Method for converting neutron time-of-flight spectrum into an energy spectrum

In neutron time-of-flight (nTOF) experiments a pulsed neutron beam travels a predetermined distance before reaching the sample under examination. The problem of reconstructing the primary neutron spectrum from the measured one with this technique is discussed in this work. A solution is presented employing the response matrix of the neutron detector and an unfolding procedure.


Introduction
When using the time-of-flight (TOF) technique, the neutron's kinetic energy is calculated from their arrival time at the measuring station.Before being captured, neutrons move at constant velocity from the production target to the detector.Inside the detector, neutrons are elastically or inelastically scattered due to the interactions with the materials in their path.These facts increase the neutrons' time of flight since there is a spread in the interaction position in the detector.This means that when a single neutron energy is sampled, the resulting TOF is a distribution rather than a single value.Then, it is inaccurate to convert from TOF to an energy spectrum using the equation of the kinetic energy of neutrons: TOF is the velocity of neutrons, L is the flight path, and c is the speed of light.A new approach is mandatory and this is the main purpose of this work.

Method
The measured TOF spectrum for a specific exposure time in the measurement of the incident neutron energy spectrum takes the shape of a histogram that contains the detected number of particles for each temporal bin.A numerical method was used to take into account the contributions of the time distributions of different monoenergetic neutrons.By convolution in the form of a matrix, the neutron TOF spectrum is transformed into an energy spectrum as follows: or in vectors form: where T is the nTOF spectrum, RM is the convolution matrix (or detector response matrix), and N e is the (yet unknown) neutron energy spectrum.In the nTOF spectrum, the scored T j for the jth time bin is composed by tallying the components of the emitted energy spectrum N emitted i times a factor equal to the contribution of the ith energy interval to the jth time bin.This factor is the corresponding element RM ij of the detector response matrix RM.To obtain the emitted neutron spectrum, the system of n equations is solved by multiplying the folded TOF spectrum by the inverse of the response matrix.The RM matrix cannot be measured directly and was determined using the Monte Carlo N Particle (MCNPX) [1] simulation code.
The efficiency of the detector and the fact that the point at which neutron capture takes place are both included in the response matrix.Each column of the matrix, named as response function of the detector, gives the probability that a neutron of energy E i ± ∆E i undergoing an interaction into the detector will be measured as a count in the measured TOF spectrum.
For monoenergetic neutron energies, the relation between the neutron energy and the most probable moderation time value is a continuum function.Due to the energy bins' continuously variation, a continuity in the temporal bins is expected.When the deconvolution method is applied with a new time-energy relation (most probable moderation time-energy), the obtained spectrum should reproduce exactly the experimental one.The response function for various neutron energies was analyzed once the TOF bins were selected.Nearly 5000 MCNPX files were examined for monoenergetic neutron sources with 0.05% energy uncertainties (E n ± 0.05%E n ).For these files, the response function of the detector using a time distribution computed from the TOF binnings was calculated.The TOF for the most probable value of the time distribution for each file was then linked to the corresponding neutron energy for that file.As a result, the time-energy association included the neutrons' moderation time inside the detector.With the new time-energy association, the response matrix of the setup is calculated with MCNPX, implementing precisely the geometry of the experimental setup.Once the RM was obtained, the inverted matrix is calculated.As input in the MCNPX file a neutron source was placed at target position and the neutron time distributions in the detector were determined.

Experimental measurement
The experimental measurement was carried out at the CN accelerator of the National Laboratories of Legnaro of the Italian National Institute for Nuclear Physics (LNL-INFN), Padua, Italy.As neutron source, the 7 Li(p,n) 7 Be reaction was employed and to determine the neutron spectrum, the neutron time-of-flight spectrometry (nTOF) was implemented.The goal of this measurement was to validate the novel conversion method.Figure 1 shows the experimental setup of this measurement, performed with a similar setup as the one reported in the work of Lederer et al. [2].
In the experiment, a lithium metallic target was prepared and employed to produce the neutron beam.A 100 µm metallic lithium foil was attached to a semi-spherical 300 µm copper backing.The Cu backing was cooled by applying an air flow to its external face.A vacuum level of 10 −6 mbar was reached at the target position during the experiment.
The neutron time-of-flight spectrum at zero degrees relative to the proton beam direction was measured with a 6 Li-glass detector, from the Scionix company, with a crystal thickness of one-inch.The detector was placed at 71.99 ± 0.01 cm from the target.A proton pulsed beam of 600 kHz frequency was employed and a proton energy of 1912 keV was set in the accelerator.The proton energy distribution was measured with proton time-of-flight spectrometry (pTOF) and the Gaussian fit for the energy distribution gave a mean proton energy of 1911.89keV with a full width half maximum (FWHM) of 0.73 keV.During the experimental measurement, a proton beam current of about 50 nA was measured in the lithium target.

Results and discussion
Each temporal bin of the final nTOF spectrum was calculated as: where (T OF ) i is the time-of-flight count in each temporal bin, (T OF ) γ is the TOF where the γ flash peak appears and, T γ is the traveling time of γ rays over the 72 cm flight path (from the neutron production Li target to the Li-glass detector).The level of background was calculated as the average value in the TOF histogram, in the region between the γ flash and the fastest neutrons.The calculated background value was subtracted from the nTOF spectrum, obtaining the final one used in the conversion method from TOF to an energy spectrum (Figure 2).Applying the method described in Section 2, employing the inverse of the response matrix of the Li-glass detector, the neutron energy spectrum at zero degrees was obtained.The time bins were chosen to have almost the same counting statistic per bin.With the acquired statistics, it was possible to obtain the neutron energy spectrum with 40 energy bins, from 1 keV to 120 keV.The obtained neutron energy spectrum is shown in Figure .3. The most probable neutron energy is 30 keV and the maximum energy around 110 keV, as expected from the 7 Li(p,n) 7 Be reaction kinematics.Less than 1% relative uncertainty were obtained.
Following the same procedure described in Section 2, a simulation of the nTOF spectrum was performed, employing the Monte Carlo MCNPX code.The obtained neutron energy spectrum (Figure 3) was employed as neutron source and was placed at 72 cm from the detector outer face.Thus, the measurement's experimental circumstances are reproduced.In Figure 4 a comparison between the experimental neutron TOF spectrum and the one generated using the measured neutron energy spectrum (Figure 3) is presented.As observed in the figure there is good agreement between the two spectra, with a relative error of less than 1% up to 400 ns and less than 10% up to 600 ns.This comparison was done to double-check the conversion method.It is important to remark that the conversion method proposed in this work reproduces the experimental data measured in the experiment, demonstrating the accuracy of the obtained neutron energy spectrum.

Conclusions
In this work a new approach to transform the measured time-of-flight (TOF) spectrum into an energy spectrum was implemented.The suggested conversion method accounts for the time   distribution of neutrons inside the detector using the detector response matrix, allowing the accurate reproduction of the emitted neutron energy spectrum from the TOF spectrum.The presented method was validated and already employed to determine an integral from 0 to 90 degrees Maxwell-Boltzmann neutron spectrum with a thermal temperature of 28 keV in another experimental measurement.

Figure 1 .
Figure 1.Experimental setup for neutron time-of-flight measurement with a proton energy of 1912 keV.Detector located at 71.99 ± 0.01 cm from the target at zero degrees relative to the beam direction.

Figure 2 .
Figure 2. Neutron time-of-flight (nTOF) spectrum at zero degrees, acquired with a 6 Li-glass detector at 72 cm from the target with E p =1911.89 ± 0.73 keV.

Figure 3 .
Figure 3. Experimental neutron time-offlight spectrum at zero degrees obtained with proton energies of 1911.89 ± 0.73 keV.

Figure 4 .
Figure 4. Experimental neutron spectrum at zero degrees compared to the simulated one, employing the measured neutron energy spectrum as neutron source.