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 \(^{ 9 8} {\text{Mo}}({\text{n}},\gamma )^{ 9 9} {\text{Mo}}\) 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.
$$R = \int\limits_{0}^{\infty } {\varPhi \left( v \right)\sigma \left( v \right){\text{d}}v} = \int\limits_{0}^{\infty } {\varPhi \left( E \right)\sigma \left( E \right){\text{d}}E} ,$$
(1)
where v is the neutron velocity (ms−1), (v) is the neutron capture cross-section (m−2) for neutrons with velocity v and \(\varPhi \left( v \right)dv\) is the neutron flux (m−2s−1) of neutrons with velocities between v and \(v \, + \, dv\), E is the neutron energy (eV), σ(E) is the neutron capture cross-section (m−2) for neutrons with energy E, and ∅(E)dE is the neutron flux (m−2s−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,γ) 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,γ) 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 235U fission process and the (n,γ) 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
$$R = \int\limits_{0}^{0.55} {\varPhi \left( E \right)\sigma \left( E \right){\text{d}}E} + \int\limits_{0.55}^{{10^{5} }} {\varPhi \left( E \right)\sigma \left( E \right){\text{d}}E} + \int\limits_{{10^{5} }}^{\infty } {\varPhi \left( E \right)\sigma \left( E \right){\text{d}}E} ,$$
(2)
Next, it is assumed that the neutron capture cross section in the thermal region can be written as
$$\sigma \left( v \right) = \frac{{\sigma_{0} v_{0} }}{v} ,$$
(3)
where v
0
is 2200 m/s and σ
0 the neutron capture cross section (m2) for neutrons of that velocity, and that
$$\varPhi_{\text{e}} \left( E \right) = \frac{{\varPhi_{\text{e}} }}{E} ,$$
(4)
where Φ
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,γ) capture cross sections are very small for such neutrons. Equation (1) then transforms to
$$R = \int\limits_{0}^{10251} {\varPhi \left( v \right)\frac{{\sigma_{0} v_{0} }}{v}{\text{d}}v} + \int\limits_{0.55}^{{10^{5} }} {\frac{{\varPhi_{e} \sigma \left( E \right)}}{E}{\text{d}}E} ,$$
(5)
where 10,251 m/s is the neutron velocity corresponding to 0.55 eV. Now, it can be observed that
$$n = \int\limits_{0}^{10251} {\frac{\varPhi \left( v \right)}{v}{\text{d}}v},$$
(6)
where n is the neutron density (m−3) in the thermal region, and the conventional thermal neutron flux Φ
0 is defined by
$$\varPhi_{0} = nv_{0} .$$
(7)
The resonance integral I
0
(m2) is defined as
$$I_{0} = \int\limits_{0.55}^{{10^{5} }} {\frac{\sigma \left( E \right)}{E}{\text{d}}E} .$$
(8)
So that Eq. (1) transforms to the Høgdahl [14] convention:
$$R = \sigma_{0} \varPhi_{0} + I_{0} \varPhi_{e} .$$
(9)
Values for σ
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
$$R_{\text{t}} = g\left( T \right)\sigma_{0} \varPhi_{0} ,$$
(10)
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,γ)-reactions show this effect to a context-relevant 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
$$\varPhi_{\text{e}} \left( E \right) = \frac{{\varPhi_{\text{e}} \left( {E_{1} } \right)}}{{\left( {\frac{E}{{E_{1} }}} \right)^{1 + \alpha } }} ,$$
(11)
where E
1 is 1 eV and α characterizes the deviation from the perfect epithermal spectrum [12]. Accordingly, the definition of the resonance integral changes to
$$I_{0} \left( \alpha \right) = \int\limits_{0.55}^{{10^{5} }} {\frac{\sigma \left( E \right)}{{\left( {\frac{E}{{E_{1} }}} \right)^{1 + \alpha } }}dE} ,$$
(12)
which leads to an additional parameter to characterize the dependence of I
0 on α, i.e. the effective resonance energy E
r. This relation can then be expressed as
$$I_{ 0} \left( \alpha \right) = \frac{{I_{ 0} - 0.429}}{{E_{\text{r}}^{\alpha } }} + \frac{0.429}{{(2\alpha + 1)0.55^{\alpha } }} .$$
(13)
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
$$R = g\left( T \right)G_{\text{t}} \sigma_{0} \varPhi_{0} + G_{\text{e}} I_{0} \left( \alpha \right)\varPhi_{\text{e}} ,$$
(14)
where Gt and Ge are the thermal and epithermal neutron self-shielding correction factors.
The specific saturation activity A
s (Bq g−1) can then be calculated with
$$A_{\text{s}} = \frac{{R\theta N_{\text{A}} }}{M} ,$$
(15)
where θ is the isotopic abundance, N
A is Avogadro’s number and M is the molar mass of the element.
Other conventions and neutron spectrum parameters
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 Φ
T as
$$\varPhi_{T} = n\left\langle v \right\rangle = n\frac{2}{\sqrt \pi }\sqrt {\frac{T}{{T_{0} }}} v_{0},$$
(16)
and a corresponding Maxwell–Boltzmann flux-averaged thermal cross-section
$$\sigma_{T} = \left\langle \sigma \right\rangle = \frac{{\int\limits_{0}^{\infty } {\varPhi \left( v \right)\sigma \left( v \right)dv} }}{{\int\limits_{0}^{\infty } {\varPhi \left( v \right)dv} }}\frac{\sqrt \pi }{2}\sqrt {\frac{{T_{0} }}{T}} \sigma_{0},$$
(17)
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 = Φσ always turns out the same. The choice of convention is therefore arbitrary, as long as the corresponding matching pairs of σ and Φ definitions are used.
Because of the easy availability of literature values for σ
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
$$\varPhi_{0} = \frac{\sqrt \pi }{2}\sqrt {\frac{{T_{0} }}{T}} \varPhi_{\text{T}}.$$
(18)
Alternative definition for epithermal or epicadmium neutron flux
In the reactor physics community, the epithermal or epicadmium neutron flux is often defined as
$$\varPhi_{\text{e}}^{*} = \int\limits_{{E_{\hbox{min} } }}^{{E_{\hbox{max} } }} {\varPhi \left( E \right){\text{d}}E} ,$$
(19)
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
$$\varPhi_{\text{e}}^{*} = \int\limits_{{E_{\hbox{min} } }}^{{E_{\hbox{max} } }} {\frac{{\varPhi_{\text{e}} }}{E}{\text{d}}E} = \varPhi_{\text{e}} \left[ {\ln \left( {E_{ \hbox{max} } } \right) - \ln \left( {E_{\hbox{min} } } \right)} \right] ,$$
(20)
or
$$\begin{aligned} \varPhi_{\text{e}} = \frac{{\varPhi_{\text{e}}^{*} }}{{\ln \left( {\frac{{E_{\hbox{max} } }}{{E_{\hbox{min} } }}} \right)}} \hfill \\ \hfill \\ \end{aligned} .$$
(21)
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
$$f = \frac{{\varPhi_{0} }}{{\varPhi_{\text{e}} }},$$
(22)
So that
$$\varPhi_{\text{e}} = \frac{{\varPhi_{0} }}{f},$$
(23)
and often also with the cadmium ratio R
Cd, i.e., the ratio of the activation rate without and with cadmium cover, defined by
$$R_{\text{Cd}} = \frac{{\left( {\sigma_{0} \varPhi_{0} + I_{0} \left( \alpha \right)\varPhi_{\text{e}} } \right)}}{{I_{0} \left( \alpha \right)\varPhi_{\text{e}} }}$$
(24)
So that
$$\varPhi_{\text{e}} = \varPhi_{0} \frac{{\sigma_{0} }}{{I_{0} \left( \alpha \right)}}\frac{1}{{\left( {R_{\text{Cd}} - 1} \right)}} .$$
(25)
Typically, gold is used as the flux monitor and the tabulated values for σ
0 and I
0 for the 197Au(n,γ)198Au 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 (Φ
0, Φ
e, f, α 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 self-shielding 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].