# Bias Effects on g- and s-Factors in Westcott Convention
## 2 Formulas and Calculation Methods  
The formulas used in this paper are described in this section, together with practical calculation methods. A neutron energy spectrum for thermalized reactor neutrons by [PERSON] et al. [9], where the spectrum is approximated by the sum of Maxwellian and 1/E distributions, is expressed as \[\phi(E)=\phi_{th}(E)+\phi_{epi}(E)=\phi_{th}.\frac{2}{\sqrt{\pi}}\cdot\sqrt{\frac{ T_{n}}{T_{0}}}.\frac{E}{\left(kT_{n}\right)^{2}}\,e^{-E/kT_{n}}+\phi_{epi}\frac{ \Delta(E)}{E} \tag{2}\]  
where \(E\) is the neutron energy, \(T_{n}\) the temperature of neutrons defined in the Maxwellian distribution, \(k\) the Boltzmann constant, and \(\Delta(E)\) a joining function. The parameters \(\phi_{th}\) and \(\phi_{epi}\) are constants expressing the intensities of thermal and epithermal neutron flux components. As a joining function, several types have been proposed such as \(\Delta_{2}^{W}(E)\) and \(\Delta_{4}^{W}(E)\), as shown in [PERSON]'s reports [9]. \(\Delta_{4}^{W}\) has been recognized as the most appropriate function among various functions reported by [PERSON] et al. However, the simple \(\Delta_{2}^{W}\) shape or, even more, a sharp cut-off function expressed by a unit step function has been utilized in both NAA and neutron capture cross section measurements, since a bias for reaction rate originating in the joining function is thought to be negligible for ideal 1/v isotopes. \(\Delta_{4}^{H}\) is the joining function adjusted to reproduce a neutron energy spectrum at an irradiation position of a research reactor by using a Monte Carlo simulation with detailed configuration [7]. The formulas of \(\Delta_{2}^{W}(E)\), \(\Delta_{4}^{W}(E)\), and \(\Delta_{4}^{H}\) are given below:  
\[\Delta_{2}^{W}(E)=\frac{1}{1+\left(\frac{4.95 kT_{n}}{E}\right)^{7}} \tag{3}\]  
\[\Delta_{4}^{W}(E)=\frac{1}{1-\frac{0.26}{1+\left(\frac{E}{6.4 kT}\right)^{3}}+ \left(\frac{4.75 kT_{n}}{E}\right)^{7}} \tag{4}\]  
\[\Delta_{4}^{H}(E)=\frac{1}{1-\frac{0.26}{1+\left(\frac{E}{6.4 kT_{n}}\right)^{1.36}}+\left(\frac{5.06 kT_{n}}{E}\right)^{5.88}} \tag{5}\]  
the joining functions \(\Delta_{2}^{W}\), \(\Delta_{4}^{W}\), \(\Delta_{4}^{H}\) for neutron temperature \(T_{n}\) = 300 K are plotted in Figure 1a by solid, dotted, and dashed line, respectively. There are bumps at around 0.23 and 0.27 eV for \(\Delta_{4}^{W}\) and \(\Delta_{4}^{H}\), respectively. The cut-off energy for both \(\Delta_{2}^{W}\) and \(\Delta_{4}^{H}\) is about 0.13 eV, which is a little higher than 0.12 eV for \(\Delta_{4}^{W}\). The difference of a high-energy side tail of the bump should be noted; there exists a larger component in \(\Delta_{4}^{H}\) compared to \(\Delta_{4}^{W}\) in a neutron energy region from about 0.5 eV to a few eV. The same for neutron temperature 1000 K are plotted in Figure 1b. The cut-off energy and bump peak energy are shifted by a factor 1000/300. It is anticipated that the reaction rates for isotopes with a resonance at around the cut-off energy or bump peak energy will be affected significantly by using different joining functions. The g factor in Equation (1), defined as the ratio of pure Maxwellian flux-weighted cross section (\(\hat{\sigma}_{m}\)) to the cross section at 0.0253 eV (\(\sigma_{0}\)), is expressed as  
\[\text{g}(T_{n},\ T_{s})=\frac{\hat{\sigma}_{m}}{\sigma_{0}}=\frac{1}{\sigma_{0}} \frac{2}{\sqrt{\pi}}\sqrt{\frac{T_{n}}{T_{0}}}\int_{0}^{\infty}\frac{E}{(kT_{n })^{2}}\cdot\text{e}^{-\frac{E}{kT_{n}}}\cdot\sigma(E,\ T_{s})dE \tag{6}\]  
where sample temperatures (\(T_{s}\)) are explicitly indicated in the cross section \(\sigma(E,\ T_{s})\).