing a larger amount of heavy water to be used. In line with the assumptions of Pešić (2018), it was further assumed that the heavy water was completely pure.

7.3.1 664 uranium cubes

First, we use 664 cubes with the new dimensions in the original staggered arrangement to examine the dependence of the effective neutron multiplication factor on the distances between the cubes. Parametric calculations were performed with Serpent 2 varying the distances between cubes. In each calculation the same value was used for the horizontal (referring to the position of the Al cables) and the vertical direction. The results are shown in Figure 9.

Figure 9:

Effect of edge length and distance to the next neighbors on k_eff with 664 uranium cubes. The results show that both the cube edge length and spacing strongly influence the fuel-to-moderator ratio and the effective neutron multiplication factor For this configuration, all tested combinations result in k_eff > 1, indicating supercriticality.

It can be seen that k_eff generally increases with increasing edge length, which is due to the fact that more uranium is present in the reactor core with a larger edge length. A larger fuel mass means more fissile material, which increases the probability of neutron interactions and thus increases k_eff. However, this increase is not linear, and an interesting behavior can be observed at a distance of 15 cm. In this case, k_eff reaches its maximum at an edge length of 5.0 cm with k_eff = 1.0201 ± 0.00075. For edge lengths of 4.5 cm (k_eff = 1.0183 ± 0.00056) and 5.5 cm (k_eff = 1.0101 ± 0.00081), the effective neutron multiplication factor is lower. This shows that at an edge length of 5.0 cm, the ratio of fuel quantity and moderator volume is optimal. A smaller edge length means that there is more moderator, but the amount of fuel is too small to achieve maximum criticality. A larger edge length, as with 5.5 cm, offers more fuel, but the space for the moderator between the cubes becomes too small to efficiently slow down neutrons, which leads to a drop in k_eff.

A similar pattern can be seen at a distance of 18 cm. Here, k_eff is maximal at an edge length of 5.5 cm (k_eff = 1.0463 ± 0.00077). The larger distance allows the moderator to have enough space to optimally decelerate the neutrons, resulting in a larger effective neutron multiplication factor. For smaller edge lengths, such as 5.0 cm (k_eff = 1.0366 ± 0.00082) and 4.5 cm (k_eff = 1.0189 ± 0.00062), the effective neutron multiplication factor decreases slightly because the moderator-fuel mix is less efficient, although the larger distance allows good moderation in principle.

With an edge length of 4.5 cm, k_eff reaches its maximum value of k_eff = 1.0230 ± 0.00071 at a distance of 16 cm to the nearest neighbor. For an edge length of 5 cm, the maximum value of k_eff is k_eff = 1.0366 ± 0.00082 at a distance of 18 cm. With an edge length of 5.5 cm, k_eff reaches its maximum value of k_eff = 1.0463 ± 0.00077 at a distance of 18 cm.

It is clear that there is an optimum in the ratio of fuel quantity and moderator availability. This optimum is determined both by the edge length of the fuel cubes and by the distance between them. Yet, obviously, k_eff is well above 1, indicating that the reactor would be supercritical. Hence, obviously the number of fuel cubes can still be decreased, as investigated in the subsequent sections.

7.3.2 508 uranium cubes

When the number of uranium cubes is reduced to 508, we investigate the cases of the edge length of 4.5 cm and 5 cm arranged at different distances. This corresponds to a mass of 882 kg for an edge length of 4.5 cm and 1,210 kg for an edge length of 5 cm. The results of the effective neutron multiplication factor is shown in Figure 10. With an edge length of 4.5 cm and a distance of 18 cm, we already have a result here that, with k_eff = 0.9998 ± 0.000043, is very close to 1 and has a uranium mass of only 882 kg.

Figure 10:

Effect of edge length and distance to the next neighbors on k_eff with 508 uranium cubes. The optimal configuration with an edge length of 4.5 cm and a spacing of 18 cm results in k_eff = 0.9998 ± 0.000043, despite the significantly reduced fuel mass of 882 kg.

7.3.3 430 uranium cubes

Since cubes with an edge length of 5 cm were available and k_eff > 1 for the case of 508 cubes, it is possible to reduce the number of cubes even further. We show that a further reduction 430 cubes is possible, which corresponds to a mass of 746 kg for an edge length of 4.5 cm and a mass of 1,024 kg for an edge length of 5 cm. The results for k_eff are shown in Figure 11.

Figure 11:

Effect of the edge length and the distance to the next neighbors on k_eff with 430 uranium cubes. For a cube size of 5 cm, criticality is nearly reached at a spacing of 16 cm with k_eff = 0.9998 ± 0.00062, while smaller cubes fall short due to insufficient fuel mass.

For an edge length of 4.5 cm, there is too little uranium in the reactor, while for an edge length of 5 cm, an optimum is found near k_eff = 1 at 16 cm distance with k_eff = 0.9998 ± 0.00062.

To summarize, with the results for an edge length of 4.5 cm, 508 cubes with a total mass of 882 kg and a heavy water volume of 3,170 L, and k_eff = 0.9998 ± 0.000043, and for the configuration with an edge length of 5 cm, 430 cubes with a total mass of 1,024 kg and a heavy water volume of 3,162 L, and k_eff = 0.9998 ± 0.00062, show that both configurations are very close to the optimal k_eff = 1. Under the condition to use as little uranium and heavy water as possible, the configuration with an edge length of 4.5 cm and a distance of 18 cm would be the optimal choice, while under the condition to use the already existing cubes with an edge length of 5 cm, they would have to be arranged at a distance of 16 cm. The amount of about 3,170 L of heavy water has to be compared to the available amount of 1,360 L (Bopp et al. 1945). An amount of almost two-and-a-half times the available quantity would have been necessary to reach criticality. In both cases, we can conclude that the bottleneck of the B8 experiment was not the lack of uranium, but the lack of heavy water.

Furthermore, it must be emphasized that operating the reactor in this configuration would neither have been advisable nor permissible under today’s safety standards. Due to the reduced amount of fuel and the slightly increased amount of moderator, an unfavorable moderator-to-fuel ratio

N Mod N Fuel

results, leading to overmoderation. This may cause the reactivity to increase with rising temperature, as the density of the moderator decreases. Such behavior poses a significant safety risk, as it can lead to a positive feedback effect and, consequently, to potential reactor instability.

7.4 Optimzed B8 configuration

Consequently, we may conjecture two optimized B8 reactor configurations (set-up A and B) which should have had the following configurations:

1. Height of the reactor core/Mg vessel: 160 cm

2. Diameter of the reactor core/Mg vessel: 160 cm

3. Type of fuel: Natural uranium cubes

4. Edge length of uranium cubes: 4.5 cm (A) or 5 cm (B)

5. Number of uranium cubes: 508 (A) or 430 (B)

6. Mass of uranium cubes: 882 kg (A) or 1,024 kg (B)

7. Number of rings for cable suspension: 4

8. Arrangement of the rings: Concentric

9. Number of cables: 78

10. Number of uranium cubes per cable: 7 and 6

11. Arrangement of uranium cubes: Staggered

12. Distance between uranium cubes or cables in x, y, z direction: 18 cm (A) or 16 cm (B)

13. Type of reflector: Graphite

14. Wall thickness of the graphite reflector: 40 cm

15. Impurities in the reflector: None

16. Stacking of graphite reflector blocks: With gaps, as in the historical reactor

17. Amount of heavy water: 3,170 L (A) or 3,162 L (B)

18. Impurities of heavy water with light water: None