2.1 Production Mechanisms

At the Earth’s surface, when high-energy cosmic rays bombard HPGe crystals, they collide with germanium atoms through elastic scattering, inelastic scattering, and other processes, thereby producing a series of radioactive unstable nuclei inside the crystals. These radionuclides will decay according to their respective half-lives, accompanied by the release of gamma-rays, electrons, and other particles, and form the cosmogenic background. The production of radionuclides is an extremely complex process typically involving a wide energy range of 1 MeV to 100 GeV. If $R_i$ is considered the production rate of a radioactive isotope i, it can be expressed as follows:

where Nj is the number of stable target nuclear isotopes j, σ_kij is the production cross section of radioactive isotope j produced by cosmic-ray particle k on target isotope j, and Ф_k is the flux of cosmic-ray particle k.

After high-energy cosmic ray bombardment, numerous types of radionuclides are produced, but not all contribute to the cosmogenic background. The major contributors can be found through the decay laws. When HPGe is exposed to cosmic rays, radionuclide decay occurs. The rate of nuclide production can be expressed as follows:

where P is the rate of nuclide production, N is the number of individual nuclides at time t, and λ is a decay constant.

Here, N is given as follows:

Clearly, when t approaches infinity, i.e., the exposure time of germanium is sufficiently long, and the number of nuclides produced and the decay become balanced such that the radionuclides no longer increase in number. This process requires a half-life of 4.3 to reach 95% of the saturation value, and a half-life of 6.6 to decay to 1%. Therefore, short-lived nuclei can be eliminated simply by placing the detectors in a deep underground site for a sufficiently long time before experimental data acquisition. Only long-lived nuclei need to be focused on in this study.

2.2 Simulation tools: Geant4 and CRY

The production of radionuclides is a compound process that involves a wide energy range of particles in cosmic-ray showers and different types of interactions within the composite shielding materials^[23]. In this study, Geant4 was used to simulate the interactions between cosmic rays and the target material. CRY^[22] provides information regarding the energy spectrum and flux of the cosmic rays. Geant4 is a toolkit for both a full and fast Monte Carlo simulation of detectors in high-energy physics. The Geant4 package (V10.5p01 + Shielding physics list) was used in this study. CRY software is based on MC calculations used to predict cosmic-ray distributions when data are unavailable. This library can generate cosmic-ray particles including neutrons, protons, muons, gammas, electrons, and positrons^[22] at different altitudes (sea level, 2,100 m, and 11,300 m). In this study, we concentrate on the first four particles. The energy, direction, and position sampled from CRY are inputs for the Geant4 simulation program. Taking Chengdu and Beijing as examples, the spectrum information of the cosmic neutrons, protons, muons, and gamma particles can be obtained. Table 1 lists the cosmic ray fluxes calculated by CRY at sea level and at a height of 11,300 m in these cities in 2018.

Clearly, the impact of the altitude and latitude on the cosmic ray flux is extremely large and thus the choice of a storage and transport method is crucial. This study takes land transport as an example. Fig. 1 shows the spectra of neutrons, protons, muons, and gamma particles at sea level in Chengdu.

Fig. 1Full-screen View

(Color online) Spectra of cosmic-ray neutrons, protons, muons, and gamma particles at sea level in Chengdu.

2.3 Simulation details

We simulated two detector geometries in this study. One is a 1 kg cylindrical HPGe crystal with a diameter of 62 mm and a height of 62 mm. The crystal was constructed at the center of an air box and was similar to that used by Ma ^[23].

The other is a cylindrical HPGe crystal with an outer shielding container, as shown in Fig. 2. The HPGe crystal (Ф42 cm ×27 cm) was the same size as that used in the GERDA experiment^[17]. However, note that the geometry of the crystals has no impact on the production rates of the cosmogenic isotopes because the simulated productivity will be given normalized treatment. For convenient transport, we used a 40-foot container truck. The exterior dimensions of the container are 12.192 m × 2.438 m × 2.591 m, and the interior dimensions are 12.032 m × 2.352 m × 2.385 m. In addition, there is another shielding structure in the container truck. From outside to inside, the initial shielding structure consists of 30-cm thick polyethylene (with 100-cm thick polyethylene on top of the crystal), 30-cm thick iron, 10-cm thick lead, and a 12-cm thick copper layer. A schematic diagram of the shielding system is shown in Fig. 2.

Fig. 2.Full-screen View

(Color online) The shielding structure for transport of germanium crystal on the surface: 1, HPGe detector; 2, cavity; 3, oxygen-free copper; 4, lead; 5, iron; and 6, polyethylene.

The total mass of the shielding container is 27,873 kg, and the weights of the truck, Fe, PE, Pb, and Cu 40-ft container are 3,400, 6,448, 13,459, 3,079, and 1,487 kg, respectively. The deadweight is under 28t and the capacity is 67.5 m³.

The position of each incident particle is randomly extracted from the plane above the detector, the momentum information is randomly extracted from CRY, the energy information is randomly extracted from the CRY-provided energy spectrum, and all secondary particles generated in the detectors are collected and counted.

3.1 Shielding effect of different materials

The shielding effects of iron, lead, copper, liquid nitrogen, polyethylene, and concrete are shown in Fig. 3. Fig. 3 (a–c) shows that lead, iron, and copper have a better effect on stopping the primary protons and gamma-rays. In addition, materials such as polyethylene and concrete are better choices for neutron shielding. According to previous studies^[16,17], iron has been proven to be an optimal material owing to its lower production rate of secondary neutrons. However, our simulation shows that copper and lead are also appropriate materials, having a lower production rate than iron with a neutron energy of above 10 MeV. By contrast neutrons at below 10 MeV cannot produce radionuclides in germanium owing to a restriction of the reaction energy threshold, as mentioned in the TENDL data library. Apart from the high-energy secondary neutrons, the primary radionuclides produced in the shielding material also cannot be ignored. The radioactivity of high-purity oxygen-free copper is as low as mBq/Kg order of magnitude. Therefore, considering the cost and weight of the shielding during transport, lead, iron, copper, and polyethylene should be selected as key materials for shielding structures used in the transport of HPGe. Based on the fluxes generated by different cosmic rays from different shielding materials, we analyzed the placement and thickness of the shielding materials. The optimal thickness of each shielding material was obtained by fitting the mass thickness function.

Fig. 3:Full-screen View

(Color online) Effect of cosmic rays when using different materials. The abscissa indicates the mass thickness of the material, and the coordinates show the number of outgoing cosmic rays. For all materials applied, the number of primary particles in this simulation is 10⁷.

The results of different shielding structures are shown in Fig. 3. As indicated in Fig. 4, the particle count rate decreases from the outside to inside by the shielding. Furthermore, the numbers of protons and gamma particles decrease significantly when passing through the materials, illustrating that these particles are easy to shield. Therefore, the main cosmic ray sources are muons and neutrons. In particular, during the transportation process, a way to effectively shield the neutrons needs to be an area of focus. The simulations show that high-energy neutrons can reduce the amount of energy in polyethylene. Thus, polyethylene is necessary for the effective shielding of neutrons. As shown in Fig. 3, the total neutron count rate decreases by approximately one order of magnitude after being shielded. This demonstrates that the selection order of the shielding materials is reasonable. The proposed composite structures have a meaningful impact on stopping any cosmic rays.

Fig. 4.Full-screen View

(Color online) Count rates (cps) of particles produced by different components of cosmic rays. The numbers on the horizontal axis denote the following: 1, polyethylene; 2, iron; 3, lead; and 4, copper.

However, the results in Fig. 4 indicate that the shielding of muons is relatively weaker. Therefore, to effectively shield cosmogenic muons, additional structures are needed.

3.2 Cosmogenic production rates of germanium

The cosmogenic production in both detectors described in Section 2.3 were simulated using Geant4 and CRY. For the given input energy spectrum of the muons, neutrons, gamma particles, and protons on the Earth’s surface, the production rates of several cosmogenic isotopes in natural germanium are calculated. Considering the total production rates, half-life, and energy region of interest, isotopes ⁶⁸Ge, ⁶⁸Ga, ⁶⁵Zn, ⁶⁰Co, ⁵⁵Fe, and ³H are the important concern for germanium-based detection experiments^[24]. The simulated production rates of cosmogenic isotopes for natural germanium at sea level in Chengdu are listed in Table 2. From Table 2, neutrons contribute to approximately 90% of the production rates of cosmogenic activation. This result is consistent with that of previous studies^[23,24].

As shown in Table 2, the shielding structure significantly reduces the production rates of cosmogenic isotopes from cosmic neutrons, gamma particles, and protons, and the suppression factors are 10 (³H) to 97(⁵⁵Fe) for natural germanium. These results demonstrate that a shielding container is necessary when transporting germanium detectors on the Earth’s surface.

3.3 Comparison between this study and previous estimates and measurements

Table 3 summarizes the available studies on the calculated and experimental production rates for several isotopes in natural germanium and shows a comparison between our approach and the results of previous production calculations and experiments. As shown in Table 3, the cosmogenic production rates for ⁶⁸Ge, ⁶⁸Ga, ⁶⁵Zn, ⁶⁰Co, ⁵⁵Fe, and ³H in natural germanium vary significantly among different model estimates. Such large variations in the production rates are mainly due to the following two reasons:

(1) The difference in estimated rates is mainly due to the use of different simulation software and cross-section libraries. Simulation software can be roughly divided into two categories. One is software calculations based on empirical formulas of isotopic generation cross sections, such as YIELDX^[25] and ACTIVIA^[26]. The other is software applying a Monte Carlo method to simulate the interactions between incident particles and the target nucleus, such as TALYA^[27] and Geant4^[16,23,28]. Geant4 is more suitable for simulating the generation of cosmic rays owing to its wide energy range and comprehensive particle cross-section data. In addition, when we use the same simulation tool, large differences will occur owing to the different versions applied, e.g., Geant4 versions 3.21^[29], 9.5.02^[24], and 4.10.02^[23]. In addition, the production rates of the cosmogenic isotopes vary significantly among the different Geant4 versions, ranging from a factor of ∼2 (³H, ⁵⁵Fe, and ⁶⁵Zn) to ∼3 (⁶⁰Co and ⁶⁸Ge) for germanium.

(2) The second reason is that they input a different neutron spectra of cosmic ray. Numerous estimates^[17,19,30] have come from calculations using historical cosmic ray neutron spectra,^[31] which are less precise than modern estimates. In addition, calculated or experimentally determined production rates of radioisotopes by cosmogenic activation in natural germanium have been published. Only EDELWEISS^[32] and SuperCDMS^[33] have published measurements of the production rate of tritium in natural germanium.

As shown in Table 3, our results agree quite well with previous available measurements^[23]. The differences among these calculations are mainly due to the use of a cosmic ray flux in different places and the use of different versions of the simulation tools.

As shown in Table 4, it is worth mentioning that our shield container achieves a higher reduction of the production rates of cosmogenic isotopes in germanium. Compared with iron shields, our shield exhibits an improvement in suppression by one order of magnitude between these isotopes for natural germanium. They are mostly produced by an inelastic scattering of neutrons. The reduction of ⁶⁰Co and ⁶⁸Ge has an actual meaning in 0νββ experiments, and ³H may have a significant impact on the sensitivity of germanium-based detectors for future ton-scale experiments in the direct detection of low-mass dark matter particles. The end point energy of ³H beta decay is only ∼18.6 keV. This implies that the entire energy spectrum of ³H can contribute to the background of a low-energy region for low-mass dark matter detection^[24]. The lifetimes of ³H, ⁶⁰Co, and ⁶⁸Ge are particularly long, which makes it difficult to eliminate these radioactive isotopes by simply placing the detectors deep underground, and thus the only way to reduce these backgrounds is to limit the exposure of the crystal when it is grown and transported^[34]. Therefore, reducing the production rates of ³H, ⁶⁰Co, and ⁶⁸Ge using a shield is extremely meaningful for rare event search experiments.