# Comparison of Methods to Determine the Fluence of Monoenergetic Neutrons in
the Energy Range from 30 keV to 14.8 MeV

R. Nolte1111retired and B. Lutz1 1 Physikalisch-Technische Bundesanstalt,
Bundesallee 100, 38116 Braunschweig, Germany<EMAIL_ADDRESS>
###### Abstract

The primary reference instruments for neutron fluence measurements used at the
Physikalisch-Technische Bundesanstalt (PTB), are based on the primary standard
for neutron measurements, which is the differential neutron-proton scattering
cross section. Such instruments require considerable effort for their
operation and analysis. Therefore, routine measurements are carried out using
a transfer instrument to facilitate the efficient provision of services to
customers. A series of measurements was conducted to compare the transfer
device to the primary reference instruments and ensure the traceability of
neutron fluence measurements. This resulted in an improved characterization of
the instrument and new analysis procedures.

††:

Keywords: fluence of monoenergetic neutrons, recoil proton detectors, De
Pangher long counter

##  1  Introduction

At the PTB Ion Accelerator Facility (PIAF), neutron fields are generated for
the characterization of neutron detectors and for various other applications.
These range from measurements of nuclear data to the irradiation of samples
for radiobiological studies and radiation hardness testing. The basic quantity
to be determined for these services is the neutron yield, that is, the number
of neutrons
$Y_{\mathrm{n}}=(\mathrm{d}N_{\mathrm{n}}/\mathrm{d}\Omega)(\Theta_{\mathrm{n}})$
emitted per unit solid angle from the production target at an angle
$\Theta_{\mathrm{n}}$ relative to the direction of the ion beam. Over a period
of forty years, various methods have been implemented for measurements in
monoenergetic and quasi-monoenergetic neutron fields at PIAF: the recoil
proton proportional counter (RPPC) P2 [1], the recoil proton telescope (RPT)
T1[2] and the precision long counter (PLC) LC1 [3]. In addition, 235U and 238U
fission ionization chambers, several liquid scintillation detectors and
6LiGlas scintillation detectors are in use at PIAF.

The P2 RPPC and the T1 RPT are primary reference instruments that allow the
determination of the neutron fluence from the mechanical dimensions of the
instruments, the amount of hydrogen nuclei in the sensitive volumes and the
differential neutron-proton (n-p) scattering cross section. This cross section
is the primary standard for neutron metrology [4]. For fast neutrons, elastic
scattering is the only open channel in the neutron-proton system because the
neutron capture cross section is more than four orders of magnitude smaller
than the elastic scattering cross section. The n-p differential scattering
cross section can be determined with a relative measurement of the angular
distributions of the scattered neutrons and the recoil protons together with
another relative measurement of the total hydrogen cross section. Therefore,
‘absolute’ neutron measurements can be carried out using these instruments.
‘Absolute’ here means that, in principle, only counting of events is needed.
In reality, however, corrections are always required that compromise the ideal
image.

In contrast to the primary reference instruments, the LC1 PLC is a transfer
instrument which must be calibrated, for example using a 252Cf neutron source
with known emission rate. In addition, the relative energy dependence of the
fluence response must be calculated with the Monte Carlo method. The emission
rate of 252Cf neutron sources can be measured using a manganese bath, which is
a primary reference instrument as well. The PLC can therefore be used to link
two primary methods of neutron measurement. The uncertainty of the fluence
response of the primary reference instruments is largely determined by the
uncertainty of the n-p scattering cross section at the energy of the incident
neutron. In contrast, the fluence response of the long counter is sensitive to
the uncertainty of the hydrogen and carbon cross sections over the whole
energy range from the energy of the incident neutron to thermal energies. A
comparison of the primary reference instruments can therefore be regarded as
an integral experiment testing the overall uncertainty of the carbon and
hydrogen cross sections.

The operation and analysis of the primary reference instruments imposes
restrictions. This is especially true in terms of the technical effort, the
long measurement times required and the more complex analysis, which makes the
routine use of these instruments less favourable. In contrast, a PLC is a very
stable and easy-to-use instrument for the time-efficient characterization of
the neutron fields which helps users to make the best use of the allocated
beam time. It is therefore important to demonstrate the traceability of LC1
measurements to the primary reference instruments.

An initial attempt to compare P2, T1 and LC1 was made during the last key
comparison CCRI(III)-K11 [5]. At this time, the PLC was employed in parallel
to the RPPC and the RPT where possible, or it was used as a transfer
instrument when the RPPC could not be employed for technical reasons [6]. The
present work reports on a more extensive series of measurements aiming at a
comparison of the LC1 transfer instrument and the primary reference
instruments P2 and T1 in the neutron energy range from 30 keV to 14.8 MeV.
This extends the measurements already performed within CCRI(III)-K11 and
provides data for crosschecking of the relative energy dependence of the PLC
fluence response calculated with the Monte Carlo method.

The analysis procedures reported here deviate from the ‘standard’ procedure
discussed in the PTB report [6] for CCRI(III)-K11. This pertains to enhancing
the characterization of instruments, calculating corrections, and utilizing
state-of-the-art Monte Carlo codes to simulate the instruments.

In the present work, the differential n-p scattering cross section
$\sigma_{\mathrm{np}}$ of the ENDF/B-V evaluation [7] is still used for the
analysis of the measurement with the primary reference instruments. This
follows an agreement within the CCRI(III) committee to not accept the
unmotivated increase of the differential n-p scattering cross section at
backward angles in the energy range between 10 MeV and 15 MeV introduced in
the ENDF/B-VI evaluation and later removed again to large extend in the
ENDF/B-VII evaluation.

As shown in the upper panel of figure 1, the deviation between n-p cross
sections from the most recent evaluation ENDF/B-VIII [4] and ENDF/B-V
evaluation is smaller than the uncertainty of the ENDF/B-V data used in the
present study. The lower panel compares the backward angular distribution
$f($$)=2\pi\sigma^{-1}_{\mathrm{np}}(\mathrm{d}\sigma_{\mathrm{np}}/\mathrm{d}\Omega)($$)$
of the differential scattering cross section in the center-of-mass system that
is required for the analysis of the T1 measurements (see 3.2.3).
Unfortunately, an estimate of the uncertainty of the differential center-of-
mass scattering cross section at $None$ is only available for the ENDF/B-VIII
evaluation [8]. Therefore, the relative uncertainty of the backward angular
distribution $f($$)$ was obtained by quadratically subtracting the relative
uncertainty of the ENDF/B-VIII scattering cross section $\sigma_{\mathrm{np}}$
from that of the ENDF/B-VIII backward differential cross section
$(\mathrm{d}\sigma_{\mathrm{np}}/\mathrm{d}\Omega)($$)$. This is a lower bound
for the uncertainty of the ENDF/B-V angular distribution at $None$. Also here,
the deviation between the two evaluations is smaller than the estimated
uncertainty of the backward scattering cross section.

Figure 1:  The red solid lines show the ratios of the n-p scattering cross
section $\sigma_{\mathrm{np}}$ (upper panel) and backward angular
distributions $f($$)$ (lower panel) from the ENDF/B-VIII and ENDF/B-V
evaluations. The uncertainty band (grey) in upper panel shows $1\pm
u_{\mathrm{np,rel}}$, where $u_{\mathrm{np,rel}}$ denotes the relative
uncertainty of the ENDF/B-V cross section used in the present study. Due to a
lack of covariance data for the ENDF/B-V differential cross section, the
uncertainty band in the lower panel was obtained from the relative uncertainty
of the ENDF/B-VIII backward diffferential cross section [8].

This study reports the results for the energy region between 30 keV and 15 MeV
222All uncertainties indicated in this publication are standard measurement
uncertainties with a coverage factor $k=1$.. The energy range from 15 MeV to
19 MeV will be covered in a forthcoming publication. The monoenergetic neutron
fields are described in section 2. Section 3 gives a summary of the properties
of the instruments and the analysis methods. The results of the comparison are
presented and discussed in the final sections.

##  2  Monoenergetic Neutron Fields

###  2.1  Neutron Production

The measurements reported here were carried out in the PIAF low-scatter hall
during two periods from 2014 to 2016 and 2020 to 2023. Monoenergetic neutrons
were generated using proton and deuteron beams from the 4 MV Van-de-Graaff
accelerator (2014 – 2016), the 2 MV Tandetron accelerator (2020 – 2023) and
the CV28 cyclotron together with the nuclear reactions 7Li(p,n)7Be (30 keV -
750 keV), 3H(p,n)3He (750 keV - 4 MeV), D(d,n)3He (4 MeV - 8 MeV) and
3H(d,n)4He (14.8 MeV - 20 MeV).

The proton and deuteron energies of the beams produced with the Van-de-Graaff
and the Tandetron accelerators were determined with a $None$ analysing magnet
that was calibrated using a set of (p,$\gamma$) resonances and (p,n) reaction
thresholds. Nevertheless, deviations between the preset and actual beam energy
of 2 keV to 6 keV were observed frequently during routine operation of the
accelerators and the beam transport system. These deviations would lead to an
unacceptable uncertainty in the mean neutron energy when the beam energy is
close to the reaction threshold. One such instance would be the production of
neutrons with energies between 140 keV and 500 keV using the 7Li(p,n)7Be
reaction. For these neutron fields, the mean neutron energy
$\bar{E}_{\mathrm{n}}$ of the monoenergetic peak was determined with a
cylindrical 3He proportional counter (outer diameter: 50 mm, outer length: 346
mm, 3He pressure: 0.1 MPa). The neutron energy is determined from the
$Q$-value of the 3He(n,p)3H reaction and the location of two peaks in the
pulse-height distribution originating from thermalized room-return neutrons
and from the direct neutrons. This method yields the mean neutron energy with
a relative uncertainty of about 1 %, that is, 1.4 keV for a mean neutron
energy of 144 keV.

The energies of the deuteron beams from the cyclotron were analysed using two
$None$ dipole magnets. The width of the deuteron energy distributions was
about 20 keV for these beams.

###  2.2  Targets

The lithium targets were prepared at the PIAF target laboratory by physical
vapour deposition (PVD) of natLi layers on 0.5 mm tantalum backings. To
prevent the lithium surface from corroding, the lithium targets were handled
without exposing the target to air from production to mounting on the beam
line. Neutron fields with energies above 140 keV were always produced at a
neutron emission angle of $None$ with respect to the ion beam direction. For
energies between 140 keV and 30 keV the 7Li(p,n)7Be reaction was used at
neutron emission angles up to $None$. In these cases, the mean zero-degree
neutron energy was always 144 keV. By using lithium targets instead of the
more readily available lithium fluoride targets, the intense production of
parasitic high-energy photons by the 19F(p,$\alpha\gamma$)16O reaction is
avoided. This helps to reduce the interference of photon and neutron-induced
events in the RPPC.

The tritium targets consisted of tritium gettered in titanium layers that were
produced by PVD on 0.5 mm silver or aluminum backings. These targets were
obtained from a commercial supplier and stored in vacuum when not in use.

To spread out the heat deposited in the target by the ion beam, the solid-
state targets perform an oscillatory movement perpendicular to the ion beam at
a frequency of about 1 Hz. Due to the curved target surface, this movement
induces a variation of the distance from a reference point to the beam spot on
the reactive target layer. The distribution of the distances has a standard
deviation of less than 0.3 mm. This is of relevance for the use of the RPT T1
at small distances from the target (see 3.2.1).

Neutron fields with energies between 5 MeV and 8 MeV were generated with a D2
gas target. The deuterium gas target is a 30 mm long stainless-steel cell with
a diameter of 11 mm and a wall thickness of 0.2 mm. The entrance window is
covered by a 5 m molybdenum foil which can stand gas pressures up to 0.2 MPa.
The deuteron beam is stopped in a 0.5 mm thick gold foil at the rear end of
the cell.

###  2.3  Target Scattering

The quantity to be measured in a neutron field is the fluence
$\Phi_{\mathrm{dir}}$ of neutrons leaving the target without further
interaction (direct neutrons) [9]. Therefore, the contribution of neutons
interacting with the target setup must be included in the calculation of the
response of the reference instruments and in the analysis of measurements. It
should be noted here that this includes also that part of the energy
distribution $(\mathrm{d}\Phi_{\mathrm{sc}}/\mathrm{d}E_{\mathrm{n}})$ of the
scattered neutrons which falls into the energy region covered by the direct
neutrons.

At PIAF, a low-mass setup is used for the solid-state targets which restricts
the relative number of neutrons scattered in the target setup (scattered
neutrons) to less than 10 %. Because of the high-density backings, the most
problematic targets in this respect are the lithium targets on 0.5 mm tantalum
backings. The energy distribution of the direct neutrons is determined from a
simulation of the the ion beam transport and the neutron production in the
target layer using the evaluated differential neutron production cross
sections. This is followed by the simulation of neutron scattering in the
target setup.

The TARGET Monte Carlo code [10] provides the energy distributions of direct
and scattered neutrons in a very time-efficient way by using point detectors,
but it has several known shortcomings. Therefore, the energy distributions of
scattered neutrons were crosschecked using MCNP5 with ENDF/B-VII cross section
data. Figure 2 shows two typical cases. While the relative number of scattered
neutrons predicted by TARGET and MCNP5 are quite close for the fields produced
with tritiated titanium targets, larger deviations were observed for the
fields generated with the lithium targets. It is most probable that the
discrepancy is due to deficient differential cross section data for elastic
scattering from tantalum. As discussed in 3.1, this procedure is supported by
results of measurements using the RPPC.

For lithium targets, TARGET was therefore only used to calculate the energy
distribution of the direct neutrons. The energy distribution of scattered
neutrons was obtained with MCNP5 [11], using the differential neutron
production cross section data from the DROSG2000 database [12] to model the
source term. This source term had a unique relation between neutron energy and
the emission angle, meaning that the energy loss of the ions in the reactive
layer was neglected. A comparison with more elaborate source descriptions [13]
showed that this was fully sufficient for the energy distribution of the
scattered neutrons. The two calculations were normalized to the same yield of
direct neutrons at zero degree.

For the tritiated titanium targets on silver and aluminum backings, TARGET was
used to calculated both distributions. The uncertainty of this procedure was
estimated to be about 50 % of the difference between TARGET and MCNP5.

Figure 2:  Energy distributions for the 250 keV (upper panel) and 1.2 MeV
(lower panel) neutron fields calculated using TARGET (black: direct and
scattered neutrons, grey: scattered neutrons only) and MCNP5 (red: scattered
neutrons only). For the 250 keV field, the ratios of the scattered and the
direct neutron fluence is 2.6 % (TARGET) and 5.4 % (MCNP5). For the 1.2 MeV
field, the ratios is 3.4 % (TARGET) and 4.3 % (MCNP5).

Due to the forward-peaked differential cross section of the D(d,n)3He reaction
and the geometry of the gas target, the relative contribution of scattered
neutrons in the neutron fields produced with the D2 gas target is
significantly less than for the solid-state targets. Consequently, the TARGET
code does not provide the energy distribution of the scattered neutrons for
these fields. For the present study, the relative contribution of the
scattered neutrons has been calculated in the same way as for the lithium
targets. It varied between 1.5 % for the 5 MeV field and 1.1 % for the 8 MeV
field. Figure 3 shows the neutron energy distribution for the 5 MeV neutron
field.

Figure 3:  Energy distributions for the 5 MeV neutron field target calculated
using TARGET (black histogram) and MCNP5 (red histogram, scattered neutrons
only). The ratios of the scattered and the direct neutron fluence calculated
with MCNP5 is 1.5 %. The TARGET code does not provide the energy distribution
of scattered neutrons. The neutron field was produced with the D2 gas target.

The energy distributions of the direct neutrons calculated with the TARGET
code are averaged over the sensitive cross-sectional area of the detector. The
corresponding opening angles covered by the instruments at their nominal
positions vary between $None$ (P2) and $None$ (LC1 at 247.3 cm). For the
calculation of the response of P2, T1 and LC1 using MCNP5 and MCNPX, the
angular distributions of scattered and direct neutrons were therefore assumed
to be isotropic over the range of neutron emission angles covered by the
instruments. Only for the D(d,n)3He reaction the angular distribution of the
direct neutrons was modelled using the differential cross section at the
deuteron energy corresponding to the mean energy of the direct neutrons at
$None$.

###  2.4  Monitors

Measurements with different detectors were related to each other by
simultaneously recording the integrated beam current $Q$ and the number of
events from a set of three neutron monitors: a 3He long counter (NM) at
$None$, a BF3 long counter (PLM) at $None$ and a 3He counter (3He) in a cover
of borated polyethylene at an angle of about $None$. The distances of the
neutron monitors NM, PLM and 3He from the target were 6.05 m, 5.30 m and 0.80
m, respectively. For most beam settings, the relative standard deviation of
the ratio of the number of neutron monitor events to the integrated beam
current was less than 0.5 % during one measurement period with the same ion
beam and neutron production target. For the majority of the measurements, the
NM monitor was the most stable one. Therefore this monitor was usually used to
analyse the measurements. The scattering of neutrons from other detectors into
the monitors was corrected for by performing short subsequent runs with the
respective detector at its measurement position and with the detector removed
from the neutron field. During these runs, the integrated beam current $Q$ was
used as a monitor to determine the number of additional monitor events induced
by neutrons scattered from the detector. For the detectors employed in this
study, the relative effect of inscattering ranged between 1 % and 2 % at
maximum and was rather independent of the neutron energy. Figure 4 shows two
sets of normalized monitor ratios for measurements at 250 keV and 14.8 MeV.

Figure 4:  Normalized ratios of the number of neutron monitor events to the
integrated beam current $Q$. For the measurements marked with ‘no det.’, all
reference instruments were removed from the neutron field to suppress
scattering from these instruments into the monitor detectors. Measurements
with the shadow cone in place ar denoted by ‘+SC’. The relative standard
deviations of the NM/Q ratios for the data sets at 250 keV and 14.8 MeV were
0.22 % and 0.18 %, respectively

##  3  Methods

###  3.1 Recoil Proton Proportional Counter P2

####  3.1.1 Design of the Instrument

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[14]. 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 aluminum 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.

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 polyethylene shadow cone placed between the
RPPC and the neutron production target.

####  3.1.2 Electrical Field

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 [15]. 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
$k_{\mathrm{V}}=0.985(8)$ was therefore applied to the geometrical volume
$V_{\mathrm{geo}}=4.680(10)\,\times 10^{2}\,\mathrm{cm}^{3}$.

Figure 5:  The P2 recoil proton proportional counter is used as the primary
reference instrument for the neutron energy range from 24 keV to 2 MeV. The
geometrical sensitive volume S is shaded in grey; the cathode is denoted by C
and the anode wire by A. Field and guard tubes are marked by F and G,
respectively. The guard tubes are held at anode potential while the field
tubes are at ground potential. The ratio of anode and cathode potentials is
1.402. The anode wire is a gold-plated tungsten wire 100 m in diameter. Figure
6:  Equipotential surfaces (light blue lines) inside P2 and the electrical
field line (red line) touching the end of the field tube. This field line
defines the surface of the sensitive volume. The axial and radial coordinates
$Z$ and $R$ are the distances from the center of the sensitive volume and the
counting wire, respectively. The electrical field was calculated with COMSOL
Multiphysics [15].

####  3.1.3 Counting Gas

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 ionization 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[16]
and on two measurements of the relative $W$-value $W(E_{\mathrm{p}})/W_{0}$
with the P2 RPPC, where $W_{0}$ 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 [17], [18], that is,
$W(E_{\mathrm{p}})/W_{0}=1$ for the hydrogen-methane mixtures.

Figure 7:  Energy dependence of the relative W-value for protons in propane.
The red circles show measurements of the absolute W-value by Posny et al[16].
The data shown by the blue triangles are relative W-values normalized to the
Posny data at 200 keV. They were obtained with the P2 RPPC by measuring the
variation of the recoil proton edge with neutron energy. The neutron energy
was varied by changing the angular position of the RPPC. The linear fit on the
logarithmic energy scale (black solid line) was used for analysing the PTRAC
file produced by MCNPX.

####  3.1.4 Monte Carlo Simulation

The neutron response of the counter was calculated using MCNPX, v2.7 [19].
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 $d_{0}$ of
115.25 cm from the nominal center 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 $None$. The PTRAC files
produced by MCNPX were analyzed 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. 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 normalized to the
unit fluence of direct neutrons $\Phi_{\mathrm{dir}}$ in vacuum at the
position of the center 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.

Figure 8:  Measured (black histogram) and calculated (lines) pulse height
distributions calculated including the energy distribution of neutrons
scattered in the target. The simulations were carried out with MCNP5 (red
solid line) and TARGET (blue dashed line). The dashed-dotted line shows the
contribution from the direct neutrons only. The blue arrow marks the lower
bound of the fit region.

Figure 9 shows the effect of an energy-dependent $W$-value for propane on the
pulse-height distribution for 1.2 MeV 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.

Figure 9:  Measured (black histogram) and calculated (lines) pulse-height
distributions calculated using the energy-dependent relative $W$-value data
shown in figure 7 (red solid line) and an energy-independent $W$-value (blue
dashed-dotted line). The fit was restricted to the pulse height region above
270 keV. The edge at about 100 keV is due to carbon recoil nuclei. Recoil
nuclei with $Z>2$ are not transported by MCNPX. Hence, this structure is
missing in the calculated pulse height distribution.

####  3.1.5 Analysis of P2 Measurements

The neutron fluence measurements were analyzed 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 tim 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’$\gamma$).

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

${Y_{\mathrm{n,dir}}=\frac{k_{\tau}\,N_{\mathrm{p}}}{k_{V}\,k_{pT}\,k_{\mathrm{fit}}\,C_{\mathrm{p}}}d^{2}.}$
(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 $k_{\tau}$ 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 $p_{0}$
and $T_{0}$ 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 center 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 $d_{0}$, 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,

${C_{\mathrm{p}}=k_{E}\,k_{\mathrm{sc,f}}\,G_{\mathrm{MC}}\,n_{\mathrm{H}}\,V_{\mathrm{geo}}\,\sigma_{\mathrm{np}}(E_{\mathrm{n,dir}}).}$
(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 $p_{0}$ 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 fielsds produced with the 3H(p,n)3He reaction,
the relative uncertainty of $\sigma_{\mathrm{np}}$ is smaller than $2\times
10^{-3}$.

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.
---|---|---|---
$N_{\mathrm{p}}$ | 0.0030 \- 0.010 | $\times$ |
$k_{\tau}$ | 0.0002 | $\times$ |
$k_{V}$ | 0.0074 |  | $\times$
$k_{pT}$ | 0.0020 | $\times$ |
$k_{\mathrm{fit}}$ | 0.0030 \- 0.015 |  | $\times$
$d$ | 0.0020 | $\times$ |
$k_{E}$ | 0.0040 \- 0.008 a | $\times$ |
$k_{\mathrm{sc,f}}$ | 0.0040 \- 0.014 |  | $\times$
$G_{\mathrm{MC}}$ | 0.0050 \- 0.013 |  | $\times$
$n_{\mathrm{H}}$ | 0.0020 |  | $\times$
$V_{\mathrm{geo}}$ | 0.0020 |  | $\times$
$\sigma_{\mathrm{np}}$ | 0.0050 \- 0.009 |  | $\times$
$C_{\mathrm{p}}$ | 0.0150 \- 0.023 b |  | $\times$
$Y_{\mathrm{n,dir}}$ | 0.0150 \- 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 $\sigma_{\mathrm{np}}$

###  3.2 Recoil Proton Telescope T1

####  3.2.1 Design of the Instrument

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 detector (SB). 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 s. 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
$10^{3}$. Accidental coincidences produced by these events are measured in
separate background runs with the radiator turned by $None$, meaning towards
the neutron source. The upper panel of figure 11 shows foreground and
background pulse height distributions measured for 2.5 MeV neutrons.

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 characterized 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/cm2 to 10.0855(28) mg/cm2 was
used to cover the energy range 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.

Figure 10:  The recoil proton telescope T1 is used as the primary reference
instrument for the neutron energy range from 2 MeV to 20 MeV. The radiator
with the tristearine layer is marked by R; the proportional counters and the
silicon surface barrier detector are denoted by P and D, respectively. The
tantalum aperture A in front of the silicon detector and a mask directly in
front of the tristearine layer define the solid angle for the detection of the
recoil protons. The radiator is shown in the intermediate $None$ position. For
the foreground and background runs it is turned such that the tristearine
layer faces the detectors ($None$) or the neutron source ($None$),
respectively. Depending on the single event rates, T1 is used at two distances
between the tristearine layer and the neutron source:
$d_{1}=22.61(2)\,\mathrm{cm}$ and $d_{1}=35.00(2)\,\mathrm{cm}$. The distance
$d_{1}$ is adjusted using calibrated gauges.

Figure 11:  The upper panel shows foreground (black histogram) and background
(red histogram) pulse height distributions in the SB detector of T1 measured
for 2.5 MeV neutrons, using a 0.345 mg/cm2 tristearine radiator. In the lower
panel, the experimental net pulse height distribution is compared with the
MCNPX simulation of T1. For better visibility of the tail region, two bins
were grouped together and the distributions were offset by one event per bin.
The solid and dashed-dotted line show all simulated events and those caused by
neutrons scattered in the target assembly, respectively. The blue arrows mark
the peak and tail regions. The number of simulated events in the tail region
amounts to 77 % of the measured ones.

####  3.2.2 Monte Carlo Simulation

The use of the RPT T1 with the D(d,n)3He source at the PTB TOF spectrometer
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 [19] 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 %. 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,$\alpha$)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.

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, 77 % of the measured tail events were reproduced by
the simulation. This excess background is probably due to accidental
coincidence 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 for by extrapolating the excess events into the peak
region, assuming a linear decrease to zero within the peak region.

####  3.2.3 Analysis of T1 Measurements

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,

$Y_{\mathrm{n,dir}}=\frac{k_{\mathrm{tail}}\,k_{\tau}\,N_{\mathrm{p}}}{k_{\mathrm{d}}\,k_{\mathrm{fit}}\,k_{\mathrm{rel}}\,C_{\mathrm{p}}}.$
(3)

Here, the correction factor $k_{\mathrm{d}}$ 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 $d_{1}$ 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

$k_{\mathrm{rel}}=\frac{(\mathrm{d}\sigma_{\mathrm{np}}/\mathrm{d}\Omega_{\mathrm{p,rel}})}{(\mathrm{d}\sigma_{\mathrm{np}}/\mathrm{d}\Omega_{\mathrm{p,n-rel}})}=\frac{\gamma_{0}^{2}(1+\mathrm{tan}^{2}(\Theta_{\mathrm{p}}))}{1+\gamma_{0}^{2}\mathrm{tan}^{2}(\Theta_{\mathrm{p}})}$
(4)

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

$\gamma_{0}=\frac{m_{\mathrm{n}}+m_{\mathrm{p}}+E_{\mathrm{n}}}{(m^{2}_{\mathrm{n}}+m^{2}_{\mathrm{p}}+2m_{\mathrm{p}}(m_{\mathrm{n}}+E_{\mathrm{n}}))^{1/2}}$
(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,

$C_{\mathrm{p}}=k_{E}\,k_{\mathrm{sc,p}}\,G_{\mathrm{MC}}\,\epsilon_{\mathrm{geo}}\,n^{\prime\prime}_{\mathrm{H}}\,\sigma_{\mathrm{np}}(E_{\mathrm{n,dir}}).$
(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 et al[25], $\epsilon_{\mathrm{geo}}$ can be calculated
numerically by integrating over the surfaces of the radiator and the aperture
in front of the surface barrier detector, using the differential proton
emission cross section as a weight. Thus, the uncertainty of the angular
distribution around $None$ enters into the uncertainty of
$\epsilon_{\mathrm{geo}}$ together with the uncertaintes 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 $100\,\unit{\micro
g/cm^{2}}$ 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 $None$ in the center-of-mass system.

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 $d_{1}$. The uncertainty is
the statistical uncertainty of the Monte Carlo calcuation.

  * $E_{\mathrm{n}}$ (MeV) |  $d_{1}$ (cm) | $k_{\mathrm{rel}}$ | $k_{\mathrm{rel}}(N_{\mathrm{p}}/Y_{\mathrm{n}})_{\mathrm{MC}}/(N_{\mathrm{p}}/Y_{\mathrm{n}})_{\mathrm{num}}$
---|---|---|---
01.2 | 22.539 | 1.0006 | 0.9955(27)
02.5 | 22.539 | 1.0013 | 1.0006(27)
05.0 | 37.239 | 1.0027 | 0.9973(31)
08.0 | 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 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 3:  Uncertainty contributions to the yield $Y_{\mathrm{n,dir}}$ of
direct neutrons measured with T1.

  * Quantity | Rel. uncertainty | Stat. | Non-stat.
---|---|---|---
$N_{\mathrm{p}}$ | 0.0100 \- 0.02 | $\times$ |
$k_{\tau}$ | 0.0005 - 0.005 | $\times$ |
$k_{\mathrm{tail}}$ | 0.0050 \- 0.008 | $\times$ |
$k_{d}$ | 0.0020 \- 0.003 | $\times$ |
$k_{\mathrm{fit}}$ | 0.0050 \- 0.008 | $\times$ |
$k_{E}$ | 0.0010 \- 0.006 | $\times$ |
$k_{\mathrm{sc,p}}$ | 0.0000 \- 0.003 | $\times$ |
$G_{\mathrm{MC}}$ | 0.0070 |  | $\times$
$n^{\prime\prime}_{\mathrm{H}}$ | 0.0080 \- 0.014 |  | $\times$
$\epsilon_{\mathrm{geo}}$ | 0.0030 a |  | $\times$
$\sigma_{\mathrm{np}}$ | 0.0100 \- 0.018 b |  | $\times$
$C_{\mathrm{p}}$ | 0.0160 \- 0.022 c |  | $\times$
$Y_{\mathrm{n,dir}}$ | 0.0190 \- 0.031 |  |

  * a only uncertainty of mechanical dimensions,

  * b including the uncertainty of the angular distribution at $None$

  * c cumulated contributions from $k_{E}$, $k_{\mathrm{sc,p}}$, $G_{\mathrm{MC}}$, $\epsilon_{\mathrm{geo}}$, $n^{\prime\prime}_{\mathrm{H}}$ and $\sigma_{\mathrm{np}}$

###  3.3 De Pangher Precision Long Counter LC1

####  3.3.1 Design of the Instrument

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 polyethylene (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.

The long-term stability of LC1 is regularly checked with an Am-Be neutron
source inserted into a channel in the outer 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.

Figure 12:  The De Pangher long counter LC1 of the PTB. Neutrons are incident
from the right. The reference surface for the measurement of the distance $d$
to the source is the front surface of the inner cylindrical moderator layer
surrounding the 10BF3 proportional counter.

####  3.3.2 Calculation of the Fluence Response

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 $R_{\Phi}$ of the LC1 PLC 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 $\rho_{\mathrm{nom}}=9.5414\times 10^{-4}\,\mathrm{g/cm}^{3}$ 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 %. The relative uncertainty of the fluence response 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 also adopted for the
present study.

Figure 13:  Fluence response $R_{\Phi}$ of the PLC calculated with MCNP5 for
two compositions of the borated PE of the thermal shield. The data for 5 % 10B
per unit mass (red closed circles) are the original NPL data from [28] while
the data for 5 % natB (blue closed squares) were calculated for the PTB’s LC1
PLC. The open blue squares show the results obtained for the same distribution
of source neutrons but for borated PE with 5 % 10B. The calculations were
carried out for a unidirectional broad neutron beam, a BF3 density of
$9.5414\times 10^{-4}\,\mathrm{g/cm}^{3}$ and an 10B enrichment in the BF3 of
96 %. The relative statistical uncertainties are smaller than 1 % (NPL data)
and about 0.4 % (PTB data), respectively.

####  3.3.3 Calibration

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 (PFNS),

$k_{\mathrm{Cf}}=4\pi\frac{\dot{Y}_{\mathrm{nom}}}{k_{\mathrm{an}}($$)\,B},$
(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 generalized 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
$9.4818\times 10^{-4}\,\mathrm{g/cm}^{3}$. This brings the correction factor
to unity, $\bar{k}_{\mathrm{Cf}}=1.000(8)$. Its uncertainty indicates the
uncertainty of the normalization 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 $\rho_{\mathrm{nom}}=9.5414\times
10^{-4}\,\mathrm{g/cm}^{3}$. In the energy range between 100 keV and 20 MeV,
the residual energy dependence of the response ratio amounts to about 0.5 %.

Table 4:  Correction factor $k_{\mathrm{Cf}}$ for the response of the LC1 long
counter in the 252Cf PFNS. The distance from the center of the 252Cf source to
the surface of the moderator surrounding the BF3 counter is denoted by
$d_{0}$. 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 |  $d_{0}$ (cm) |  $B(10^{5}\,\mathrm{s}^{-1}$) |  $\dot{N}_{\alpha}(\mathrm{s}^{-1}$) | $k_{\mathrm{Cf}}$
---|---|---|---|---|---
2012-03-12 | 3524NC | 227.7 |  07.28(11) |  03.713(15) | 0.990(17)
2012-03-14 | 3524NC | 372.7 |  07.27(11) |  01.363(8) | 0.983(17)
2023-01-22 | NS0520 | 240.8 | 38.68(27) | 17.833(12) | 0.998(9)
2023-01-29 | NS0520 | 372.7 | 38.49(27) |  07.295(9) | 0.994(9)

Figure 14:  Ratio of the fluence response $R_{\Phi}$ for the PTB’s LC1 PLC
(thermal shield: PE(natB), BF3 density: $9.4818\times
10^{-4}\,\mathrm{g/cm}^{3}$) to that for the NPL’s PLC (thermal shield:
PE(10B), BF3 density: $9.5414\times 10^{-4}\,\mathrm{g/cm}^{3}$). The data for
the NPL’s PLC are shown by the red solid circles in figure 13. In order to
reduce the statistical uncertainty of the ratios, the two MCNP5 calculations
were carried out for the same distribution of source neutrons. The error bars
shown for representive energies indicate the relative statistical uncertainty
of about 0.4 %.

####  3.3.4 Analysis of LC1 Measurements

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 $\Phi$ at a particular point in space. Therefore, it is not necessary
to use the concept of the effective center 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 $d_{0}$ 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 $N_{\alpha}$ of detected 10B(n,$\alpha$0+$\alpha$1)7Li events and the
number $C_{\alpha}$ of simulated events per unit yield
$Y_{\mathrm{n}}^{\mathrm{sim}}$ of source neutrons,

$Y_{\mathrm{n,dir}}=\frac{k_{\tau}\,N_{\alpha}}{k_{\mathrm{sc}}\,\bar{k}_{\mathrm{Cf}}\,C_{\alpha}}.$
(8)

The quantity $C_{\alpha}$ is related to the fluence response
$R_{\Phi}=C_{\alpha}\,d^{2}$ where $d$ denotes the distance from the neutron
source to the effective center of the long counter. 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.

The uncertainty contributions related to $C_{\alpha}$, $\bar{k}_{\mathrm{Cf}}$
and $k_{\mathrm{sc}}$ are treated as non-statistical contributions. The total
relative uncertainty of $Y_{\mathrm{n,dir}}$ measured with LC1 ranges between
0.021 and 0.032. 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
normalized 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.

##  4 Results and Discussion

For selected energies, the reproducibility of the measurements for the period
from 2016 to 2022 is demonstrated in figure 15. The error bars shown here
comprise only those uncertainty components which are expected to vary from
measurement to measurement and are not correlated over neutron energy. The
empirical relative standard deviation $\sigma$ of the yield ratios ranges
between 0.4 % and 1.9 % and is consistent with the error bars of the data
points.

Figure 15:  Ratio of the target yields $(Y/M)$ of direct neutrons per event in
the NM monitor measured with P2 (upper panel) and T1 (lower panel) to those
measured with LC1 as a function of the date of the measurement. The error bars
show the statistical uncertainty contribution. The solid and dashed lines
exhibit the mean and the standard deviation of the data points, respectively.
For P2, the empirical standard deviation $\sigma$ of the data points ranges
between 0.004 and 0.017 and for T1 between 0.014 and 0.019.

Figure 16 shows the ratio of the target yields $(Y/M)$ per event in the NM
monitor measured with P2 and T1 to those measured with LC1 for the same
neutron field. The neutron energy ranges from 30 keV to 14.8 MeV. As discussed
above, the neutron fields with energies below 144 keV were produced at neutron
emission angles $\Theta_{\mathrm{n}}>$$$. All other measurements were made at
$None$. The maximum neutron energy covered with P2 was 2 MeV and the minimum
neutron energy for T1 was 1.2 MeV. The error bars indicate the total
uncertainties and the dashed-dotted lines show the cumulated non-statistical
uncertainty contributions (see tables 1 and 3) averaged over larger energy
bins.

Figure 16:  Ratio of the target yields $(Y/M)$ of direct neutrons per event in
the NM monitor measured with P2 (upper panel) and T1 (lower panel) to those
measured using LC1 for the same neutron field. The error bars show the total
uncertainties, including the non-statistical components which are correlated
over neutron energies. The dashed-dotted lines show the non-statistical
component of the uncertainty of the measured yield ratios averaged over larger
energy bins.

The data measured with P2 and T1 match around 2 MeV. Below 2 MeV, the T1 data
show a decrease while the P2 results are almost constant between 1 MeV and 2
MeV. This indicates a problem with the RPT measurements at very low recoil
proton energies. It could be caused, for instance, by the effect of
electromagnetic interference at the low signal amplitudes found at these
energies. It is also interesting to note that the yield ratios measured with
P2 using the H2/CH4 mixture as counting gas seem to be systematically lower
than those measured using C3H8. This tendency needs further investigation.

For neutron energies above 144 keV most of the present data are located
between the non-statistical uncertainty margins indicated by the dashed-dotted
lines. There is nonetheless a clear trend with neutron energy which continues
at lower energies where the data fall below the lower uncertainty margin.
Thus, the estimated uncertainties appear to be realistic for the measurements
at an emission angle $\Theta_{\mathrm{n}}=$$$ . At angles greater than $None$,
however, there must be unrecognized sources of uncertainty. Possible
explanations could be a contribution of scattered neutrons greater than that
calculated with MCNP5 or an effect related to the increased kinematical
sensitivity of the neutron yield and neutron energy on the emission angle
$\Theta_{\mathrm{n}}$ for angles much larger than $None$. This would also
explain the increased scatter of the data below 144 keV.

A reduced data set with error bars exhibiting the statistical uncertainty
component only is shown in figure 17. The T1 measurements below 2 MeV were
discarded from this data set. In addition, it does not include the deviating
P2 measurement at 30 keV that was determined without photon subtraction. Nor
does it include the T1 data points at 2.95 MeV and 3.03 MeV that are too close
to the prominent dip in the 12C elastic scattering cross section around 2.98
MeV which is also visible in the long counter response shown in figure 13.

The red solid line shows a fit of a sigmoid function on a logarithmic energy
scale,

$S(E_{\mathrm{n}})=\delta
s\,\Bigl{(}\frac{2}{1+\exp(-u/\sigma)}-1\Bigr{)}+s_{0},$ (9)

with $u=\ln(E_{\mathrm{n}}/E_{0})$, $E_{0}=0.27(5)\,\mathrm{MeV}$,
$\sigma=0.38(10)$, $\delta s=0.034(4)$ and $s_{0}=0.988(4)$.

Figure 17: Ratio of the target yields $(Y/M)$ (red and blue closed symbols)
fitted by a sigmoid function on a logarithmic energy scale (red solid line).
Data points measured with T1 at neutron energies below 2.5 MeV and three other
deviating data points at 30 keV, 2.95 MeV and 3.03 MeV were excluded from the
fit. The error bars show the statistical uncertainty component only. The
dashed-dotted lines exhibit the non-statistical component of the uncertainty
of the measured yield ratios averaged over larger energy bins.

Under the tentative assumption that the observed energy dependence of
$(Y_{\mathrm{P2,T1}}/M_{\mathrm{P2,T1}})/(Y_{\mathrm{LC1}}/M_{\mathrm{LC1}})$
is due to the calculated energy dependence of the fluence response $R_{\Phi}$
of the long counter LC1, $S(E_{\mathrm{n}})$ describes the correction for the
energy dependence of $R_{\Phi}(E_{\mathrm{n}})$ required to match the results
obtained with P2 and T1.

Therefore, it is interesting to compare $S(E_{\mathrm{n}})$ with the results
of measurements carried out with the NPL De Pangher long counter and a set of
radionuclide neutron sources which had been calibrated in the NPL manganese
bath. Figure 18 shows a comparison of the ratio of calculated and measured
(C/M) fluence response values for the various neutron sources in use at the
NPL [33] with the spectrum-averaged correction factor

$\bar{S}(\bar{E}_{\mathrm{n}})=\frac{\int\limits_{0\,\mathrm{MeV}}^{20\,\mathrm{MeV}}S(E_{\mathrm{n}})\,R_{\Phi}(E_{\mathrm{n}})\,\Phi_{E}\,\mathrm{d}E_{\mathrm{n}}}{\int\limits_{0\,\mathrm{MeV}}^{20\,\mathrm{MeV}}R_{\Phi}(E_{\mathrm{n}})\,\Phi_{E}\,\mathrm{d}E_{\mathrm{n}}}.$
(10)

Here, $\Phi_{E}$ and $\bar{E}_{\mathrm{n}}$ denote the neutron energy
distribution of the radionuclide source and its mean energy, respectively.
This correction factor was calculated using the original MCNP model of the
NPL, that is, with the thermal shield of PE(10B) and a BF3 density of
$9.5414\times 10^{-4}\,\mathrm{g/cm}^{3}$ adjusted by the NPL to give the best
match to the measurements with the NPL radionuclide neutron sources. It is
interesting to note that the correction function $S(E_{\mathrm{n}})$ for the
fluence response of the long counter seems to reproduce the results obtained
at the NPL for the Am-Li, Am-F, Cf, Am-B and Am-Be sources. The results for
the Pu-Li source and the Sb-Be photoneutron source, however, clearly deviate
from this trend.

Figure 18: The black closed squares show the ratio of calculated and measured
fluence response values for the NPL De Pangher long counter and various
radionuclide neutron sources [33]. The red solid line is the fit
$S(E_{\mathrm{n}})$ to the ratio of normalized target yields $(Y/M)$ measured
with P2 or T1 to that obtained with LC1. The red closed circles show the
spectrum-averaged value $\bar{S}(\bar{E}_{\mathrm{n}})$ for the respective
neutron sources. The error bars of the spectrum-averaged data points were
determined from the uncertainties of the fit parameters only. This means that
they do not include the uncertainties of the energy distributions.

##  5 Conclusions

In the energy region between 144 keV and 14.8 MeV, this study confirms the
consistency of neutron fluence measurements with the primary reference
instruments P2 and T1 of the PTB and the LC1 PLC. The P2 RPPC and the T1 RPT
can be used for neutron energies below and above about 2.2 MeV, respectively.
The observed energy dependence of the neutron yield ratios is within the
estimated uncertainty margins and consistent with measurements with
radionuclide neutron sources carried out at the NPL. This suggests that the
deviation is actually caused by the long counter response calculated with
MCNP. The present results confirm the chain of traceability between fast
neutron metrology based on the differential n-p scattering cross section as
the primary standard and the manganese bath technique that relates neutron
measurements to the activity standard. The fit to the measured yield ratios
varies between 0.966 and 1.023, that is, around $0.995\pm 0.029$, which is
consistent with the uncertainty budgets of the instruments and the uncertainty
of the calibration of LC1 with the PTB 252Cf neutron sources. This study is
complementary to previous work [34] using a 252Cf fission ionization chamber
to compare neutron source emission rate measurements with the manganese bath
and with a liquid scintillation detector operated in time-of-flight mode.

At energies below 144 keV, however, the results are less satisfactory. Here,
the measured ratio of target yields is only about 0.948 and is clearly outside
the calculated uncertainty margins. This deviation is currently not
understood, but is most likely due to the energy distribution of the neutron
fields produced by the 7Li(p,n)7Be reaction at neutron emission angles greater
than zero degrees. Further work is needed to characterize these fields in
detail using time-of-flight spectrometry. Alternatively, neutron fields with
energies between 8 keV and about 40 keV can be produced using the
45Sc(p,n)45Ti reaction [35] at $None$. It should, however, be noted that this
only useful for metrological investigations and is no solution for routine
applications because of the low neutron yield and the intense production of
parasitic photons. Until the problem at neutron energies below 144 keV is
solved, the uncertainty of neutron yield measurements using P2 or LC1 at PIAF
must be increased accordingly for these neutron fields.

The present study demonstrates that fast neutron fluence measurements with
relative uncertainties around 2 % are possible but challenging. This is more
than sufficient for applications in radiation protection metrology but does
not meet the most challenging demands from nuclear technology, especially for
standard cross sections.

This study has a long lasting history dating back to the first draft of a PTB-
Report from the early 1990s. Therefore, the authors are indebted to the former
and present heads of the PTB neutron departments H. Klein, H. Schuhmacher and
A. Zimbal for many discussions on this subject. The authors are grateful to
N.J. Roberts and D.J. Thomas of the NPL for many valuable comments and for
providing the MCNP model of the De Pangher long counter and the data for the
NPL radionuclide neutron sources. S. Löb, M. Thiemig and B. Schulze carried
out the numerous measurements reported in this study and developed the
preparation of the lithium targets. The PIAF accelerator staff provided the
ion beams for the experiments. Their tireless effort is gratefully
acknowledged.

## References

## References

  * [1] Schlegel D and Guldbakke S 2002 Neutron fluence measurements with a recoil proton proportional counter Laboratory report PTB-6.41-2002-03 Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany
  * [2] Schlegel D and Guldbakke S 2002 Neutron fluence measurements with a recoil proton telescope Laboratory report PTB-6.41-2002-04 Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany
  * [3] Hunt J B 1976 The calibration and use of long counters for the accurate measurement of neutron flux density NPL Report RS(EXT)5 National Physical Laboratory Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom
  * [4] Carlson A, Pronyaev V, Capote R, Hale G, Chen Z P, Duran I, Hambsch F J, Kunieda S, Mannhart W, Marcinkevicius B, Nelson R, Neudecker D, Noguere G, Paris M, Simakov S, Schillebeeckx P, Smith D, Tao X, Trkov A, Wallner A and Wang W 2018 Nuclear Data Sheets 148 143–188 ISSN 0090-3752 special Issue on Nuclear Reaction Data
  * [5] Gressier V, Bonaldi A C, Dewey M S, Gilliam D M, Harano H, Masuda A, Matsumoto T, Moiseev N, Nico J S, Nolte R, Oberstedt S, Roberts N J, Röttger S and Thomas D J 2014  51 06009
  * [6] Nolte R and Röttger S 2018 PTB contribution to the key comparison CCRI(III)-K11 ’Comparison of neutron fluence measurements for neutron energies of 27 keV, 565 keV, 2.5 MeV and 17 MeV’ PTB-Bericht PTB-N-58 Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany
  * [7] Uttley C A 1983 The H(n,n) cross section IAEA Technical Report Series 227 International Atomic Energy Agency Vienna, Austria
  * [8] Hale G 2019 private communication
  * [9] Nolte R and Thomas D J 2011 Metrologia 48 S274
  * [10] Schlegel D 2005 TARGET user’s manual Laboratory report PTB-6.42-05-02 Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany
  * [11] X-5 Monte Carlo Team 2004 MCNP - a general Monte Carlo N-particle transport code, version 5 Laboratory report LA-UR-03-1987 Los Alamos National Laboratory Los Alamos, New Mexico, United States of America
  * [12] Drosg M 2016 DROSG-2000: Codes and database for 60 neutron source reactions IAEA report IAEA-NDS-87 Rev. 10 International Nuclear Energy Agency Vienna, Austria URL https://www-nds.iaea.org/public/libraries/drosg2000/
  * [13] Birgersson E and Lövestam G 2009 NeuSDesc - neutron source description software manual Laboratory report EUR 23794 EN Joint Research Centre, Institute for Reference Materials and Measurements Geel
  * [14] Skyrme T H R, Tunnicliffe P R and Ward A G 1952 Rev. Sci. Instrum. 52 204 – 209
  * [15] COMSOL AB 2014 COMSOL - Multiphysics, version 4.4 URL https://www.comsol.com
  * [16] Posny F, Chary J and Nguyen V D 1987 Phys. Med. Biol. 32 509
  * [17] Breitung W 1972 Untersuchungen zur Energieabhängigkeit von w (Energieverlust/Ionenpaar) für Protonen in Wasserstoff, Methan und H2-CH4-Gemischen im Hinblick auf Neutronenspektrumsmessungen mit Rückstoßprotonen-Proportionalzählrohren unterhalb 25 keV Laboratory report KFK 1623 Kernforschungszentrum Karlsruhe Karlsruhe, Germany
  * [18] Grosswendt B, Willems G and Baek W Y 1997 Radiat. Prot. Dosim. 70 37–46 ISSN 0144-8420
  * [19] Pelowitz D 2011 MCNPXTM \- user’s manual, version 2.7.0 Laboratory report LA-CP-11-00438 Los Alamos National Laboratory Los Alamos, New Mexico, United States of America
  * [20] Deutsches Institut für Normung 1980 DIN 1871 Gasförmige Brennstoffe und sonstige Gase
  * [21] Bame S J J, Haddad E, Perry J E J and Smith R K 2004 Rev. Sci. Instrum. 28 997–1006 ISSN 0034-6748
  * [22] Vanleeuw D, Sibbens G and Plompen A 2015 J. Radioanal. Nucl. Chem. 305 957–962
  * [23] Siebert B R L, Brede H J and Lesiecki H 1985 Nucl. Instrum. Methodsin Phys. Res. A 235 542–552 ISSN 0168-9002
  * [24] Thomas D and Axton E 1980 Nucl. Instrum. Methods 174 321–322 ISSN 0029-554X
  * [25] Sloan D and Robertson J 1982 Nucl. Instrum. Methodsin Phys. Res. 198 365–372 ISSN 0167-5087
  * [26] De Pangher J and Nichols L L 1966 A precision long counter for measuring fast neutron flux density Laboratory report BNWL-260 Pacific Northwest Laboratory Richland, Washington, United States of America URL https://www.osti.gov/biblio/4513481
  * [27] Marshall T O 1970 Health Physics 18 427–429
  * [28] Roberts N J, Tagziria H and Thomas D J 2004 Determination of the effective centres of the NPL long counters Laboratory report DQL RN004 National Physical Laboratory Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom
  * [29] Roberts N J 2019 private communication
  * [30] Axton E, Cross P and Robertson J 1965 J. Nucl. Energy. Parts A/B 19 409–422 ISSN 0368-3230
  * [31] Roberts N J and Jones L N 2010 Appl. Radiat. and Isotopes 68 626–630 ISSN 0969-8043
  * [32] Cox M G, Eiø C, Mana G and Pennecchi F 2006 Metrologia 43 S268
  * [33] Thomas D J 2023 private communication
  * [34] Bardell A G, Zieba K J, Böttger R and Klein H 1994 Bilateral intercomparison of neutron emission rate of a 252Cf spontaneous fission source between the National Physical Laboratory Teddington, UK. and the Physikalisch-Technische Bundesanstalt, Braunschweig, Germany NPL report RSA(EXT) 47 National Physical Laboratory Hampton Road, Teddington, Middlesex, TW11 0LW, United Kingdom
  * [35] Cosack M, Lesiecki H and Hunt J B 1987 Monoenergetic neutrons from the 45Sc(p,n)45Ti reaction Technical Report IAEA-TECDOC 410 International Atomic Energy Agency (IAEA) Vienna, Austria