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Characterization of Germanium Detectors for the Measurement of the Angular Distribution of Prompt gamma-rays at the ANNRI in the MLF of the J-PARC

In this study, the germanium detector assembly, installed at the Accurate Neutron-Nuclear Reaction measurement Instruments (ANNRI) in the Material and Life Science Facility (MLF) operated by the Japan Proton Accelerator Research Complex (J-PARC), has been characterized for extension to the measurement of the angular distribution of individual gamma-ray transitions from neutron-induced compound states. We have developed a Monte Carlo simulation code using the GEANT4 toolkit, which can reproduce the pulse-height spectra of gamma-rays from radioactive sources and (n,gamma) reactions. The simulation is applicable to the measurement of gamma-rays in the energy region of 0.5-11.0 MeV.


Introduction
A germanium detector assembly is installed at the Accurate Neutron-Nuclear Reaction measurement Instruments (ANNRI), located at the neutron beamline BL04 of the pulsed spallation neutron source of the Material and Life Science Facility (MLF) operated by the Japan Proton Accelerator Research Complex (J-PARC) [1]. The performance of the assembly enables the detection of prompt γ-rays from (n,γ) reaction with a sufficient energy resolution to resolve individual γ-ray transitions. The accuracy of the resonance parameters of the compound states has been improved by the combination of the γ-ray energy resolution and the capability to determine the incident neutron energy along with the neutron time-of-flight.
Extremely large parity violation (P-violation) was found in the helicity dependence of the neutron absorption cross section in the vicinity of the p-wave resonance for a variety of compound nuclei [2]. The magnitude of the P-violating effects is 10 6 times larger than that of the nucleonnucleon interactions [3][4][5]. The large P-violation is explained as an interference between the amplitudes of the p-wave resonance and the neighboring s-wave resonance [6,7]. The enhancement mechanism is expected to be applicable to P-and T-violating interactions and to enable highly sensitive studies of CP-violating interactions beyond the Standard Model of elementary particles [8]. Theoretically, the interference between partial waves in the entrance channel causes an angular distribution of the individual γ-rays from the compound state [9]. The ANNRI can measure the angular distribution of the γ-rays by using germanium detectors installed at a particular angle.
In this paper, we describe the determination of the pulse-height spectra of each germanium detector in the assembly by measuring the γ-rays from radioactive sources and 14 N(n, γ) reactions. We studied the γ-ray energy dependence of the detection efficiencies and the angle of the germanium detectors by simulations.

Germanium Detector Assembly in the ANNRI
A schematic of the experimental setup of the ANNRI is shown in Fig. 1. The configuration of the germanium detector assembly (F in Fig. 1) is shown in Fig. 2. The beam axis is represented by the z-axis, the vertical direction is the y-axis and the x-axis is defined such that they form a right-handed coordinate system. The origin of the coordinate system is the position of the nuclear target. The pulsed neutron beam is transported through a series of collimators (A in Fig. 1) to the target position located 21.5 m from the moderator surface. A T0-chopper (B in Fig. 1) and a disk chopper (D in Fig. 1) are installed at 12 m and at 17 m, to eliminate fast neutrons and cold neutrons, respectively [1]. Several neutron filters (C in Fig. 1) are also inserted between the T0-chopper and the disk chopper, to adjust the beam intensity in the energy region of interest. There are two kinds of germanium detectors: type-A and type-B. The shape and dimension of type-A and type-B crystals are shown in Fig. 3. The front-end of the type-A crystal has a hexagonal shape, and the back-end has a hole for the insertion of the electrode. The seven type-A crystals form a detector unit as shown in Fig. 4. Germanium detectors are covered by two neutron shields (F and H in Fig. 4) made of LiH with a thickness of 22.3 mm and 17.3 mm, respectively [10]. The neutron shield G in Fig. 4 is made of LiF and its thickness is 5 mm. The central crystal is directed toward the target center, while the surrounding six detectors are directed farther beyond the center. Therefore, they have different solid angles of 0.010 × 4π sr (central) and 0.0091 × 4π sr (one of the surrounding six detectors), respectively. The side and back of the assembly of the type-A crystals are surrounded by bismuth germanate (BGO) scintillation detectors, illustrated by D in Fig. 4. The polar angle in the spherical coordinate is denoted by θ and the azimuthal angle is ϕ. Two assembly of the type-A crystals are placed at  upper (lower) type-A assembly is denoted by d1 (d8) and the other surrounding six detectors are denoted by d2-d7 (d9-d14), as shown in Table 1. The front-end of the type-B crystal has a circular shape and the back-end has a hole for the insertion of the electrode. Eight type-B crystals are assembled, as shown in Fig. 5. The surrounding of the type-B assembly is shown in Fig.6. Detectors are denoted by d15-d22. It should be noted that the germanium crystal of detector d16 is smaller than that of the other detectors. All type-B crystals are directed toward the target center. Therefore, except for detector d16, they all have the same solid angle of 0.0072 × 4π sr. The solid angle of detector d16 is 0.0048 × 4π sr. Each type-B crystal is surrounded by a BGO scintillator (A in Fig.6). A conical-shape γ-ray collimator made of Pb is located between each type-B crystal and the nuclear target. The diameter of the collimator at the front-end of the type-B crystal is 60 mm. The inside of the collimator is filled with LiF powder, which is encapsulated in an aluminum case, for absorbing the scattered neutrons. The beam duct consists of two layers. The outer layer is made of aluminum with the thickness of 3 mm. The cross-sectional dimension is 86 mm × 96 mm. The inner layer is made of LiF with a thickness of 10.5 mm for absorbing scattered neutrons.

Pulse-Height Spectra using Radioactive Sources and Melamine Target
The linearity of the detector response was ensured by measuring the pulse-height of prompt γ-rays from the neutron capture reactions with target nuclei of 27 Al. We calibrated the γ-ray energy from the analog-to-digital converter (ADC) channel, by using 17 peaks of the 27 Al(n,γ) reaction from 511 keV to 7724 keV. The energy calibration was verified for several peaks of the 27 Al(n,γ) reaction. The values on the vertical axis are expected to scatter around 0. We confirmed good linearity with less than 1 keV of deviations, as shown in Fig. 7. The deviations are acceptable, as they are smaller than the energy resolution of the germanium detector.

Energy resolution of germanium detectors for γ-rays
We simulated the energy spectrum of the germanium detector assembly, by using GEANT4 9.6 [12]. The simulation of the energy resolution was implemented as follows. Ideally, the shape of the full absorption peak of the γ-ray is Gaussian, with a low energy tail (see Fig. 8). Therefore the following function was used: where F gauss , F skew and F erfc are the Gaussian function, skewed Gaussian function, and the complementary error function, respectively. These are expressed as follows: where c is the peak position, σ is the standard deviation of F gauss , and β represents the extent of the low energy tail. Functions F skew and F erfc are introduced to account for the low energy tail.  [11] and calibrated values for the 27 Al(n,γ) reaction. The horizontal axis is the energy of γ-ray.
The obtained σ and β as a function of the γ-ray energy are shown in Fig. 9. The obtained σ as a function of the energy is fitted by using the following formula: where σ D , σ X , and σ E denote the Fano statistics, collection efficiency of charge carriers, and electrical noise, respectively [13]. These are defined as follows: where F is the Fano factor (F = 0.112 ± 0.001 [14]) and ǫ is the energy of the creation of an electron-hole pair in the germanium. The obtained β as a function of the energy is fitted by the following empirical formula:

Comparisons of measurement and Monte Carlo simulation in the order of 1 MeV
In order to confirm the reproducibility of the simulation, we measured the energy spectra of γ-rays of the germanium detector from 137 Cs and 152 Eu radioactive sources placed at the nuclear target position. The spectra obtained are shown in Fig. 10 represented by black dots, together with the simulated spectra. Each simulated pulse-height spectrum was adjusted to fit to the corresponding measured spectrum with a single parameter of the average efficiency, as shown in Fig. 10. . Pulse-height spectra of γ-rays from 137 Cs (left) and 152 Eu (right) radioactive sources. The spectra were measured by detectors d8, d22, and d16, shown in the top, middle, and bottom figures, respectively. The black dots and shaded histogram represent the measurement and the simulation, respectively.

Comparison of measurement and Monte Carlo simulation up to 11 MeV
The prompt γ-rays from the (n,γ) reaction have γ-ray energies of approximately 5-9 MeV. Therefore, the γ-ray from the 14 N(n, γ) reaction with a γ-ray energy of up to 11 MeV is useful to verify the reproducibility of the simulation. The γ-ray energy spectrum of the 14 N(n, γ) reaction was obtained by using a melamine target with a diameter of 10.75 mm and a thickness of 1 mm. Figure 11 shows comparisons of the melamine measurements and simulations in the detectors d8, d22, and d16. The spectrum of the measurement was reproduced by summing up the simulated spectra of monochromatic γ-rays. Here, in addition to γ-rays from C, H, and N with melamine target, Li, F, Al, Fe, and Ni were also considered.  Figure 11. Pulse-height spectra of γ-rays from the (n, γ) reaction of the melamine target. The spectra were measured by detectors d8, d22, and d16 are shown in the top, middle, and bottom figures, respectively. The black dots and shaded histogram represent the data and simulation, respectively.

Peak Efficiency of Individual Germanium Detectors
We define the probability of a γ-ray emitted to the direction of Ω γ = (θ γ , ϕ γ ) from the target position with a γ-ray energy of E = E γ at the dth detector as The distribution of the energy deposit is defined asψ where Ω d is the geometric solid angle of the dth detector. The efficiency of the emitted γ-rays at each detector is defined as are the upper and lower limits for the region of the full absorption peak.
We used w = 1/10 (full width of 10th maximum of the peak of the probability) as an integration interval. These definitions are summarized in Fig. 12. The γ-ray distribution can be expanded by a sum of Legendre polynominals ∞ p=0 c p P p (cos θ γ ) as The function ψ ′ d (cos θ, E γ ) was calculated by the simulation, as shown in Fig. 13. The dip structure of the peak shown in Fig. 13 is due to the opening on the back of germanium crystal where the electrode is inserted. The vertical dotted lines in Fig. 13 represent the angle of the detector center as indicated in Table 1.
In the left figure in Fig. 13, the dotted lines for θ = 70.9 • and θ = 109.1 • deviate from the center of each front-face, as the surrounding six detectors in the type-A assembly are not directed toward the target. This effect is more visible as E γ becomes higher. In the right figure in Fig. 13, the peak shapes for θ = 36.0 • and θ = 144.0 • are different from the others due to the different acceptance angles of the collimator (see D in Fig. 5). The peak for θ = 108.0 • is smaller than that of θ = 72.0 • , due to the size of detector d16. The calculated ǫ pk,w d (E γ ) andP d, p (E γ ) values are listed in Table 2-9 for 1 ≤ p ≤ 6 and shown in Fig. 14 for detector d3.
In the ǫ pk,w d (E γ ) column, the values are decreasing as the γ-ray energy increases, due to the punch-through effect. Nevertheless, the deviations from the averaged ǫ pk,w d (E γ ) values also decrease as the γ-ray energy increases, because the Compton scattering due to the materials between the target and the germanium detector is smaller for higher-energy γ-rays. Figure 15 shows the energy dependence of ǫ pk . Here, ǫ pk is defined as follows The values of ǫ pk,w d (E γ ) and ǫ pk decrease depending on E γ due to the loss of full absorption.   Figure 16 shows comparisons of the literature values [15] of the γ-rays intensities from the 14 N(n,γ) reaction and our values, which are calculated by using measurements and the simulation.
The vertical axis is normalized to unity at 10829.1 keV. The errors in Fig. 16 are given by the errors of literature values and the statistics from the measurement for the melamine target. Our expectation was that the data points scatter linearly and can be fitted with a constant parameter, as represented by the solid line in Fig. 16. The mean of the χ 2 /ndf of the fittings was about 1.5. We were able to verify the simulation in the wide γ-ray energy range up to 11 MeV.
Calculating Eq.4.2 for the 661.7 keV peak of the 137 Cs source, the difference of detection efficiency between measurement and simulation is about 10% on average for all detectors. This, represented by a solid line in Fig. 16, deviated from unity, and the difference was about 8% on average. The main reasons of the deviation of the absolute detection efficiency are the inaccuracy of the detector position, the difference in the size of germanium crystals due to crystal growth, and the difference in the sensible volume inside the crystals due to the difference in the cooling performances of each detector. However, when the energy dependence is reproduced, it is possible to extrapolate an efficiency to arbitrary energy region by using radiation sources or targets which have not been observed angular distribution, such as 14 N(n, γ).

Summary
In this paper, a Monte Carlo simulation with GEANT4 toolkit is presented, to reproduce the measurements using radioactive sources and a melamine target at the ANNRI. Using a simulation, we calculated the energy dependence of ǫ pk andP d, p . This novel method can be applied in the study of the partial neutron widths of p-wave resonances in the entrance channel to the compound states, which is required in the study of the discrete symmetry breaking in compound states. The germanium detector assembly installed at the ANNRI was characterized for the measurement of the angular distribution of individual γ-rays.