Multiscale modeling of the low-temperature electron irradiation of beryllium

Multiscale modeling of the low-temperature electron irradiation of beryllium

Presented are the methodology and the results of the multiscale modeling of radiation defects primary production and time evolution for 2.5 MeV cryogenic (77 K) irradiation of highly deformed to ~ 10¹² cm⁻² dislocations density beryllium at the NSC KIPT electron linac ELIAS. It is shown that the app...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Вопросы атомной науки и техники
Datum:2017
Hauptverfasser: Bratchenko, M.I., Dyuldya, S.V.
Format: Artikel
Sprache:English
Veröffentlicht: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2017
Schlagworte:
Чистые материалы и вакуумные технологии
Online Zugang:https://nasplib.isofts.kiev.ua/handle/123456789/137263
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Multiscale modeling of the low-temperature electron irradiation of beryllium / M.I. Bratchenko, S.V. Dyuldya // Вопросы атомной науки и техники. — 2017. — № 6. — С. 30-38. — Бібліогр.: 26 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-137263
record_format dspace
spelling Bratchenko, M.I.
Dyuldya, S.V.
2018-06-17T09:33:50Z
2018-06-17T09:33:50Z
2017
Multiscale modeling of the low-temperature electron irradiation of beryllium / M.I. Bratchenko, S.V. Dyuldya // Вопросы атомной науки и техники. — 2017. — № 6. — С. 30-38. — Бібліогр.: 26 назв. — англ.
1562-6016
PACS: 61.80.-x, 61.82.Bg, 61.72.Cc, 61.80.Fe, 07.05.Tp, 02.70.Ns, 02.70.Uu
https://nasplib.isofts.kiev.ua/handle/123456789/137263
Presented are the methodology and the results of the multiscale modeling of radiation defects primary production and time evolution for 2.5 MeV cryogenic (77 K) irradiation of highly deformed to ~ 10¹² cm⁻² dislocations density beryllium at the NSC KIPT electron linac ELIAS. It is shown that the application of low-temperature e⁻-irradiation of prestrained targets allows efficient suppression of vacancy-interstitial recombination due to escape of freely mi-grating self-interstitial atoms to dislocation sinks and results in abnormally high (~ 10⁻³ per atom) vacancy yield comparable with that of primarily produced Frenkel pairs at a reasonable (≤10³h)e⁻-beam exposure.
Представлено методологію та результати багатомасштабного моделювання первинної продукції та еволюції у часі радіаційних дефектів за криогенного (77 К) опромінення сильнодеформованого до густини дислокацій ~10¹² cм⁻² берилію на електронному лінаці ELIAS ННЦ ХФТІ. Показано, що застосування низькотемпературного електронного опромінення попередньо напружених мішеней дозволяє ефективно пригнічувати рекомбінацію пар Френкеля через витік вільно мігруючих власних міжвузельних атомів до дислокаційних стоків і призводить до аномально високих (~10⁻³ на атом) концентрацій вакансій, які добре порівнянні з концентраціями первинних пар Френкеля за прийнятної (≤10³ год) тривалості електронного опромінення.
Представлены методология и результаты многомасштабного моделирования первичной генерации и временной эволюции радиационных дефектов при криогенном (77 К) облучении сильнодеформированного до плотности дислокаций ~ 10¹² cм⁻² бериллия на электронном линаке ELIAS ННЦ ХФТИ. Показано, что применение низкотемпературного облучения предварительно напряженных мишеней позволяет эффективно подавлять рекомбинацию пар Френкеля за счет ухода свободно мигрирующих собственных межузельных атомов на дислокационные стоки и приводит к аномально высоким (~ 10⁻³ на атом) выходам вакансий, сопоставимых с концентрациями первичных пар Френкеля при разумной (≤10³ ч) длительности e⁻-облучения.
The authors are very grateful to A.S. Bakai for drawing their attention to the problem, putting forward the basic idea of future experiments, and for the continuous support and critical discussion of this study. We acknowledge K.V. Kovtun and V.N. Borisenko for valuable data on the prestrained Be targets preparation and the ELIAS linac irradiation e – -beam parameters, opportunities and limitations. We thank I.I. Papirov and A.A. Nikolayenko for provision of their expertise on structural and mechanical properties of beryllium. We also always keep in memory the outstanding role of Yu.T. Petrusenko in providing cryogenic irradiations in NSC KIPT.
en
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
Вопросы атомной науки и техники
Чистые материалы и вакуумные технологии
Multiscale modeling of the low-temperature electron irradiation of beryllium
Багаторівневе моделювання електронного опромінення берилію за низьких температур
Многоуровневое моделирование электронного облучения бериллия при низких температурах
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Multiscale modeling of the low-temperature electron irradiation of beryllium
spellingShingle Multiscale modeling of the low-temperature electron irradiation of beryllium
Bratchenko, M.I.
Dyuldya, S.V.
Чистые материалы и вакуумные технологии
title_short Multiscale modeling of the low-temperature electron irradiation of beryllium
title_full Multiscale modeling of the low-temperature electron irradiation of beryllium
title_fullStr Multiscale modeling of the low-temperature electron irradiation of beryllium
title_full_unstemmed Multiscale modeling of the low-temperature electron irradiation of beryllium
title_sort multiscale modeling of the low-temperature electron irradiation of beryllium
author Bratchenko, M.I.
Dyuldya, S.V.
author_facet Bratchenko, M.I.
Dyuldya, S.V.
topic Чистые материалы и вакуумные технологии
topic_facet Чистые материалы и вакуумные технологии
publishDate 2017
language English
container_title Вопросы атомной науки и техники
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
format Article
title_alt Багаторівневе моделювання електронного опромінення берилію за низьких температур
Многоуровневое моделирование электронного облучения бериллия при низких температурах
description Presented are the methodology and the results of the multiscale modeling of radiation defects primary production and time evolution for 2.5 MeV cryogenic (77 K) irradiation of highly deformed to ~ 10¹² cm⁻² dislocations density beryllium at the NSC KIPT electron linac ELIAS. It is shown that the application of low-temperature e⁻-irradiation of prestrained targets allows efficient suppression of vacancy-interstitial recombination due to escape of freely mi-grating self-interstitial atoms to dislocation sinks and results in abnormally high (~ 10⁻³ per atom) vacancy yield comparable with that of primarily produced Frenkel pairs at a reasonable (≤10³h)e⁻-beam exposure. Представлено методологію та результати багатомасштабного моделювання первинної продукції та еволюції у часі радіаційних дефектів за криогенного (77 К) опромінення сильнодеформованого до густини дислокацій ~10¹² cм⁻² берилію на електронному лінаці ELIAS ННЦ ХФТІ. Показано, що застосування низькотемпературного електронного опромінення попередньо напружених мішеней дозволяє ефективно пригнічувати рекомбінацію пар Френкеля через витік вільно мігруючих власних міжвузельних атомів до дислокаційних стоків і призводить до аномально високих (~10⁻³ на атом) концентрацій вакансій, які добре порівнянні з концентраціями первинних пар Френкеля за прийнятної (≤10³ год) тривалості електронного опромінення. Представлены методология и результаты многомасштабного моделирования первичной генерации и временной эволюции радиационных дефектов при криогенном (77 К) облучении сильнодеформированного до плотности дислокаций ~ 10¹² cм⁻² бериллия на электронном линаке ELIAS ННЦ ХФТИ. Показано, что применение низкотемпературного облучения предварительно напряженных мишеней позволяет эффективно подавлять рекомбинацию пар Френкеля за счет ухода свободно мигрирующих собственных межузельных атомов на дислокационные стоки и приводит к аномально высоким (~ 10⁻³ на атом) выходам вакансий, сопоставимых с концентрациями первичных пар Френкеля при разумной (≤10³ ч) длительности e⁻-облучения.
issn 1562-6016
url https://nasplib.isofts.kiev.ua/handle/123456789/137263
citation_txt Multiscale modeling of the low-temperature electron irradiation of beryllium / M.I. Bratchenko, S.V. Dyuldya // Вопросы атомной науки и техники. — 2017. — № 6. — С. 30-38. — Бібліогр.: 26 назв. — англ.
work_keys_str_mv AT bratchenkomi multiscalemodelingofthelowtemperatureelectronirradiationofberyllium
AT dyuldyasv multiscalemodelingofthelowtemperatureelectronirradiationofberyllium
AT bratchenkomi bagatorívnevemodelûvannâelektronnogoopromínennâberilíûzanizʹkihtemperatur
AT dyuldyasv bagatorívnevemodelûvannâelektronnogoopromínennâberilíûzanizʹkihtemperatur
AT bratchenkomi mnogourovnevoemodelirovanieélektronnogooblučeniâberilliâprinizkihtemperaturah
AT dyuldyasv mnogourovnevoemodelirovanieélektronnogooblučeniâberilliâprinizkihtemperaturah
first_indexed 2025-11-27T02:25:12Z
last_indexed 2025-11-27T02:25:12Z
_version_ 1850791240523579392
fulltext ISSN 1562-6016. PASТ. 2018. №1(113), p. 30-38. MULTISCALE MODELING OF THE LOW-TEMPERATURE ELECTRON IRRADIATION OF BERYLLIUM M.I. Bratchenko, S.V. Dyuldya National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine E-mail: sdul@kipt.kharkov.ua Presented are the methodology and the results of the multiscale modeling of radiation defects primary production and time evolution for 2.5 MeV cryogenic (77 K) irradiation of highly deformed to ~ 10 12 cm –2 dislocations density beryllium at the NSC KIPT electron linac ELIAS. It is shown that the application of low-temperature e – -irradiation of prestrained targets allows efficient suppression of vacancy-interstitial recombination due to escape of freely mi- grating self-interstitial atoms to dislocation sinks and results in abnormally high (~ 10 –3 per atom) vacancy yield comparable with that of primarily produced Frenkel pairs at a reasonable ( 10 3 h) e – -beam exposure. PACS: 61.80.-x, 61.82.Bg, 61.72.Cc, 61.80.Fe, 07.05.Tp, 02.70.Ns, 02.70.Uu 1. INTRODUCTION The NSC KIPT sited charged particles accelerators are widely used to support R&D of radiation damage (RD) physics and radiation material science (RMS). Major applications of the ion beam (IB) machines con- cern the ‘simulation irradiation’ (SI) concept targeted on the prediction of reactor (n, γ) irradiation induced ef- fects in materials. The required IB exposure of SI ex- periments is planned on the basis of calculation of the primary RD (PRD) dose measured in the number of ato- mic displacements per atom, Ndpa. The established stan- dard practice [1] considers dpa only as “a unit of radia- tion exposure”: Ndpa shall be calculated coherently for both the targeting reactor and the accelerator irradiation environments in a strict accordance with the Norgett- Robinson-Torrens proposed ‘NRT standard’ model [2]      dPKADPKA NRT dpa 2EEEEN   (1) which is a generalization of the famous Kinchin-Pease model primarily developed to estimate the number of point defects in an atomic collision cascade (ACC) initi- ated by the primary knock-on atom (PKA) of a given energy EPKA. Here  = 0.8, ED(EPKA) is the “damage en- ergy”, the total elastic energy loss of a PKA available to produce displacements, and Ed ~ 10 1 …10 2 eV is the ma- terial-specific displacement threshold energy. The validity of the NRT standard is confined in the area of comparative dosimetry of reactor/accelerator ba- sed neutron/ion damage of metals and alloys [3]. The number of NRT dpa cannot be directly measured experi- mentally and thus does not represent an observable phy- sical quantity. Instead, one must consider the actual RD metrics, the atomic concentrations CFP of Frenkel pairs (FPs) as well of their constituents, CV of vacancies (V) and CI of self-interstitial atoms (SIAs). The observable CV,I are orders of magnitude smaller than the NRT calculated dpa because of several reasons [4, 5]. First, FPs annihilate inside the ‘hot’ ACC core during its athermal quenching stage, ~ 10 –(13…11) s. This is essential [4] for fast neutron and ion impact when successive atomic displacements occur at a distance ~ a, the lattice constant of a material. But it is not so topical for the moderate (~ MeV) energy electron beams (EBs) impact when only isolated spatially separated FPs appear. Next, the PRD stage produced defects evolve in space and time through the complex short-term (~ 10 –(11…6) s) kinetic stage of their recovery and agglomeration into complexes till the thermal diffusion stage (> 10 –6 s) of their recombination and interaction with sinks. Finally, this is asymptotically forming the saturated damaged state of a material which results in the observed irradiation induced changes of macroscopic properties. The complexity of the self-consistent description of the whole RD picture (which covers 13…20 orders of magnitude of temporal development at a spatial scale 10 –8 …10 2 cm) gave birth to the novel Multiscale Modeling & Simulation (MSMS) paradigm of the RMS computer modeling studies [6]. It involves the problem- specific sequence of diverse methods and input/output data concordant codes: ab initio quantum mechanics (QM), classical Molecular Dynamics (MD), kinetic Monte-Carlo (kMC) and reaction rate theory [5, 7] calculations, dislocation dynamics [7], and Finite Element Method (FEM). Each of them simulates its own scale of the spatiotemporal evolution to transfer outputs to the subsequent scale. Recently we have proposed [8] the MSMS program incorporation into the computational support of the NSC KIPT accelerator based irradiations and identified the appropriate simulation software toolbox [9]. The present paper encompasses the results of our first attempt to push the rationale and planning of NSC KIPT irradiations of materials ‘beyond NRT’. We pre- sent the results of the trans-NRT MSMS of the accumu- lation and evolution of point defects in a highly defor- med high-purity hcp Beryllium, a structurally complex anisotropic functional material, for the case of its cryo- genic irradiation at the NSC KIPT EB linac ELIAS. 2. PROBLEM STATEMENT Beryllium manifests the non-trivial properties and effects (e.g., superplasticity) promoted by its pronoun- ced anisotropy [10, 11]. This also applies to Be low- temperature physics regarding its electronic structure and transport properties, and particularly the structural sensitivity of its superconducting transition temperature Tc. Without going into details that go beyond the scope of this paper, let’s declare that the experimental investi- gation of this lattice defects density-of-state dependent effect requires high concentrations of the cryogenically ‘frozen’ vacancies, CV ~ 10 –3 per Be atom [12], much greater than the thermal equilibrium CV. The extra non- equilibrium vacancies can be injected by e − -irradiation. However, the elastic displacement cross-section d [13] based calculation shows that even the most conservative estimate of CV as the concentration of as-irradiated pri- mary FPs yields CFP ~ 10 –(5…4) per atom per day of exposure to ~ MeV energy EB. The thermal diffusion activated recombination V + I =  does diminish CFP by orders of magnitude. So, the problem looks insoluble. Nevertheless, it has been put forward [14] an idea to solve it by means of the V–I recombination suppression with the following experimental setup: (i) to apply the nitrogen temperature T = 77 K e − -irradiation when only SIA are diffusively mobile; (ii) to use the high-purity Be target which is beforehand highly deformed up to the dislocation densities d ~ 10 11…12 cm –2 granting the highest possible concentration of SIA sinks, and (iii) to transfer the unheated irradiated sample into the helium temperature appliance for subsequent measurements. Despite of the extreme conditioning of each step of this scenario, it is in fact feasible by means of the NSC KIPT developed processes of high-purity Be samples preparation and cryogenic e − -irradiation at the accelera- tor ELIAS. It is worth noting the NSC KIPT ELIAS team advanced capability to provide immediate post-ir- radiation examination of samples (without their warm- ing-up) at the same (or lower) temperature. The goal of the present work is the computational substantiation of the proposed technique by means of the MSMS calculation of the utmost value of CV obtain- able at the ELIAS e − -irradiation. For this purpose, we apply the subset of our MSMS software toolkit [9] em- bracing (i) the in-house developed GEANT4 Toolkit ba- sed radiation transport (RT) Monte-Carlo (MC) code RaT 3.1 [15], (ii) the LAMMPS MD [16], and (iii) the SPPARKS kMC [17] packages developed and freely distributed by the U.S. Sandia Nat. Lab. team. 3. PRIMARY RADIATION DAMAGE Without loss of generality, we consider a semi-infi- nite planar target of Be = 1.85 gm/cm 3 dense polycrys- talline beryllium irradiated by a broad parallel e − -beam with the nominal parameters of the ELIAS linac: the EB energy Ee = 2.5 MeV, the current density j = 10 μA/cm 2 , the e − -flux  = j/e= 6.24·10 13 e − ·cm –2 ·s –1 . An adequate computer modeling of the shortest-time ballistic stage of the as-irradiated PRD was conducted by means of the RaT 3.1 MC code using the GEANT4.9.5 supplied algorithms and data for simulation of energy losses and multiple scattering of relativistic electrons in matter. Relatively rare events of the strong electron– atom displacing collisions were sampled according to the Mott elastic scattering [13] cross-section d calcula- ted for the NRT standard recommended value of the dis- placement threshold energy Ed = 31 eV of beryllium. Trajectories of electrons and -quanta of the radiati- on cascade as well as those of all Be recoils (both PKAs and secondary knock-ons, SKAs, of the ACC) were tra- ced until a ‘thermalization’ i. e. down to the energy Emin, ~ 100 eV for e – , and ~ 1 eV for cascade atoms. Due to fairly short ranges, Emin are small enough to not affect the calculated spatial distributions vs. the EB penetrati- on depth z. Note that the ACC was modeled explicitly i. e. using the RaT code specific algorithm and data [15, 3] validated against those of the SRIM package, a practical standard of the IB induced PRD evaluation. Primary results of simulation concerning the EB pe- netration into a target are presented in Fig. 1. The depth profile of thermalized electrons spreads up to the EB extrapolated projected range Rp  1 cm. Only the brems- strahlung produced secondary Compton electrons are present at z > Rp. At z < Rp, the profile is drawn (roughly exponentially) until a broad maximum, at z  ¾Rp, which is due to the complex interplay of energy loss, straggling, and multiple scattering of electrons. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 10 -3 10 -2 10 -1 10 0 D ep o si ti o n r at e (1 0 1 5 s -1 ·c m -3 ) Depth z (cm) e  -beam electrons Be recoil atoms 2.5 MeV e   Be 10 A/cm 2 RaT 3.1 0 5 10 15 20 25 D o se r at e (W /g m ) e  -beam deposited power Fig. 1. The Monte-Carlo calculated depth profiles of the EB specific power deposition () and thermalized par- ticles, electrons () and Be recoil atoms () 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 P ro d u ct io n r at e (1 0  9 /s /a to m ) Depth z (cm) displacements (d) Frenkel pairs (FP) replacements (r) 2.5 MeV e  10 A/cm 2 Be E d = 31 eV RaT 3.1 explicit cascade N FP = N d  N r 0 2 4 6 8 P ro d u ct io n r at e (1  5 /d ay /a t. ) Fig. 2. The atomic displacement (dpa) rate depth profile subdivided into contributions of the actually formed Frenkel pairs and the Be–Be atoms replacements We supplemented Fig. 1 with the standard dose (or the mass specific power deposition) rate profile Pdep(z). In itself, the EB deposited power does not affect the RD of metal targets [4, 5]. But it is essential for the mainte- nance of the thermal regimes of cryogenic e − -irradia- tion. Note that Pdep is high as compared with the -hea- ting relevant ~ 0.1…1 W/gm of in-pile test channels. The Pdep(z) maximum is shifted to lower z  ½Rp where electrons are energetic enough to heat the target and to displace the lattice atoms. Be recoils (both subthreshold ones, and PKAs) are distributed almost uniformly right up to this depth, as shown in greater details in Fig. 2. The maximal atomic displacement rate occurs at the depth z = 4 mm  0.4·Rp and exceeds the target surface (z = 0) value only by  14%. But only  80% of displacements (EPKA > Ed) are actual FPs. The rest  20% of them do not form a point defect but result in Depth z, cm Depth z, cm D ep o si ti o n r at e, 1 0 1 5 s-1 ∙c m -3 D o se r at e, W /g m P ro d u ct io n r at e, 1 0 -9 /s /a t. P ro d u ct io n r at e, 1 -5 /d ay /a t. the replacement of one Be atom with another. They are due to the Ed <EPKA < 2·Ed range of the PKA energy spectrum. The explicit modeling of an ACC and the simplified NRT standard approach (1) treat the partitioning of dis- placements onto FPs and replacements somewhat diffe- rently [3]. This results in the systematic bias of the ac- tual NFP = NI  NV and    NRT dpa NRT FP NN  shown in Fig. 3. One can see that the NRT standard always underesti- mates the number of actually produced point defects. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.5 MeV e  10 A/cm 2 F P p ro d u ct io n r at e (1 0  9 /s /a t. ) Depth z (cm) RaT 3.1 calculation models: explicit cascade NRT standard 2.5 MeV e  10 A/cm 2 Be E d = 31 eV 0 1 2 3 4 5 6 F P p ro d u ct io n r at e (1 0  5 /d ay /a t. ) Fig. 3. The FP depth profiles of Be PRD by EB of dif- ferent energies MC calculated in the NRT standard ap- proximation () and by the explicit ACC modeling () 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 F P p ro d u ct io n r at e (1 0  9 F P p a/ s) Depth z (cm) total PKA SKA 2.5 MeV e  10 A/cm 2 Be E d = 31 eV RaT 3.1 explicit cascade  d · 0 2 4 6 8 F P p a ra te ( 1 0  5 p er d ay ) Fig. 4. The Be target PRD depth profile () partitioned into contributions of primary (solid curve) and secon- dary cascade (dashed curve) knock-on atoms The ratio   1NRT FPFP  NN is only weakly depth dependent in the highly damaged area (z < ½Rp) of a tar- get;  = 1.2…1.25 for both 1.5 MeV and 2.5 MeV e − -ir- radiations of Be. But  is known to grow with energy (and mass) of a projectile when the PKA spectra beco- me harder, the secondary displacement function (SDF) increases, and ACCs become more and more developed. For the LPE-10 linac 10 MeV e − -irradiation of Ni–Cr al- loy 690,   1.5, and amounts   2 for heavy ions irra- diation of structural steels [3]. This is significant enough to opt for the explicit cascade modeling for estimations of the actual concentrations of the V–I Frenkel pairs. For not to get it mixed up with the conventional NRT dpa, it is expedient to introduce the special unit of measurement, FPpa = (NRT dpa). We shall use it below for the topical case of the current study. In Fig. 4, the dash-dotted arrow indicates the FPpa rate calculated analytically using the total displacement cross-section [13] d (Ee,Ed) = 7.44 barn for the initial EB energy. It is fairly consistent with the zero-depth ex- trapolation of the MC simulation data for the contribu- tion of PKAs immediately displaced by electrons. How- ever, it covers only two thirds ( 66%) of the actually modeled total FPpa rate (markers) in a thin target. The remaining one third is due to the ACC development. The atomic collision cascades are frequently ignored in a planning of the EB PRD production. Sporadic ACCs are considered irrelevant to the electron impact and a simple formula CFP = d (Ee, Ed)··t0 is applied for the EB exposure duration t0. The Fig. 4 data show the drawbacks and quantitative limitations of this approxi- mation namely for the case of thin targets. The exact MC modeling is favorable to obtain the reliable data. For the subsequent MSMS calculations, we adopt, from Fig. 4, the conservative zero-depth estimate of the PRD FP production rate K0 = 7∙10 –10 FPpa/s. 4. THE REACTIONS RATE THEORY ANSATZ AND MODEL ESTIMATES The CV,I(r,t) spatiotemporal evolution at the next ti- mescale of the RD development and subject to (V,SIA) thermal diffusion and reactions with each other and with sinks is generally described by the partial derivative master equations of the RD rate theory [7]. For the limi- ted purpose of the current study, we neglect gradient terms and confine ourselves to the mean-field approxi- mation of the reaction rate theory [5, 7] taking into ac- count the problem specific simplifications. In this case, the time derivatives  tC IV,  are governed by the follow- ing system of the ordinary differential equations (ODE)                   )2.2( )1.2( II 2 VI VIIRV tCDktCtC tCtCDtKtC    where the source term K(t) is the FPpa production rate, K(t) = K0 for t  t0 (EB on), K(t) = 0 otherwise (EB off); DI, cm 2 ·s –1 , is the SIA diffusion coefficient; R, cm –2 , is the V–I recombination constant and k 2 , cm –2 , is the total sink strength [5] for SIAs. In Eq. 2, it is assumed that only SIAs are mobile (DV = 0). The ODE system (2) was readily solved numerically for the initial conditions CV,I(0) = 0 (we neglect the equilibrium CV(0) << 10 –6 for a cryogenic irradiation) and benchmarked against the asymptotic solution [5]               )2.3(, )1.3(, 000FP00 00FP00 0V tttttCttK tttCtK tC where the characteristic time t * =  /K0 scales with the dimensionless ratio  = k 2 /R rating the sink absorption of a SIA to its recombination with a vacancy. For short t0 < t * of ballistic PRD irradiation exposure, CV  t0. For t0 > t * , the diffusion limited buildup of vacancies is slo- wing down very notably; asymptotically, 0V tC  . First, we estimated CV(t) by applying some synoptic RMS models and reference data for Be. Assuming that the only SIA sinks are edge disloca- tions, we evaluated the dislocation sink strength k 2 ac- cording to the Nichols isotropic model [18]          4 3 ln2 c d d 2 r R k  (4) , , , Depth z, cm Depth z, cm F P p ro d u ct io n r at e, 1 0 -9 /s /a t. F P p ro d u ct io n r at e , 1 0 -5 /d ay /a t. F P p ro d u ct io n r at e, 1 0 -9 F P p a/ s F P p a ra te , 1 0 -5 p er d ay where Rd = (d) –½ is the dislocation cylindrical unit cell radius and rc is the dislocation core radius, ~ a = 2.3 Å for Be. This yields k 2  10 12 cm –2 for d = 10 11 cm –2 . The recombination constant R  4·reff/Be was es- timated from the effective SIA capture radius reff of a vacancy and the atomic volume Be. Since the MD data derived reff  2a [7], R  10 17 cm –2 >> k 2 , and  << 1. The SIA self-diffusion coefficient DI is strongly temperature T dependent       Tk E DD B m 0I exp , (5) where kB is Boltzmann’s constant and Em is the SIA mi- gration barrier energy. In Be, the pre-exponential factor D0  0.62 cm 2 /s [11] while Em is highly anisotropic. Vladimirov and coworkers carried out an extensive ab initio QM MD simulation of the SIA diffusion in Be [19, 20] by the density-functional (DFT) code VASP. They found Em = 0.12 eV for the preferential basal pla- ne (BP) diffusion and twice as much, Em = 0.27 eV, for the diffusion out of BP [20]. For T = 77 K, this yields DI(0.12 eV)  8.4·10 –8 cm 2 /s while DI(0.27 eV)0. For a reasonable mean Em = 0.2 eV, DI  4.8·10 –14 cm 2 /s. According to Eq. (2), the saturation of CI(t) occurs at a diffusion-to-sink time  = (k 2 ·DI) –1 which varies from ~ 10 –4 s for Em = 0.12 eV up to ~ 10 2 s for Em = 0.2 eV. In our case, K0 ~ 10 –9 s –1 , the saturated maximal values of CI() = K0· are ~ 10 –13 and ~ 10 –7 , respectively. The- refore, CI << CV for t0 >> , the V–I recombination rate becomes negligible for t > t0, and the SIA diffusion in- dependent asymptotic (3.2) is fairly applicable. We applied it to estimate the ELIAS EB exposure ti- me  tKCt 2 0 2 V0 required to inject the requested CV, say, 10 –3 . Having t *  10 4 s, we obtained t0 ~ 10 8 s which is more than 3 years long, and is inadmissible. This issue impels a revision of the models and data used above in this section. There is no vagueness about the values of R, Rd, rc, and D0 the more so as the result is DI independent. Only the dislocation sink strength k 2 remains open to questions. The model (4) neglects the dislocation elastic field which can enhance the SIA cap- ture rate. It also disregards the hcp lattice inherent ani- sotropy. We suggested the k 2 evaluation (4) looks un- derestimated, and proceeded to its refined MSMS calcu- lation by means of the MD and kMC simulations. 5. MULTISCALE CALCULATIONS OF THE DISLOCATION SINK STRENGTH 5.1. MD EVALUATION OF THE SIA-EDGE DISLOCATION ELASTIC INTERACTION FIELD We started from the linear theory of elasticity deri- ved [21] potential energy Ud(r, ) of a SIA first-order size interaction with an edge dislocation of Burgers vec- tor b(b,0,0). In a cylindrical (r, ,z) frame of reference with the dislocation line L aligned axis z,   r bUrU   sin , 0d  , (6) where the interaction strength factor U0 has the form VU          1 1 3 0 , (7)  is the elastic shear modulus;  is Poisson’s ratio; V is the dilative volume change of an isotropic elastic me- dium due to the presence of a SIA. Hereinafter we ne- glect the presence of screw dislocations since, for them, Ud is vanished to a first-order of magnitude [21]. The ab initio simulations [19, 20] have shown that the basal-octahedral (BO) interstitial position is highly preferential for Be SIAs at low temperatures. Thus, a simple estimate of the SIA dilatation volume difference is  3 BO 3 Be3 4 RRV   , where RBe = a/2  1.14 Å is the maximal radius of the hexagonally closely packed hard spheres representing Be atoms, RBO  0.41·RBe  0.47 Å is the corresponding inner radius of the BO spherical void. This yields V  5.82 Å 3 = 0.72·Be for the refer- ence atomic volume of beryllium Be = 8.08 Å 3 . Substituting this value of V into (7) together with the up-to-date measured data [22] on the elastic cons- tants of beryllium,  = 150.1 GPa and  =0.050, we ob- tained the interaction energy factor U0 = 0.64 eV. Other (,) data taken from the compendium [22] result in the rather close values of U0 = 0.58…0.66 eV. However, notably smaller U0  0.5 eV can be also derived from the earlier measured (,) reference data [10, 11]. In order to refine the proposed heuristic hard-sphere model, we proceeded to the atomic-scale MD simulation. The general formula V = tr(Pij)/3B [21] expresses the SIA dilatation volume change in terms of the trace of the point defect dipole-force elastic tensor Pij and the bulk modulus B of an anisotropic medium. We applied the MD code LAMMPS [16] and the Be–Be interatomic potential taken from ref. [23] to calculate Pij according to the following method [24] and algorithm. To evaluate Pij atomistically, one has to calculate the tensor ij of internal stress produced by a solitary point defect in an equilibrated fixed volume V crystallite with periodic boundary conditions (p.b.c.). Then Pij = –ij·V. The rectilinear p.b.c. N1N2N3 size 3D simulation box of N = 4N1·N2·N3 hcp-Be atoms is first MD equili- brated, by energy minimization at zero temperature, to find its ground state structure, size, and volume V. After that, and with these parameters fixed, the extra Be SIA is inserted into the BP BO position, and the system is MD re-relaxed to its minimal energy at a fixed V. Fi- nally, ij is calculated by the LAMMPS code intrinsic routine, and the dipole tensor Pij is found as – ij·V. To avoid unphysical effects of the crystallite size li- mitation, this procedure was repeated for the simulation box expanded from 223 up to 121219 hcp lattice units. It was found that the ij components become the box size independent for sufficiently large N > 10 3 . For the representative 10944 atoms 121219 box, the BO SIA dipole-force tensor Pij has a diagonal form with principal components Pxx = 2.33 eV, Pyy = 6.98 eV, Pzz = 3.24 eV, and tr(Pij) = 13.25 eV. Using the consis- tently measured [22] elastic constants B = 116.8 GPa,  = 150.1 GPa,  = 0.05 of Be, we readily obtained V = tr(Pij)/3B = 5.73 Å 3 = 0.709·Be and, from Eq. (7), U0 = 0.63 eV. This agrees well with the hard-sphere mo- del estimate, 0.64 eV, and thus is quite reliable. 5.2. THE SINK STRENGTH kMC CALCULATION We treated the diffusion-timescale evolution of point defects statistically as their random walks (RW), mutual annihilations, and absorptions by sinks. The object kMC (okMC) method [25, 26] was applied as the following problem-specific algorithm for the Sandia SPPARKS kMC package [17] to sample all these events. Vacancies are randomly placed into the hcp Be latti- ce sites with probability CFP. According to the ab initio results [19, 20], only the most energy profitable BO in- terstitial sites, the centers of the basal plane octahedral voids, are randomly filled by the same number of SIAs. Edge dislocations with Burgers vector ]1102[ 3 1b are modeled as the rc = 2a core radius cylinders centered in the 10, 2 1 d    , p.b.c. simulation box. For the thermally activated diffusion kMC modeling, the frequency ij of a defect transition from i th to j th po- sitions ri,j is assumed to follow the Arrhenius rule ij = 0exp[–Ea(rirj)/kBT] , (8) where 0 is the transition attempt frequency, Ea is the transition activation energy Ea(rirj) = Em(rirj) + Ud(rj) – Ud(ri), (9) Em is the dislocation elastic field Ud(r) (6) independent migration barrier energy ([20], see in sec. 4). The reactions events are governed by the following rules. The recombination occurs if the V–SIA distance r  reff = 2a; both are excluded from simulation. The dislocation trap occurs if the freely migrating SIA is fo- und inside the core, rd  rc; then, it is immobilized. The sketches of this algorithm outputs are shown in Fig. 5 for the same RW step #50. Three L-directions, }0001{ , }1110{ and }1011{ , relevant to Be major slip systems were modeled. Cause of the ab initio predicted bias in SIA Em, 0.12 eV in BP vs. 0.27 eV out of it, the major feature of the diffusion anisotropy is clearly seen. SIAs are readily trapped onto the }0001{ and }1110{ oriented dislocations while no absorption occurs for the BP parallel }1011{ sink. The reason is the BP confined SIA diffusion which effectively happens in 2D. Fig. 5. SPPARKS kMC simulation of Be SIA () diffusi- on, recombination with vacancies (), and trapping by differently directed edge dislocation sinks, T = 77 K This okMC algorithm is easily extendable to sinks of other kind and dimensionality (voids, grain boundaries, etc.). In general, it can model the solutions of the rate theory ODE system (2) explicitly by the incorporation of the FP source K(t) [25, 26] and at the expense of sig- nificant computational efforts. But for the specific semi- analytical framework of our study we shall use it only for calculations of the sink strength k 2 . In Eq. (2.2), k 2 DI = 1/ is the characteristic frequency of SIA absorption by a dislocation sink. Thus, by definition, k 2 = (DI) –1 is a scalar (not a tensor) quantity. The okMC tally of k 2 within the scope of the d-di- mensional RW in  d has been proposed in ref. [25]: k 2 = 2d / (l 2 n) where l is the rirj jump length, n is the mean number of jumps each SIA performs before being trapped by a sink (such that  = n/0). For the isotropic RW (e.g., in bcc-Fe [26]), l = l(a) is a unique lattice unit dependent constant. Case of hcp-Be, l||  l (the sub- scripts || and  denote intra- and inter-BP jumps, respec- tively) and the RW is generally characterized by the (unknown) anisotropic diffusion coefficient tensor D. This implies the 2 k tensor which is out of the scope of the rate theory ansatz (2). To score the properly avera- ged k 2 as a scalar okMC tally, one has to apply, to it, a certain appropriate probabilistic measure p(l). The hcp-Be SIA BP-BO lattice has N|| = 6 intra-BP closest-neighbor and N = 14 next-neighbor inter-BP BO sites. Following the refined okMC approach [26], we assume the attempt frequency 0 to be isotropic, and measure the intra/inter-BP jumps with the probabilities p||, = N||,||,/(N|||| + N), ||, = 0exp(–E||,/kBT), where E||, are the anisotropic jumps activation energies (E >> E|| for Be [20]). The averaging operator of this measure is x = p||x|| + px. Let’s define the Eq. (2) consistent sink strength as k 2 = (Deffeff) –1 with the effective diffusion coefficient Deff = l 2  / (2d· t) of the isotropic RW jumps of mean square (m.s.) length l 2  and duration  t. The effective mean diffusion-to- sink time eff = n||· t|| + n· t is composed from the numbers n||, and durations  t||, of intra/inter-BP jumps, n|| + n = n. Since n||, = p||,n, eff = n·(p||· t|| + p· t) = n· t and, consequently, k 2 = (Deffeff) –1 = 2d / (l 2 n). Therefore, the ‘isotropic’ k 2 tally [25] is still applicable to score the anisotropic case sink strength, but at a redefined temperature dependent m.s. jump length l 2 . Having a = 2.286 Å, c = 3.584 Å, c/a = 1.568, we obtain l|| = a = 2.286 Å, l = [( c /2) 2 + 6 /7a 2 ] ½ = 2.773 Å. In a very high temperature limit kBT >> E||,, ||,  0, the r.m.s. l = l 2  ½  lmax = ( 9 /10·a 2 + 7 /40·c 2 ) ½ = 2.636 Å. This is an our study irrelevant limiting case of a 3D iso- tropic SIA diffusion. At low temperatures, p << p|| and l  lmin = a = 2.286 Å also independently on 0 and T. This corresponds to an entirely 2D BP SIA diffusion. Calculations show the 3D diffusion RW component sha- rply appearing just at room temperature, T  293 K, and gradually increasing at elevated temperatures. We applied the developed okMC algorithm to calcu- late k 2 by means of the SPPARKS kMC code and omit- ted vacancies as irrelevant to the SIA k 2 evaluation to speed-up the k 2 tally convergence. The per-dislocation normalized dimensionless sink efficiency  = k 2 /d kMC calculation results are presented in Figs. 6 and 7. To uncover the sink strength qualitative regularities, the kMC simulations of Figs. 6, 7 were performed for the ]1102[ 3 1b , L = {0001} dislocation of Be in wide ranges of temperature T (incl. the topical T = 77 K), dislocation density d = 10 10 …10 12 cm –2 and the dislocation elastic field strength parameter U0 = 0.0….0 eV (incl. the quite realistic value U0 = 0.5 eV). We compared the results with the simplistic T and U0 independent estimates (4) labeled, in Figs. 6, 7, as the ‘isotropic theory’. One can see that the SIA diffusion anisotropy itself (U0 = 0) has only a small ( 50%) temperature indepen- dent effect on the SIA trapping efficiency . In contrast to this, the SIA-dislocation interaction affects its absor- ption very considerably, and results in a drastically (by 1...2 orders of magnitude) increasing . This enhance- ment is mainly a low-temperature effect, see Fig. 6. It is the most pronounced at kBT << U0 while tends to disap- pear at high kBT ~ U0 when the SIA-to-sink drift is com- peting with its thermally enhanced chaotic diffusion. 10 -3 10 -2 10 -1 10 0 10 1 10 2 U 0 = 5.0 eV U 0 = 0.5 eV U 0 = 0.1 eV U 0 = 0.0 eV S in k ef fi ci en cy  = k2 / d (r el .u n. ) Temperature T (eV)  d = 10 11 cm 2 77 K isotropic theory, Eq. 4 SPPARKS kMC 10 K 100 K 1000 K Fig. 6. Temperature dependencies of the normalized ef- ficiency k 2 /d of the {0001} edge dislocation sink at dif- ferent SIA-dislocation interaction parameters U0 10 10 10 11 10 12 1 10 100 SPPARKS kMC U 0 = 0 eV isotropic theory, Eq. 4 5.0 eV 1.0 eV 0.5 eV 0.1 eV U 0 = 0 eV S in k ef fi ci en cy  = k2 / d (r el .u n. ) Dislocation density  d (cm 2 ) T = 77 K Fig. 7. Dislocation density dependence of the SIA- {0001} sink efficiency for different values of U0 Fig. 7 clearly shows that the per-dislocation sink ef- ficiency increases with d. Abt. twofold gain of (d), at d  10 1012 cm –2 , is predicted already by a field-free isotropic theory (4). For U0 = 0.5 eV and T = 77 K, it optimally amounts to  5. One can see that the kMC cal- culations confirm both of the proposed features which enhance the SIA sink efficiency, the cryogenic tempera- ture and the highest possible d of an irradiated sample. The very significant (d) anisotropy is found in the Fig. 8 data calculated for three dislocation line directions at the sec. 5.1 MD modeling evaluated U0 = 0.63 eV. The {0001} dislocations capture SIAs most effici- ently. The }1110{ system is  (25…50)% less efficient, esp. at higher d. The difference is mainly due to the  -dependence of Ud (6) and, thus, is of the hcp lattice geometry nature. The BP parallel }1011{ dislocations do not capture SIAs at all ((d)  0 at a SIA low-T 2D diffusion, see in Fig. 5). With respect to our main goal, the mobile SIA removal maximization, this system is lost. Properly textured targets with lowered content of the BP parallel dislocation are favorable. But for the subsequent final calculations, we adopted the cautious hypothesis of an equipartitioning of these slip systems in a Be target. 10 10 10 11 10 12 0 10 20 30 40 50 60 70 SPPARKS kMC {0 0 0 1} {0 1 1 1} {1 1 1 0} Be S in k ef fi ci en cy  = k2 / d (r el .u n. ) Dislocation density  d (cm 2 ) T = 77 K U 0 = 0.63 eV Fig. 8. Anisotropy of the SIA-dislocation sink efficiency 6. RESULTS AND DISCUSSION The consolidated results of the MSMS calculations of the residual concentrations CV of surviving vacancies are shown in figures below. They were calculated, ac- cording to the rate theory Eqs. (2)–(3), using the Be target parameters (R, k 2 , etc.) evaluated in sections 4 and 5. 10 -6 10 -5 10 -4 10 -3 10 -2 10 -6 10 -5 10 -4 10 -3 10 -2 3 2 FPpa C V ( v ac an ci es p er a to m ) Primary FP concentration (FPpa) Be Eq. 4 theory kMC 1. d =10 10 cm 2 2.  d =10 11 cm 2 3.  d =10 12 cm 2 K 0 = 7·10 10 FPpa/s 1 T = 77 K U 0 = 0.63 eV FPpa 1 h 10 h 100 h 1000 h 2.5 MeV e  10 A/cm 2 t 0 = Fig. 9. The residual vacancy concentrations plotted vs. the initial FPpa produced at a EB exposure t0 (top axis) in Be targets of various dislocation densities d In Fig. 9, it is seen that the increase of d allows to prolongate effectively, in exposure time t0, the ballistic PRD stage (3.1) (when CV  FPpa) and to delay the oc- currence of the kinetic V–I recombination dominance stage (3.2), CV  (FPpa) ½ , to much greater t0. Note that this prediction relies entirely on the results of the ade- quate MD and kMC calculation of k 2 (sec. 5) since the isotropic field-free theory (4) results in the much more pessimistic data plotted with dashed curves of Fig. 9. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t 0 = 1000 h Be 2.5 MeV e  10 A/cm 2 C V ( 1 0  3 v ac an ci es p er a to m ) Depth z (cm) primary FPpa d = 10 10 cm 2 10 11 cm 2 10 12 cm 2 Fig. 10. Depth profiles of the residual CV in Be targets of different dislocation densities d (,,) in com- parison with the initial FPpa depth profile () Fig. 9 can be used as a diagram for the evaluation of the ELIAS linac EB exposure duration t0(CV) needed to obtain the required CV. The top-right corner arrows indi- cate this paper topical t0(10 –3 )  10 3 h for the highest considered d = 10 12 cm –2 . The depth profiles CV(z, t0) were calculated by combining the Fig. 9 data with the primary FPpa rate depth profile K0(z) of Fig. 4 (sec. 3). In Fig. 10, they are shown for the 1000 h long exposure. One can see that the most essential is the strong dis- location density d impact on the residual CV. In a hypo- thetical SIA-sink-free beryllium (k 2 = 0), the maximal CV = (K0 /RDI) ½ (see Eq. (2.1)) of the irradiation induced vacancies drops to zero within a FP recombination time (RDI) –1 < 10 –4 s just after switching off the EB. In a well annealed beryllium, d~10 8 cm –2 , the residual CV is of ~ (10 –6 …10 –5 ) even at t0 = 10 3 h. However, Be targets prestrained to d ~ 10 11 …10 12 cm –2 can retain 10 –4 …10 –3 vacancies per atom that is sufficient for subsequent cryogenic measurements of their impact on the electronic properties of beryllium. Sinks of all other kinds will promote the SIA outflow enhancing the V–I imbalance in favor of vacancies. Therefore, this work model gives only a lower estimate of the residual CV, and thus is a conservative evaluation. The estimated EB exposure, 1000 h  42 days, is rather challenging but is not so impossible bearing in mind the already gained experience of long-time (500…700 h) e – -irradiation of materials at the NSC KIPT operating electron accelerators. CONCLUSIONS In the present work, the developed multiscale com- puter modeling technique was successfully applied to the characterization and planning of cryogenic irradiati- ons at the NSC KIPT sited electron accelerator ELIAS in order to study experimentally the impact of point de- fects on the superconductivity of beryllium. The primary radiation damage rate in a target was calculated using the e – -beam transport Monte-Carlo mo- deling code. The quantitative distinction of the results of its explicit atomistic simulation from the NRT standard model predictions and the considerable contribution of atomic collision cascades into the spatial distributions of primarily produced Frenkel pairs have been revealed. The later stages of the primary damage time evolu- tion were modeled by different simulation methods bas- ing on the ab initio calculated and other reference data on the structure and migration of point defects in Be. Molecular dynamics modeling has been applied to evaluate the parameters of the elastic interaction of Be self-interstitial atoms with dislocations sinks. Reliable estimates of the dipole-force tensor Pij and the interacti- on energy factor U0 = 0.63 eV have been obtained. Kinetic Monte-Carlo modeling has been used for calculation of the dislocation sink strength k 2 basing on ab initio and MD data with due account of the hcp-Be anisotropy and the elastic strain of dislocations. The sig- nificant growth of k 2 with a decrease in temperature and an increase in the dislocations density has been found. The k 2 anisotropy has been revealed and explained by the preferentially two-dimensional basal plane confined diffusion of self-interstitial atoms at low temperatures. The concluding data on the multiscale calculated ef- ficiency of the introduction of vacancies into the e – -irra- diated Be target were obtained within the scope of the mean-field reaction rate theory. It has been shown that the application of cryogenic (77 K) e – -irradiation of Be targets prestrained up to ~ 10 12 cm –2 dislocation density results in the abnormally high (~ 10 –3 per atom) yield of residual vacancies which is comparable, to within a half, with that of primarily produced Frenkel pairs at a rea- sonable ( 10 3 h) ELIAS linac e – -beam exposure. In conclusion, it should be noted that the presented MSMS technique and software are flexible enough to be applied, in future, for the computational support of the other NSC KIPT sited accelerators driven irradiations, including the RMS ‘simulation irradiations’ with ion beam machines and neutron sources. ACKNOWLEDGMENTS The authors are very grateful to A.S. Bakai for draw- ing their attention to the problem, putting forward the basic idea of future experiments, and for the continuous support and critical discussion of this study. We ackno- wledge K.V. Kovtun and V.N. Borisenko for valuable data on the prestrained Be targets preparation and the ELIAS linac irradiation e – -beam parameters, opportuni- ties and limitations. We thank I.I. Papirov and A.A. Ni- kolayenko for provision of their expertise on structural and mechanical properties of beryllium. We also always keep in memory the outstanding role of Yu.T. Petrusen- ko in providing cryogenic irradiations in NSC KIPT. REFERENCES 1. ASTM E521-96. Standard practice for neutron radiation damage simulation by charged-particle irra- diation. Annual Book of ASTM Standards, v. 12.02. American Society for Testing and Materials, Philadelphia, PA, USA, 1996, p. 141-160. 2. M.J. Norgett, M.T. Robinson, I.M. Torrens. A proposed method of calculating displacement dose rates // Nucl. Eng. Design. 1975, v. 33, N 1, p. 50-54. 3. M.I. Bratchenko, V.V. Bryk, S.V. Dyuldya, A.S. Kalchenko, N.P. Lazarev, V.N. Voyevodin. Com- ments on DPA calculation methods for ion beam driven simulation irradiations // PAST. Ser. “Rad. Dam. Phys. Rad. Mat. Sci.” 2013, N 2(84), p. 11-16. 4. R.E. Stoller. Primary radiation damage formation / R. Konings (ed.). Comprehensive Nuclear Materials. Amsterdam: Elsevier, 2012, v. 1, p. 293-330. 5. G.S. Was. Fundamentals of radiation materials science: metals and alloys. Berlin, Heidelberg: Sprin- ger-Verlag, 2007, 827 p. 6. J.A. Elliott. Novel approaches to multiscale modelling in materials science // Int. Materials Reviews. 2011, v. 56, N 4, p. 207-225. 7. S.I. Golubov, A.V. Barashev, R.E. Stoller. Radiation damage theory / R. Konings (ed.). Compre- hensive Nuclear Materials. Amsterdam: Elsevier, 2012, v. 1, p. 357-391. 8. S.V. Dyuldya, M.I. Bratchenko. Multiscale modeling of radiation damage in the computational support of accelerator driven simulation studies // Proc. of the 3 rd Int. Conf. “High-purity materials: production, application, properties”. Kharkov: NSC KIPT, 2015, p. 60. 9. S.V. Dyuldya, M.I. Bratchenko. Methodology and software for multiscale modeling of radiation impact on materials // Abstr. of XV Conf. “HEP. Nucl. Phys. Accelerators.” Kharkov: NSC KIPT, 2017, p. 51- 52. 10. I. Papirov, A. Nikolayenko, Yu. Tuzov. Beryllium: monograph. M.: MEPhi NNRU, 2017, 344 p. 11. I.I. Papirov. Handbook on the structure and properties of beryllium alloys. M.: “Energoatomizdat”, 1981, 368 p. 12. A.S. Bakai, A.N. Timoshevskii, S.A. Kalkuta, A. Moeslang, V.P. Vladimirov. On the influence of vacancies on the electronic properties of beryllium // Low Temp. Phys. 2007, v. 33, issue 10, p. 889-891. 13. A. McKinley (Jr.), H. Feshbach. The Coulomb scattering of relativistic electrons by nuclei // Phys. Rev. 1948, v. 74, N 12, p. 1759-1763. 14. A.S. Bakai, M.I. Bratchenko, S.V. Dyuldya. Impact of the highly deformed Be dislocation structure on the point defects kinetics at the NSC KIPT ELIAS li- nac cryogenic electron irradiation // Abstr. of the XIII Conf. “HEP. Nucl. Phys. Accelerators”. Kharkov: NSC KIPT, 2015, p. 85. 15. S.V. Dyuldya, M.I. Bratchenko. Refined model and code for calculation of point defects concentrations in multicomponent heterogeneous materials // Proc. of the ICPRP-XX, Alushta, Sept. 10–15, 2012. Kharkov: NSC KIPT, 2012, p. 42-43. 16. S. Plimpton. Fast parallel algorithms for short- range molecular dynamics // J. Comput. Phys. 1995, v. 117, N 1, p. 1-19; http://lammps.sandia.gov. 17. S. Plimpton, C. Battaile, M. Chandross, et al. Crossing the mesoscale No-Man’s Land via parallel Kinetic Monte Carlo. SANDIA Report SAND2009- 6226, Oct. 2009; http://spparks.sandia.gov. 18. F.A. Nichols. On the estimation of sink- absorption terms in reaction-rate-theory analysis of radiation damage // JNM. 1978, v. 75, issue 1, p. 32-41. 19. M.G. Ganchenkova, P.V. Vladimirov, V.A. Borodin. Vacancies, interstitials and gas atoms in beryllium: ab initio study // JNM. 2009, v. 386-388, N 1, p. 79-81. 20. V.A. Borodin, P.V. Vladimirov. Self- interstitial diffusion in beryllium: elementary jumps and overall dynamics // Proc. of the 10 th IEA Int. Workshop on Beryllium Technology, Sept. 19–21, 2012, Karlsruhe, Germany. KIT Sci. Publ., 2012, p. 318-326. 21. C. Teodosiu. Elastic models of crystal defects. Berlin Heidelberg: Springer-Verlag, 1982, 336 p. 22. A. Migliori, H. Ledbetter, D.J. Thoma, T.W. Darling. Beryllium’s monocrystal and polycrystal elastic constants // J. Appl. Phys. 2004, v. 95, N 5, p. 2436-2440. 23. C. Björkas, K.O.E. Henriksson, M. Probst, K. Nordlund. A Be–W interatomic potential // J. Phys.: Cond. Matter. 2010, v. 22, N 35, 352206 (6 p.). 24. E. Clouet, S. Garruchet, H. Nguyen, M. Perez, C.S. Becquart. Dislocation interaction with C in -Fe: A comparison between atomic simulations and elasticity theory // Acta Materialia. 2008, v. 56, N 14, p. 3450- 3460. 25. L. Malerba, C.S. Becquart, Ch. Domain. Object kinetic Monte-Carlo study of sink strengths // JNM. 2007, v. 360, N 2, p. 159-169. 26. A.B. Sivak, V.M. Chernov, V.A. Romanov, P.A. Sivak. Kinetic Monte-Carlo simulation of self- point defect diffusion in dislocation elastic fields in bcc iron and vanadium // JNM. 2011, v. 417, N 1-3, p. 1067- 1070. Article received 02.11.2017 МНОГОУРОВНЕВОЕ МОДЕЛИРОВАНИЕ ЭЛЕКТРОННОГО ОБЛУЧЕНИЯ БЕРИЛЛИЯ ПРИ НИЗКИХ ТЕМПЕРАТУРАХ М.И. Братченко, С.В. Дюльдя Представлены методология и результаты многомасштабного моделирования первичной генерации и временной эволюции радиационных дефектов при криогенном (77 К) облучении сильнодеформированного до плотности дислокаций ~ 10 12 см –2 бериллия на электронном линаке ELIAS ННЦ ХФТИ. Показано, что применение низкотемпературного облучения предварительно напряженных мишеней позволяет эффективно подавлять рекомбинацию пар Френкеля за счет ухода свободно мигрирующих собственных межузельных атомов на дислокационные стоки и приводит к аномально высоким (~ 10 –3 на атом) выходам вакансий, сопо- ставимых с концентрациями первичных пар Френкеля при разумной ( 10 3 ч) длительности e – -облучения. http://lammps.sandia.gov/ http://spparks.sandia.gov/ БАГАТОРІВНЕВЕ МОДЕЛЮВАННЯ ЕЛЕКТРОННОГО ОПРОМІНЕННЯ БЕРИЛІЮ ЗА НИЗЬКИХ ТЕМПЕРАТУР М.І. Братченко, С.В. Дюльдя Представлено методологію та результати багатомасштабного моделювання первинної продукції та ево- люції у часі радіаційних дефектів за криогенного (77 К) опромінення сильнодеформованого до густини дис- локацій ~10 12 см –2 берилію на електронному лінаці ELIAS ННЦ ХФТІ. Показано, що застосування низько- температурного електронного опромінення попередньо напружених мішеней дозволяє ефективно пригнічу- вати рекомбінацію пар Френкеля через витік вільно мігруючих власних міжвузельних атомів до дислокацій- них стоків і призводить до аномально високих (~10 –3 на атом) концентрацій вакансій, які добре порівнянні з концентраціями первинних пар Френкеля за прийнятної ( 10 3 год) тривалості електронного опромінення.

Ähnliche Einträge