Simulation of wall materials with isotope concentration gradient for corrosion degree calibration in ITER

In the ITER reactor, the degree of corrosion of the wall is monitored by detecting the concentration of the isotope injected into the wall to ensure safe operation. Therefore, a wall material with an isotope concentration gradient that can be easily monitored must be developed. In this study, we adopted TRIM, Monte Carlo (M-C), and N (X) to predict the concentration distribution of isotopes injected into wall materials. The concentration peak and depth range of the isotope concentration distribution curve calculated by the TRIM program were very different, and the deviation was as high as 2.70%. Combined with the Monte Carlo (M-C) calculation method and the modified longitudinal static stability theory (LSS), the simulated isotope concentration distribution curve was in good agreement with the actual detection curve. However, the result was discontinuous, so the deviation could not be calculated. The N (X) simulation calculation exhibited a high degree of agreement, and the deviation was only 0.67%, so it may be considered suitable for the simulation of the concentration distribution of ion implantation in wall materials under various conditions.


Introduction
Plasma Facing Materials (PFM) are structure that directly contact high-energy plasma beams in the ITER device [1][2][3]. The wall material inside the fusion device is bombarded by plasma and various particles (including D, T, other and electron, neutron and impurity particles, etc), resulting in reduced service life due to transient and steady-state thermal loads and electromagnetic effects damage (such as irradiation, activation, evaporation, spatter, and bubbling) [4][5][6]. Hence, methods to detect the corrosion degree of the wall to ensure safety are being rapidly developed, and considerable effort has been invested to develop new technologies to monitor the corrosion degree of wall materials in real time.
Therefore,the concentration of ion implantation in samples must be predicted. Graphite is a traditional wall material of nuclear power plant. Here,we adoped three simulation methods to calculate the isotope concentration in graphite to provide a theoretical basis for the selection of injection parameters to prepare samples with isotopic carbon concentration gradient.

Experimental details
Samples of isotope 13 C were of analytical grade (purity of 99 wt%, Cambridge Isotope Laboratorics,Inc.) were used without further purification (figure 1). The graphite sample was a block with a high purity of 99.9% purity. A 2 × 3 MV tandem accelerator (Sichuan University)was used in the ion implantation experiment. The injection parameters were as follows:the ion energy E was 9 MeV, and the ion implantation dosage Φ was 5.5 × 10 15 atoms cm −2 . Carbon isotopes are injected into graphite as C 3+ . A laser-induced breakdown spectrometer was adopted to detect the concentration distribution of isotope 13 C in graphite [18].

SRIM-2013 simulation method
The TRIM (Transport of Ions in Matter) program in SRIM-2013 can be used not only to calculate the recoil cascade of particles in solids, but also to calculate the damage caused by the recoil cascade to the target by using external information such as the dynamics information and energy of the target atom. All kinetic phenomena related to ion energy loss, such as target damage, sputtering and ionization, are tracked in detail to obtain the 3D distribution of incident particles and ion energy loss, which is more accurate than the stopping range (SR) program.
where N T is the ion concentration distribution simulated by TRIM, the unit is atoms cm −3 ; N 0 = 1.07228 × 10 21 atoms cm −3 is the natural abundance of 13 C; and N TRIM is the depth-concentration data of the 13 C ion simulated by TRIM.
Compared with the depth-concentration distribution curve of 13 C ion actually detected (red line in figure 2), the results showed a large difference between the two curves [19]. The deviation between the calculation results of the TRIM program calculated and the actual experimental result was 2.70% calculated by equation (2),  indicating that the SRIM-2013 simulation method cannot be used to simulate the concentration distribution of isotopes in graphite walls.
where V c is the peak value of simulation, V m is the peak value of detection , V d is the deviation between the measured peak value and the calculated peak value.

Monte Carlo simulation method
The TRIM program is based on the LSS theory, but the calculated results were far from the experimental measurements and cannot be directly used to simulate the depth-concentration distribution of 13 C graphite injection. Therefore, the LSS theory is modified and the Monte Carlo method (Monte Carlo, M-C) to calculate the concentration of 13 C isotope in injected graphite [20].
The depth range after injecting 13 C ions with energy of 9 MeV and dose of 5.5 × 10 15 atoms cm −2 into graphite is known to be less than 10 μm. For the convenience of calculation, the graphite sample was divided into 50 layers from plane facing the incoming ion beam, each with a thickness of 0.2 μm, and we assumed that the target atom density within the graphite sample to be uniform within a given layer. To directly apply the LSS theory, the graphite sample size was set as 50 × 10 × 10 mm 3 , the weight was set as 11.925 g, and the density is calculated as follows: 11.925 50 10 10 10 cm 2.3825 g cm 3 Calculation of the average density of the target material as the actual density of the graphite sample was performed as follows: where n is the total number of layers was 50, ρ(l) is the graphite internal density of the layer. There were 3 nm holes in the graphite sample, which accounted for about 12% of the total volume. Considering this factor, an approximate physical model of graphite material was established by concentrating the holes and layering distribution processing, assuming that 6 of the 50 layers of the graphite sample were empty layers.
According to LSS theory, the energy loss of 13 C ion implantation into graphite includes elastic collision energy loss and inelastic collision energy loss. The energy loss per unit distance of ion implantation into target material is given as Where x is the incoming ion beam propagation direction, In the first order approximation, the electron is a free electron gas and has no correlation with the energy at a position as given below : At the same time, S e (E) is proportional to the incident energy of ion injection as follows: where Z 1 , M 1 , Z 2 and M 2 are the atomic number and mass number of 13 C ions and sample materials. However, in contrast to the first approximation in Formula (6), S e (E) is not a constant, but is related to energy. After fitting the data calculated by TRIM program, the functional relationship with ion energy is established as given below: Considering the energy loss of 13 C, the physical processes of mass precipitation and charge exchange are very complex and no mature theory yet. Hence to simplify the calculation, a correction coefficient was introduced to correct the energy loss per unit distance, which depends on the initial energy ion, ion species and target material. Because the mass number of the injected 13 C was slightly larger than that of the sample, the deflection angle of 13 C after collision was very small and the direction of motion remained largely straight, we assumed that the collision process to be one-dimensional collision. In one-dimensional approximation, the advance distance after the collision is random. To simplify the calculation, only the energy loss of the injected ion 13 C was considered. In addition, it is assumed that the distance of bounce of each collision was assumed to be a fixed value.
The average atomic number, mass number, injection energy, initial position and graphite density parameters of 13 C ion and graphite target atoms are shown in table 1.
According to the above modified LSS theory, the energy loss per unit distance is: The total atomic number of the sample per unit volume is given as: where N A = 6.022 × 10 23 atoms mol −1 Because the energy loss of ion and atom collisions is random, we used the M-C method in the experiment to count the implanted ions statistically: where Rand () is a MATLAB program that can use random numbers in the range of 0 ∼ 1. After the process of 13 C ion implantation and energy loss is simulated by the M-C method, the number of 13 C ions residing in each layer is obtained, and the thickness between layers is known to be 0.2. The number of 13 C ions residing in the layers can be converted to the depth distribution of 13 C ions. The depth-concentration relationship of 13 C ions can be obtained by normalized processing in Formulas (12) and (13). The M-C method was used to simulate the depth-concentration distribution of 13 C ions after the specification( figure 3). It is only applicable to the calculation of discontinuous depth-concentration distribution after low dose 13 C ion implantation.

Determination of projection range parameters
For the sputtering effect, the energy loss of 13 C ion implantation -dE dx is mainly the energy loss caused by the collision between a 13 C ion and a nucleus

Determination of sputtering coefficient parameters
We then determine the sputtering coefficient parameters where Y is the sputtering coefficient, and U 0 is the sublimation energy of graphite target atom:  Equation (23) is modified as follows for greater accurate and applicable to a wider range of applicability [24].
where η is also related to the constant M The calculation flow chart of sputtering coefficient and injection energy of 13 C ion implantation is shown in figure 5.

Simulation of concentration distribution of 13 C ions after implantation
High dose 13 C ion implantation causes a sputtering effect, and the concentration distribution after 13 C ion implantation no longer follows the normal Gaussian distribution as given below:

=´-
where n 0 can be calculated by Formula (10), R p and ΔR p can be known by table 2, d is the depth, and the unit is μm, S is the thickness of construction layer removal: x t Owing to the instability of the sputtering effect, the correction coefficient r was introduced to modify the formal equation (30):  ( ) The calculation error of N (X) was only 0.67% , and it exhibited a continuous curve ( figure 6) that can be used to calculate the depth-concentration relationship after 13 C ion implantation under various conditions.

Conclusion
In this study, we have adopted three simulation calculation methods to simulate the concentration distribution of 13 C isotopes in the 13 C ion implanted graphite. The results showed a large deviation of 2.7% between the carbon distribution calculated by the TRIM method and the actual detected distribution. The modified LSS theory and M-C method simulation results were in good agreement with the detection results, but were not continuous and are not suitable for calculating high-dose 13 C ion implantation. The calculation deviation of the new concentration distribution formula N (X), was only 0.67%, and it exhibited a continuous curve. Therefore, the N (X) method can be used to simulate the concentration distribution of the isotopes at various injection doses.