Comparison of methods to determine the fluence of monoenergetic neutrons in the energy range from 30 keV to 14.8 MeV

The P2 RPPC has been in use at PTB since the 1980s. As shown in figure 5, its design closely follows the original layout of Skyrme et al [15]. The instrument is employed in the energy region between 24 keV and about 2 MeV. It consists of a 0.5 mm thick cylindrical stainless-steel housing, 76 mm in diameter and 360 mm in length. The thickness of the stainless-steel entrance window is 0.5 mm as well. The mechanical size of the sensitive volume within this housing is defined by a cylindrical aluminium cathode, 55.52(5) mm in diameter and 0.3 mm in thickness. The height of the volume is 193.3(2) mm and it is defined by the end of the field tubes surrounding the anode wire with a diameter of 100 μm. The field tubes are held at ground potential while the anode and cathode potentials are kept at a voltage ratio $U_+ / U_-$ of 1.402, as calculated from the diameters of the anode wire, field tubes and cathode.

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The counter is positioned with the cylindrical axis on the zero-degree direction of the neutron field and at a distance of about 115.5 cm between the neutron production target and the geometrical centre of the sensitive volume. The contribution of neutrons scattered by air or by structural materials of the low-scatter facility was subtracted experimentally by conducting separate measurements with a 30 cm long PE shadow cone placed between the RPPC and the neutron production target.

The efficiency of the instrument is defined by the size of the volume from which electrons can drift to the amplification region around the anode wire. When neglecting lateral diffusion during the drift of the electrons along the field lines, this volume is defined by the electric field at the axial heights of the ends of the field tubes. The electric field distribution was calculated numerically using COMSOL Multiphysics [16]. Figure 6 shows a field map. In the vicinity of the end of the field tubes the equipotential surfaces are not parallel to the anode wire. Therefore, the separatrix is not a plane surface perpendicular to the axis of the geometrical volume. A correction factor $k_{V} = 0.985(8)$ was therefore applied to the geometrical volume $V_\mathrm{geo} = 4.680(10)\,\times 10^2\,\mathrm{cm}^3$.

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The gas types and pressures used in the RPPC are selected such that the leakage of recoil protons from the sensitive volume is kept within acceptable limits. For neutron energies below 400 keV, a mixture of 96.5% hydrogen and 3.5% methane per unit volume was used, and propane was employed at energies above 400 keV. The pressure p varied between 300 hPa and 1000 hPa, depending on the gas type and neutron energy.

The conversion of proton energy deposited in the gas to ionisation charge is described by the W-value. For propane an energy dependence $W(E_\mathrm{p})$ was considered, where $E_\mathrm{p}$ denotes the kinetic energy of the recoil protons. This was based on the data of Posny et al [17] and on two measurements of the relative W-value $W(E_\mathrm{p})/W_0$ with the P2 RPPC, where W₀ denotes the W-value for 600 keV protons. These measurements were made by varying the neutron energy and observing the change of the location of the recoil proton edge in the pulse height distribution. For the simulations, the data in figure 7 were approximated by a linear relation on the logarithmic energy scale. This fit excluded the Posny data above 200 keV because the re-increase of these data could not be verified in the measurements with P2. In contrast to propane, there is no evidence of a significant energy dependence of the W-value for pure hydrogen and hydrogen-methane mixtures for protons energies above 10 keV [18, 19], that is, $W(E_\mathrm{p})/W_0 = 1$ for the hydrogen–methane mixtures.

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The neutron response of the counter was calculated using the Monte Carlo code MCNPX, v2.7 [14]. Cross section data from the ENDF/B-VII evaluation were used except for the n–p cross section that was taken from ENDF/B-V. The simulated neutron energy distributions discussed in section 2 were used to model the source term. The simulations were carried out for a divergent source at a distance d₀ of 115.25 cm from the geometrical centre of the RPPC and with a layer of air between the source and the RPPC to account for outscattering of neutrons in air. The angular distribution of the source was assumed to be isotropic because the opening angle covered by the sensitive volume is only 1.5^∘. The PTRAC files produced by MCNPX were analysed using a dedicated filter code to determine the 'pulse height' $(W(E_\mathrm{p})/W_0) E_\mathrm{p}$ produced by recoil protons in the sensitive volume. In this way all neutron and proton transport effects were correctly modelled. The finite pulse-height resolution of the RPPC was accounted for by folding the simulated pulse height distributions with Gaussians of constant relative widths. Unfortunately, recoil nuclei with Z > 2 are not transported by MCNPX. Therefore, the contribution from carbon recoil nuclei at a low pulse height could not be modelled and had to be excluded from the analysis. The simulated pulse height distributions were normalised to the unit fluence of direct neutrons $\Phi_\mathrm{dir}$ in vacuum at the position of the centre of the sensitive volume.

The effect of scattered neutrons on the pulse height distribution is exhibited in figure 8 for 250 keV neutrons produced with a lithium target on a tantalum backing. As discussed in section 2, the energy distribution calculated with MCNP5 improves the agreement of the measured and the calculated pulse height distribution in the region below 100 keV.

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Figure 9 shows the effect of an energy-dependent W-value for propane on the pulse-height distribution for 565 keV neutrons. The slope of the experimental distribution is only reproduced when the energy-dependent W-value is employed. This makes it possible to use the entire pulse-height region above the carbon recoil edge for the analysis and reduces the uncertainty due to the selection of the fit region.

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The neutron fluence measurements were analysed by fitting a calculated pulse height distribution to the measured one after correcting for the contribution of neutrons scattered in air and on structural materials. The loss of events due to the dead time was also included in the corrections. For measurements with propane, the fit region was restricted to the pulse height region above the carbon recoil edge, corresponding to a proton energy of about $0.29\,E_\mathrm{n}$. With hydrogen–methane mixtures, the lower bound of the fit region was set to about $0.15\,E_\mathrm{n}$ to avoid interference from events produced by 478 keV photons from ⁷Li(p,p'γ).

The yield $Y_\mathrm{n,dir} = (\mathrm{d}N_\mathrm{n,dir}/\mathrm{d}\Omega)$ of direct neutrons emitted from the target is

$$\begin{align} Y_\mathrm{n,dir} = \frac {k_\tau \, N_\mathrm{p}} {k_{V} \, k_{pT} \, k_\mathrm{fit} \, C_\mathrm{p}} d^2. \end{align} \tag{ 1 }$$

The net number of detected recoil proton events in the fitting region after subtraction of the events registered with the shadow cone in place is $N_\mathrm{p}$ and $C_\mathrm{p}$ denotes the number of simulated events in the same region per unit fluence $\Phi_\mathrm{n,dir}$ of simulated direct neutrons in vacuum at the nominal distance $d_0 \approx d$ from the source. The correction factors k_τ and $k_{V}$ account for dead time losses and for the deviation of the effective sensitive volume from the geometrical sensitive volume $V_\mathrm{geo}$ and $k_{pT} = (p / p_0) (T_0 / T)$ denotes the correction for the deviation of the temperature and pressure of the counting gas from the reference values p₀ and T₀ used for the simulation. The correction factor $k_\mathrm{fit} = 1$ is used to introduce the uncertainty resulting from the selection of the fit region. The actual distance of the centre of the sensitive volume from the target is d.

To evaluate the uncertainty of the Monte Carlo simulation, $C_\mathrm{p}$ can be broken down into the contribution from an ideal point-like detector at distance d₀, with volume $V_\mathrm{geo}$ and hydrogen density $n_\mathrm{H}$, multiplied by a kernel factor $G_\mathrm{MC}$ that describes the deviation of the Monte Carlo simulation from the result for an ideal detector,

$$\begin{align} C_\mathrm{p} = k_{E} \, k_\mathrm{sc,f} \, G_\mathrm{MC} \, n_\mathrm{H} \, V_\mathrm{geo} \, \sigma_\mathrm{np}\left(E_\mathrm{n,dir}\right). \end{align} \tag{ 2 }$$

The n–p scattering cross section at the mean energy $E_\mathrm{n,dir}$ of the direct neutrons is denoted by $\sigma_\mathrm{np}$. The hydrogen density $n_\mathrm{H}$ is calculated for the reference conditions p₀ and $T_0 = 293.15\,\mathrm{K}$ using an equation-of-state for real gases [20]. The correction factor $k_\mathrm{sc,f}$ accounts for the additional simulated recoil proton events in the fit region produced by neutrons scattered in the target. The correction factor $k_{E} = 1$ is used to introduce the uncertainty of the n–p cross section $\sigma_\mathrm{np}(E_\mathrm{n})$ due to the uncertainty of the mean energy $E_\mathrm{n,dir}$ of the direct monoenergetic neutrons. For the reasons discussed in 2.1, this contribution is only relevant for measurements with P2 in neutron fields produced using the ⁷Li(p,n)⁷Be reaction. For the fields produced with the ³H(p,n)³He reaction, the relative uncertainty of $\sigma_\mathrm{np}$ is smaller than $2\times10^{-3}$.

The uncertainty of $G_\mathrm{MC}$ was assessed by voiding all elements of the model except for the counting gas and those elements for which the effect on the number of recoil proton events was investigated, for example the entrance window, the cylindrical housing, the cathode or the anode support. These numbers were compared with those for a model that was fully voided except for the counting gas. The relative difference was multiplied by the relative uncertainty of the total or differential cross section for the respective material to obtain the uncertainty contribution to $G_\mathrm{MC}$, which was added quadratically to those for the other elements of the model. Table 1 shows the relative uncertainties for the quantities appearing in (1) and (2).

Table 1. Uncertainty contributions to the yield of direct neutrons measured with P2. The uncertainty contributions marked 'Stat.' are related to variations from measurement to measurement while the contributions marked 'Non-stat.' are constant or are strongly correlated over neutron energies.

Quantity	Rel. uncertainty	Stat.	Non-stat.
$N_\mathrm{p}$	0.003–0.010	×
kτ	0.0002	×
$k_{V}$	0.0074		×
$k_{pT}$	0.0020	×
$k_\mathrm{fit}$	0.003–0.015	×
d	0.002	×
$k_{E}$	0.004–0.008 a	×
$k_\mathrm{sc,f}$	0.004–0.014		×
$G_\mathrm{MC}$	0.005–0.013		×
$n_\mathrm{H}$	0.002		×
$V_\mathrm{geo}$	0.002		×
$\sigma_\mathrm{np}$	0.005–0.009		×
$C_\mathrm{p}$	0.015–0.023 b		×
$Y_\mathrm{n,dir}$	0.015–0.028

^a Only for neutron fields produced with ⁷Li(p,n)⁷Be. ^b Cumulated contributions from $k_{E}$, $k_\mathrm{sc,f}$, $G_\mathrm{MC}$, $n_\mathrm{H}$, $V_\mathrm{geo}$ and $\sigma_\mathrm{np}$.

The T1 RPT follows the design for RPTs for use in non-collimated ('open') neutron fields introduced by Bame et al [21]. As shown in figure 10, the instrument consists of a hydrocarbon sample, the radiator, and a stack of two proportional counters (PC1 and PC2) using CO₂ as a counting gas together with a 1.5 mm thick silicon surface barrier (SB) detector. Depending on the neutron energy, the CO₂ pressure ranges between 60 hPa and 300 hPa. Neutrons incident on the radiator generate recoil protons that are detected in the detector stack. Valid events produce triple coincidences in PC1, PC2 and SB within a coincidence resolving time of about 2.5 μs. The solid angle for the detection of the recoil protons is defined by a tantalum aperture in front of the SB detector and a mask directly in front of the tristearine layer that defines the effective diameter of the radiator. In addition to valid recoil proton events originating from the radiator, neutron interactions with the counting gas and other parts of T1 produce single event rates in the three detectors which exceed the rate of triple-coincidence events by a factor of more than 10³. Accidental coincidences produced by these events are measured in separate background runs with the radiator turned by 180^∘, meaning towards the neutron source. The upper panel of figure 11 shows foreground and background (radiator at 0^∘ and 180^∘, respectively) pulse height distributions measured for 2.5 MeV neutrons.

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The radiators consist of tristearine ($\mathrm{C}_{57}\mathrm{H}_{110}\mathrm{O}_{6}$) layers produced by PVD on 0.5 mm tantalum backings. The masses per unit area of the layers were determined by differential weighing and measuring the aperture used to define the diameter of the layers during the evaporation. The stoichiometric composition was characterised by combustion analysis, $m_\mathrm{H}/m = 0.1235(10)$ by weight [22].

The width of the energy distribution of the valid recoil proton events depends on the thickness of the radiator. Therefore, a set of four radiators with masses per unit area ranging from 0.756(6) mg cm⁻² to 10.0855(28) mg cm⁻² was used to cover the energy region 1.2 MeV to 14.8 MeV. As an example, the pulse height distribution measured with the SB detector for 2.5 MeV neutrons is shown in figure 11.

The use of the RPT T1 with the D(d,n)³He source and the PTB gas target has been reported in [23]. The dedicated Monte Carlo code SINENA [23] developed specifically for measurements with this source was replaced by a full simulation of the neutron and charged-particle transport using MCNPX, v2.7 [14] for all neutron fields. In this way, the effect of neutron scattering in the RPT housing could be included consistently.

As for the P2 RPPC, the measurements with the T1 RPT were analysed by fitting a calculated pulse height distribution to the measured net pulse height distribution. The MCNPX model used for the simulation includes all relevant details of the instrument. Detailed field maps for the proportional counters were not available. Therefore, the effective sensitive volumes of the counters were modelled as ellipsoidal cylinders oriented parallel to the counting wires. Their sizes were adjusted such that the experimental pulse height distributions of the two proportional counters were reproduced.

For measurements carried out with solid-state targets, the neutron sources were modelled in the same way as for the P2 measurements, using the energy distributions of direct and scattered neutrons calculated with the TARGET code [10]. Only for the measurements with the D(d,n)³He reaction and the deuterium gas target were the variation of the source yield along the axis of the gas volume and the anisotropy of the angular distribution included in the source description.

Forced neutron interactions [11] were induced in the thin tristearine layers to obtain statistical uncertainties of less than 0.5% in the Monte Carlo simulations. Protons and alpha particles were transported with energy cutoffs of 1 keV and 10 keV, respectively. This makes sure that also events from ¹⁶O(n,α)¹³C reactions in the counting gas are properly tracked. The triple-coincidence events were reconstructed from the PTRAC files event by event using a dedicated filter code. The finite pulse-height resolution of the silicon detector was accounted for by folding the simulated pulse height distributions with Gaussians of constant relative widths.

As demonstrated in the lower panel of figure 11, the simulation also describes the tail on the left of the recoil peaks. This tail contains events produced by neutrons scattered in the target setup, the radiator backing or in the T1 housing together with events resulting from protons that lost energy in the counting wires or the edges of the apertures. These events do not contribute to the determination of the yield $Y_\mathrm{n,dir}$ of direct neutrons. In this particular example, the calculated number of events in the tail underestimates the measured number by 23%. This excess background is probably due to accidental coincidences and is assumed to extend into the peak region. For the very intense T(d,n) and D(d,n) sources, it can can be close to 50% of the tail events. It is corrected by extrapolating the excess events into the peak region, assuming a linear decrease to zero within the peak region.

The yield $Y_\mathrm{n,dir}$ of direct neutrons is obtained from the net number $N_\mathrm{p}$ of events in the peak region of the net SB pulse height distribution and the number $C_\mathrm{p}$ of events per unit source yield of direct neutrons in the corresponding region of the simulated pulse height distribution,

$$\begin{align} Y_\mathrm{n,dir} = \frac{k_\mathrm{tail} \, k_\tau \, N_\mathrm{p}}{k_{d} \, k_\mathrm{fit} \, k_\mathrm{rel} \, C_\mathrm{p}}. \end{align} \tag{ 3 }$$

Here, the correction factor $k_{d}$ accounts for the difference between the actual distance between the target and the radiator and that assumed for the simulation. Its uncertainty includes the variation of the distance d₁ from the radiator to the beam spot on the solid-state target and the uncertainty due to the bending of the molybdenum foil of the gas target. The correction factor $k_\mathrm{fit} \approx 1$ is introduced to include any effect of the fit region on the results and $k_\mathrm{tail} \approx 1$ denotes a correction factor for an excess of events in the tail region which is extrapolated into the peak region. The correction factor

$$\begin{align} k_\mathrm{rel} = \frac {\left(\mathrm{d}\sigma_\mathrm{np}/\mathrm{d}\Omega_\mathrm{p,rel}\right)}{\left(\mathrm{d}\sigma_\mathrm{np}/\mathrm{d}\Omega_\mathrm{p,n-rel}\right)} = \frac {\gamma_0^2 \left(1 + \mathrm{tan}^2\left(\Theta_\mathrm{p}\right)\right)} {1 + \gamma_0^2 \mathrm{tan}^2\left(\Theta_\mathrm{p}\right)} \end{align} \tag{ 4 }$$

corrects the effect of the non-relativistic kinematic formulas used in MCNPX for the transformation of the differential scattering cross section in the centre-of-mass system to the recoil proton emission cross section in the laboratory system [24]. Here,

$$\begin{align} \gamma_0 = \frac {m_\mathrm{n}+m_\mathrm{p}+E_\mathrm{n}} {\left(m^2_\mathrm{n}+m^2_\mathrm{p}+2m_\mathrm{p}\left(m_\mathrm{n}+E_\mathrm{n}\right)\right)^{1/2}} \end{align} \tag{ 5 }$$

denotes the Lorentz factor for n–p scattering. The correction factor $k_\mathrm{rel}$ is evaluated at the mean proton emission angle of $\bar \Theta_\mathrm{p}$ in the laboratory system and ranges from 1.001 at 2 MeV to 1.008 at 14.8 MeV.

In the same way as for the RPPC, the number $C_\mathrm{p}$ of simulated events per unit source yield is broken down into the number of events expected for an ideal mass-less RPT and a kernel factor $G_\mathrm{MC}$ that describes the neutron and proton transport effects in the simulation of the RPT,

$$\begin{align} C_\mathrm{p} = k_{E} \, k_\mathrm{sc,p} \, G_\mathrm{MC} \, \epsilon_\mathrm{geo} \, n^{^{\prime\prime}}_\mathrm{H} \, \sigma_\mathrm{np}\left(E_\mathrm{n,dir}\right). \end{align} \tag{ 6 }$$

Here, $n^{^{\prime\prime}}_\mathrm{H}$ denotes the number of hydrogen nuclei per unit area of the radiator and $\epsilon_\mathrm{geo}$ the geometric neutron detection efficiency of the RPT. The correction factor $k_\mathrm{sc,p} \approx 1$ accounts for neutrons scattered in the target and producing recoil proton events in the peak region. The correction factor $k_{E} = 1$ is used to introduce the uncertainty of $\sigma_\mathrm{np}$ due to the uncertainty of the mean energy of the direct neutrons. This correction is only relevant for the neutron fields produced with the D(d,n)³He reaction.

As shown by Sloan and Robertson [25], $\epsilon_\mathrm{geo}$ can be calculated numerically by integrating over the surfaces of the radiator and the aperture in front of the SB detector, using the differential proton emission cross section as a weight. Thus, the uncertainty of the angular distribution around 180^∘ enters into the uncertainty of $\epsilon_\mathrm{geo}$ together with the uncertainties of the mechanical dimensions. In order to separate the contributions resulting from mechanical dimensions and from nuclear data, the contribution from the relative angular distribution was added quadratically to that from the integrated cross section $\sigma_\mathrm{np}$, taken from the ENDF/B-V evaluation.

Table 2 shows a comparison of geometrical efficiencies calculated numerically and with MCNPX using identical geometric parameters. The Romberg integration scheme was used for the numerical calculation, and the MCNPX model was simplified accordingly by voiding all volumes except for a $100\,\unicode{x03BC}$g cm⁻² hydrogen radiator. The good agreement between the $N_\mathrm{p}/Y_\mathrm{n}$ values confirms the validity of the Monte Carlo calculations, especially the accuracy of the tabular cross section data at scattering angles of 180^∘ in the centre-of-mass system. Table 3 shows the relative uncertainties for the quantities appearing in (3) and (6). The uncertainty of $G_\mathrm{MC}$ was evaluated in same way as that for the RRPC.

Table 2. Ratio of the quantity $N_\mathrm{p} / Y_\mathrm{n}$ calculated with the Monte Carlo code MCNPX and using numerical integration. The relativistic correction factor $k_\mathrm{rel}$ was used to correct for the use of non-relativistic kinematics in MCNPX. The distance between the source and the radiator surface of the RPT is denoted by d₁. The uncertainty is the statistical uncertainty of the Monte Carlo calculation.

$E_\mathrm{n}$ (MeV)	d1 (cm)	$k_\mathrm{rel}$	$k_\mathrm{rel} (N_\mathrm{p}/Y_\mathrm{n})_\mathrm{MC} / (N_\mathrm{p}/Y_\mathrm{n})_\mathrm{num}$
1.2	22.539	1.0006	0.9955(27)
2.5	22.539	1.0013	1.0006(27)
5.0	37.239	1.0027	0.9973(31)
8.0	37.239	1.0043	1.0001(31)
14.8	37.239	1.0079	1.0034(31)
17.0	22.539	1.0091	1.0018(26)
19.0	22.539	1.0101	0.9964(26)

Table 3. Uncertainty contributions to the yield $Y_\mathrm{n,dir}$ of direct neutrons measured with T1.

Quantity	Rel. uncertainty	Stat.	Non-stat.
$N_\mathrm{p}$	0.01–0.02	×
kτ	0.0005–0.005	×
$k_\mathrm{tail}$	0.005–0.008	×
$k_{d}$	0.002–0.003	×
$k_\mathrm{fit}$	0.005–0.008	×
$k_{E}$	0.001–0.006	×
$k_\mathrm{sc,p}$	0.000–0.003		×
$G_\mathrm{MC}$	0.007		×
$n^{^{\prime\prime}}_\mathrm{H}$	0.008–0.014		×
$\epsilon_\mathrm{geo}$	0.003 a		×
$\sigma_\mathrm{np}$	0.010–0.018 b		×
$C_\mathrm{p}$	0.016–0.022 c		×
$Y_\mathrm{n,dir}$	0.019–0.031

^a Only uncertainty of mechanical dimensions. ^b Including the uncertainty of the angular distribution at 180^∘. ^c Cumulated contributions from $k_{E}$, $k_\mathrm{sc,p}$, $G_\mathrm{MC}$, $\epsilon_\mathrm{geo}$, $n^{^{\prime\prime}}_\mathrm{H}$ and $\sigma_\mathrm{np}$.

The De Pangher long counter [26] LC1 of PTB is one out of a series of identical instruments manufactured with very small tolerances [27]. Other long counters of this series are used for neutron fluence measurements at several national metrology institutes. As shown in figure 12, the long counter consists of a cylindrical PE moderator surrounding a proportional counter filled with BF₃ enriched in ¹⁰B. An intermediate cylindrical layer is made of borated PE and acts as a shield against external thermal neutrons.

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The long-term stability of LC1 is regularly checked with an Am–Be neutron source inserted into a channel in the inner PE moderator layer. After correction for the decay of ²⁴¹Am, the event rates measured with this source showed a relative standard deviation of 0.0035 over the period from 2010 to 2022.

The analysis of the measurements performed with LC1 is largely based on the extensive studies of long counters carried out at the National Physical Laboratory (NPL) in Teddington (UK) [28]. Therefore, the MCNP model developed at the NPL was also adopted to model the LC1 PLC for the present study. A recent review of this model by N J Roberts (NPL) showed that the borated PE of the thermal shield was modelled assuming a boron content of 5% of ¹⁰B per unit mass [29]. This is not consistent with the available information [29] which suggests a mass fraction of 5% of ^natB per unit mass instead. Therefore, the model for LC1 used for this study was modified accordingly. In addition, a minor change was made for the PTB-specific mechanical support and the divergent source models also used for P2 and T1 were implemented.

Figure 13 shows the fluence response $R_\Phi = N_\alpha / \Phi$ of the LC1 PLC, where N_α denotes the number of ¹⁰B(n,$\alpha_1+\alpha_2$) events. The fluence response data were calculated for a unidirectional extended neutron field and the two isotopic compositions of the borated PE in the thermal shield. All calculations were made for a nominal BF₃ density $\rho_\mathrm{nom} = 9.5414 \times 10^{-4}\,\mathrm{g}\,\mathrm{cm}^{-3}$ and a ¹⁰B enrichment of 96%. On average, the response calculated for the thermal shield containing ^natB is about 2.1% higher than that for the original composition assumed at the NPL. At energies below 1 MeV, the deviation increases to 2.4%.

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The fluence response of a long counter is calibrated by adjusting the nominal BF₃ density in the proportional counter such that measurements in known neutron fields are reproduced on average. At the NPL, a set of six radionuclide neutron sources was used. These sources had been calibrated in the NPL manganese bath [30, 31]. At PTB, only ²⁵²Cf sources are available for this purpose. The emission rates of these sources are traceable to the NPL manganese bath and the correction factors $k_\mathrm{an}(\Theta)$ for the angular anisotropy of the sources were measured at PTB. Measurements with two different sources were carried out in 2012 and 2023 in the PIAF low-scatter hall. For each source, two measurements were made at different distances. Table 4 shows the correction $k_\mathrm{Cf}$ for the response of LC1 in the ²⁵²Cf prompt fission neutron spectrum,

$$\begin{align} k_\mathrm{Cf} = 4\pi \frac {\dot Y_\mathrm{nom}} {k_\mathrm{an}\left(90^{\circ}\right)\,B}, \end{align} \tag{ 7 }$$

where B denotes the emission rate of the ²⁵²Cf source and $\dot Y_\mathrm{nom}$ the source yield rate calculated using the nominal BF₃ density $\rho_\mathrm{nom}$. The generalised weighted mean [32] of the four measurements is $\bar{k}_\mathrm{Cf} = 0.995(8)$. It accounts for correlations induced by using the same ²⁵²Cf source for the measurements at the two distances, by the MCNPX simulation and by the long-term stability of LC1. To properly account for the dependence of the partial self-shielding in the BF₃ counter on the energy of the incident neutrons, the correction factor was not applied directly to the response calculated for the nominal BF₃ density $\rho_\mathrm{nom}$, but the BF₃ density was reduced to $9.4818 \times 10^{-4}\,\mathrm{g}\,\mathrm{cm}^{-3}$. This brings the correction factor to unity, $\bar k_\mathrm{Cf} = 1.000(8)$. Its uncertainty indicates the uncertainty of the normalisation of the LC1 response to the ²⁵²Cf emission rate. Figure 14 shows the ratio of the fluence response calculated for LC1 to that calculated for the NPL's PLC, that is, for a thermal shield made of PE(¹⁰B) and the nominal BF₃ density $\rho_\mathrm{nom} = 9.5414 \times 10^{-4}\,\mathrm{g}\,\mathrm{cm}^{-3}$. In the energy range between 100 keV and 20 MeV, the residual energy dependence of the response ratio amounts to about 0.5%.

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Table 4. Correction factor $k_\mathrm{Cf}$ for the response of the LC1 long counter in the ²⁵²Cf PFNS. The distance from the centre of the ²⁵²Cf source to the surface of the moderator surrounding the BF₃ counter is denoted by d₀. The uncertainties indicated for $k_\mathrm{Cf}$ comprise the statistical uncertainty of the measured event rate $\dot N_\alpha$, the effect of the field divergence and the total uncertainty of the emission rate B of the ²⁵²Cf sources.

Date	Source	d0 (cm)	$B (10^5\,\mathrm{s}^{-1}$)	$\dot N_\alpha (\mathrm{s}^{-1}$)	$k_\mathrm{Cf}$
12 March 2012	3524NC	227.7	7.28(11)	3.713(15)	0.990(17)
14 March 2012	3524NC	372.7	7.27(11)	1.363(8)	0.983(17)
22 January 2023	NS0520	240.8	38.68(27)	17.833(12)	0.998(9)
29 January 2023	NS0520	372.7	38.49(27)	7.295(9)	0.994(9)

As for T1 and P2, the measurements with LC1 aim at determining the source yield $Y_\mathrm{n} = (\mathrm{d}N_\mathrm{n}/\mathrm{d}\Omega)$ and not the fluence Φ at a particular point in space. Therefore, it is not necessary to use the concept of the effective centre for the analysis of measurements at arbitrary distances. Instead, the response of LC1 is calculated with MCNP5 for two standard distances using a divergent source model and including a cone filled with air between the source and the long counter. The two standard distances d₀ between the reference surface of LC1 and the target are 379.3 cm and 247.3 cm for neutron energies above and below 500 keV, respectively. For these distances, dedicated shadow cones are available for subtracting the events induced by neutrons scattered in air or from structural materials.

The yield of direct neutrons $Y_\mathrm{n,dir}$ is calculated from the net number N_α of detected ¹⁰B(n,α₀+α₁)⁷Li events and the number C_α of simulated events per unit yield $Y_\mathrm{n}^\mathrm{sim}$ of source neutrons,

$$\begin{align} Y_\mathrm{n,dir} = \frac {k_\tau \, N_\alpha} {k_\mathrm{sc}\, \bar{k}_\mathrm{Cf} \, C_\alpha}. \end{align} \tag{ 8 }$$

The correction factor $k_\mathrm{sc} = 1 + Y_\mathrm{n,sc}^\mathrm{sim} / Y_\mathrm{n,dir}^\mathrm{sim}$ accounts for the simulated events induced by scattered neutrons.

Table 5 shows the relative uncertainties for the quantities appearing in (8). The relative uncertainty of the fluence response $R_\Phi$ was investigated in detail by Roberts et al [28]. The first result of 0.014 [28] was later increased to 0.020 [5] and this value is adopted for relative uncertainty of the quantity C_α . The uncertainty of the correction factors for the effect of scattered neutrons are fully correlated for P2 and LC1, meaning that the covariance $\mathrm{cov}(k_\mathrm{sc,f},k_\mathrm{sc})$ must be subtracted from the total variances when the uncertainty of the ratio of normalised yields $(Y_\mathrm{P2}/M_\mathrm{P2})/(Y_\mathrm{LC1}/M_\mathrm{LC1})$ is calculated. Here, $M_\mathrm{P2}$ and $M_\mathrm{LC1}$ denote the corrected numbers of monitor events for the respective measurements.

Table 5. Uncertainty contributions to the yield of direct neutrons measured with LC1.

Quantity	Rel. uncertainty	Stat.	Non-stat.
Nα	0.001–0.003	×
kτ	0.001	×
$\bar k_\mathrm{Cf}$	0.008	×
$k_\mathrm{sc}$	0.005–0.024		×
Cα	0.020 a		×
$Y_\mathrm{n,dir}$	0.022–0.032

^a Relative uncertainty adopted from the NPL contribution to [5].