Estimation of 99Mo production rates from natural molybdenum in research reactors

Molybdenum-99 is one of the most important radionuclides for medical diagnostics. In 2015, the International Atomic Energy Agency organized a round-robin exercise where the participants measured and calculated specific saturation activities achievable for the 98Mo(n,γ)99Mo reaction. This reaction is of interest as a means to locally, and on a small scale, produce 99Mo from natural molybdenum. The current paper summarises a set of experimental results and reviews the methodology for calculating the corresponding saturation activities. Activation by epithermal neutrons and also epithermal neutron self-shielding are found to be of high importance in this case.


Introduction
Molybdenum-99 and its daughter nuclide 99m Tc are of paramount importance in medical diagnostics: the NRG 2014 annual report of the High Flux Reactor in Petten states that there, about 1/4 of the world's supply is being produced, and that 1000 people per hour are subjected to diagnostics using Petten-produced 99 Mo, every hour, every day of the year. At the same time, the 99 Mo supply is at risk because it is being produced at only a few locations, and mostly as a 235 U fission product. This has the advantage of very high 99 Mo specific activity after the chemical separation from the other fission products, but also the disadvantage of generating large amounts of radioactive waste. Recently, the producers are also hampered by the restricted use of highly-enriched uranium targets as an initial material.
The International Atomic Energy Agency (IAEA) [1,2] and others [3,4] propose or consider a more distributed network of 99 Mo production sites, where it would then be produced through the 98 Mo(n,c) 99 Mo reaction. The achievable specific activities are less favourable that way, but the absolute amounts can be amply sufficient for local needs. Subsequently, this would also improve the utilization of national research reactors and would ease the issues associated with international imports [5]. Hot-atom chemistry methods have been developed to separate 99 Mo from the targeted 98 Mo atoms to improve the specific activity of the product [6].
In 2012, a feasibility study was perfomed at the Atominstitut in Vienna, involving small samples of MoO 3 [7]. Ryabchikov et al. [8] reported on the complex relationship between the 99 Mo production rate, the neutron spectrum and neutron self-shielding effects at various resonance energies in natural and enriched Mo, concluding that the use of enriched Mo has a minor effect of the activation rate through the self-shielding of epithermal neutrons by the other Mo isotopes. Both papers stress the dominance of the epithermal contribution to the activation rate. Wolterbeek et al. point out that the epithermal contribution is sometimes overlooked when this production route is compared to others [9].
In Japan, 99 Mo production from 98 Mo appears to be well on its way, both solid and dissolved Mo having been investigated as targets [10].
El Abd reports on measurements of the 98 Moðn; cÞ 99 Mo cross section and resonance integral, performed on metal foils (0.15 mm thick for natural Mo), confirming the generally accepted values [11].
In order to allow the various research reactors in the world to assess their 99 Mo production capabilities, the IAEA organised a round-robin in 2015 where 16 reactor institutes from all over the globe irradiated sizeable natural molybdenum samples, measured the induced radioactivity and attempted to predict the experimental results quantitatively by performing model calculations. In this paper, the lessons learned are presented, as well as the approximate production capacity of the reactors involved.

Theory
In the following, often-used existing methods of modelling the neutron activation rate are related to each other. This should make it easy to estimate 98 Moðn; cÞ 99 Mo production rates, no matter what set of neutron spectrum parameters is available. It is a useful exercise to perform because various definitions of thermal neutron flux (also known as ''thermal neutron fluence rate'') and epithermal flux are in use in different places and contexts, and the confusion that might ensue can lead to erroneous production capacity estimates. More in-depth information on the subject and the basic equations given here can be found in [12,13].
If, in a neutron activation experiments, the neutron flux can be considered homogeneous in the sample, then the activation rate R (s -1 ) per atom is given by.
where v is the neutron velocity (ms -1 ), (v) is the neutron capture cross-section (m -2 ) for neutrons with velocity v and U v ð Þdv is the neutron flux (m -2 s -1 ) of neutrons with velocities between v and v þ dv, E is the neutron energy (eV), r(E) is the neutron capture cross-section (m -2 ) for neutrons with energy E, and [(E)dE is the neutron flux (m -2 s -1 ) of neutrons with energies between E and E ? dE.
If a nuclear reactor is used as the neutron source, it is helpful to distinguish three energy regions: the thermal region (where the neutrons are in thermal equilibrium with the moderator, their velocities are represented by the Maxwell-Boltzmann distribution and the (n,c) neutron capture cross sections are mostly inversely proportional to the neutron velocity), the epithermal region (where the neutrons are slowing down, the neutron flux is roughly inversely proportional to the neutron energy and the (n,c) capture cross sections exhibit resonances), and the fast region (where the neutrons have the energy distribution as dictated by the emission of fast neutrons during the 235 U fission process and the (n,c) capture cross sections are very small).
The thermal region is taken here to range from 0 to 0.55 eV, where 0.55 eV is the ''Cd cut-off energy'' and the corresponding neutron flux is then called the ''subcadmium flux'', the epithermal region from 0.55 eV to 100 keV as the ''epicadmium flux'', and the fast region from 100 keV to several MeV. Equation (1) can then be written as Next, it is assumed that the neutron capture cross section in the thermal region can be written as where v 0 is 2200 m/s and r 0 the neutron capture cross section (m 2 ) for neutrons of that velocity, and that where U e is the epithermal flux (m -2 s -1 ) at 1 eV. Also, the contribution of the fast neutrons is considered negligible because the (n,c) capture cross sections are very small for such neutrons. Equation (1) then transforms to where 10,251 m/s is the neutron velocity corresponding to 0.55 eV. Now, it can be observed that where n is the neutron density (m -3 ) in the thermal region, and the conventional thermal neutron flux U 0 is defined by The resonance integral I 0 (m 2 ) is defined as So that Eq. (1) transforms to the Høgdahl [14] convention: Values for r 0 and I 0 are tabulated and widely available in the literature [15] as well as in on-line databases.
Refinements for non-1/v thermal reactions, epithermal activation and self-shielding In this section, a few refinements often encountered in activation rate modelling are discussed. If the nucleus that will result from the capture of a thermal neutron has a resonance energy close to the excited state it will be produced in, the capture cross section will not be inversely proportional to the neutron velocity in the thermal range. Then the thermal capture rate R t is approximated by where g(T) is the Westcott factor, for which tabulated values as a function of the temperature T of the Maxwell-Boltzmann neutron velocity distribution are available [16]. Only a few (n,c)-reactions show this effect to a contextrelevant degree in research-reactor conditions. In the neutron activation analysis community, where the highest possible accuracy is of interest, is has been established that the epithermal neutron spectrum shape is better described by where E 1 is 1 eV and a characterizes the deviation from the perfect epithermal spectrum [12]. Accordingly, the definition of the resonance integral changes to which leads to an additional parameter to characterize the dependence of I 0 on a, i.e. the effective resonance energy E r . This relation can then be expressed as When objects are irradiated that are not transparent to neutrons due to scattering or absorption, the neutron flux inside the material will be affected. This phenomenon is called neutron self-shielding and depends on sample composition, size, and shape as well as on incident neutron energy. With all these refinements, the sample-volume averaged capture rate is then given by where G t and G e are the thermal and epithermal neutron self-shielding correction factors. The specific saturation activity A s (Bq g -1 ) can then be calculated with where h is the isotopic abundance, N A is Avogadro's number and M is the molar mass of the element.

Alternative definitions for thermal or subcadmium flux
In other conventions, similar expressions are derived, but the thermal neutron flux may be defined differently. Beckurts and Wirtz [13], for example, define a thermal flux U T as and a corresponding Maxwell-Boltzmann flux-averaged thermal cross-section where T is the temperature associated with the Maxwell-Boltzmann velocity distribution, and T 0 is 293.6 K. In each convention, the definitions of the thermal cross section and the thermal neutron flux match, so the product R = Ur always turns out the same. The choice of convention is therefore arbitrary, as long as the corresponding matching pairs of r and U definitions are used. Because of the easy availability of literature values for r 0 and I 0 , Eq. (14) with its parameters is taken as the convention to relate the other models to in this paper, in this case by writing Alternative definition for epithermal or epicadmium neutron flux In the reactor physics community, the epithermal or epicadmium neutron flux is often defined as where E min and E max are the energy limits chosen for the integration-various values are used to this end. Using Eq. (4), the two epithermal fluxes are related by or Alternative spectrum parameters: cadmium ratio, thermal/epithermal ratio The neutron spectrum shape can be characterized with the parameter f, i.e. the thermal-to-epithermal flux ratio, defined as So that and often also with the cadmium ratio R Cd , i.e., the ratio of the activation rate without and with cadmium cover, defined by So that Typically, gold is used as the flux monitor and the tabulated values for r 0 and I 0 for the 197 Au(n,c) 198 Au reaction are to be used as a consequence. Imperfect shielding of subcadmium neutrons by the cadmium cover as well as neutron self-shielding in the flux monitor have been disregarded here.

Determination of neutron spectrum parameters
The neutron spectrum parameters (U 0 , U e , f, a etc.) can be determined in a variety of ways, ranging from theoretical Monte Carlo calculations, where the reactor and the irradiation facility are modelled in their entirety, to experimental irradiation and measurement of appropriate combinations of elements (such as Zr?Au, or Cr?Mo?Au), possibly with and without cadmium cover.

Neutron self-shielding calculation
Both the thermal and the epithermal neutron flux tend not to be homogeneous throughout the sample. The ratio of the volume-averaged flux within the sample and the flux in the same location in the absence of the sample is the selfshielding correction factor. In the thermal region, equations to calculate these factors are readily available for various sample shapes. A good overview is given by De Corte [12].
In the epithermal region, the situation is more complex because of the presence of resonance energies where neutron absorption may be extremely high. Two approaches were employed in this work: the MATSSF software developed by Trkov [17] and the method of Martinho [18] and Chilian [19].

Experimental
The participating countries and reactors are presented in Table 1.
The IAEA sent 3 molybdenum samples to each participant in the exercise: approx. 1 g of Mo 2 O 3 in a polyethylene capsule (9 mm internal diameter 9 9 mm height), and two rectangular pieces of Mo metal of 10 9 10 9 1 and 50 9 10 9 1 mm, weighing about 1 and 5 g, respectively.
From the neutron spectrum parameters of the irradiation facility used, as reported by the participants, the capture rates and the specific saturation activities for a very small natural molybdenum sample (i.e. with negligible neutron self-shielding) were calculated, as well as for the actual bulky molybdenum samples. Both epithermal self-shielding correction approaches were applied.
Most participants experimentally activated the three molybdenum samples and measured the induced activities using a (HP)Ge detector.
The irradiation, decay and measurement times were to be chosen so that the precision of the irradiation time would be better than 1 %, the decay time would be long enough to allow the counting of the samples at dead time below 10 %, and the measurement would provide a Poisson uncertainty in the observed peak area of less than 1 %.
The characteristics of the Ge detectors used are shown in Table 2. Full-energy detection efficiency curves were determined with mixed-radionuclide sources emitting multiple gamma-ray energies or combinations of calibrated point sources in all cases, at the same distances used for sample measurements. Each participant used their own, unspecified nuclear data source to this end. All participants used the same gamma-ray yields for the 99 Mo gamma rays.
From the observed peak areas, the capture rates during activation were calculated (in the process correcting for dead time, coincidence summing effects and counting geometry differences), using where N p is the net peak area after correction for dead time during measurement, t ir , t d and t m are irradiation time, decay time and counting time (s), k is the decay rate (s -1 ), c is the gamma mission probability, e is the full-energy detection efficiency and w is the sample mass (g). The detection efficiency for the molybdenum samples differed from the detection efficiencies as measured with small calibration sources, necessitating corrections for gamma-ray self-absorption and geometrical effects. The participants who estimated these corrections used solid angle approaches, where a numerical integration is carried out over the sample volume, taking the pathlength through the sample towards the detector, as well as the distance from the detector, into account at the same time. The result is a single correction factor, rather than two separate ones for geometry and self-absorption. Also, 99 Mo often emits the 740 keV photon in coincidence with the 181 keV, so that true-coincidence summing corrections must be applied. These correction factors and the dead-time correction factors or methods, if made available by the participant, are shown in Table 3.
The resulting specific Eq. (15). Finally, the measured specific saturation activities were compared to the calculated ones to assess the quality of the various methods used.
During a follow up workshop at the IAEA in Vienna, in December 2015, the participants compared notes, corrected their results for a number of issues found, and decided on the best nuclear data and methods to be used for 99 Mo production rate estimation. Not all discrepancies were resolved during this meeting-however all results presented in this paper are the ones obtained from the participant-supplied information during or after this meeting, using the agreed-upon methods and nuclear data. The equations used are the ones given above in this paper. The nuclear data used, all taken from De Corte et al. [15], are as shown in Table 4.

Neutron spectrum characterization
In Table 5, the neutron spectrum parameters as reported by the participants are shown (the participants were not instructed on which parameters to supply in the roundrobin process). From these parameters, the parameters needed as input for Eq. 14 were calculated using the equations given above as needed. The resulting values are also shown in Table 5. The Westcott factor g(T) was assumed to be unity in all cases for the 98 Mo(n,c) 99 Mo reaction. In case of unspecified values for the a parameter, a value of 0 was assumed. Also in Table 5, the specific saturation activities for a very small Mo target are shown, as calculated from the neutron spectrum parameters and the nuclear data for the 98 Mo(n,c) reaction, assuming absence of neutron selfshielding, using Eqs. 13, 14 and 15. Table 6 shows the thermal and epithermal neutron selfshielding factors, the latter as calculated using Trkov's and Chilian's methods, for the three sample types.
In Table 7, the measured specific saturation activities are shown, as well as the ratios of these over the values that were calculated with the two epithermal neutron selfshielding methods.

Discussion
Even though many experimental over calculated ratios as reported in Table 7 are close to unity, as they should be, the entire set of results indicates that the calculation and/or measurement of specific saturation activities for a reaction like 98 Mo(n,c) is challenging. The contribution of epithermal neutrons to the total activation rate is high-up to 80 % of the total rate in an in-core facility like the BigBeBe in the Netherlands. This, in turn, necessitates accurate epithermal neutron self-shielding corrections.
Disregarding the epithermal contribution entirely leads to gross underestimation of the saturation activity, as occurred in the case of Kazakhstan. In the neutron trap position in the Vietnam Dalat reactor, the epithermal flux was not disregarded but perhaps underestimated in the characterization process.
Not all participants achieved the desired overall precision of 1 % or better in the measured activation rates. In view of the standard deviations of the resulting ratios between measured and predicted activation rates of about 20 % (Kazakhstan excluded), all reported results were still deemed relevant.
In the cases of Malaysia, Peru, Romania and Vietnam (beam position), it appears that confusion may have occurred between the thermal fluxes U 0 , U T and the corresponding capture cross sections. It is indeed unfortunate that both U 0 and U T are known by the same name, i.e. ''thermal flux'', when the definitions are different by a factor of Hp/2 = 0.886-a value that appears to occur a number of times as the experimental-over-theoretical ratio in Table 7.   All factors are stated so that an uncorrected measured activation rate has to be multiplied with these to obtain the activation rate in a very small sample TCC stands for true coincidence summing, ZDT denotes zero-dead time counting Unity values indicate that the participant deemed the correction negligible, blank values that no data were provided by the participant Various participants reported measured specific saturation activities quite close to the theoretical values, like Hungary and Egypt. The Hungarian results strongly suggest that Chilian's method for epithermal neutron selfshielding correction performs better than Trkov's. The Egyptian results were obtained with a very thermalized neutron spectrum and do not offer conclusive evidence in this respect. The results from Slovenia and especially The Netherlands, where the least thermal spectrum was employed, suggest that Trkov's method may overestimate the severity of the epithermal self-shielding, where Chilian's may underestimate it to a smaller extent.
For neutron spectrum characterization, the Monte Carlo methods used by various participants appear to perform equally well as the experimental methods.
To assist the reader in quickly estimating the 99 Mo production capability in a given irradiation facility, Fig. 1 shows the specific saturation activities to be expected in an irradiation facility with a neutron flux of 10 13 cm -2 s -1 , as a function of thermal/epithermal flux ratio, for typical Table 4 Nuclear data used in this work (1 b = 10 -28 m 2 )   Values in bold italics were calculated from the other data in the table with the equations given in this paper

Conclusions
The results presented in this paper demonstrate that the specific saturation activity for the 98 Mo(n,c) 99 Mo reaction in natural molybdenum can be estimated with good accuracy, i.e. better than 10 %, if the epithermal activation contribution is taken into account properly and the epithermal neutron self-shielding is corrected for. Both Trkov's and Chilian's method give satisfactory results.
In-core irradiation facilities where the epithermal neutrons are strongly present are to be favoured over more thermalized facilities when the goal is to produce as much 99 Mo as possible.
Metallic molybdenum exhibits more neutron selfshielding than molybdenum oxide, leading to a higher specific saturation activity in the molybdenum itself. On the other hand, due to the lower density, less material can be introduced to the irradiation facility. Finally, post-  irradiation processing capabilities available at a given site will determine which material is to be favoured in practice.