Method of experimental data analysis
Note
The following section describes the overall principles of analysis for context. More detailed description can be found in numerous references from past thesis and articles [1], [2], and [3].
The analysis method for Grapheme [4] and [5] (or other similar setups [6]) combines precise \(\gamma\)-ray spectroscopy with Germanium detectors, neutron detection with a fission chamber, time-of-flight measurements (for neutron energy determination), and the Gauss method for solid angle integration [7].
The Grapheme setup
For the recording of the 183W data, Grapheme was consisting of :
Two fission chambers, upstream from the target. Both use a highly enriched (\(\geq 99.5~\%\)) 235U, thin (\(\approx 600~\text{nm}\)), in the form of UF4 and U3O8 deposits [1].
Four planar High Purity Germanium detectors with diameters between 3.5 and 6 cm, and thickness 2 to 3 cm. The detectors are named after colors in French: Bleu, Vert, Rouge and Gris.
Figure 12 gives a schematic view of the setup. Reference [1] extensively describes the detectors.
Figure 12 Schematic representation of the Grapheme setup. (Adapted from [1].)
The detectors are connected to a digital acquisition [8] which records the events in list mode, without condition of multiplicity, but requiring a time coincidence (within 7 microseconds) with the beam pulse signal (that is also connected to the acquisition input).
The isotopically enriched (80.9 %) 183W target, with a 7.1 centimeters diameter (39.6 cm2 area) (compared to \(\approx~55~\text{mm}\) diameter of the neutron beam [1]) and 1.2 millimeters thickness, was placed in the beam axis, at the center of the HPGe detectors (the distance from the target center to the front of the HPGe detectors were between \(13\) and \(21\) centimeters).
The data is processed into 2D histograms: time-of-flight vs. \(\gamma\) energy (Figure 13).
Selection of the reaction channel with \(\gamma\) rays
The use of High Purity Germanium detectors in the setup allows for a clear selection of the reaction channel. (Although, it may become tricky for highly radioactive and/or fissile isotopes because of a large number of contaminant \(\gamma\) lines. Elements like 183W, with many low-intensity transitions, also require a precise selection work in the \(\gamma\) spectrum.) Using the structure information [9], we know what particular \(\gamma\) ray to look for in the spectrum from the HPGe detector, signing unequivocally a particular \((\text{n},~x\text{n}~\gamma)\) channel (see in Figure 13).
Figure 13 Example of time-of-flight vs. \(\gamma\) energy matrix, from the detector “Vert“. The different reaction channels can be identified. Two \((\text{n},~2\text{n}~\gamma)\) transitions are identified in green, and four \((\text{n},~\text{n}'~\gamma)\) in orange. The big concentration of events at the earliest time of flight is the \(\gamma\) flash. The low energy, continuous lines at the bottom are X rays from the environment (in particular, the lead shielding around Grapheme).
In the cases of non isotopically pure targets, the \(\gamma\) from a \((\text{n},~\text{n'})\) reaction on one isotope may overlap with the same \(\gamma\) from the \((\text{n},~2\text{n})\) reaction on the one neutron heavier isotope in the material, but both contribution can usually be disentangle with incident neutron energy consideration. In some very rare cases, the level scheme presents two \(\gamma\) rays with energies so near that they cannot be separated, in that case, a sum cross section will be extracted.
Determination of the incident neutron energy by Time of flight
There is no direct measurement of the energy of the neutron that induced the reaction.
We only detect the outgoing \(\gamma\) rays.
Furthermore, the neutron beam at the Gelina Facility [10], [11] and [12] is white, i.e. there is a continuous spectrum of neutron energy (read more in [12]).
To determine (with a given accuracy) the energy of the neutron that induced the inelastic scattering for which a \(\gamma\) ray is detected, we rely on the neutron time of flight. Indeed, the accelerator that beams electrons on the neutron production target provides a pulse signal that is inputted in our digital acquisition setup and used as start signal.
The detection of events in our HPGe detectors is considered as the stop and from the time elapsed between the two (after a correction for electronic processing and signal travel, that is determined using the \(\gamma\) flash detected in our setup), one can easily find the energy of the neutron \(E_\text{n}\), using the formula \(E_\text{n} = m_\text{n} c^2 \left( \left[1 - \left(\frac{DoF / ToF}{c}\right)^2\right]^{-1/2} - 1 \right)\) (where \(m_\text{n}\) is the mass of the neutron in MeV/c2, DoF and ToF the distance and time of flight, respectively).
Note
Because of this formulation, a constant uncertainty on the time (or distance), will have an energy dependent impact on the neutron energy uncertainty. For a distance of about 30 meters, and a time resolution of 10 nanoseconds, the energy resolution will go from 1 keV for a 100 keV neutron, up to around 2.7 MeV for 20 MeV neutrons.
Below is a table giving a few reference uncertainties.
Neutron Energy (keV) |
Neutron energy uncertainty (keV) |
|---|---|
100 |
1 |
500 |
10 |
1000 |
30 |
5000 |
350 |
10000 |
1000 |
20000 |
2700 |
Normalization to neutron flux with fission chamber
The absolute number of neutron going through the target is determined using a fission chamber. It is, in fact, a ionization chamber with a thin (\(\approx 600~\text{nm}\)) deposit of Uranium 235 [1]. The neutron passing through will, with a small probability due to the low amount of material, induce fission of the Uranium in the deposit. The fission fragment (usually only one of the two will get out of the material, for geometric reasons) will then escape into the gas within the high voltage bias region and induce an electric signal that will be detected. Counting the number of recorded signals with an amplitude high enough to be the signature of a fission fragment (against a constant background of \(\alpha\) particle emission), also combined with the time of flight determination, gives us a yield of fission events as a function of neutron energy, after correction for the fission chamber efficiency (\(\varepsilon_\text{FC}\)) which is of the order of 90 %. Normalization by material quantity (\(N_{^{235}U}\)) and the reference cross section of neutron induced fission of 235U (\(\sigma_{^{235}\text{U(n,~f)}}(E_n)\)) allows us to reconstruct the spectrum of incident neutrons, both in distribution of energy and absolute value (i.e. the number of neutrons that went through our setup during the time of the experiment).
The number of neutrons at and energy \(E_\text{n}\) that traversed the fission chamber can be expressed as :
To get the proper flux, one just needs to divide \(N_\text{n} (E_\text{n})\) by the beam spot area and the time of recording.
Angle integration of the partial cross sections
After determining the number of \(\gamma\) rays for given neutron energy intervals (which translates to time-of-flight intervals) at a given angle (the determination is usually down by fitting a peak shape on the \(E_\gamma\) histogram, in order to remove background and possible parasitic peaks contributions), the \(N_\gamma(E_n; \theta)\) values are combined with the neutron flux and normalization factor related to the amount of material in the target (\(N_\text{target}\)), as well as efficiencies (\(\varepsilon(E_\gamma)\)), or pile up corrections, we can obtain a partial cross-section at the given angle of the detector, following the formula:
The angle integrated value is then obtained by using the Gauss method, that uses the decomposition of the spatial emission probability of the \(\gamma\) rays (according to their multipolarity) in Legendre polynomials to perform a solid angle integration via a simple weighted sum of partial cross section at specific angles [7].
With two detectors at 110 and 150 degrees in respect to the beam axis [7], the angle integrated cross-section is
With \(w_{110^{\circ}} = 0.6521\) and \(w_{150^{\circ}} = 0.3479\).
This relation is generally true for \(\gamma\) rays connecting levels with well defined parity, it breaks down only if the \(\gamma\) multipolarity is strictly larger than 3 and the \(\gamma\) decays for a state with spin larger than 3 [7].
Footnotes