Preamble

Radiation tolerance test and damage of single-crystal CVD Diamond sensor under high fluence particles Jialiang Zhang, 1, 2, Shuo Li, Yilun Wang, Shuxian Liu, Guojun Zifeng Xu, Lifu Hei, Fanxiu Lv, Lei Zhang, and Ming Qi 1, 2, 1 National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China School of Physics, Nanjing University, Nanjing 210093, China School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China Single-crystal chemical vapor deposition (CVD) diamond is a promising material for radiation detectors op- erating in extreme environments, owing to its outstanding radiation hardness. As nuclear and high-energy physics applications demand particle detectors that withstand higher radiation fluences, understanding the dam- age thresholds and degradation mechanisms of diamond-based detectors is essential for their practical operation.

In this study, Synthetic single-crystal CVD diamond sensors were exposed to fast neutron irradiation at fluences up to , one of the highest test doses for evaluating radiation tolerance in diamond detectors.

Modules exhibited stable signal output, retaining approximately 5% of their initial response after irradiation, confirming potential for application in future high-dose radiation environments. Fast neutron induced damage in the diamond crystals was characterized using photoluminescence and scanning electron microscopy. The dominant defects were identified as point defects including self interstitials, vacancies, and lattice disor- der. In addition, macroscopic defects on the crystal surface, including nanocavities and cracks, were observed with areal densities approaching . The impact of

100 MeV

proton irradiation on diamond detector re- sponse was quantified by extracting a damage constant of

100 MeV

proton from a linear carrier drift degradation model. Moreover, the mean free path of carriers was found to exhibit sat- uration behavior beyond a fluence of under

100 MeV

proton irradiation. Monte Carlo together with molecular dynamics simulations were performed to assess irradiation induced defect production and its in- fluence on carrier transport. The results indicate that saturation arises when local frenkel defect densities exceed , at which defect interactions and clustering begin to dominate during irradiation. By considering saturation effects and defect-interaction corrections, we develop an enhanced carrier-drift degradation model that accurately captures detector response under high-dose irradiation. Furthermore, the simulation framework was applied to evaluate damage induced by protons and pions on diamond at various energies, yielding results that show better agreement with experimental data than conventional NIEL based estimates.

Keywords

Diamond detector, Radiation tolerance, Radiation damage, Defects simulation (Contents for review process) Contents

3.2. Radiation induced crystal damage in diamond

3.2.1. Luminescence centers associated with radiation-induced defects

Supported by the International Science & Technology Cooperation Pro- gram of China (No. 2015DFG02100), the Ministry of Science and Tech- nology of the People’s Republic of China.

Contact author, Jialiang Zhang:

3.2.2. Irradiation-Induced Surface Morphology Characterization

3.3. Evaluation of the radiation hardness and damage

in diamond with damage model 3.3.1. Damage constant of single-crystal diamond under 100 MeV proton with linear damage model 3.3.2. Multiscale modelling of radiation damage

by combining Monte Carlo and molecular 20

dynamics simulations 3.3.3. Modification of the simple damage model at high radiation dose

1 Introduction

Over the past decades, particle physics has entered a trans- formative era, marked by unprecedented precision and dis- covery. The 2012 observation of the Higgs boson [ ] sig- naled a milestone, pushing experimental physics toward prob- ing the Standard Model at increasingly finer scales while ex-

ploring potential new physics beyond it. These pursuits have driven accelerator and collider upgrades along both the lumi- nosity and energy frontiers as well as the foundation of con- ceptual design for next-generation experiments[ ], setting the stage for demanding operational conditions where detector radiation tolerance becomes critical[ ]. In parallel, applica- tions in nuclear fusion systems are approaching environments of comparable or even greater radiation severity[ ], further amplifying the need for robust detection technologies.

Among candidate materials, synthetic single-crystal dia- mond has emerged as a leading contender for next-generation radiation detectors. It combines exceptional material prop- erties, including high charge carrier mobilities large than 3000 cm ], a wide bandgap of 5.45 eV that enables excellent signal-to-noise performance, and superior thermal conductivity reaching

2000 W

]. Most critically, its radiation hardness surpasses silicon by factors of three for low-energy incident particles and by more than a factor of ten at high energies[ ], which is attributed to its high displace- ment energy of up to 43.5 eV[ ]. The advent of high-quality

chemical vapor deposition (CVD) techniques has made large- 52

area, defect-minimized single-crystal diamond (scCVD) sub- 53

strates increasingly accessible[ ], making their a promis- ing choice for applications in detectors subjected to high ra- diation levels. scCVD diamond detectors have been success- fully employed in previous and ongoing experiments and fa-

cilities for beam monitoring and particle tracking, including 58

the LHC, SuperKEKB, and the EAST tokamak system[ Despite these advances, the upper radiation dose limits that scCVD diamond detectors can tolerate without critical signal loss remain an open question. Prior studies, including work from RD42[ ], have demonstrated the viability of dia- mond detectors under proton and neutron fluences approach- . However, future environments are expected to push beyond these thresholds[ More- over, irradiation sources often include mixed particle fields with broad energy spectra, making it essential to understand not only how damage accumulates but also how it impacts charge transport at the microscopic level and consequently affects detector performance. Detector performance degrada- tion is commonly understood as a two-step process: irradi- ation introduces lattice defects, and these defects in turn act as traps or recombination centers, reducing carrier lifetimes and thus degrades charge collection efficiency and detector performance[ Although diamond is renowned for its exceptional radia- tion hardness, a detailed understanding of how irradiation- induced defects influence charge transport is still lacking. To address this gap, pioneering studies have carried out system- atic irradiation experiments using protons and pions at mul- tiple energies. These efforts produced quantitative data on signal degradation, which in turn enabled the development of simplified damage models[ ]. Such models have been applied to comparatively assess the effects of different parti- cles and energy levels on diamond detector degradation[ ]. Beyond these experimental insights, predictive under- standing requires complementary modeling approaches. Two main strategies have been developed. The first relies on the

concept of non-ionizing energy loss (NIEL)[ 25 ], where the 91

energy deposited into the lattice leads to atomic displace- ments and crystal damage. The NIEL cross section provides

a convenient and widely used metric for estimating radiation 94

damage. The second strategy adopts a material-centric per- spective, explicitly evaluating the number and types of de- fects generated in the lattice[ ]. This defect-informed methodology offers a more physically grounded framework for predicting detector degradation.

In this study, we present a comprehensive investigation

into the radiation tolerance and damage mechanisms of sc- 101

CVD diamond sensors exposed to high fluence fast neutron and protons. Four scCVD diamond sensors were fabricated and irradiated at nuclear facilities for a fast neutron radia- tion experiment. The cumulative neutron fluence achieved , one of the highest doses reported for single-crystal diamond to date. The sensors exhibited a sus- tained signal response, underscoring the potential of scCVD diamond as a viable candidate for mitigating the limited op- erational lifetime of silicon-based detectors, especially in the innermost layers of next-generation harsh radiation experi- ments.

To elucidate the microscopic origins of radiation- induced performance degradation, the neutron irradiated di- amonds were systematically characterized using photolumi-

nescence spectroscopy and scanning electron microscopy. 115

Damage was observed to generate atomic-scale point defects, crystalline lattice disorder, and macroscopic defects including voids and microcracks.

To assess the correlation between defect generation and transport degradation, we analyzed signal response data from scCVD diamond detectors previously irradiated with 100 MeV protons. Using a simplified carrier drift degradation model, we extracted a damage constant and normalized it to radiation damage of 24 GeV protons via established scaling relations[ ], allowing direct comparison with results from other irradiation studies. This analysis contributes quantita- tive insight into an energy regime that has remained relatively unexplored. Notably, we observed that the carrier mean free path exhibited saturation behavior at high fluence levels, sug-

gesting a shift in the dominant damage mechanisms. To in- 130

terpret this effect, a combined simulation framework includ- ing Monte Carlo simulations and molecular dynamics mod- eling incorporating adiabatic recombination (arc-DPA)[ were employed. The results indicate that when local defect densities exceed, interactions among defects begin to dom- inate over isolated point defect formation, driving a transi- tion toward saturation in performance loss. Based on this, we refined the traditional carrier drift degradation model to account for saturation effects at extreme doses. Finally, the combined simulation framework was applied to assess radia- tion damage from protons and pions across a range of ener- gies. The predictions show closer agreement with experimen- tal data than conventional NIEL-based estimates, highlighting the importance of defect-level modeling for accurate perfor- mance forecasting. Together, these results expand our under- standing of diamond detector behavior under extreme radia- tion and provide actionable insights for their deployment in

high-luminosity colliders, nuclear science such as fusion re- actors, and space-based instruments. 2 Experiment and Methods

2.1 Fabrication of DUT modules

The single crystal diamonds material were synthetic using a commercial 30 kW DC arc plasma jet chemical vapor de- position (CVD) system operated in gas recycling mode. On the substrates of commercial high pressure high temperature (HPHT) type-Ib (100) single-crystal diamond, a number of high quality large-sized single-crystal diamond plates were

fabricated using the CVD homoepitaxial growth technique. In 158

preparation for the fast neutron irradiation experiment, self- supporting synthetic scCVD diamond plates were separated

from the substrates by laser cutting. Then, mechanical pol- 161

ishing and boiling with a combination of acids were also used in order to get rid of any potential contaminations and dam- aged layers on the surfaces of scCVD diamond. Following the

cutting, polishing, and cleaning procedures, scCVD diamond 165

plates have the final size with a surface area of and thickness around , as depicted in Fig. (a). Raman spectroscopy was carried out on the surfaces of the plates to examine the purity and perfection of the single-crystal dia- mond, the results[ ] of strong first-order peak at 1332 cm with a narrow full wave at half maximum (FWHM) of demonstrating the obtain of high quality synthetic scCVD di- amond plates prepared for radiation detection sensors. Pla- nar Ti–W–Au electrodes were deposited on both surfaces of the scCVD diamond plates by magnetron sputtering, form- ing efficient metal–insulator–metal (MIM) detection sensors used as the Device Under Test (DUT) sensors, as illustrated in Fig. (a). The electrical properties of these sensors were assessed through I-V curve measurements, as depicted in proximately 0.4 nA under a 500 V bias voltage with good linearity, indicating robust ohmic contact between the elec- trodes and the diamond. Subsequently, DUT sensors were in- corporated into Rogers ceramic base high-frequency PCBs as modules, featuring planar electrodes connected to the readout

electronics through gold wire bonding, as show in Fig. 1 [FIGURE:1] (b). 186

Long Kapton-insulated coaxial cables were utilized for elec-

tronic communication, facilitating a connection to the remote 188

data acquisition (DAQ) system.

2.2 Radiation tolerance experiment

The fast neutron irradiation was undertaken at the IBR-2M reactor in the Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Dubna, Russia. The experiment took place in the beamline specifically designed for neutron irradi- ation experiments[ ]. The DUT modules were placed inside a container, which was positioned at one end of an extended conduit and located 30 centimeters away from the reactor moderator, to get high fluence of fast neutrons. Connected by long cables passing through a nuclear radiation shielded area, DUT modules were linked to the DAQ system. The DAQ system consists of a high voltage supply module, a volt-

age monitoring module, and a digital multimeter. The high- 202

voltage supply module is responsible for providing and mon- itoring bias voltage up to 260 V on the DUT sensors during the experiment, supplied by the Keithley 6487 module. The multi-channel multi-meter Keithley 2700, equipped with the Keithley 7703 relay card, is used to read out and record DC

current ionization detection signals triggered by fast neutrons, 208

and transmit them to the computer. The schematic diagram of the entire process is shown in Fig. (a). One HV line, not con- nected to the detector, was also read out to measure dark cur- rent and noise. The entire irradiation experiment accumulated for approximately 280 hours. During this period, the reactor’s operating power was maintained at around

2 MW

, as shown in Fig. (b). The accumulated fast neutron irradiation fluence at the location of DUT modules reached with a flux of

Shield HV Supply Moderator Modules Active Core

Voltage Monitor

Cables (To PC) 3 Results and discussion 3.1 Radiation tolerance of scCVD diamond detector The data acquisition system recorded the continuous DC current response from each of the four single-crystal CVD diamond detector modules under a bias of 260 V, with mea- surements taken every 30 minutes. As shown in Fig. a sixty hour interruption occurred due to a reactor halted, leading to data interruptions. the relationship between the DC signal and the fast neutron fluence was reconstructed, as presented in Fig. . Following exposure to a fluence of up to fast neutrons, the results indicate that four DUT modules maintained a signal response around

5% of their initial signal, confirming the modules’ robustness 231

against extreme neutron irradiation. The observed DC sig- nal originates from electron–hole pairs generated by neutron- induced atomic displacements in diamond. Although neu- trons are electrically neutral, they induce nuclear collisions governed by a hard-sphere potential, displacing carbon atoms and producing primary knock-on atoms (PKAs) that generate charge carriers. The resulting radiation damage introduces displacement type defects as trapping centers whose density defects scales with the accumulated fluence. Assuming that the signal degradation is inversely proportional to the number of traps, the detector response as a function of neutron fluence follows[

S ( ϕ ) = 1 1 S 0 + kϕ + c (1) 244

where S 0 is the initial signal, k is the damage constant, and c 245

accounts for baseline offsets, as fitted lines shown in Fig.

After irradiation, the DC signal remains at 5% of its initial 247

value with sustained responsiveness, suggesting that sc-CVD diamond is a promising candidate to overcome the lifetime limitations of silicon-based materials in harsh high-radiation environments, particularly in the innermost detector layers of collider experiments.

Signal degradation can be corrected through dedicated calibration procedures. In early simula-

tion studies of the HL-LHC mini-Forward Calorimeters em- 254

ploying CVD diamond sensors, such procedures were devised to account for radiation-induced charge and current losses, thereby ensuring the stability of the energy resolution[ 3.2 Radiation induced crystal damage in diamond After the irradiation experiment, DUT modules were re- trieved from the irradiation facility when their radiation level had sufficiently decreased to an acceptable threshold for hu- man exposure. High dose fast neutron radiation induced dam- age in sc-CVD diamonds was characterized using photolu-

minescence (PL) spectroscopy and scanning electron micro- 264

scope (SEM). For the purpose of investigating crystal dam- age, sc-CVD diamonds were extracted from the test modules

and subjected to a cleaning procedure to remove the metal 267

electrodes. This procedure involved immersing the diamonds

in a boiling mixture of hydrochloric acid (HCl) and nitric acid 269

), followed by a hydrofluoric (HF) acid treatment to eliminate the Ti-W-Au metal and oxide layer. Additionally, sc-CVD diamonds were sequentially immersed in solutions of acetone, alcohol, and ultra-pure water, respectively, for ul-

trasonic cleaning to ensure a pristine surface free from any 274

residual organic or inorganic contaminants. 275

3.2.1 Luminescence centers associated with radiation-induced defects The optical image of the sc-CVD diamond after de-

metalization and cleaning has been included in the subfig- 279

ures of Fig. 4 [FIGURE:4]. It can be clearly observed that the sc-CVD

diamond has turned brown-black, significantly reducing its 281

optical transparency. This phenomenon originates from an increased presence of luminescent centers. Additionally, the incorporation of graphitic regions, nanoscale domains, and interface boundaries into the diamond structure can also con-

tribute to the darkening of diamonds[ 33 , 34 ]. To investigate 286

these radiation induced luminescent centers, PL spectroscopy was performed using a 488 nm blue laser.

The tempera-

ture was lowered to

77 K

in a liquid nitrogen environment, 289

aimed at mitigating the influence of first-order Raman scat- tering within the diamond. The PL results of neutron irra- diated diamonds are present in the Fig. . Aside from the first-order Raman excitation in 522 nm , additional lumines- cent centers within the diamond lattice had been induced due to neutron radiation. In Fig. , the 503 nm zero-phonon line (ZPL) can be seen, which originates from two distinct com- ponents: the 3H defects[ ] and the H3 defects[ ], both resulting in a 46 eV energy level transition. The 3H de- fects center consists of single, isolated self-interstitial defects, a direct consequence of radiation-induced atomic dis- placement within the diamond lattice. The H3 structure, in-

clusive of the nitrogen vacancy N-V-N defect, may arise from 302

nitrogen atoms occupying several adjacent vacancies or void 303

defects created by radiation upon the surface of sc-CVD di- amond. The presence of the 550 nm center suggests the the occurrence of plastic deformation[ ] and shear stress[ in diamond crystal, frequently seen in brown diamonds[ The ZPL at 649 nm is assigned to disordered regions in the di- amond lattice induced by neutron irradiation[ ]. The small

ZPL at 678 nm can be refered to nitrogen induced fluores- 310

cence peak caused by 83 eV photon[ ]. The well-known GR1 center in diamond is represented by a ZPL at 741 nm This center is a common occurrence in diamonds when ex- posed to radiation and can be attributed to neutral defects of carbon atom vacancies ( ), which exhibit a tetrahedral ( symmetry. Meanwhile, Jahn-Teller interaction caused by dis- tortion in the GR1 center would result in the degeneracy split- ting of the ground state into two sub-levels, , with an energy difference of 8 meV 673 eV of energy level tran- sition between lowest ground state 1E and first excited state volving state and state leads to the 744 nm side peak, which has a photon energy of 665 eV 3.2.2 Irradiation-Induced Surface Morphology Characterization

Scanning electron microscopy (SEM) was employed to in- 326

vestigate the surface morphology of neutron-induced damage

in sc-CVD diamond samples. A Zeiss Gemini SEM 500, 328

equipped with both an in-lens detector and a backscattered electron (BSE) detector, was used to acquire high-resolution secondary electron (SE) and backscattered electron images.

To enhance sensitivity to surface features and defect con- trast, a low accelerating voltage was applied during imag- ing. The surface morphologies of the (001) diamond plane after neutron irradiation are shown in Fig. . In comparison

with the unirradiated diamond surface in Fig. 5 [FIGURE:5] (a), neutron 336

irradiation has clearly resulted in significant surface dam- 337

age. Fig. (b)–(d) present secondary electron (SE) images acquired under low accelerating voltage conditions. In par- ticular, Fig. (b) reveals intersecting crystal cracks accompa-

nied by cavities distributed along the crack paths. The crystal 341

cracks extend over several micrometres, shown in Fig. (c), as

the magnified view of the red-boxed region in Fig. 5 (b). At 343

the same time, cavities ranging from a few tens to several

hundreds of nanometres are evident in the magnified image 345

in Fig. (d). BSE imaging in Fig. (e) provides further in- sight into the subsurface damage morphology by revealing the distribution of cavities, as backscattered electrons origi- nate from deeper regions at higher accelerating voltages, ac-

cording to the Kanaya–Okayama formula[ 44 ]. The magnified 350

BSE image in Fig. (f) shows shallow, layered fringe patterns surrounding the cracks, representing the extension of crack- induced damage and indicating the presence of possible stress or graphite layers[ ] induced by radiation. Based on SEM analysis, the estimated surface defect densities are ap-

proximately for cracks and for cavities. The cracks, with widths on the order of sev- eral nanometres, can be regarded as two-dimensional planar defects analogous to grain boundaries, intersecting the (001) crystal surface. The cavities, in contrast, are thought to be bulk defects emerging at the surface due to the aggregation of vacancies and voids during irradiation, or stress accumulation around point defects leading to localized fracture.

(a) Unirradiated diamond surface. (b-d) Surface defect morphology via in-lens secondary electron imaging. (e-f) Surface topography revealed by backscattered electron imaging.

3.3 Evaluation of the radiation hardness and damage in diamond with damage model The irradiation-induced defects in the detector crystal orig- inate from interactions between incident particles and lattice atoms. In this section, we evaluate the impact of crystal dam- age on the performance of diamond detectors using 100 MeV proton irradiation, and simulate the defect formation process. 3.3.1 Damage constant of single-crystal diamond under 100 MeV proton with linear damage model In diamond detector irradiation experiments, a simplified linear model is often employed to estimate radiation-induced damage[ ]. This model assumes that the total number of radiation-induced defects in a crystal scales linearly with the irradiation fluence:

These defects introduce energy levels that trap charge carri- ers generated by incident particles. Moreover, following a simplified analysis by Kramberger et al.[ ], the carrier life- time ( ) is inversely proportional to the defect concentration ), derived from defect-mediated trapping behav-

τ L = 1 �

Here, denotes the mean drift velocity, represents the interaction (trapping) cross-section for a specific type of charged carrier, and is the concentration of a given de- fect/impurity species. Given that the mean free drift distance

1 /λ of carriers scales linearly with their lifetime λ = τ × v = 390

, it follows that the inverse drift distance is propor- tional to the defect concentration, , rewrite into linear relationship as

1 λ = k · ϕ + 1

λ 0 denotes the initial mean free path (MFP) of carriers prior 395

to irradiation, and is the damage constant, which depends on the type and energy of incident particles, can be expressed . This parameter quantifies the radiation hardness of a material under specific particle species and monoener- getic flux.

As the irradiation fluence increases, radiation- induced damage (e.g., crystallographic defects) accumulates, reducing the MFP and degrading sensor performance (e.g., decreased signal response). For a given particle type and en- ergy, a smaller k corresponds to slower MFP degradation, in- dicating superior radiation resistance. Conversely, for a fixed detector, a smaller k under varying particle types or ener- gies implies less damage induced by those irradiation con- ditions. In an ideal single-crystal diamond, should ap-

proach zero due to minimal intrinsic defects, whereas poly- 409

crystalline diamonds exhibit a finite λ 0 owing to grain bound- 410

aries and intrinsic defects. Building on this framework, we utilized results from our previous high-fluence experiment to determine the damage constant of single-crystal diamond un- der 100 MeV proton irradiation. In previous study[ ], sc- CVD diamond sensors were subjected to fluence of , with the full set of transient current responses recorded. In planar electrode radiation detectors, the mea- sured charge signal generated by incident particles is directly related to the mean free path of charge carriers, which can be described using the Hecht[ ] model as two representations of charge collection distance (CCD):

CCD = Q meas Q gen /d = �

i = e,h λ i

4 also know as Messenger-Spratt equation[ 48 , 49 ]: ∆ 1 = 1 irr − 1 0 =

where represents the theoretically generated signal from incident particles, and d is the detector thickness. Using the

signal response current relationship I = d Q mea / d t , together 425

with the CCD expression in equation (5), we define a cali- bration factor that links the CCD to the mea- sured current gen/d

CCD ( λ ( ϕ )) =

For high-quality CVD-grown single crystal diamond, the ini- 430

tial CCD can approach the full detector thickness, as demon- strated by the RD42 collaboration[ ]. In our analysis, we

therefore assume an initial CCD close to the detector thick- 433

ness, i.e., CCD 0 /d ∼ 1 . From the initial current response 434

, we extract the calibration factor , which is then applied to irradiated signals to determine the CCD at each fluence step. The corresponding mean free path is obtained by solv- ing the CCD expression and subsequently fitted as a function of fluence using the linear degradation model of equation (4).

The fitting procedure, illustrated in Fig. , yields the damage constant for single-crystal diamond under irradiation as:

k 100 MeV proton = 1 . 451 ± 0 . 006( stat ) × 10 − 18 cm 2 ( p µm ) − 1 (7) 442

To enable direct comparison of radiation damage in dia- mond across different particle types and energies, a scaling approach[ ] was employed. Specifically, the damage con- stant measured under 100 MeV proton irradiation was nor- malized to that for 24 GeV protons, yielding a relative dam-

age coefficient κ i = k p (100 MeV) /k p (24 GeV) .

A sum- 448

mary of results from this work alongside data from previous studies[ ] is provided in Table 1 [TABLE:1]. The results show that protons at 100 MeV produce approximately 2.34 times more damage in single-crystal diamond than those at 24 GeV, under equivalent fluence conditions. This scaling enables the esti- mation of fluence equivalence across irradiation conditions

using ϕ eq = κ i ϕ i . Such an approach offers a unified frame- 455

work for quantifying and comparing radiation hardness, al- lowing the conversion of experimental fluence to equivalent damage levels across a wide range of energies and particle species.

Particles Damage constant Relative coefficient

800 MeV proton

1.67/1.85

70 MeV proton

2.48/2.6/2.5

200 MeV pion

This work. 3.3.2 Multiscale modelling of radiation damage by

combining Monte Carlo and molecular dynamics 461

To assess the performance degradation of diamond detec- tors under irradiation, we previously established that crys- tal damage influences carrier transport properties, ultimately leading to signal attenuation. While radiation-induced lat- tice damage is often analyzed from the perspective of en- ergy transfer by incoming particles, specifically the non-

ionizing energy loss (NIEL) that leads to atomic displace- 469

ments and phonon excitations[ ]. In this work, we focus on understanding how the diamond crystal lattice it- self evolves under irradiation. In diamond, irradiation dam-

age evolves through a two-stage mechanism. Primary dam- 473

age occurs when energetic incident particles displace carbon atoms from their lattice sites, producing primary knock-on

atoms (PKAs)[ 55 ]. These PKAs then initiate secondary colli- 476

sion cascades, generating secondary knock-on atoms (SKAs) and extended structural defects.

The distribution of PKA

energies plays a pivotal role in determining the final defect 479

morphology and density, thereby controlling the evolution of radiation-induced damage in the crystal. We employed a hy- brid simulation strategy, Monte Carlo (MC) simulations us- ing Geant4[ ] were used to compute the PKA energy spec- tra resulting from interactions between incident particles and the diamond lattice, while molecular dynamics (MD) using LAMMPS[ ] was applied to model the subsequent defect cascades induced by PKAs. As shown in Fig. , PKA energy spectra in diamond were calculated for proton beams with en- ergies of 24 GeV, 800 MeV, 100 MeV, and 70 MeV, as well as for 200 MeV , revealing how the incident particle type

and energy determine the initial damage state in diamond. 491

Based on Monte Carlo simulations, the probability distri- bution function of PKA energy within the interval

p ( E ) = ρ N

representing the probability of PKA generation within an

infinitesimal energy interval dE . Here, ρ N denotes the 497

atomic number density, represents the incident parti- cle energy spectrum, and is the differential cross- section for producing a PKA with energy when the in- cident particle energy is This process is governed by the interaction cross-sections between incident parti- cles and carbon atoms in diamond, as implemented in GEANT4 simulations. According to the law of large num-

bers, lim n →∞ �� f A n − p � < ε � = 1 the simulated event fre- 505

quency converges to the true probability distribution when the sampling size is sufficiently large. Thus, the probability dis- tribution function can be treated as the actual probabil- ity density function Subsequently, the concentration of accumulated displaced

atoms resulting from cascade interactions initiated by PKAs 511

is evaluated using the athermal recombination-corrected displacement per atom (arc-DPA) model[ This re- fined framework, which builds upon the traditional Norgett- Torrens-Robinson (NRT)[ ] formalism, enables a more ac- curate estimation of defect production from PKA-induced displacements. The arc-DPA model is formulated as:

0 for T d < E d 1 for E d ≤ T d < 2 E d 0 . 8 0 . 8 T d 2 E d · ξ arc ( T d ) for 2 E d 0 . 8 ≤ T d < ∞ (9) 518

N d , arc ( T d ) =

is defined as the efficiency function, expressed as:

ξ arc ( T d ) = 1 − c arc − dpa (2 E d / 0 . 8) b arc − dpa · T b arc − dpa d + c arc − dpa (10) 520

Here, represents the damage energy, i.e., the kinetic energy of the PKA, while denotes the average dis- placement threshold energy of the lattice. The parameters are determined through fitting the MD simulations[ ]. which describes how the penetration depth of a PKA scales with its energy, and characterizes the satu- ration value of the defect survival probability within regions of high defect density, particularly relevant at high PKA en- ergies.

By combining MC-generated spectra (equation 8) within 530

the arc-DPA model (equation 9), we derive the fluence- dependent defect concentration Once the PKA energy spectra in diamond are obtained for dif- ferent types and energies of incident particles, it becomes fea- sible to quantify and compare the resulting radiation damage within the crystal lattice. By correlating the total number of defects with the particle fluence, in a manner analogous to the experimental extraction of the damage constant k, we define

a simulation-derived parameter k sim = N arc ( ϕ ) /ϕ . This quan- 540

tity, extracted using a multiscale approach combining MC and 541

MD simulations, characterizes the irradiation-induced degra- dation of diamond under various radiation conditions.

MeV, 100 MeV, 70 MeV protons, and 200 MeV pion+, compar- ing experimental measurements (orange), NIEL calculations (pur- ple; from SR-NIEL dataset[ ]), and combined Monte Carlo/molec- ular dynamics predictions (green; arc-DPA model).

As shown in Fig. , the simulated damage constants denote in green markers exhibit closer agreement with experimental data than traditional estimates based on the NIEL(purple markers) of the incident particles. This result demonstrates that a crystal-structure-based simulation frame- work provides a accurate and physically grounded assessment of radiation damage in diamond detectors across different ir- radiation scenarios.

3.3.3 Modification of the simple damage model at high radiation dose In Section 3.3.1, we applied a linear damage model to de- scribe the degradation of single-crystal diamond under 100

MeV proton irradiation. The model captures the initial trend 556

of radiation-induced performance loss and enables cross- comparison of damage constants. At very high fluences, how- ever, the linear form diverges. In real crystals, atomic den-

sity and spatial volume are finite, placing an upper bound on 560

the number of defects that can be generated. At the same time, the continued accumulation of irradiation-induced de- fects may ultimately drive structural transitions such as amor- phization. Under such extreme conditions, the model is ex- pected to break down. At high doses, irradiation data reveal that for 100 MeV protons, the previously observed linear rela- tionship between carrier mean free path and particle fluence becomes invalid beyond approximately shown in Fig. , the mean free path begins to saturate, indi- cating that further damage accumulation no longer leads to proportional degradation in carrier transport.

In light of the observed deviation behavior, two types of

nonlinear damage mechanisms are proposed. The first in- 573

volves a saturation model of effective defect accumulation.

Here, effective defects are defined as those that significantly 575

contribute to the degradation of carrier lifetime, as opposed to the total population of structural disruptions within the crystal lattice.

At high irradiation fluences, where the spatial density of

energy deposition events increases significantly within local- 580

ized regions of the crystal, certain areas may undergo early- stage amorphization or the formation of extended defect clus- ters, such as the defect clusters and phase tran- sitions observed in irradiated diamond[ ]. Subsequent en- ergetic particles traversing heavily damaged and structurally disordered regions predominantly interact with existing amor- phous networks or defect clusters that have already formed, depositing their energy within these disordered structures. In

such amorphous domains, the electronic density of states ex- 589

tends into the gap, forming band tails and localized states due to disorder[ ]. As a result, additional defects introduced into these regions tend to merge into existing localized states, contributing little further trapping or scattering. By contrast, when irradiation occurs in regions where the crystal lattice remains relatively ordered, newly generated defect levels act as efficient electrically active traps. These lattice defects are thus identified as the primary contributors of carrier transport degradation.

saturation model of effective defect within a unit volume 599

is proposed, we assume the existence of a saturation defect density , beyond which additional damage has a negligi- ble effect on charge transport. When the local density of ef- fective defects, , remains below , energy deposition

by incident particles generates crystal damage that signifi- 604

cantly degrades the carrier mean free path. Once exceeds , further energy deposition in these already disordered or cluster-rich regions is assumed to contribute little additional impact on drift behavior. Across the entire diamond crystal, the local effective defect density is expressed as a spa- tially averaged quantity . The evolution of the effective defect density is then described by:

N eff = N sat · � 1 − e − N ( ϕ ) /N sat � (12) 612

Using a combined Monte Carlo and molecular dynamics ap- proach, the total defect number can be approximated by the

relation N = N arc ( ϕ ) = kϕ , as given by Equation 11. In the 615

low-fluence regime, this expression naturally reduces to the previously linear damage model:

� = kϕ (13) 618

At excessively high fluence conditions, the effective defect concentration approaches a saturation limit:

lim N ≫ N sat N eff → N sat · (1 − 0) = N sat (14) 621

By substituting into Equation 4, we obtain the dam-

age expression under the effective defect model as 1 /λ = 623

k · N eff + 1 /λ 0 . This approach provides a significantly im- 624

proved description of the degradation behavior at high flu- ences, as seen by the blue curve in Fig. . From the fit- ting, we extract the saturation defect density. For 100 MeV protons incident on diamond, this corresponds to approxi-

mately 2,036.8 defects within a 100 nm 3 unit volume as used 629

in Geant4 simulations, which is equivalent to a bulk defect density of . When the number of defects gen- erated by radiation in a local region exceeds this threshold, saturation effects are expected to occur. If we estimate the

saturation defect density as N sat = N arc ( ϕ 0 ) , then the onset 634

of saturation effects is expected to emerge at an equivalent fluence of approximately

Inspired by defect interactions and built on the estima- tion of overlapping irradiation-induced defects, we propose

a second nonlinear damage mechanism, as detailed below. 639

Molecular dynamics simulations were employed to estimate the spatial distribution of defects generated by PKAs in dia- mond. The resulting defect distributions for PKAs with vary- ing recoil energies, obtained using the LAMMPS package, are shown in Fig. . As evident from the visualizations, higher-energy PKAs produce increasingly clustered defects.

To quantify this behavior, we define a cutoff distance of between carbon atoms and use it to statistically evaluate the size and number of defect clusters. The results are presented in Fig. . These simulations reveal that a single PKA not only forms localized defect clusters but also establishes a characteristic spatial range over which its damage extends.

As shown in Fig. , the cumulative number of clusters produced by a PKA increases with recoil energy, and dislo- cation defects are also observed in the simulations. These findings indicate that each PKA generates a distinct dam- age volume comprising energy-dependent cluster regions and extended defects.

At high fluences, the probability that a new PKA’s damage volume overlaps with pre-existing dam- age regions rises.

We hypothesize that within these over- lapping regions, interactions between clusters dominate over the formation of isolated point defects. Furthermore, we as-

sume that such overlapping volumes contribute minimally 662

to additional carrier degradation. Consequently, only non- overlapping PKA damage regions are considered effective in reducing carrier transport. We define the effective damage

volume as V eff ( ϕ ) = V tot ( ϕ ) − V overlap ( ϕ ) , and introduce a 666

fluence-dependent damage reduction factor:

k overlap ( ϕ ) = V tot − V overlap V tot (15) 668

Cumulative defect clusters and dislocations induced by PKAs at different energies in diamond crystal (excluding counts of 1 Wigner-Seitz unit cell size).

In the Monte Carlo simulations, the spatial positions of individual PKAs were recorded. The total damage volume, , was obtained by summing the effective defect volumes produced by each PKA over the entire irradiation period. Si- multaneously, the overlapping volume, overlap , was deter- mined by calculating the intersection between newly gener- ated PKA defect volumes and those already existing. For each PKA, the associated damage volume was derived by fitting molecular dynamics simulation results (Fig. ) for various recoil energies. Finally, by replacing the constant damage parameter in Eq. (4) with the fluence-dependent damage re- duction factor at each fluence, we obtained the carrier mean

free path as a function of fluence according to the second non-

linear damage mechanism, as shown by the orange curve in 682

reproduces the damage behavior observed at high fluence.

To further illustrate this effect, we consider a diamond crys- tal with a volume of 300 nm . At different proton fluences, denoted as

ϕ = 1 . 2 × 10 17 , the number of PKAs and their corresponding 690

spatial damage ranges are visualized in Fig. . In the figure, the circular mappings represent the extent of defect-affected regions, while different colors correspond to the defect vol- umes generated by different PKAs. Lighter colors indicate larger affected volumes. It is evident that as the fluence ex- ceeds around , the over- lapping regions between PKA-induced damage volumes in-

crease significantly, leaving less undisturbed crystal volume 698

for the formation of isolated point defects. From the per- spective that each PKA generates multiple defect clusters, the increasing dominance of defect–defect interactions becomes apparent at high fluences.

The degradation behavior is better described by incorporating effec- tive defect saturation (blue curve) and defect-overlap effects consid- ering interactions between defects (orange curve).

Through the effective defect saturation model and the de- fect interaction model, this study establishes a connection be- tween irradiation-induced defect evolution and carrier trans- port degradation in diamond, providing a phenomenological framework that captures the nonlinear behavior of diamond detectors under high radiation fluence. The first model as- sumes a saturation of electrically active defects capable of trapping carriers, while the second considers the spatial over- lap and interaction among newly generated and pre-existing defects. Together, they account for the observed deviation from linearity in detector response at high doses. Both frame- works converge on a consistent interpretation that under in- tense irradiation the diamond lattice undergoes a structural transformation from a regime dominated by isolated point defects to one governed by defect clusters and locally amor- phous configurations. This structural transition in radiation damage modifies the distribution of density of states within the band structure, subsequently changing the effective car- rier trapping cross section and driving the detector response from linear to nonlinear behavior in CCD and other electrical characteristics. Monte Carlo and molecular dynamics simu- lations indicate that this transition begins to emerge when the defect density approaches , corresponding to an equivalent neutron fluence of , where linear scal- ing with fluence no longer holds. In this regime, comprehen- sive understanding of these nonlinear processes will require integrated first principles calculations and advanced experi- mental probes capable of resolving the atomic-scale mecha-

nisms underlying defect clustering and amorphization in dia- 731

mond. , based on effective defect volumes derived from MD simulations.

4 Conclusion

This study demonstrates that single-crystal CVD diamond sensors retain functional signal response under exception-

ally high radiation fluence, maintaining approximately 5% 736

of their initial output after fast neutron irradiation up to 737

among the highest levels tested to date. These results confirm the feasibility of applying such sensors in extreme radiation environments.

Spectroscopic and electron microscopy analyses revealed that both bulk and surface defects induced by irradiation, such as self-interstitials, vacancies, and nanoscale surface cracks play a central role in the observed degradation of de- tector performance. Following the linear carrier-drift degra- dation framework, we experimentally extracted the quan- titative damage constant for 100 MeV protons based on

low fluences experimental data as k

100 MeV

proton = 1 . 452(6) × 748

, providing essential reference data for ra- diation damage assessment in diamond under medium-energy proton irradiation conditions.

To explore the underlying mechanisms of damage satura- 752

tion, we performed multiscale simulations that couple Monte Carlo particle transport with molecular dynamics modeling.

This combined approach yields damage estimates that align more closely with experimental observations than conven- tional NIEL predictions, offering a new perspective for study- ing radiation damage in diamond detectors.

Furthermore, the framework provides a phenomenological means to in- vestigate the observed saturation in carrier transport at high doses and to refine nonlinear degradation models. At high fluences, interactions between defects may become the dom-

inant mechanism of lattice modification, gradually replacing 763

isolated point defect formation. This transition leads to the emergence of a local effective saturation defect density, be-

yond which additional damage has a diminishing effect on 766

the carrier drift length. In summary, these findings establish a fundamental under- standing of radiation-induced damage in diamond at high flu- ence, and offer practical guidance for the design and deploy- ment of diamond-based detectors in future high-radiation par- S. Chatrchyan, V. Khachatryan, A.M. Sirunyan, et al ., Ob- servation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. Sect. B Nucl.

Elem. Part. High-Energy Phys. 716 (2012) 30–61. G. Aad, T. Abajyan, B. Abbott, et al ., Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. Sect. B Nucl. Elem. Part. High-Energy Phys. 716 (2012) 1–29. ticle physics experiments and advanced nuclear technologies.

Acknowledgements The authors would like to acknowledge M. Bulavin, A.

Cheplakov, L. Kurchaninov, V. Kukhtin, J. Ye, C. Li, H. 775

Ding, J. Du, Z. Zou and F. Miao for their testing facilities and helpful discussions. They would also like to acknowl- edge the ATLAS-LAr Collaboration for their strong supports and beneficial comments. This work was supported by the “International Science & Technology Cooperation Program

of China”(Contract No. 2015DFG02100), The Ministry of 781

Science and Technology of the People’s Republic of China.

Author contributions All authors contributed to the work. Jialiang Zhang, Shuo Li, Guojun Yu, and Zifeng Xu performed characterization ex- periments, data analysis, simulations, and manuscript writ- ing. Yilun Wang and Shuxian Liu contributed to model anal- ysis. Lifu Hei and Fanxiu Lv provided materials. Ming Qi

contributed to experimental design, planning, conducted ir- 789

radiation experiments, and provided funding support. Ming Qi and Lei Zhang were responsible for manuscript revision and supervision. The first draft of the manuscript was writ- ten by Jialiang Zhang, and all authors commented on previ- ous versions. All authors have read and approved the final manuscript.

Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of interests The authors declare that they have no known competing interests or personal relationships that could have appeared to influence the work reported in this paper.

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